Perspectives in quantum Hall effects: novel quantum liquids in low-dimensional semiconductor structures

PERSPECTIVES IN QUANTUM HALL EFFECTS

PERSPECTIVES IN QUANTUM HALL EFFECTS Novel Quantum Liquids in Low-Dimensional Semiconductor Structures

Edited by

Sankar Das Sarma Aron Pinczuk

WILEYVCH

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.

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Printing and Bookbinding buch biicher dd ag, Birkach ISBN-13: 978-0-471-11216-7

ISBN-10: 0-471-1 1216-X

CONTENTS xi xiii

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

1

S. Das Sarma

Introduction 1.1.1. Background 1.1.2. Overview 1.1.3. Prospectus 1.2. Two-Dimensional Localization: Concepts 1.2.1. Two-Dimensional Scaling Localization 1.2.2. Strong-Field Situation 1.2.3. Quantum Hall Effect and Extended States Scaling Theory for the Plateau Transition 1.2.4. 1.2.5. Disorder-Tuned Field-Induced Metal-Insulator Transition 1.3. 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-InducedMetal-Insulator Transitions 1.4. Related Topics 1.4.1. Universality 1.4.2. Random Flux Localization References 1.1.

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

2.1. 2.2.

Introduction Spin and the FQHE 2.2.1. Tilted Field Technique Phase Transition at v = 815 2.2.2. The v = 512 Enigma 2.2.3.

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

FQHE in Double-Layer 2D Systems 2.3.1. Double-Layer Samples 2.3.2. The v = 1/2 FQHE 2.3.3. Collapse of the Odd Integers 2.3.4. Many-Body v = 1 State 2.4. Summary References

2.3.

3 Properties of the Electron Solid

49 50 51 56 58

66 67 71

H. A. Fertig

3.1.

Introduction 3.1.1. Realizations of the Wigner Crystal 3.1.2. Wigner Crystal in a Magnetic Field 3.2. 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. 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 3.4. Quantum Effects on Interstitial Electrons 3.4.1. Correlation Effects on Interstitials: A Trial Wavefunction 3.4.2. Interstitials and the Hall Effect 3.5. Photoluminescence as a Probe of the Wigner Crystal 3.5.1. Formalism 3.5.2. Mean-Field Theory 3.5.3. Beyond Mean-Field Theory: Shakeup Effects 3.5.4. Hofstadter Spectrum: Can It Be Seen? 3.6. Conclusion: Some Open Questions References

4 Edge-State Transport

71 72 73 74 74 75 79 81 81 82 86 90 91 92 95 97 97 99 100 103 104 105 109

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

4.1 4.2.

Introduction Edge States 4.2.1. IQHE 4.2.2. FQHE

109 114 114 119

CONTENTS

Randomness and Hierarchical Edge States The v = 2 Random Edge Fractional Quantum Hall Random Edge Finite-Temperature Effects 4.4. 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 Summary 4.5. Appendix: Renormalization Group Analysis References 4.3.

4.3.1. 4.3.2. 4.3.3.

5 Multicomponent Quantum Hall Systems: The Sum of Their Parts and More

vu

126 127 132 135 136 138 145 151 152 154 156 157 161

S. M . Girvin and A. H . MacDonald

5.1. 5.2 5.3. 5.4. 5.5.

Introduction Multicomponent Wavefunctions Chern-Simons Effective Field Theory Fractional Charges in Double-Layer Systems Collective Modes in Double-Layer Quantum Hall Systems 5.6. Broken Symmetries 5.7. Field-Theoretic Approach 5.8. Interlayer Coherence in Double-Layer Systems Experimental Indications of Interlayer Phase 5.8.1. Coherence Effective Action for Double-Layer Systems 5.8.2. 5.8.3. Superffuid Dynamics 5.8.4. Merons: Charged Vortex Excitations 5.8.5. Kosterlitz-Thouless Phase Transition 5.9. Tunneling Between the Layers 5.10. Parallel Magnetic Field in Double-Layer Systems 5.11. Summary References 6 Fermion Chern-Simons Theory and the Unquantized Quantum Hall Effect B. I . Halperin 6.1. 6.2. 6.3. 6.4.

Introduction Formulation of the Theory Energy Scale and the Effective Mass Response Functions

161 165 169 169 172 180 185 192 193 196 199 203 206 209 213 216 218 225 225 227 230 233

viii

CONTENTS

Other Fractions with Even Denominators Effects of Disorder Surface Acoustic Wave Propagation Other Theoretical Developments 6.8.1. Asymptotic Behavior of the Effective Mass and Response Functions 6.8.2. Tunneling Experiments and the One-Electron Green’s Function 6.8.3. One-Particle Green’s Function for Transformed Fermions 6.8.4. Physical Picture of the Composite Fermion 6.8.5. Edge States 6.8.6. Bilayers and Systems with Two Active Spin States 6.8.7. Miscellaneous Calculations 6.8.8. Finite-System Calculations 6.9. Other Experiments 6.9.1. Geometric Measurements of the Effective Cyclotron Radius R: 6.9.2. Measurements of the Effective Mass 6.9.3. Miscellaneous Other Experiments 6.10. Concluding Remarks References 6.5. 6.6. 6.7. 6.8.

7 Composite Fermions

238 24 1 243 247 247 249 25 1 252 253 254 254 254 255 255 256 257 258 259 265

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. Composite Fermions on a Sphere 7.4.4. Band Structure of FQHE 7.4.5. Lowest Band 7.4.6. Incompressible States 7.4.7. CF-Quasiparticles

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 7.5. Quantized Screening and Fractional Local Charge 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. Fermi Sea of Composite Fermions 7.7.9. Resonant Tunneling 7.8. Concluding Remarks References

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

8 Resonant Inelastic Light Scattering from Quantum Hall Systems

307

A . Pinczuk

8.1. 8.2. 8.3.

Introduction 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 8.4. Experiments in the Fractional Quantum Hall Regime 8.5. Concluding Remarks References 9

Case for the Magnetic-Field-Induced Two-Dimensional Wigner Crystal

307 311 317 319 326 331 337 338

343

M. Shayegan

9.1. 9.2.

Introduction Ground States of the 2D System in a Strong Magnetic Field 9.2.1. Ground State in the v << 1 Limit and the Role of Disorder 9.2.2. Properties of a Magnetic-Field-Induced 2D WC 9.2.3. Fractional Quantum Hall Liquid Versus WC 9.2.4. Role of Landau Level Mixing and Finite Layer Thickness

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. Insulating Phase Reentrant Around v = 1/3 9.7.3. Disappearance of the Reentrant Insulating Phase at High Density 9.7.4. Wigner Crystal Versus Hall Insulator Bilayer Electron System in Wide Quantum Wells 9.8.1. Details of Sample Structure 9.8.2. Reentrant Insulating States Around v = 1/3 and 1/2 Concluding Remarks References

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 DASSARMA, Department of Physics, University of Maryland, College Park, MD 20742 J. P. EISENSTEIN, Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125 H. A. FERTIG, Department of Physics and Astronomy and Center for Computational Sciences, University of Kentucky, Lexington, KY 40506 MATTHEW P. A. FISHER, Institute for Theoretical Physics, University of California, 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 Brook, Stony Brook, NY 11794 C. L. KANE, Department of Physics, University of Pennsylvania, Philadelphia, PA 19104

A. H. MACDONALD,Department of Physics, Indiana University, Bloomington, IN 47405 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, 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 discoveriesin 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 fascinatingquantum 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 comprehensivepersonal 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 accessibleto experimentalists 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

