perspectives in quantum hall effects · 2013-07-23 · 7.7.9. resonant tunneling fermi sea of...

PERSPECTIVES IN QUANTUM HALL EFFECTS

Novel Quantum Liquids in Low-Dimensional Semiconductor Structures

Edited by

Sankar Das Sarma

Aron Pinczuk

WILEY- VCH

Wiley-VCH Verlag GmbH & Co. KGaA

This Page Intentionally Left Blank


Novel Quantum Liquids in Low-Dimensional Semiconductor Structures

Edited by

Sankar Das Sarma

Aron Pinczuk

WILEY- VCH

Wiley-VCH Verlag GmbH & Co. KGaA

Cover illustration Sample of silicon on which the Quantum Hall Effect was verified by Klaus von Klitzing in 1980 (courtesy of Deutsches Museum, Bonn)

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Library of Congress Card No.: Applied for

British Library Cataloging-in-Publication Data: A catalogue record for this book is available from the British Library

Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>.

0 1997 by John Wiley & Sons, Inc. 0 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form - nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.

Printed in the Federal Republic of Germany Printed on acid-free paper

Printing and Bookbinding buch biicher dd ag, Birkach

ISBN-13: 978-0-471-1 1216-7 ISBN-10: 0-471-1 1216-X

CONTENTS

Contributors Preface

xi xiii

1 Localization, Metal-Insulator Transitions, and Quantum Hall Effect 1 S. Das Sarma

2

1.1.

1.2.

1.3.

1.4.

Introduction 1.1.1. Background 1.1.2. Overview 1.1.3. Prospectus Two-Dimensional Localization: Concepts 1.2.1. Two-Dimensional Scaling Localization 1.2.2. Strong-Field Situation 1.2.3. 1.2.4. 1.2.5. Disorder-Tuned Field-Induced Metal-Insulator

Strong-Field Localization: Phenomenology 1.3.1. Plateau Transitions: Integer Effect 1.3.2. Plateau Transitions: Fractional Effect 1.3.3. Spin Effects 1.3.4. Frequency-Domain Experiments 1.3.5. Magnetic-Field-Induced Metal-Insulator Transitions Related Topics 1.4.1. Universality 1.4.2. Random Flux Localization

Quantum Hall Effect and Extended States Scaling Theory for the Plateau Transition

Transition

References

Experimental Studies of Multicomponent Quantum Hall Systems J . P. Eisenstein

2.1. Introduction 2.2. Spin and the FQHE

2.2.1. Tilted Field Technique 2.2.2. 2.2.3.

Phase Transition at v = 815 The v = 512 Enigma

1 1 2 4 5 5 7 9

12

16 18 18 21 22 23 23 28 28 30 31

37

37 38 39 40 45

V

vi CONTENTS

2.3. FQHE in Double-Layer 2D Systems 2.3.1. Double-Layer Samples 2.3.2. The v = 1/2 FQHE 2.3.3. 2.3.4.

Collapse of the Odd Integers Many-Body v = 1 State

2.4. Summary References

3 Properties of the Electron Solid H. A. Fertig

3.1.

3.2.

3.3.

3.4.

3.5.

3.6.

Introduction 3.1.1. 3.1.2. Some Intriguing Experiments 3.2.1. Early Experiments: Fractional Quantum

Hall Effects 3.2.2. Insulating State at Low Filling Factors:

A Wigner Crystal? 3.2.3. Photoluminescence Experiments Disorder Effects on the Electron Solid: Classical Studies 3.3.1. Defects and the State of the Solid 3.3.2. Molecular Dynamics Simulations 3.3.3. Continuum Elasticity Theory Analysis 3.3.4. Effect of Finite Temperatures Quantum Effects on Interstitial Electrons 3.4.1.

3.4.2. Photoluminescence as a Probe of the Wigner Crystal 3.5.1. Formalism 3.5.2. Mean-Field Theory 3.5.3. 3.5.4. Conclusion: Some Open Questions

Realizations of the Wigner Crystal Wigner Crystal in a Magnetic Field

Correlation Effects on Interstitials: A Trial Wavefunction Interstitials and the Hall Effect

Beyond Mean-Field Theory: Shakeup Effects Hofstadter Spectrum: Can It Be Seen?

References

4 Edge-State Transport

4.1 Introduction 4.2. Edge States

C. L. Kane and Matthew P. A . Fisher

4.2.1. IQHE 4.2.2. FQHE

49 50 51 56 58 66 67

71

71 72 73 74 74

75 79 81 81 82 86 90 91

92 95 97 97 99

100 103 104 105

109

109 114 114 119

CONTENTS vu

4.3. Randomness and Hierarchical Edge States 4.3.1. The v = 2 Random Edge 4.3.2. Fractional Quantum Hall Random Edge 4.3.3. Finite-Temperature Effects Tunneling as a Probe of Edge-State Structure 4.4.1. Tunneling at a Point Contact 4.4.2. Resonant Tunneling 4.4.3. Generalization to Hierarchical States 4.4.4. Shot Noise

4.4.

4.5. Summary Appendix: Renormalization Group Analysis References

5 Multicomponent Quantum Hall Systems: The Sum of Their Parts and More S. M . Girvin and A. H . MacDonald

5.1. 5.2 5.3. 5.4. 5.5.

5.6. 5.7. 5.8.

5.9. 5.10. 5.11.

Introduction Multicomponent Wavefunctions Chern-Simons Effective Field Theory Fractional Charges in Double-Layer Systems Collective Modes in Double-Layer Quantum Hall Systems Broken Symmetries Field-Theoretic Approach Interlayer Coherence in Double-Layer Systems 5.8.1.

5.8.2. 5.8.3. Superffuid Dynamics 5.8.4. Merons: Charged Vortex Excitations 5.8.5. Kosterlitz-Thouless Phase Transition Tunneling Between the Layers Parallel Magnetic Field in Double-Layer Systems Summary

Experimental Indications of Interlayer Phase Coherence Effective Action for Double-Layer Systems

References

6 Fermion Chern-Simons Theory and the Unquantized Quantum Hall Effect B. I . Halperin

6.1. Introduction 6.2. Formulation of the Theory 6.3. 6.4. Response Functions

Energy Scale and the Effective Mass

126 127 132 135 136 138 145 151 152 154 156 157

161

161 165 169 169

172 180 185 192

193 196 199 203 206 209 213 216 218

225

225 227 230 233

viii CONTENTS

6.5. 6.6. Effects of Disorder 6.7. Surface Acoustic Wave Propagation 6.8. Other Theoretical Developments

Other Fractions with Even Denominators

6.8.1.