SANKARDASSARMA ARONPINCZUK

PERSPECTIVES IN QUANTUM HALL EFFECTS: Novel Quantum Liquids in Low-DimensionalSemiconductorStructures Edited by Sankar Das Sarma, Aron Pinczuk Copyright0 2004 WILEY-VCH Verlag GmbH & Co. KGaA

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 confined 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 effectivescattering 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 relationship 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 oxyto the next occurring exactly at E = E,. At T = 0, therefore, C T ~ ~ ( ~ Tis ~ , ,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 existenceof extended states in two dimensions. The quantized Hall effect is then a manifestly nonperturbative effect where extended states appear in a disordered two-dimensionalelectron 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-dimensionalelectron 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, concretecalculations(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 perturbation 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 differencebetween 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

4

LOCALIZATION,METAL-INSULATOR TRANSITIONS

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-induceddirect 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 experimentalevidencein 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 correspondinglyhigher 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 electronicsystem).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-dimensionalstrong-field localization and metal-insulator transitions as they relate to the phenomena of quantum Hall effects. While the experimentaland 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 twodimensional 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-dimensionalphase, which originated in the late 1970sand early 1980sand 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 twodimensional semiconductor structures with decreasing temperature at lowenough 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 lowtemperature transport is often masked by much stronger temperature dependence arising from phonon scattering and Coulomb screening effects.) Application of an external magnetic field suppressesthe weak-localizationeffect but does not eliminate it. The nonexistence of true extended states in a disordered twodimensional 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 fl function allows one to compute how the system conductance (9) behaves in the thermodynamic limit.

6

LOCALIZATION,METAL-INSULATOR TRANSITIONS

The nonlinear sigma model-based field-theoretic treatment of the oneelectron 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 /Ifunction 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 logarithmicallylocalized 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 gomay be the dimensionless mean-field Drude conductance calculated,for example, in the self-consistent Born approximation. Note that for weak disorder gocan be very large, making the localization length exponentiallylarge, 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 dimensionlessunits 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-dimensionalmetal 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.]

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TWO-DIMENSIONALLOCALIZATIONCONCEPTS

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 perturbation 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 threedimensionalorthogonal ensembleand the two-dimensionalsymplecticensemble. Currently, it is not known whether this is a fundamental disagreement associated with &-expansionor 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 timereversal 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

8

LOCALIZATION, METAL-INSULATOR TRANSITIONS

incorporate the possibility of the existence of extended states into the structure of the field theory, one needs to add a topologicalterm to the effectiveLagrangian 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(essentiallythe same as the dimensionlessconductance 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 oxyThe 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 oxyis 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 oxyis 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 frameworkmay 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 oxywhen 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 criticalexponent and the critical value of ox, at the saddle point based on the field theory. Unfortunately, such concrete calculations have been singularlylacking [30) within this topological field theory

TWO-DIMENSIONAL LOCALIZATION: CONCEPTS

9

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 ofthe 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 introducing Landau quantization into the standard weak-localization self-consistent diagram technique, which, however, by definition cannot access the strongcoupling situation. 1.2.3. Quantum Hall Effect and Extended States

As emphasized in Sections 1.1and 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 zerotemperature values of n,, and oxYare given by oxx= 0

oxy

=

(7)

ve

h

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

10

LOCALIZATION,METAL-INSULATOR TRANSITIONS

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 everywhereexcept 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 twoparameter flow diagram [lS] as well as analogies to other (e.g., superconductorinsulator [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 2h

axx= -

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,, = e2/2hat the critical point, but the situation is by no means conclusive.

TWO-DIMENSIONAL LOCALIZATION:CONCEPTS

11

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-sizesystem. In a qualitative physical picture, all of these modifications (i.e., introduction of finite temperature, frequency, or system size) bring an effectivelength 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 effectiveextended 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 identicallyzero in the localized regime (uxxin 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.

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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 terminologysingle-parameter scaling theory in the context of Eq. (13)is totally disconnected from the terminology two-parameter scaling [15,161used 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 oxxand ox,,. The meaning of the two terminologiesis 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 Schrodingerequation is solved directly in the presence of random quenched disorder (e.g.,the two-dimensionalAnderson 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-sizenumerical 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-DIMENSIONALLOCALIZATION 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 twodimensional 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 [20,21,35,39-421of 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 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. f ( x >> 1)

14

LOCALIZATION. METAL-INSULATOR TRANSITIONS

Several different groups [20,21,35,39-421 have carried out finite-size scaling calculations using somewhat differentmodels 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) ~=2.3fO.1 w 5.5 k 0.5

for N = O for N = 1

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 xN= 1. While the possible Landau level (in)dependenceof 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-levelcalculations.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 semiclassicalequipotential 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 delocalizationtransition at E,, onegets x = 4/3. It turns out that quantum tunnelingeffectscan be included in this percolation picture through a quantum network model where the nodes of

-

-

-

-

TWO-DIMENSIONALLOCALIZATION CONCEPTS

15

the network represent saddle points in the disorder potential and the links represent the equipotentials. Transfer matrix calculations [43,44] based on such a quantum percolation network model yield x N 2.5 k 0.5, in agreement with direct calculation of the lowest-Landau-level Lyapunov exponent in the uncorrelated white-noise disorder potential. There are theoretical argument [37], not entirely convincing, which also claim to show that the classical two-dimensional percolation exponent x = 4/3 is modified by quantum tunneling effects to a quantum percolation exponent of x = 4/3 + 1 = 7/3 = 2.33.. .,in agreement with the numerical simulation results both for the quantum percolation network model and the uncorrelated white-noise disorder Anderson model. The numerical results [443 for the percolation network model give an exponent x N 1.3 x 4/3, in agreement with the theoretical [36] classical percolation exponent in the limit of vanishing quantum tunneling. This classical fixed point is, however, unstable to quantum effects, which change the exponent to x x 2.5, as discussed above. The fact that two seeminglyvery different models of disorder (i.e., uncorrelated white noise and smooth long range) lead to identical critical behavior [45] is extremely interesting and argues in favor of the localization exponent x being universal in the quantum Hall plateau transition. It should be noted that the numerically obtained scaling functions in the lowest Landau level are also consistent in the uncorrelated white-noise disorder and quantum percolation network models. This lends additional support for universality in the Landau level localization problem, at least for the lowest Landau level. It should be mentioned that in a different context, in three-dimensional situations involving zero magnetic fields, it has been claimed [46] that the quantum percolation transition belongs to a different universality class than that of white-noise nonlinear sigma model localization. Quantum critical phenomena are characterized [25] by the inherent mixing of statics and dynamics due to quantum fluctuation effects, and the dynamical exponent z enters quantum critical problems in a natural way, describing the dispersion of the critical-mode fluctuations. At finite temperatures/frequencies the scaling behavior as a function of temperature or frequency involves the exponent K = (xz)- l , which is a composite of the localization exponent x and the dynamical exponent z. The easiest way to see this is to consider a finitetemperature experiment where the magnetic field B is being varied to make the chemical potential move through E, to cause delocatization-localization transition. The scaling part of the resistivity p(B, T) behaves in two dimensions as p(B, T )= p(b''X 6B, bZT)

where b is a scale factor and 6 B = ( B - BJB, is the scaling variable. The B and T dependence are combined by the choice b = T - ' I z , leading to p(B, T ) = p(GB/T"),

where K = (xz)-

Thus the temperature scaling occurs according to the exponent IC = l/xz.

16

LOCALIZATION,METAL-INSULATOR TRANSITIONS

Currently, there is no true theory for the dynamical exponent z in the quantum Hall plateau transitions. It has been argued [33] that z = 1 in all quantum phase transitions involving Coulomb systems, based essentially on the fact that Coulomb interaction scales as the inverse of a distance. No real calculations exist establishing z = 1 in quantum Hall plateau transitions. It should be noted that taking z = 1 and x = 7/3, one obtains K = 1/xz = 317 = 0.43, which agrees with temperature (and frequency) scaling experiments discussed in Section 1.3, 1.2.5. Disorder-Tuned Field-Induced Metal-Insulator Transition

As has been emphasized throughout this chapter, all two-dimensionalsystems(in orthogonal and unitary ensembles) are localized in the absence of any magnetic field according to the scaling theory of localization and the associated nonlinear sigma model-based field theory. For weak disorder (k,l>> 1, where k, and 1 are the electron Fermi wavevector and mean free path, respectively)the localization is very weak (only logarithmic correction to the mean-field resistance), with a crossover to strong localization as disorder becomes strong, k,l<< 1. The localization length in the weak-localization situation is exponentially large, and the system behaves like a good metal except at very low temperatures, when the effective system size (defined, for example, by the coherence length) exceeds the localization length. In the strongly localized (k,l<< 1) limit, however, the localization length 5 is much smaller than the effective system size (typically,5 lattice sizes)and finite-temperatureconduction is via an activated and/or variable-range hopping mechanism. From a scaling localization field-theoretic perspective, the zero-field transition between two-dimensional weak and strong localization (i.e., the transition between k,l>> 1 and k , l c 1) is only a disorder-tuned crossover effect; however, operationally and from a practical experimental viewpoint [28,47], increased disorder causes a rather sharp metal-to-insulator transition in two-dimensional semiconductor-based heterostructures, where the mobility drops abruptly to zero at low temperatures, and the conduction on the insulating strongly localized (i.e., high disorder/low density k,l << 1) side of this transition is manifestly very strongly temperature dependent (activatedlvariable range hopping, etc.), whereas it is essentially independent of temperature on the metallic (weak disorder/high density, k , l x 1) side. In fact, for a given sample with a fixed amount of disorder, experimental metal-insulator transitions of this type have been studied extensivelyboth in Si/SiO, MOSFETs [48] and in GaAs/Al,Ga, - 4 s heterostructures [28,49] by tuning the electron density (made possible by varying a gate voltage) so that the chemical potential moves into the strongly localized tail of the density of states. Such disorder-tuned (usually achieved by varying the electron density or the chemical potential by changing a gate voltage) metal-insulator transitions in two-dimensional finite-temperature zero-field situations operationally [47] do not look much different from the corresponding three-dimensional finite-temperature metal-insulator transitions even though the scaling theory (and the associated nonlinear sigma model-based field theory) predicts a zero-temperature three-dimensional metal-insulator quantum phase

-

TWO-DIMENSIONALLOCALIZATION: CONCEPTS

17

transition and only a crossover in two dimensions. Experimentally, this transition (k,l>> 1 to k , l c 1)looks very much like a finite-temperature quantum phase transition in both two- and three-dimensional systems. The usual samples in which quantized Hall effect experiments are carried out are metallic (low disorder, weakly localized) at zero field, with their effective scaling theory-based localization length being many orders of magnitude larger than the effective system size. Thus the fact that the application of an external magnetic field gives rise to delocalized states at Landau level centers, while being very interesting and intriguing theoretically, is not a great experimental surprise because the system even at zero magnetic field has effectiveextended states at the Fermi level. A far more interesting situation would be to pick a system that is strongly localized (k,l<< 1,5 << effectivesystem size) at zero magnetic field and to create delocalized states at the chemical potential via the application of a strong external magnetic field. The natural question then arises about whether one would observe at finite fields the quantized Hall conductance phenomenon in a high-disorder strongly localized (at zero field) two-dimensional system as the magnetic field is increased. Much recent theoretical and experimental work has focused on this question. The specific question is whether and how application of a strong external magnetic field produces delocalized states, leading to a quantum Hall liquid, starting from a highly disordered two-dimensional system which is a strongly localized insulator at zero magnetic field. (The corresponding issue in three-dimensional disordered systems is well understood phenomenologically-application of a strong external magnetic field typically enhances localization effects, due to the wavefunction shrinkage effect if the starting zero-field system is already a strongly localized insulator.) In two-dimensional systems, experiments [49-561 suggest the possibility of a direct transition from a strongly localized insulator phase at B = 0 to a quantum Hall state at finite fields. The central theoretical construct in this direct zero-field insulator to the strong-field quantum Hall liquid transition is the idea of the floating [15-171 of extended states that exist at Landau level centers in the strong-field situation and are responsible for producing the quantized Hall effect phenomenon as the external magnetic field is reduced. The floating idea is based [16] on the persuasive physical argument that as the magnetic field is reduced, the extended states cannot disappear, and therefore to be consistent with the nonlinear sigma model results, the extended states must rise to infinite energy in the zero-field limit. While the floating argument is quite reasonable, it is, of course, logically possible for the extended states to just disappear at some (Landau-level-dependent) critical values of the magnetic field, with the critical magnetic field being smaller for the lower Landau levels. There is no explicit theoretical calculation that compellingly establishes the floating concept. There is some limited numerical support [21,25-591 for floating, and there is at least one numerical simulation [60] that demonstrates the disappearance of extended states at critical magnetic fields. It should be emphasized that it is difficult for numerical simulations to establishing floating decisively because the primary microscopic mechanism

18

LOCALIZATION, METAL-INSULATOR TRANSITIONS

for floating must be strong Landau level coupling at weak magnetic fields, which is difficult to handle in exact numerical simulations. Finally, the issue of the global phase diagram [17] for insulator-quantum Hall liquid phases, which is a direct consequence of the extended-state floating concept, has remained controversial [61,62], with experimental claims both in its favor [SO, 53,551 and against it [56,63,64]. In particular, experimental support for the global phase diagram concept [17] is weak in the fractionally quantized situation [63], where a correlation-driven Wigner solid phase probably becomes a significant factor. 1.3. STRONG-FIELDLOCALIZATION. PHENOMENOLOGY 1.3.1. Plateau Transitions: Integer Effect

Transitions between integer quantum Hall plateaus have been investigated [20] extensively by several experimental groups by studying the temperature dependence of the widths, AB, of the resistivity peaks and of the maximum slopes, (dp,,( T)/dB),,,, of the Hall resistivity between neighboring Hall plateaus. As explained earlier, the idea is that finite temperature smears the zero-temperature quantum phase transition, and the resistivity width (as well as the slope of the Hall resistivity) between quantized plateaus should reflect the underlying quantum phase transition by showing a scaling behavior in temperature:

where K = (xzT)- is the effective exponent characterizing temperature scaling, with zT as the thermal dynamical exponent. The most complete experimental analysis [65] of temperature scaling in the plateau transitionhasbeencarriedout inratherlowmobility(-34,000cm2 V-ls-') InGaAs/InP heterostructures (with the electrons in InGaAs), where the spin-split Landau levels N = O J , l t , 1 1 were studied. Extremely compelling T" scaling behavior was observed [65] in the impressive temperature range 0.1 to 4 K with K = 0.42 If: 0.04 for all the plateau transitions studied. Although this study [65] stands as one of the most compelling and careful experimentaldemonstrations of quantum critical behavior, several puzzles remain. It seems that one needs rather poor quality samples (i.e., very low mobilities) with strong scattering to see the scaling behavior over a wide temperature range. The choice of the material (In GaAs/InP) has been explained on the basis of the short-range nature of the alloy scattering (as against high-mobility GaAs/AlGaAs heterostructures, where the dominant scattering is long ranged, arisingfrom the remote dopants), which is claimed to produce a wider scaling region in temperature. One problem is that the poor-quality InGaAs/InP samples, which show excellent temperature scaling in the plateau transitions, have oEYkvalues substantially different from the theoreti-