6.8.2.

6.8.3.

6.8.4. 6.8.5. 6.8.6. 6.8.7. 6.8.8.

Asymptotic Behavior of the Effective Mass and Response Functions Tunneling Experiments and the One-Electron Green’s Function One-Particle Green’s Function for Transformed Fermions Physical Picture of the Composite Fermion Edge States Bilayers and Systems with Two Active Spin States Miscellaneous Calculations Finite-System Calculations

6.9. Other Experiments 6.9.1.

6.9.2. 6.9.3. Miscellaneous Other Experiments

Geometric Measurements of the Effective Cyclotron Radius R: Measurements of the Effective Mass

6.10. Concluding Remarks References

7 Composite Fermions J . K . Jain

7.1. 7.2

7.3

7.4

Introduction Theoretical Background 7.2.1. Statement of the Problem 7.2.2. Landau Levels 7.2.3. Kinetic Energy Bands 7.2.4. Interactions: General Considerations Composite Fermion Theory 7.3.1. Essentials 7.3.2. Heuristic Derivation 7.3.3. Comments Numerical Tests 7.4.1. General Considerations 7.4.2. Spherical Geometry 7.4.3. 7.4.4. Band Structure of FQHE 7.4.5. Lowest Band 7.4.6. Incompressible States 7.4.7. CF-Quasiparticles

Composite Fermions on a Sphere

238 24 1 243 247

247

249

25 1 252 253 254 254 254 255

255 256 257 258 259

265

265 267 267 268 269 270 270 270 273 275 278 278 279 280 28 1 282 283 284

CONTENTS ix

7.4.8. Excitons and Higher Bands 7.4.9. Low-Zeeman-Energy Limit 7.4.10. Composite Fermions in a Quantum Dot 7.4.11. Other Applications Quantized Screening and Fractional Local Charge 7.5.

7.6. Quantized Hall Resistance 7.7. Phenomenological Implications

7.7.1. FQHE 7.7.2. Transitions Between Plateaus 7.7.3. Widths of FQHE Plateaus 7.7.4. FQHE in Low-Zeeman-Energy Limit 7.7.5. Gaps 7.7.6. Shubnikov-de Haas Oscillations 7.7.7. Optical Experiments 7.7.8. 7.7.9. Resonant Tunneling

Fermi Sea of Composite Fermions


8 Resonant Inelastic Light Scattering from Quantum Hall Systems A. Pinczuk

8.1. Introduction 8.2. 8.3.

Light-Scattering Mechanisms and Selection Rules Experiments at Integer Filling Factors 8.3.1. Results for Filling Factors v = 2 and v = 1 8.3.2. Results from Modulated Systems Experiments in the Fractional Quantum Hall Regime 8.4.


9 Case for the Magnetic-Field-Induced Two-Dimensional Wigner Crystal M. Shayegan

9.1. Introduction 9.2. Ground States of the 2D System in a Strong

Magnetic Field 9.2.1.

9.2.2. 9.2.3. 9.2.4.

Ground State in the v << 1 Limit and the Role of Disorder Properties of a Magnetic-Field-Induced 2D WC Fractional Quantum Hall Liquid Versus WC Role of Landau Level Mixing and Finite Layer Thickness

285 288 290 292 293 294 295 295 297 297 297 297 298 298 298 299 300 302

307

307 311 317 319 326 33 1 337 338

343

343

347

347 348 352

352

x CONTENTS

9.3.

9.4. 9.5.

9.6.

9.7.

9.8.

9.9.

Low-Disorder 2D Electron System in GaAs/AlGaAs Heterostructures Magnetotransport Measurement Techniques History of dc Magnetotransport in GaAs/AlGaAs 2D Electron Systems at Low v Summary and Discussion of 2D Electron Data 9.6.1. Reentrant Insulating Phase 9.6.2. Nonlinear Current-Voltage and Noise

Characteristics 9.6.3. Normal Hall Coefficient 9.6.4. Finite Frequency Data: Pinning and the Giant

Dielectric Constant 9.6.5. Washboard Oscillations and Related Phenomena 9.6.6. Melting-Phase Diagram and the WC-FQH

Transition Summary and Discussion of the 2D Hole Data 9.7.1. Sample Structures and Quality 9.7.2. 9.7.3.

9.7.4. Bilayer Electron System in Wide Quantum Wells 9.8.1. Details of Sample Structure 9.8.2. Concluding Remarks References

Insulating Phase Reentrant Around v = 1/3 Disappearance of the Reentrant Insulating Phase at High Density Wigner Crystal Versus Hall Insulator

Reentrant Insulating States Around v = 1/3 and 1/2

10 Composite Fermions in the Fractional Quantum Hall Effect H. L. Stormer and 0. C. Tsui

10.1. 10.2. 10.3. 10.4. 10.5. 10.6. 10.7. 10.8. 10.9. 10.10. 10.11.

Index

Introduction Background Composite Fermions Activation Energies in the FQHE Shubnikov-de Haas Effect of Composite Fermions Thermoelectric Power Measurements Optical Experiments Related to Composite Fermions Spin of a Composite Fermion How Real Are Composite Fermions? Transport at Exactly Half Filling Conclusions References

353 355

355 357 357

358 361

362 365

368 370 370 370

372 372 375 376 377 380 380

385

385 387 390 392 395 400 404 405 410 415 418 419

423

CONTRIBUTORS

SANKAR DAS SARMA, Department of Physics, University of Maryland, College Park, MD 20742

J. P. EISENSTEIN, Division of Physics, Mathematics and Astronomy, Califor- nia Institute of Technology, Pasadena, CA 91 125

H. A. FERTIG, Department of Physics and Astronomy and Center for Com- putational Sciences, University of Kentucky, Lexington, KY 40506

MATTHEW P. A. FISHER, Institute for Theoretical Physics, University of Califor- nia, Santa Barbara, CA 93 106