STRONG-FIELDLOCALIZATION PHENOMENOLOGY

19

-

cally expected e2/2h universal value. The reason for this inconsistency is not understood. In corresponding low-quality (mobility 50,OOO cmz V.s) AlGaAs/GaAs samples, the quality of scaling [66] is much poorer and has only been seen at much lower (T c 200 mK) temperatures, the explanation being that the crossover temperature is lower for long-range disorder potential. Lowmobility GaAs heterostructures are also dominated by short-range disorder scattering intrinsic to the GaAs layer, and therefore the striking difference in the temperature-dependent scaling behavior [65,66] between InGaAs/InP and AlGaAs/GaAs system cannot simply be attributed to crossover effects, as has been values in the GaAs samples approach done rather uncritically. In addition, ox","" [66] the theoretical value of 0.5e2/h around T x 1 K, precisely the temperature region where scaling is poor. An additional problem arises in the universality of the temperature exponent K. While detailed experimental analysis of InGaAs/InP- based (low mobility) systems produces a reasonably disorder-independent universal K for different spin-split Landau levels (with K also beixig insensitive to the sample mobility, with the caveat that all the samples had rather low mobilities), the corresponding AlGaAs/GaAs sample results give K values which clearly depend [66-681 on the Landau level index for T > 0.2 K. Studies of AlGaAs/GaAs heterostructures [67-691 and Si/SiO, MOSFET [70] structures by other groups also produce Landau-level-dependent K values, with K varying from 0.2 and 0.9 among the lowest few (i.e., Ol,lt, 13) levels. Systematic dependence of K on disorder in GaAs-based samples has been reported in at least two experimental studies [68,69], whereas in Si MOS systems,K for the excited Landau levels is found [70] to be much larger than that in the lowest level. Although it is certainly possible that some of these studies suffer from crossover problems and are not really in the asymptotic scaling regimes, the experimental situation, as it currently stands, cannot be definitively accepted as overwhelmingly supporting absolute universality of K. (It is, however, absolutely clear that there is temperature scaling in the plateau transitions-the quantitative value of K is the issue here.) More experimental work will be helpful in resolving this issue. To understand the actual value of K, one notes that zT = ( x K ) - x 1 if one takes x = 7/3 x 2.3 and K x 0.4. This value of the dynamical exponent is consistent [33] with two-dimensional Coulomb systems and has been speculated to be universal. One also arrives [71] at zT = 1 by assuming that transport in the plateau transition is dominated by variable range hopping in the presence of Coulomb interaction. Why such hopping-based strong localization theory should have validity in the scaling regime is not clear. An alternative physical way [65] to obtain the exponent K is to argue that temperature introduces an effective finite system size equal to the phase breaking length L, and that the transition regime is defined by

-

Noting that r ( E ) IE - EcI-X

-

<>L, 1B - BcI-X

-

(19)

1ABI-X and that usually L, scales

20

LOCALIZATION, METAL-INSULATOR TRANSITIONS

with temperature as L,

-

T - , , one gets

-

( A B ( T,IX

that is,

x=-4

x

Within this temperature-induced effective length cutoff picture, therefore, zT =.4-', where is the temperature exponent for the phase breaking length. Again, taking x x 2.3, one gets 4 N 1. Currently, no theory is available for the phase breaking length or, equivalently, the exponent 4( = z; ') in the strong-field quantum Hall effect region, even though there have been several attempts at calculating [38,72,73] the exponent p, which determines the temperature dependence of the inelastic scattering length L, :

4

L, a T - P

(22)

In the clean limit, if the electrons move ballistically, the inelastic length L, and the phase breaking length L, are the same, making p E 4 = z; '. But in the diffusive limit one has L,

-

T-PI~

(23)

where the elastic scattering length L, is essentially temperature independent. Thus in the diffusive case, 4 = p/2 = z; Experimental value of K x 0.4 implies that either p = 1 (clean limit) or p = 2 (diffusivelimit) if one takes the localization exponent x x 2.3. These values ofp are exactly opposite to those obtained in a zero-field two-dimensional system (up to logarithmic corrections), assuming electron-electron interaction to be the important inelastic scattering mechanism [74]. There have been several recent attempts at understanding this puzzle using the electron-phonon interaction [38,72] to be the primary phase-breaking mechanism. The results of these calculations are unconvincing. None of the theories can account for the systematic variation in the experimental K value as a function of disorder (i.e., zero-field mobilities) and Landau level index, which is observed in only some of the experiments [67-701. There has also been one ambitious [75] attempt at direct measurement of the localization exponent x by studying the plateau transition in GaAs/AlGaAs samples as a function of the system size L at very low fixed temperatures, where L, is presumably much larger than L and the relevant cutoff length scale is the system size L itself with the crossover from localized to extended states occurring at L = t 1ABI-X. Using four different system sizes varying from 10 to 64pm and using a low T x 25 mK (so that Ld > L), the experimental scaling of the

'.

-

STRONG-FIELD LOCALIZATION: PHENOMENOLOGY

21

peak width AB against the system size produced [67,68,75] a best fit x x 2.3, essentially independent of the Landau level (01,l and 11) index and disorder (mobility 15,000 to 50,000 cm2V - s- ') provided that the spin splitting was resolved. Careful temperature-dependent scaling analysis in the same samples at higher temperatures (so that L, < L), where L, introduces the finite-size cutoff, also produced values of K = 4/x = l/z,x, so that 4(z), and x can be determined independently. The resulting values of 4 = z; show some variations with Landau level index and disorder with 4 x 1.2 to 1.7, which is quite different from the value 4 x 1 obtained in the earlier [65,66] temperature-dependent studies. The main difference is that K x 0.64 in these experiments [67,68,75] as against K x 0.43 in the earlier ones [65,66]. The reason for this discrepancy is not understood and is attributed uncritically to variations in the dynamical exponent zp There are two possible problems with the length-dependent study [75]. First, the scaling IABI ILI-''X is obviously very limited because there are only four values of L available. Second, the phase-coherent regime L, > L should exhibit fluctuations in transport properties because of the lack of ensemble averaging, which could complicate considerably the scaling analysis of the plateau transitions. For reasons not completely understood, conductance fluctuations in the experiments (in the L, > L regime) were small. Finally, the measured u,","^at the conductivity peak was substantially lower (in the range 0.1 to 0.2e2/h)than the theoretically expected 0.5e2/h value. (The basis of this theoretical expectation, except for some limited numerical simulations [34,76], is essentially a dimensional argument [15- 171.) Temperat ure-dependent scalinganalysis of strong-field magnetotransport has recently been extended [77] to the temperature dependenceof the u!$ima value in the quantized Hall plateau regime. In the activated transport regime, this [77] analysis gives a value of K x 0.3 to 0.7, which is consistent with the K values obtained by studying the plateau transitions.

r,

-

-

1.3.2. Plateau Transitions: Fractional Effect There has so far been only one reported experimental study [78] of temperature scaling in the fractional Hall plateau transition. The study was carried out for the plateau transition between v = 1/3 and v = 2/5 fractional quantized Hall states in high-mobility GaAs/AlGaAs systems with a reported value of K x 0.4 very close to that in the integer plateau transition. Because of the order-of-magnitudesmaller energy gap in the fractional state compared with the integral effect, the temperature range for fractional scaling is necessarily quite limited, and the value of K x 0.4 is not as reliable as it is for the integer effect. It should also be emphasued that temperature scalingin the GaAs/AlGaAsinteger plateau transition has not yet led to a decisive universal value of K, presumably due to long-range disorder fluctuations inherent in high-mobility modulation-doped structures. Thus the universality in K between integer and fractional plateau transitions is not really unambiguously established because the low-mobility samples [66],