S. M. GIRVIN, Department of Physics, Indiana University, Bloomington, IN 47405

B. I. HALPERIN, Department of Physics, Harvard University, Cambridge, MA 02138

J. K. JAIN, Department of Physics, State University of New York at Stony

C. L. KANE, Department of Physics, University of Pennsylvania, Philadelphia,

A. H. MACDONALD, Department of Physics, Indiana University, Bloomington,

ARON PINCZUK, Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974

M. SHAYEGAN, Department of Electrical Engineering, Princeton University, Princeton, NJ 08544

H. L. STORMER, Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974

D. C. TSUI, Department of Electrical Engineering, Princeton University,

Brook, Stony Brook, NY 11794

PA 19104

IN 47405

Princeton, NJ 08544

xi

PREFACE

The unexpected observations of the quantized Hall phenomenon in 1980 and of the fractional quantum Hall effect in 1982 are among the most important physics discoveries in the second half of the twentieth century. The precise quantization of electrical resistance in the quantum Hall effect has led to the new definition of the resistance standard and had major impact in all of science and technology. From a fundamental viewpoint, studies of quantum Hall phenomena are among the most active research areas of physics, with vigorous contributions by researchers in condensed matter, low temperature, semiconductor materials science and devices, and quantum field theory. Striking new behaviors of two-dimensional electron systems in semiconductor quantum structures continue to be discovered, as researchers strive to achieve better and purer materials, higher magnetic fields, lower electron densities, lower temperatures, and new experimental methods.

Examples of new discoveries in the quantum Hall regimes are the even denominator fractional quantum Hall effects, the anomalies at half-filled Landau level, and the long-elusive, spectroscopic finding of the fractional excitation gap. Theoretically, a number of new concepts and paradigms have emerged from studies of quantum Hall phenomena that are both elegantly beautiful and powerfully relevant to experiment.

A basic picture that emerges from studies of quantum Hall effects over the past ten years is that a low-dimensional electron system, as occurring in semiconductor quantum structures in a high external magnetic field, is a fascinating quantum system for the exploration of fundamental electron interactions. Studies of such many-electron systems have already offered remarkable new insights, and the results of intense current research will continue to suprise us.

The first phase in the study of the quantum Hall effect phenomena is the 1980-1986 period where the basic integer and the fractional quantization phenomena were observed and their fundamental understanding consolidated. This “classical” era is well covered by the book The Quantum Hall Effect, edited by R. E. Prange and S. M. Girvin (New York: Springer-Verlag, 1987). During the period 1987-1995 the subject entered its second phase when a large number of spectacular new developments have occurred that have considerably expanded and enhanced the scope of the field. In this second phase the fractional quantum Hall effect has emerged as an archetype of the novel electron quantum fluids that may exist in low-dimensional systems of man-made quantum structures. These studies have also cemented the conceptual links between quantum Hall phenomena and several areas at the current frontiers of physics-in particular, quantum field theory.

xiii

xiv PREFACE

The purpose of our book is to cover milestones of this second exciting phase by having a leading expert in each topic give a comprehensive personal perspective. The ten chapters in this book cover, necessarily somewhat interconnected, major developments. One key feature of the book is our effort to present both experimental and theoretical perspectives more or less on equal footings. In deciding the contents of the various chapters we were guided by two considerations: (1) current impact and intrinsic interest of the subject matter, and (2) the anticipated interest in the topic in the future. In a fast-developing field it is not always easy to anticipate which interesting discovery or theory of today will remain interesting years from now. We have tried our best to include topics that we perceived to possess some lasting intrinsic value. Only time can tell the extent of our success in this respect. Finally, it should be emphasized that the various chapters are personal views or perspectives of individual authors rather than conventional reviews. We feel that in a book on a subject as important as this one, it is critical that the viewpoints and perspectives are expressed without the constraints of conventional reviews.

The book is meant for graduate students as well as for experienced researchers. We have made particular efforts to make each chapter accessible to experimental- ists and theorists alike. Each chapter is self-contained, with its own set of references guiding the reader to the original papers on the subject as well as to further reading on the topic.

This book would have been impossible without prompt support from all the authors. We wish to thank them for their great care in preparing their chapters.

College Park, Maryland Murray Hill. New Jersey

SANKAR DAS SARMA ARON PINCZUK

1 Localization, Metal-Insulator Transitions, and Quantum Hall Effect

S. DAS SARMA Department of Physics, University of Maryland, College Park, Maryland

1.1. INTRODUCTION

1.1 .l. Background

The close connection between the strong-field localization problem and the phenomenon of the quantum Hall effect was in some curious sense already appreciated before von Klitzing’s celebrated discovery [ 11 of the quantized Hall effect phenomenon in 1980. Klaus von Klitzing was, in fact, investigating the strong-field galvanomagnetic properties [2] of two-dimensional electrons con- fined in Si/SiO, MOSFETs with the specific goal of elucidating the nature of activated transport at the localized tails of Landau levels when he discovered the quantized Hall effect. (To be more precise, one of the issues being investigated [2] by von Klitzing was to ascertain the specific cause for the vanishing of the longitudinal conductivity, crxx, at the Landau level tails-in particular, whether oxx = 0 at T = 0 is caused by the number n of mobile carriers vanishing at the chemical potential or by a vanishing of the effective scattering time z in a semiclassical Drude-type formula: cr = ne2z/m.) A number of “interesting” observations (which became significant on hindsight after the discovery of the quantized Hall effect) on two-dimensional quantum magnetotransport properties [3], which predate von Klitzing’s discovery and were discussed in light of strong-field metal-insulator localization transition, can now be understood on the basis of the interplay between localization and quantum Hall effect. Thus, the rela- tionship between metal-insulator localization transitions and the phenomenon of the quantum Hall effect is an old subject of considerable fundamental significance.

Perspectives in Quantum Hall Effects, Edited by Sankar Das Sarma and Aron Pinczuk ISBN 0-471-1 1216-X 0 1997 John Wiley & Sons, Inc.

1

2 LOCALIZATION, METAL-INSULATOR TRANSITIONS

1.1.2. Overview

It is now well accepted that the existence of localized states at Landau level tails (and of extended states at Landau level centers) is essential to the basic quantization phenomenon. Quantum Hall plateau transitions, where one goes from one quantized value of pxy to another by tuning the electron density or the magnetic field with a concomitant finite value of pxx [and, therefore, of oxx= p,Jp:, + p:y)- '1, is now understood to be a double localization transition through a very narrow band of extended states (the bandwidth of the extended states vanishing at T + 0). In fact, the consensus is that at T = 0 the plateau transition is a quantum critical phenomenon at E = E,, where E, is the critical energy at the middle of a Landau level where the extended state is located, with the localization length ((E) diverging as ((E) - 1 E - Ecl-x, where x is the localization exponent [4]. All states away from E, (ie., for E # E,) are localized, and as the chemical potential sweeps through these localized states (E # E,), oxy is quantized and ox, is vanishingly small, with a quantum phase transition from one quantized value of oxy to the next occurring exactly at E = E,. At T = 0, therefore, C T ~ ~ ( ~ T ~ , , ) is zero (quantized) everywhere (i.e., for all values of the chemical potential) except at isolated energies E = E,", where E," is the critical energy at the Nth Landau level center, which is a set of measure zero. At nonzero temperatures (or, equivalently, at finite frequencies) the plateau transition is no longer infinitely sharp, and ox, (and consequently, pxx) is finite over a finite range of the chemical potential around E,".

The foregoing scenario describing the quantum Hall plateau transition to be a quantum critical phenomenon is remarkably well verified experimentally. A fundamental detailed field-theoretical understanding of the quantum critical phenomenon, however, still eludes us even though a great deal of numerical work and experimental data provide a rather compelling phenomenology in its support. The quantized Hall effect requires the existence of a mobility gap and thereby directly implies the existence of localized and extended states with metal-insulator transitions between them at discrete values of the chemical potential, E = E:. Our current theoretical understanding of the two-dimensional localization problem, which is based on a perturbative renormalization group analysis of the nonlinear sigma model, rules out the existence of extended states in two dimensions. The quantized Hall effect is then a manifestly nonperturbative effect where extended states appear in a disordered two-dimensional electron gas in a strong external magnetic field. It is a curious coincidence that the discovery of the quantized Hall effect phenomenon occurred around the time (1979- 1982) when a consensus [S-71 was developing in the physics community that no true extended states can exist in a disordered two-dimensional electron system. It was obvious [S-141 right after von Klitzing's discovery that both localized (at the Landau level tails) and extended (at Landau level centers) states are needed to explain the quantization, and the idea of an additional strong field nonperturbative topological term, which is allowed by symmetry in the usual nonlinear sigma model Lagrangian, was put forth [lS] as the reason for the existence of

INTRODUCTION 3

extended states. Much has been written on this subject, but unfortunately, concrete calculations (e.g., leading to critical exponents) including the topological term do not exist.

The current understanding of the quantized Hall effect is thus based on the existence of localized states almost evergwhere except near the Landau level centers, with the plateau transition being the delocalization transition caused by the chemical potential passing through the extended state. Since there are no two-dimensional extended states in zero or weak magnetic fields (at least within the prevailing scaling theory or, more generally, within a weak-coupling pertur- bation treatment of the nonlinear sigma model), whereas the quantized Hall effect implies the essential existence of two-dimensional strong-field extended states, a question naturally arises about what happens to the extended states as the magnetic field is decreased with the eventual disappearance of the quantized Hall effect. There are two possible distinct scenarios consistent with the experimental situation:

1. The extended states float up in energy [10,16] as the magnetic field is decreased and eventually when the extended state corresponding to the lowest (N = 0) Landau level at E , N = O passes above the chemical potential, no quantized Hall effect can be observed and the system is localized because the chemical potential is necessarily in localized states. In this scenario the strong-field extended states float up to infinite energy as the magnetic field vanishes, making the zero-field (or, weak-field) two-dimensional system completely localized, consistent with the scaling theory of localization,

2. The second possibility is that the extended state at the middle of each Landau level eventually disappears (without floating up) at a characteristic magnetic field B:, and when the lowest extended state at vanishes, the quantized Hall effect disappears, with the system being completely localized without any delocalized states whatsoever.

Note that in the first scenario the extended states float up in energy (eventually to infinity), and in the second scenario they disappear (without floating) at (nonuniversal) critical magnetic fields. Currently, much of the community seems to be in favor of the first scenario, which, in fact, was proposed [lo, 16) right after the discovery of the quantized Hall effect (and has since been extended and resurrected under the rubric of a global phase diagram [17]) to reconcile the essential existence of extended states in strong magnetic fields as necessitated by the quantized Hall effect and the nonexistence of true two-dimensional extended states (based on weak-localization experiments and theories) in weak magnetic fields. In turns out, however, that despite a substantial fundamental theoretical difference between the two scenarios (i.e., the floating of extended states to infinity and the disappearance of extended states at finite magnetic fields without floating), it is not easy experimentally to distinguish directly between the two scenarios. It should be noted that from a practical viewpoint both scenarios lead to the existence of a (nonuniversal) critical magnetic field where a system that is


strongly localized in a zero magnetic field makes a transition to the quantum Hall state with delocalized states; in the second scenario the critical magnetic field corresponds to the field at which the extended state first appears, whereas in the first scenario the critical field corresponds to the floating down of the extended states from infinite energy (at zero field) to the chemical potential. There is now experimental evidence for such a field-induced direct transition [ 181 between an insulator (at zero field) and a quantum Hall liquid (at finite field). The critical field for this transition is nonuniversal and depends on the system parameters, particularly the amount of disorder in the sample (which may not necessarily be characterized by a single experimental quantity). It has been claimed [ 171 that such field-induced metal-insulator transitions [ 181, where the system goes from an insulator at zero field to a “metal” at some finite fields, also belong to the same universality class (ie., the same critical exponents) as the quantum Hall plateau transitions. Current experimental evidence in support of this claim is not entirely conclusive.