22

LOCALIZATION,METAL-INSULATOR TRANSITIONS

which show excellent integer scaling, do not exhibit any fractional quantized Hall state. More experiments are clearly needed to clarify this unsatisfactory state of affairs. If the universality in K between integer and fractional effects holds up and leads to universality in the localization exponent x (assuming z to be the same for both effects), then interaction seems to be an irrelevant perturbation for the quantum Hall localization transition (assuming, of course, that the value of the exponent x is indeed around 2.3, as given by numerical simulations based on noninteracting models). This would be a rather remarkable result because in most (if not all) field theories [25] of the quantum localization problem, interaction is found to be a relevant perturbation which nontrivially changes the noninteracting localization expoents. There has been one numerical calculation [79] of the exponent 1, including interaction effects in a mean-field HartreeFock theory, which, not surprisingly, finds x = 2.3, the same exponent as for the corresponding noninteracting theory. This result is expected because the Hartree-Fock theory is an effective one-electron theory that should not change the exponent. There have also been some heuristic corresponding states arguments supporting the universality of the localization transition in integer and fractional cases, based on the composite fermion picture [80]. 1.3.3. Spin Effects The experimental results discussed so far are all for spin-split Landau levels. When the spin splitting is not resolved, the experimental value of K as obtained from temperature scaling of the plateau transition seems to be roughly half of that in the spin-split situation in all the experimentalstudies,(i.e., AB seems to scale as TK12in the spin-degeneratesituation) [Sl,821.Assuming the dynamical exponent z to be spin independent, this implies that the localization exponent x is twice in the spin-degenerate situation compared with the spin-resolved situation. This is very difficult to understand theoretically because within the noninteracting localization picture it is very hard to see how electron spin could be a relevant perturbation. While the experimental situation for temperature scaling in the spin-split case still remains a puzzle, a theoretical resolution has recently been suggested. It has been proposed [21] that the spin-splitting AE* between the neighboring spin levels in the spin-degeneratesituation is small but finite with A E , < k,T as well as AE* < r,where r i s typical Landau level broadening. Thus there are actually two critical energiesE,, in the spin-degeneratecase which are experimentallyunresolved because their separation, AEi, is much smaller than the experimental temperature. It is easy to see that this will lead to an effective temperature exponent ~ / if2the two unresolved transitions are interpreted as being a single localization transition. Direct numerical simulations [83] support this theoretical idea. To check this idea one needs to do temperature-dependent scaling experiments down to low temperatures (k,T < A&), whence the effective exponent ~ / should 2 change to K.

STRONG-FIELD LOCALIZATION PHENOMENOLOGY

23

1.3.4. Frequency-Domain Experiments In a remarkable recent experiment [84], the idea of quantum localization and critical scaling as applied to the plateau transition has been verified by carrying out finite-frequency( -GHz) low-temperature (-0.05 to 0.5 K) measurements of the dynamical magnetoconductance in the transitions between integer Hall plateaus in low-mobility GaAs/AlGaAs heterostructures. In these experiments the effective cutoff is provided by hw (provided that k,T << hw) and one obtains (for w > k,T/h) (ABIccwY

(24)

instead of (LIB(K T" as in the temperature scaling experiments (which are recovered here in the k,T > hw regime). In analogy with the finite-temperature measurement, one has

Y = (xz,)where z , is the frequency-domain dynamical exponent. For quantum localization phenomena, temperature and frequency should behave similarly and one expects z,

=ZT =z

(26)

where z is the dynamical exponent for the localization transition. All these expectations were verified spectacularly in this important experiment [84], where the frequency scaling exponent y = 0.41 f 0.04 was found in the spinresolved situation and an exponent y/2 N 0.21 was found in the spin-degenerate situation, in excellent agreement with the temperature scaling exponent K z 0.43. If the localization exponent 1, which should be the same in temperature and frequency scaling experiments, is taken to be the accepted value of 7/3, this implies that z N 1 for quantum Hall plateau transitions.

1.3.5. Magnetic-Field-Induced Metal-Insulator Transitions There has been a great deal of recent experimental activity in studying the direct transition between a zero-field two-dimensionalinsulating phase and a finite field quantum Hall phase in highly disordered low-density samples. Because of the essential requirement of the existence of extended states for the quantized Hall effect phenomenon, these experiments can be thought of as a field-induced two-dimensional delocalization transition [IS). Several groups have carried out such experimentsin the last few years, and the interpretation of these experiments is a matter of some controversy at the present time [50-56,61-641. The most spectacular and convincing experiments [50-53,641 of this type start with a zero-field highly disordered and low-density strongly localized insulating system which eventually exhibits a quantum Hall phase at finite magnetic

24

LOCALIZATION,METAL-INSULATOR TRANSITIONS

fields before becoming strongly insulating again at very high fields. There are, therefore, at least two nonuniversal critical fields, B,, and B,, > B,,,in these experiments: For B < B,, and B > B,, the system is strongly insulating, as manifest in the temperature dependence of pxx(T), with pxx(T) increasing (exponentially) strongly with decreasing temperature (typically in the T 0.05 to 1K regime, but higher temperatures have also been used), usually the high-field regime (B> B,,) being even more strongly localized (with very high pxx values) than the low-field regime (B < B,,); for B,,< B < BE,, the magnetoresistance pxx(T)shows a distinct minimum at a particular magnetic field value, with the minimum becoming sharper with decreasing temperature as pxx(7')increases at higher temperature, and p,,(T) shows a quantized Hall plateau corresponding to the pxx(T)minima. The intermediate quantum Hall phase (for B,,< B < B,,), which clearly arises from field-induceddelocalization at B = Bcl,is novel and has not been seen in three-dimensional systems. The nonuniversal critical fields B,, and B,, which define the delocalized state locations depend on the disorder strength, and pxx(T)is essentially temperature indepedent at B = Bcl,B,,. The quantization of the Hall resistance pxyin the B,, < B < B,, regime is quite good, but perhaps not as perfect as in samples of better quality. These field-induced metal-insulator transition experiments are important because they establish beyond any reasonable doubt that nonperturbative delocalization (not found within the prevailing nonlinear sigma model localization theories) indeed occurs (even) in a (strongly) disordered two-dimensional system, due to the application of an external magnetic field. Unambiguous observation of a quantum Hall phase straddled on both low- and high-field sides by strongly insulating phases is a rather spectacular phenomenon, sometimes referred to as a reentrant insulating transition. Such reentrant insulating transitions have been observed in both integer [50,53,64] and fractional [63,85-871 quantum Hall states. Developing a unified quantitative theory for a field-induced metal-insulator transition (or, more precisely, quantum Hall conductor-Anderson insulator transition) has been difficult and controversial, partly because the transport measurements described above allow for competing and conflicting physical interpretations. A popular and physically appealing explanation [15-17] is based on the idea of the floating of strong-field extended states as the magnetic field is lowered. In particular, the extended-state E," associated with the Nth Landau level is postulated [16] to float up in energy according to the heuristic formula

-

E," = EN[l + ( ~ , z ) - ~ ] = hw,[l

+(WCz)-2](N

+ 1/2)

(27)

where z is an effective scattering time related to the effective Landau level broadening r = h/22. No microscopic derivation of Eq. (27) is available, but presumably a strong Landau level coupling effect in low magnetic fields (particularly at high disorder, whence q z << 1) produces extended-state floating. As