All of the foregoing localization issues, which so far have been discussed in the context of the integer quantized Hall effect, have their counterparts in the fractional quantized Hall situation, with the plateau transitions between fractional incompressible states behaving similar to the integer plateau transitions (albeit at lower temperatures and correspondingly higher magnetic fields because the energy gap associated with the fractional effect is typically much smaller than the cyclotron gap for the integer effect), and the reentrant metal-insulator transition occurring around a low fraction filling factor value v (v x 1/5 usually for the electronic system). There is, however, a significant additional complication in the localization problem for the fractional state because of the theoretical possibility of the existence of a strong-field quantum Wigner solid phase at low filling factor (v 6 1/5). Indeed, the very high field reentrant metal-insulator transition [18] in the fractional situation (around v x 1/5 for electrons and v x 1/3 for holes) has been interpreted as a quantum phase transition between the Laughlin incompressible liquid and the quantum Wigner solid. This identification is based on theoretical calculations [19] which predict the Wigner solid phase as having lower energy than the Laughlin liquid phase for v 6 1/5, and on the expectation that a strongly pinned (due to disorder) Wigner solid should behave as an insulator with defect-mediated activated transport. (This issue is discussed in more detail in Chapters 3 and 9.)

1.1.3. Prospectus

The goal of this chapter is to provide a broad and critical perspective on the subject of two-dimensional strong-field localization and metal-insulator transitions as they relate to the phenomena of quantum Hall effects. While the experimental and theoretical status of the subject is reviewed, no attempt is made to discuss the various topics in detail; that is left to the original articles (and several detailed reviews [20-231) cited throughout this chapter. The emphasis here is on providing a critical (and necessarily somewhat subjective) perspective

TWO-DIMENSIONAL LOCALIZATION: CONCEPTS 5

on what is understood and what is not and the extent to which theory and experiment combine to form a unified description of localization phenomena in the quantum Hall effect.

The introduction essentially sets the tone and provides an overview of the topics discussed in subsequent sections. In Section 1.2 the basic ideas of two- dimensional localization are discussed, emphasizing the underlying theoretical concepts. In Section 1.3 the phenomenology is discussed by describing the current status of experimental studies on localization aspects of the quantum Hall effect phenomena, with a critical assessment of the agreement between experiment and theoretical concepts. The issues of universality in two-dimensional Landau level localization and random flux localization are discussed in Section 1.4, based on a critical evaluation of the experimental results and various numerical studies.

1.2. TWO-DIMENSIONAL LOCALIZATION: CONCEPTS

1.2.1. Two-Dimensional Scaling Localization

It is now almost [24] universally accepted [7,25] that all states in a disordered noninteracting two-dimensional electron system are (weakly) localized (i.e., no true extended states) independent of how weak the disorder is. This concept of the nonexistence of a truly “metallic” two-dimensional phase, which originated in the late 1970s and early 1980s and has been discussed and reviewed extensively in the literature, is often referred to as the scaling theory of localization or weak localization and is based on a weak-coupling perturbative renormalization group treatment of the nonlinear sigma model as well as on a great deal of perturbative self-consistent diagrammatic calculations. There is also experimental evidence showing weak logarithmic rise in the resistance of thin-metal films and two- dimensional semiconductor structures with decreasing temperature at low- enough temperatures. (The effect is generally extremely small, and at least in GaAs-based two-dimensional electron systems, where most of the quantum Hall effect experiments are carried out, the weak-localization correction to low- temperature transport is often masked by much stronger temperature dependence arising from phonon scattering and Coulomb screening effects.) Application of an external magnetic field suppresses the weak-localization effect but does not eliminate it. The nonexistence of true extended states in a disordered two- dimensional electron gas is usually characterized by a beta function, fl(g), which depends only on the dimensionless system conductance g(L), which, in turn, depends on the length scale L:

Thus a knowledge of the f l function allows one to compute how the system conductance (9) behaves in the thermodynamic limit.


The nonlinear sigma model-based field-theoretic treatment of the one- electron localization problem is a mature subject [6,7,25,26]. The most complete results have been obtained by a five-loop perturbative renormalization group calculation [27] which exploits certain connections of the theory to heterotic superstring models. In two dimensions the perturbative /I function is [27]

where E = d - 2 with d as the spatial dimension, and the subscripts u, 0, and s refer to unitary, orthogonal, and symplectic ensembles, respectively. The symplectic ensemble applies to the situation where the disorder is associated with spin-flip and spin-orbit scattering, and is, therefore, not of much relevance to us. In the absence of an external magnetic field, there is time-reversal symmetry and the orthogonal ensemble applies, whereas the application of an external magnetic field destroys time-reversal symmetry. It follows from Eq. (3) that the zero-field two-dimensional situation is logarithmically localized with /3(g) - g - ', implying that the resistance R = g- ' has logarithmiccorrections, R(L) = R, + Cln(L/L,).

The zero-field orthogonal ensemble can be characterized by a localization length, <, in the weak disorder limit, which is exponential in the mean-field conductance go:

< - ego ( 5 )

where go may be the dimensionless mean-field Drude conductance calculated, for example, in the self-consistent Born approximation. Note that for weak disorder go can be very large, making the localization length exponentially large, implying that the weak-localization correction to conductance is logarithmically small. For strong disorder, go is extremely small, also making <( - 1) small, which is the usual strong localization situation. [Note, however, that Eq. (5 ) does not apply when go < 1 in dimensionless units because only the leading order dependence in g-' has been kept.] It is important to realize that within the nonlinear sigma model scenario (or, within the scaling theory of localization in a narrower sense) the transition from a weakly localized two-dimensional metal at weak disorder to a strongly (exponentially) localized insulator at strong disorder is only a crossover phenomenon with no quantum critical point separating weak and strong localization phases. [This last statement is true only for orthogonal and unitary ensembles-in a symplectic ensemble, as can be seen from Eq. (4), there is a metal-to-insulator phase transition in two dimensions as the disorder strength is increased.]

TWO-DIMENSIONAL LOCALIZATION CONCEPTS 7

The consensus that all two-dimensional electron states are at least logarithmically localized (orthogonal ensemble), no matter how weak the disorder is based on weak-coupling perturbative renormalization group arguments [25-271. There is no exact or rigorous argument (even in a narrowly defined context) to rule out potential strong-coupling problems in the renormalization group pertur- bation expansion (which, despite being carried out [27] to an amazing five-loop order exploiting heterotic superstring theoretic techniques, is somewhat ill behaved). Although there is definite experimental evidence [7) in favor of weak logarithmic increase in the sheet resistance of thin-metal films and Si MOSFETs at very low temperatures, it is probably correct to state that experimental evidence supporting weak localization in good-quality GaAs heterostructures is not abundant. On the other hand, there is a great deal of experimental data [28] showing strong localization behavior in GaAs heterostructures at low electron densities (equivalently, at high disorder). Thus some questions remain about the universal applicability of Eq. (3) to all two-dimensional situations in the orthogonal ensemble. It may also be relevant in this context to mention that the localization exponents calculated by direct numerical simulation [29] do not agree with those obtained from the nonlinear sigma model theory for the three- dimensional orthogonal ensemble and the two-dimensional symplectic ensemble. Currently, it is not known whether this is a fundamental disagreement associated with &-expansion or a subtle crossover effect in the simulations [29].

1.2.2. Strong-Field Situation

Application of an external magnetic field, among other things, breaks the time- reversal symmetry in a two-dimensional electron system so that, at least in the weak-field situation (assuming the only effect of the magnetic field to be the breaking of the time-reversal symmetry), the unitary ensemble nonlinear sigma model result, Eq. (2), applies. Thus all states are still localized except that the Cooperon channel associated with the maximally crossed backscattering diagram is suppressed, and the leading order term (the diffusion channel) in the /? function instead of being -g- ' is -g- ' , implying the unitary ensemble localization to be even weaker than that in the zero-field orthogonal situation. The effective weak disorder localization length in the unitary ensemble behaves as

and can be macroscopically large for large values of the mean-field conductivity go. But the states are, in principle, localized for any disorder, albeit with extremely large localization lengths for weak values of disorder (large values of go).

The unitary ensemble, therefore, does not allow quantization of Hall conductance, which requires the existence of true extended states (along with bands of localized states). It is not surprising that a perturbative renormalization group theory misses the basic essence of the quantized Hall effect phenomenon, which is manifestly a nonperturbative (topological) macroscopic quantum effect. To


incorporate the possibility of the existence of extended states into the structure of the field theory, one needs to add a topological term to the effective Lagrangian of the nonlinear sigma model in the strong-field situation. This topological term (also called the theta term) is related to the theta vacuum in four-dimensional gauge theories and is intrinsically nonperturbative. The topological term in the effective Lagrangian is postulated to be proportional to the Hall conductance, and the field theory in the strong-field situation therefore depends on two parameters: the regular longitudinal conductivity oxx (essentially the same as the dimensionless conductance g introduced in Section 1.2.1) and the Hall conductivity oxy (which determines the topological term).

Thus the strong-field localization theory, in contrast to the usual weak-field (orthogonal or unitary ensemble) nonlinear sigma model situation [which is a one-parameter scaling theory with the fl function, fl(g), being dependent only on a single parameter g], is claimed [lS, 161 to be a two-parameter scaling theory which depends on the renormalization flow for ox, and oxy The renormalization group flow is postulated to be characterized by two different fixed points. One is a stable fixed point connected with the localized states where oxx = 0 and oxy is quantized, and the other is a saddle-point fixed point connected with the extended states that carry the Hall current, where oxx is nonzero and oxy is intermediate between two successive quantized Hall values. The saddle-point fixed point associated with the extended states is partly attractive and partly repulsive, and is therefore semistable (and semi-unstable). There is a true quantum metal-insulator phase transition in this picture as one goes through the saddle-point fixed point. Note that the stable fixed point in this scenario (oxx = 0, ox,, quantized) is the usual nonlinear sigma model result in the absence of the topological term (except that the quantum number for the Hall current is zero without the extended states), whereas the topological term introduces the extended states and the saddle-point fixed point in the strong-field situation, On dimensional grounds one hypothesizes that (T, is universal at the unstable fixed point, whose value is taken to be e2/2h.

While the field-theoretic description involving the topological term and the saddle-point fixed point has a certain elegant attractiveness, it is at best a framework of a theory. The theory itself has not yet been worked out and in fact does not look particularly promising [30]. (Questions [31] have even been raised about the appropriateness of the framework itself.) In some sense, the entire framework may be considered somewhat of a tautology where the introduction of the topological term (and the consequent saddle-point fixed point) into the Lagrangian is equivalent to postulating the existence of strong-field extended states, which experiments unambiguously demonstrate to be there by virtue of the nontrivial quantization of oxy when ox, = 0. For the field theory to be a statement stronger than the mere statement of the existence of extended states (which is implied by the experiments anyway), there must be concrete calculational results such as the localization critical exponent and the critical value of ox, at the saddle point based on the field theory. Unfortunately, such concrete calculations have been singularly lacking [30) within this topological field theory


framework, and until that happens, it is not clear that this theoretical framework is much more than a formal statement about the existence of extended states in the strong-field situation.

There is an important fundamental issue in the strong-field localization problem which requires elucidation in this context. Presumably, the unitary ensemble /3 function applies in some weak-field situation when the magnetic field breaks the time-reversal symmetry but Landau quantization effects of the magnetic field are unimportant. In the strong-field situation, however, there must be extended states at Landau level centers. The question therefore arises as to how these extended states originate with increasing magnetic field, the zero (or the weak)-field situation being exactly localized. Whether there is a critical magnetic field separating orthogonallunitary ensemble nonlinear sigma model results from the strong-field two-parameter scaling situation is not known. The most plausible scenario is described by a heuristic argument [ 161 which asserts that the extended states in the strong-field situation essentially rise in energy as the magnetic field decreases, and eventually, in the zero-field limit, they float out to infinite energy, leaving out only the localized states at and below the Fermi level. The tuning parameter in this scenario is o C z , where o, = eB/mc is the cyclotron frequency, with ho, defining the Landau energy (gap) and T the mean-field scattering time (defining the conductivity n = ne2t/m, where n is the electron density). For o,z >> 1, one is in the strong-field situation, and o , ~ << 1 is the weak-field situation. Unfortunately, there is no concrete calculation that describes the crossover between these two regimes. A drawback of the theoretical framework of the field theory is that this important question cannot be addressed within its context since the topological term, which is put in by hand in an ad hoc manner, is either there (strong field) or not (weak field). There was at least one early inconclusive attempt [32] to tackle this issue diagrammatically by intro- ducing Landau quantization into the standard weak-localization self-consistent diagram technique, which, however, by definition cannot access the strong- coupling situation.