STRONG-FIELD LOCALIZATION PHENOMENOLOGY

-

25

B-rO, the extended states float up to infinite energy according to Eq. (27) as E," (h/wcr2)(N 1/2), leaving only localized states at the Fermi level in the wcz<< 1 limit. Starting from the zero-field limit, increasing magnetic field lowers the extended states from infinite energy with a quantum Hall state showing up when the N = 0 extended state crosses the Fermi level (presumably at B = Bcl), causing the field-induced delocalization transition. Note that the floating scenario inevitably implies that the very first weak field quantized Hall plateau to form (in the spin-split or spinless situation) must necessarily be the N = O quantum Hall phase. This specific prediction has never been verified experimentally in high-mobility systems [88,89], where a large number of quantized Hall plateaus are invariably seen, starting with some large value of N in the weak-field limit, and the v = 1 plateau shows up only in the strong-field (rather than the weak-field) limit. A direct observation of the v = 1 plateau in both weak- and strong-field regimes, which seems experimentally unlikely to happen, is needed for unambiguous verification of the floating [16] and the associated global phase diagram [171concept. The problem is that in good-quality samples w,z becomes larger than unity for very small values of w, (i.e., the magnetic field), and therefore the weak-field transition to the v = 1 quantum Hall phase occurs at very small values of B, making it impossible to observe the state except at unattainably low temperatures. The question arises about the universality class of the field-induced insulatorquantum Hall liquid delocalization transition. The issue has been investigated experimentally by carrying out temperature-dependent scaling studies [52, 53, 901 around the critical magnetic fields for the metal-insulator transitions. While the details for these temperature-dependent analyses differ somewhat from the corresponding plateau transition studies, the basic idea of a quantum phase transition is the same in both cases. These studies have led to differing values of the temperature exponent K characterizing the field-induced metalinsulator transition. Both K x 0.4 and K x 0.2 have been reported, which agree respectivelywith the values of K for the spin-resolved and spin-degenerateplateau transition situations, respectively. While the quantitative reliability of the measured K is substantially less in field-induced transitions than in plateau transition, universality among the two transitions would require K x 0.2 because the fieldinduced transition experiments [42,53,90] determining the scaling exponent K observed transitions from a zero-field insulator to a spin-degenerate v = 2 quantum Hall state, which then went into a strongly insulating state as the magnetic field is increased. The reason for the experimental discrepancy in the value of K is not understood, and whether the field-induced transition and the plateau transition fall in the same universality class is not yet absolutely clear from the experimental standpoint. There are heuristic theoretical arguments [17] as well as a recent numerical simulation 1591claiming that the plateau transition and the field-induced insulator-metal transition are in the same universality class. One glaring conceptual difficulty with the field-induced insulator-quantum Hall liquid transition has been that all of the hitherto published experimental

+

26

LOCALIZATION,METAL-INSULATOR TRANSITIONS

work observes only the transition from insulator to v = 2 quantum Hall state (as the magnetic field is increased from zero), without seeing any trace of the v = 1 lowest quantum Hall phase. In the conventional extended-state floating-global phase diagram picture [16,17], the first weak-field transition should be to the N = 0 quantum Hall state (instead of the experimentally observed N = 1 state) because that is the lowest-energyfloating extended state, which should be the first one to pass through the chemical potential. This conceptual difficulty has been explained away by involving spin degeneracy, assuming that the observed v = 2 quantum Hall state is really the two spin-degenerate N = 0 Landau level states (not the N = 1 spin-split Landau level state) whose spin splitting is unresolved at the experimental magnetic field values. This is somewhat surprising because at magnetic field values as high as 10T one would have expected the N = 0 Landau level to be spin resolved. The issue is of great importance because the extendedstate floating concept (and the closely related global phase diagram idea) are based on the insulator-quantum Hall liquid transition occurring through the v = 1 quantum Hall liquid state. It is, therefore, of considerable significance that a very recent experiment [64] observes a zero-field insulator-v = 2 quantum Hall state-v = 1 quantum Hall state-insulator transition with increasing magnetic field in highly disordered GaAs/AlGaAs heterostructures. The expected insulator-1-2-1-insulator transition based on the floating-global phase diagram picture is simply not seen in this (or, for that matter, any other) experiment. Increasing disorder in this experiment first destroys the v = 2 quantum Hall phase and eventually, both quantum Hall phases. A good understanding of this particular experiment is needed to make further progress on the field-induced metal-insulator transition question. It is possible that interaction is playing a rather crucial role in this transition [64]. The issue of floating of extended state has been investigated numerically in a number of recent publications [21,58-601. Most calculations [21,58,59] obtain small upward shifts in the critical energies E, as the Landau level coupling becomes significant with decreasing magnetic field, but the shifts are typically small, and whether they approach infinite energy in the weak-field limit is completely unknown. One recent numerical study [60] on a lattice concludes that there is no extended-state floating whatsoever; instead, the extended states simply .disappear at finite critical magnetic fields, with the lower N extended states disappearing at lower critical fields. Another numerical study [59] using a similar lattice model arrives at a different conclusion,claiming to find extendedstate floating as well as universality between field-inducedtransitions and plateau transitions. Finally, the reentrant field induced insulating transition has also been seen [63,85-871 in fractional quantum Hall systems (at high fields and in highmobility structures) where increasing field takes one through the insulatorfractional Hall state-insulator phase transitions. The phenomenology of these transitions has many similarities with that of the field-induced transitions in the integer situation. The possible existence of a high-field Wigner solid phase, however, considerably complicates the theoretical picture, and it is perhaps fair

STRONG-FIELDLOCALIZATION PHENOMENOLOGY

27

to say that a complete theoretical understanding of the problem does not yet exist. A number of recent field-induced insulator-quantum Hall metal transition experiments have concentrated on measuring the critical conductance, a,, at the transition point to check its universality. One experiment [89] finds a universal pxxz e2/h for transitions to the insulating phase from both the v = 1 quantum Hall phase and the v = 1/3 fractional quantum Hall phase. Another experiment [90] finds similar values for the critical B,, in the integer (v = 2) and the fractional (v = 1/3) situations, but the matrix inversion involved in the pxx-to-a, transformation is always a problematic issue. For the plateau transitions, universality of ,a at the critical field B, has remained essentially unverified, with ,:a values ranging from 0.05e2/h to 0.5ez/h, depending on the experiment (in general, experiments showing better scaling behavior obtain B:, values very far from e2/2h). A very recent Corbino geometry experiment [9l], which measures a, directly, very clearly demonstrates that ::a values are substantially below the expected universal value of e2/2h. This has been attributed to interaction effects [91]. Thus the universality of the magnitude of a,, at the transition point E, has remained an open question, particularly for the plateau transitions. Part of this problem may very well be arising from finite temperature effects-the theoretical expectation of a universal = e2/2h is based on a T = 0 quantum phase transition picture, and finite temperature corrections to this universal conductance are not easy to calculate theoretically. In particular, it is possible ( :for :a T) to have nonuniversal finite temperature corrections from disorder and interaction effects, and an extrapolation to T = 0 in estimating o‘,, based on finite temperature data may not necessarily work. In concluding this section it should be emphasized [18] that while the plateau transition involves a,, p,, = 0; a,,,, p,,, = quantized to a,, p,, = 0; a,,,, pX, = quantized phases, the field-induced insulating transition involves a,, = 0, p,, = 00; a,,,, p,,, = unquantized to a,,, p,, = 0; ox,,, px,, = quantized to B,, = 0, p,, = 00, a,,,, pxy= unquantized phases through intermediate “unstable metallic” (a,,, pxx# 0) phase transition points. Since all two-dimensional systems are, in principle, insulating at B = 0, and since all plateau transitions eventually terminate in a transition to an insulating phase (p,, = m, a,, =0) from some low v (usually v = 1,2,1/3,1/5, etc.) quantum Hall phase, one can think of all two-dimensional quantum Hall systems as essentially exhibiting field-induced reentrant insulating transitions with quantum Hall phases and unstable metallic transition points in between the B = 0 and high-B insulating phases [l8]. What is interesting is that some of these systems, particularly the very low disorder/high mobility samples [63,85-88], also exhibit some nontrivial intermediate insulating phases, probably arising from a correlation-driven Wigner solid phase. Obviously, disorder and interaction both play significant roles in this high field/ low v Wigner phase. However, our knowledge of this Wigner “glass” or disordered electron solid phase is quite limited at the present time.

ex

28

LOCALIZATION, METAL-INSULATOR TRANSITIONS

1.4. RELATED TOPICS 1.4.1. Universality

The great deal of experimental and numerical work carried out in quantum Hall plateau transitions and in field-inducedreentrant insulating transitions (through intermediate quantum Hall phases) during the last ten years or so establishes beyond any doubt the existence of localization-delocalization phase transitions in strong-field two-dimensional electron systems. These transitions are now established to be bulk quantum critical phenomena-the edges do not play any roles here because ox.. # 0 at the transition. The fact that these phase transitions are quantum critical phenomena, characterized by the localization exponent x and the dynamical exponent z, is also well established. The absence of a quantitatively predictive field theory for the problem (as emphasized before, the two-parameter scaling theory based on the nonlinear sigma model and the nonperturbative topological theta term is, at best, only a framework for a theory that has not yet led to any concrete calculated results) does not detract from these conclusions. An important theoretical issue for the quantum Hall localizationdelocalization transition is the question of universality [20,21]. The extreme view, which is shared by many, that all plateau transitions (independent of Landau level index, the nature of disorder potential, and the electron-electron interaction effect) and all field-induced insulator-to-metal transitions (with one exception)belong to a single localization universality class with the localization exponent x = 7/3 N 2.3 (and the dynamical exponent z = 1) is not inconsistent with the experimental results (with one exception), which most often yield K = (zx)- N 0.4 for spin-split plateau transitions, any variation in IC with disorder/ Landau level index being attributed to crossover effects. The exception is the high-field reentrant insulator-fractional quantum Hall phase-insulator transition observed in several experiments (for both electrons and holes) involving extremely high mobility/low disorder samples [63,85-881. These transitions are most likely to be Wigner solid-quantum Hall fluid phase transitions rather than disorder-driven continuous transitions. Theoretically,the issue of universality is subtle and difficult to settle without a quantitative calculational theory. There are three primary aspects to the universality question involving the two critical exponents x and z, and the zerotemperature maximum peak value of the longitudinal conductivity = u;? exactly at the critical energy E, (or, equivalently, at the critical magnetic field B,). Note that at T = 0, crxx= 0 everywhere except for E(B)= E,(B,), where oXx= a,”,”x = e2/2h is the expected universal conductivity. The current status of the universality question with respect to x,z, and 0;: is summarized below. The most extensively numerically studied quantity is the noninteracting localization exponent x for the integer plateau transition, which has been studied by at least three different techniques: quantum percolation network model [43,44], recursive Green’s function (or transfer matrix)-based [20,21] whitenoise disorder Anderson model, and Chern numer calculations [45]. All of these

RELATED TOPICS

29

techniques given an exponent x N 2.3 N 7/3 in the lowest Landau level, which is also consistent with a crude theoretical argument [37] for tunneling correction to the percolation transition. The white-noise disorder calculations in the lowest Landau level have been extended to include a finite range in the disorder potential and Landau level coupling effects, leading essentially to the same value of the exponent. This has led to the viewpoint that the exponent x = 7/3 2: 2.3 is universa1,independent of the Landau level index as well as the range and strength ofthedisorder potential [20]. Recently,the same exponent ( x 7/3) is claimed [59] to have been found in a numerical Chern number calculation of the field-induced insulator-quantum Hall state transition, making the localization transition completely universal. The only discrepancy arises from the fact that every numerical calculation [39-421 of the localization exponent in higher Landau levels assuming short-range white-noise disorder potential unambiguously produces an exponent x x 5, calling into question the Landau level independence of x for zero-range disorder. For finite-range disorder (with the range being comparable to or larger than the magnetic length) potential [21,41] and/or in the presence of Landau level coupling effects [Zl], the exponent x seems to be universal, with x z 7/3. Since some Landau level coupling is always present in the experimental situation, it is safe to assume that x is universal ( x x 7/3) in the experiments. The theoretical question, however, remains as to whether for uncoupled Landau levels x really depends on the Landau level index for short-range disorder potential. In a recent publication [92] it has been claimed that the Landau level dependence of x for short-range potentials is a very slow crossover effect arising from the existence of irrelevant scaling field operators in the underlying field theory, whose existence is postulated in an ad hoc manner. Introduction of this irrelevant operator (with its own scaling exponent) allows one to manipulate almost arbitrarily the limited available numerical data, leading to the conclusion that the asymptotic exponent is indeed the same in the higher Landau level as in the lower level even for zero-range disorder, restoring universality. Careful analysis [93] of the numerical work shows that the numerics associated with the irrelevant variable is nonunique, and the whole exercise is essentially a tautology where one asserts universality,which leads immediately to the conclusion that there must be some very long irrelevant length scales in the problem causing this crossover. While the issue is somewhat academic in nature, the actual value of x in higher Landau levels for short-range disorder in the Landau level uncoupled situation remains an open question. Although it is quite possible that x for short-range disorder in higher Landau levels is indeed 7/3 x 2.3, it is fair to say that at the present time this universality has not yet been established theoretically or numerically. Note that similar problems exist even in three dimensions, where the theory is on much firmer grounds [29]. The universality of the dynamical exponent z is not well understood at the present time except for the theoretical argument [33] that z = 1 for all quantum phase transitions in two dimensions involving Coulombic systems. Note that if K = (zx)- x 0.4 experimentally,then z x 1 if x x 2.3. The argument that z = 1 for Coulomb systems in two dimensions essentially arises from the V, l/r nature of N

30

LOCALIZATION, METAL-INSULATOR TRANSITIONS

the Coulumb interaction. It has therefore been suggested [21,71] that one should carry out experiments in very thick two-dimensionalsamples where V, In r or in the presence of metallic back gates whence V, l/r2 so as to modify the dynamical exponent. There has recently been a current heating experiment [94] which measures the inelastic scattering exponent p directly with the finding that p w 2, which again implies that z = 2/p x 1. It seems that this current heating experiment may be an ideal candidate for gated experiments that modify z by changing the Coulomb interaction. Current heating experiments [95] are also useful for studying electron-phonon interaction mechanisms in the quantum Hall regime. In the noninteracting system z = 2 is expected. Finally, the issue of the universality ,:of;a which is expected to be e2/2h at the transition at T+O, has remained a puzzle because the measurements exhibiting the best localization scaling behavior invariably show large deviations in ::a from the expected e2/2h value. While several experiments and numerical calculations claim the universal value : : a = e2/2h for both the plateau transition and the field-induced transition, it is best to concede that most experimentsshow a,","" e2/2h based on an extrapolation of finite-temperature results. Recently, a direct Corbino geometry measurement of axxhas been carried out [Sl], showing clear deviation from the universal e2/2h value, which is attributed to interaction effects. More work is needed to resolve this question. N