1.2.3. Quantum Hall Effect and Extended States

As emphasized in Sections 1.1 and 1.2.2, the existence of extended states is implied directly by the quantized Hall effect phenomenon. When the chemical potential resides in the localized states away from Landau level centers ( p # Ey), the zero- temperature values of n,, and oxY are given by

oxx = 0

ve o x y = h

(7)

where v, the filling factor, is the number of completely filled Landau levels, which


is, in fact, the same as the number of filled extended states since there is exactly one extended state (EF) at the center of each Landau level. As the chemical potential passes through the critical energy (p = Ef), there is an insulator-metal-insulator transition (at T = 0) where the system is insulating for p = Ef & 6, where 6 is infinitesimal and is metallic precisely at p = E:. The Hall conductance is quantized everywhere (i.e., all values of chemical potential) except at p = E r , where it jumps from one quantized plateau to the next, and the longitudinal conductance u,, is zero everywhere except at E,“. The quantized values of Hall conductance are given by

with 6 as an infinitesimal energy. Equations (9) and (10) apply at T = 0 whenever the chemical potential is in the localized state (except in the lowest Landau level N = 0, which is the fractional quantization regime, v < 1, discussed below). At the extended state energy a,, # 0 is finite and the system is “metallic”(at a discrete set of energies E,” of measure zero at zero temperature). At E r , the Hall conductance jumps from one plateau to another, and its value is intermediate between the two adjacent quantized Hall conductances. The expectation, based on the two- parameter flow diagram [lS] as well as analogies to other (e.g., superconductor- insulator [MI) two-dimensional quantum phase transitions, is that the value of u,, at the plateau transition point (i.e., at Ef) is universal. This leads to the conjucture that for Ef - 6 < p < E,N + 6 with 6 -0,

e2 axx = -

2h

a,,, = ( v + i) f Note that Eqs. (9), (lo), and (1 1) characterize, respectively, the stable and saddle fixed points. It must be emphasized that while Eq. (9) about the quantization phenomenon itself is absolutely beyond any doubt (and in fact now serves as the definition of the unit of resistance), the same cannot be said about Eq. (1 1). The experimental support for it is at best weak (many experimental results flatly contradict the universality of a,, at the peak, but the zero-temperature limit is always a problematic issue). There is some numerical support [34] for a universal value a,, = e2/2h at the critical point, but the situation is by no means conclusive.


The picture above for a zero-temperature thermodynamic limit quantum phase transition at E," where the system undergoes a quantum Hall liquid- "metal"-quantum Hall liquid transition with a concomitant plateau transition for uxy (with the two localized phases having adjacent integral quantized values of the Hall conductance and the metallic phase being the transition from one plateau to the next) is modified in an obvious manner at finite temperatures (or frequencies) or for a finite-size system. In a qualitative physical picture, all of these modifications (i.e., introduction of finite temperature, frequency, or system size) bring an effective length scale, L , into the problem, which now competes with the diverging localization length (5 ) at the critical point. When 5 2 L , the system behaves as a metal because the localization length has exceeded the effective system size. (For the true phase transition in the thermodynamic limit, 5 -, 00

exactly at the critical point where the extended states of measure zero reside.) Thus at finite temperatures (and/or finite system sizes, frequencies, etc.) the effective extended state at E = E," now smears out into a narrow band of extended states around E,N (the bandwidth 26 increases with decreasing Li or increasing 7') and the metallic phase around the Landau level centers now occupies a finite region of energy instead of being a set of measure zero. The net effect of having a finite temperature is, therefore, to smear out the plateau transition, which instead of being infinitely sharp exactly at E," now occurs over a small but finite range of energy around E; (which is basically the same range of chemical potential where ax,#O and the system is metallic in the sense that increasing temperature weakly increases the value of pxx). Thus 6 in Eq. (9) is now a small and finite energy instead of being infinitesimal. In addition, finite temperature introduces activated and/or phonon-assisted variable-range hopping transport in the localized phase so that u,, is exponentially small but not identically zero in the localized regime (uxx in the localized regime increases exponentially with increasing temperature).

All of these considerations remain valid in the fractional quantum Hall regime except that v is now a fractional filling factor. The basic phenomenology of the quantum Hall liquid-"metal"-quantum Hall liquid plateau transition remains equally valid for the fractional quantum Hall effect even though the nature of the extended and localized states are necessarily more complicated in the fractional situation because interaction and correlation effects play essential roles.

In concluding the section it should be mentioned that E,", the location of the extended state in the Nth Landau level, coincides with the Landau level center in the electron-hole symmetric situation:

In the asymmetric situation (which could arise, for example, from unequal numbers of attractive and repulsive random scattering centers making up the disorder) E," may differ [21] from the Landau level energies of Eq. (12). The same is true when Landau level coupling effects [21] become important at high disorder or, equivalently, low magnetic field.


1.2.4. Scaling Theory for the Plateau Transition

There is currently no existing (analytical) theory for the plateau transition and the associated metal-insulator localization transition at E = E:. In analogy with quantum critical phenomena and other localization transitions, one expects a diverging localization length as E approaches the critical energy E: (with the extended state at E = E: having an infinite localization length). The most common critical behavior is the power-law divergence, where in the scaling region the localization length <(E) behaves as

where x (often referred to as v, which is used as the symbol for the Landau level filling factor in this chapter) is the localization critical exponent. Most of the experimental work on the plateau transition in quantum Hall effect can be quantitatively understood on the basis of the scaling law defined in Eq. (13). Somewhat confusingly [35], the scaling theory defined by Eq. (13) has sometimes been referred to as the one-parameter scaling theory [20,35], presumably to emphasize (somewhat redundantly) the fact that there is only one diveriging length in the problem, <(E). It is to be noted that the terminology single-parameter scaling theory in the context of Eq. (13) is totally disconnected from the terminology two-parameter scaling [ 15,161 used earlier (Section 11.2) in the context of the topological nonlinear sigma model theoretical framework, where the two parameters refer to the renormalization group flow of oxx and ox,,. The meaning of the two terminologies is totally different, and they do not contradict each other.

As emphasized above, the absolute validity of Eq. (13) for the quantum Hall effect localization transition has not yet been established from an underlying field theory. It is, in fact, by no means obvious that the quantum Hall effect localization critical behavior must necessarily be characterized by a power law divergence in the localization length-in classical two-dimensional critical phenomena the well-known Kosterlitz-Thouless transition, for example, has a different behavior. Since no rigorous or exact (or even an analytic) derivation of Eq. (13) currently exists, it may be helpful to summarize the great deal of phenomenological support [20] in its favor:

1. A great deal of experimental evidence (discussed in Section 1.3) exists supporting the scaling law.

2. A great deal of numerical simulation [20], where the noninteracting strong-field two-dimensional Schrodinger equation is solved directly in the presence of random quenched disorder (e.g., the two-dimensional Anderson model in a strong magnetic field), indicates the existence of a critical point E," where the localization length diverges according to Eq. (13). Direct finite-size numerical simulation [20,21] has emerged as a powerful tool to study the interplay between localization and quantum Hall effect phenomena and has provided the most significant evidence supporting Eq. (13).

TWO-DIMENSIONAL LOCALIZATION CONCEPTS 13

3. The only analytical support for a behavior qualitatively similar to that described by Eq. (13) comes from the conjecture that the plateau transition can be considered (or, more accurately, can be mapped onto) a two- dimensional percolation transition [36-381, where the disorder potential is extremely smooth on magnetic length scales. This then leads to a percolation length or a percolation cluster size (which can also be thought as the size of a typical quantum Hall droplet), which diverges at the percolation transition, and an identification of the localization length ( with the percolation cluster size leads to behavior similar to that in Eq. (13).

The most compelling evidence in favor of the critical scaling behavior defined by Eq. (13) for the delocalization-localization transition associated with the quantum Hall plateau transition comes from a great deal of direct numerical simulation of the strong-field two-dimensional Anderson model using various types of disorder potential. The most direct numerical technique has been the calculation [20,21,35,39-421 of localization length for a finite two-dimensional strip and then using the finite-size scaling analysis to deduce the thermodynamic limit results. Similar techniques have been utilized extensively and successfully in studying the zero-field localization problem.

In the finite-size scaling analysis, one calculates the localization length A@) for a finite two-dimensional strip of width M, and connects it to the localization length e(E) for the corresponding infinite-size system via the scaling ansatz:

M = f (&) M

where f(x) is the universal scaling function having the asymptotic forms f ( x >> 1) - l/x and f (x << 1) - constant. In calculating A,(E) numerically, one considers a disordered two-dimensional strip of length L and width M, with L typically so large that the system is self-averaging. In practice, L a 10’ (in units of magnetic length) usually suffices. Periodic boundary conditions are used in the width direction, and the largest value of the width that is accessible with the currently available massively parallel processing machines is M = 5 12, even though M = 32 - 128 is more typical. A recursive Green’s function technique or a transfer matrix technique is used to solve the Schrodinger equation, leading to the Lyapunov exponent or the inverse localization length. For a given finite system of width M, the calculation of &(E) is exact. Details of the calculational method are given in the literature [20,21].

Obtaining A,(E) numerically as a function of M and E, one can then calculate e (E) by plotting A,(E)/M against M/C;(E), with ( ( E ) being adjusted to achieve the best scaling behavior in accordance with Eq. (14). The values of ( ( E ) that give the best finite-size scaling fit [i.e., Eq. (14)] for the numerical data are also found to obey the scaling law defined by Eq. (13), providing the most direct numerical evidence for the quantum critical behavior of the plateau transition.

14 LOCALIZATION. METAL-INSULATOR TRANSITIONS

Several different groups [20,21,35,39-421 have carried out finite-size scaling calculations using somewhat different models of the random disorder potential. The agreement among the numerical results of different groups is excellent, and the localization critical exponent for short-range white-noise disorder is found to be (neglecting Landau level coupling)

~ = 2 . 3 f O . 1 for N = O

for N = 1 w 5.5 k 0.5

with E, N (N + 1/2)hw,. For white-noise disorder potential with a finite range, the critical exponent in the lowest Landau level remains essentially unchanged, whereas xN = is found to decrease appreciably, from 5.5 to roughly 2.8, when the disorder range equals or exceeds the magnetic length [21,40-42). Calculations including Landau level coupling [21] also reduce the critical exponent x N = 1. While the possible Landau level (in)dependence of x will be discussed later in the chapter, it should be emphasized that finite-size scaling calculations in higher Landau levels, and/or for finite range disorder potential, and/or in the presence of Landau level coupling are necessarily less accurate than the uncorrelated (zero range) white-noise disorder calculations in the lowest Landau level, which yield a value of x N 2.3. In fact, the localization calculations for higher Landau levels can be carried out only up to a maximum M / t - 10, whereas the lowest Landau level calculations extend to M/( - 100. The maximum system sizes one can use for higher Landau level/finite range disorder/Landau level coupling situations are typically M - 16 - 64, which is substantially less than M - 512, the maximum possible system size for short-range lowest-Landau-level calculations. Thus the calculated critical exponent xo=2.3 in the lowest Landau level is more reliable than the calculated exponent x1 = 5.5 in the first excited level. Systematic studies [20,21] of the dependence of the critical exponent on the strength and type (attractive/repulsive, etc.) of disorder have been carried out. These studies are somewhat incomplete, and their results are consistent with the conclusion that the localization exponent x is universal and independent of the strength or type of disorder potential. In practice, however, a weak dependence of the calculated exponent on the details of disorder potential is found in numerical simulations [21] and attributed (rather uncritically) to crossover effects.

The localization critical exponent for the plateau transition has also been calculated using a quantum percolation network model [43,44], where the random potential is assumed to vary slowly on the scale of the magnetic length and the electron guiding centers follow the semiclassical equipotential contours of the smooth disorder potential. If quantum tunneling effects are neglected,the problem reduces to a two-dimensional classical percolation problem, where in the symmetric situation, there is exactly one energy at the percolation threshold, where the critical percolation cluster encompasses the entire system. If one identifies [36,43,44] this percolation transition as the delocalization transition at E,, onegets x = 4/3. It turns out that quantum tunnelingeffects can be included in this percolation picture through a quantum network model where the nodes of

perspectives in quantum hall effects · 2013-07-23 · 7.7.9. resonant tunneling fermi sea of...

Documents