N

-=

1.4.2. Random Flux Localization

The localization transition discussed in this chapter so far is associated with the strong-field localization/delocalization quantum phase transition in the quantum Hall effect phenomenon. A somewhat related phenomenon has attracted a great deal of recent theoretical attention [96-1061. This is the question of the nature of the ground state of a noninteracting two-dimensional fermion in an external static random magnetic field with zero mean. Specifically,within a lattice model one assumes that the external magnetic flux varies randomly spatially, with the net averaged flux being zero. There is no random disorder potential, the only disorder arising from the random magnetic flux. The problem may be of interest in certain gauge theories of high-temperature superconductivity and in Chern-Simons theory [1071 of composite gauge fermions at half-filled Landau levels (see Chapter 6). At half-filled Landau levels the composite fermions invariably move in a random flux environment, due to the background gauge fluctuations, and the random flux localization problem may be relevant [lOS]. Theoretically,the problem is nontrivial only if it does not belong to the unitary ensemble. If it belongs [lo31 to the unitary ensemble, the nonlinea: sigma model asserts [27] that all states are localized with extremely long (( ego) localization lengths. It has been argued [lo21 that the random flux problem may not belong to the unitary ensemble because, while there is no net topological term as in the corresponding quantum Hall problem, there may be a local topological density arising from the random flux fluctuations. The problem has,therefore, been studied via direct numerical simulations by a number of groups [98-101, 104-

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31

1061. The two largest numerical simulations of the lattice model based on a transfer matrixcalculation [1041and a Chern number calculation [lo51 show the existence of a delocalization transition whose critical energies and exponents are consistent with each other. Other calculations [99] (on somewhat smaller system sizes) and analytical arguments [1031 claim that all states are localized, the system being in the unitary ensemble. This problem is of importance in the quantum Hall phenomena because of its possible relevance [108) to transport properties at half-filling. Clearly, more work is needed to clarify this issue. A direct experimental investigation of the two-dimensionalelectron eigenstates in a random static magnetic field seems to be a difficult task. Presumably, the magnetotransport behavior at v = n $, where n is an integer or zero, will manifest some aspects of the random flux localization problem because the effective composite fermions, following the Chern-Simons gauge transformation [1071, move in a zero average external field environment where gauge fluctuations associated with disorder may make [lo81 the problem equivalent to a random flux localization problem. Whether these composite fermions are localized in the unitary ensemble sense or extended at the band center has important conceptual and experimental implications that have not yet been explored in detail. Finally, if there is a random flux delocalization transition, a question arises about its universality class, the natural speculation [1043 being that it belongs to the same universality class as do quantum Hall plateau transitions.

+

REFERENCES K. von Klitzing, G. Dorda, and M.Pepper, Phys. Rev. Lett. 45,494 (1980). Th. Englert and K. von Klitzing, Sut$ Sci.73,70 (1978). S. Kawaji, Sut$ Sci. 73,46-49 (1978)and references therein. We denote the localization critical exponent as x throughout this chapter instead of the more traditional notation v, which we reserve for the Landau level filling factor. 5. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42,673 (1979); D. Vollhardt and P. Wolfle, Phys. Rev. Lett. 45,483 (1980); P. W. Anderson, E. Abrahams, and T. V. Ramakrishnan, Phys. Rev. Lett. 43, 718 (1979); L. P. Gorkov, A. I. Larkin, and D. E. Khmel'nitskii, JETP Lett. 30,248 (1979). 6. F. Wegner, Z. Phys. B 25,327 (1976);35,207 (1979);51,279 (1983); L. Schafer and F. Wegner, Z. Phys. B 38,113 (1980). 7. P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985) and references therein. 8. R. E. Prange, Phys. Rev. B 23,4802 (1981);R. E. Prange and R. Joynt, Phys. Rev. B 25, 2943 (1982); R. Joynt and R. E. Prange, Phys. Rev. B 29,3303 (1984). 9. R. B. Laughlin, Phys. Rev. B 23,5631 (1981). 10. B. I. Halperin, Phys. Rev. B 25, 2185 (1982). 11. P. Streda, J . Phys. C 15, L717 (1982). 1. 2. 3. 4.

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12. H. Aoki and T. Ando, Solid State Commun. 38, 1079 (1981). 13. S. V. Iordansky, Solid State Commun. 43,1(1982); R. F. Kazarinov and S . Luryi, Phys. Rev. B 25,7626 (1982). 14. D. J. Thouless, J . Phys. C 14,3475 (1981); see also Q. Niu, D. J. Thouless, and Y . S . Wu, Phys. Rev. B 31, 3372 (1985); Q. Niu and D. J. Thouless, Phys. Rev. B 51,2188 (1987). 15. H. Levine, S. B. Libby, and A. M. M. Pruisken, Phys. Rev. Lett. 51,1915 (1983); A. M. M. Pruisken, Phys. Rev. Lett. 61,1297 (1988). 16. D. E. Khmel’nitskii, J E T P Lett. 38, 552 (1983); Phys. Lett. lMA, 182 (1984); R. B. Laughlin, Phys. Rev. Lett. 52,2304 (1984). 17. S. Kivelson, D. H. Lee, and S. C . Zhang, Phys. Rev. B 46,2223 (1992). See also C. A. Liitken and G. G. Gross, Phys. Rev. B 48,2500 (1993). 18. Such field-induced transitions from insulator ( B = 0) to quantum Hall metal ( B > 0)

fall into three broad phenomenological categories, depending on the disorder strength in the system (as characterized, for example, by the zero-field mobility). In the most common situation of moderate- to high-quality samples, the system is weakly localized at B = 0 with extremely long localization length and behaves as essentially metallic at B=O. In such a system, application of an external magnetic field eventually produces some high-N quantized Hall plateau with progressively lower N plateaus forming at higher B fields, until at high-enough field values, the system makes a transition to an insulator from a v = 1 integer Hall plateau or a v = fractional Hall plateau. No low-field insulator-to-metal transition is seen in these systems because the zero-field system is essentially metallic in character. Also, the first quantized Hall plateaus appearing at low fields in these samples are invariably some large N plateaus, with progressively lower N plateaus appearing at higher fields, with the plateau transitions terminating through a single transition to an insulator from the w = 1 (or 1/3) plateau. In extremely high quality-low disorder (i.e., very high zero-field mobility) samples, a reentrant insulator-fractional quantum Hall-insulator transition is seen at a low value of the filling factor v( z 4 usually), with the reentrant transition becoming sharper with decreasing disorder. In highly disordered low-density samples (which are strongly localized at B = 0, with extremely low mobilities), there is a field-induced insulator-quantum Hall liquid (usually, but not always, to the w = 2 quantum Hall state) transition, followed by another transition to a high-field insulator phase at still higher fields. While phenomenologically this reentrant insulator-quantum Hall liquid-insulator transition is similar to the reentrant transition in the fractional case observed in very high mobility samples, the latter is more likely to be a transition between Laughlin liquid and Wigner solid phases, whereas the former is more likely to be a disorder-driven Anderson localization transition. A quantitative localization theory, which takes into account both Coulomb correlations and strong disorder effects, does not currently exist. It should be emphasized that all two-dimensional systems exhibiting a quantum Hall effect must necessarily fall into one of these three categories of field-induced metalinsulator transitions, the generic situation being that of the moderately disordered case where the B = O weakly localized system makes a transition to a high-field insulating state after undergoing several plateau transitions at intermediate field values (see, e.g., Refs. [88] and [89] for typical examples), whereas the more specialized reentrant insulator-metal-insulator transition occurring only in ex-

5

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24. 25. 26. 27. 28.

29.

tremely highly disordered (see Refs. [SO], [53]. and [64] for typical examples) samples for the integer case, or extremely pure samples (see Refs. [85] to [87] for typical examples) for the fractional case. P. K. Lam and S. M. Girvin, Phys. Rev. B 30,473 (1984);31,613(E);R. Price, X, Zhu, S. Das Sarma, and P. M. Platzman, Phys. Rev. B 51,2017 (1995). B. Huckestein, Rev. Mod. Phys. 67, 357 (1995) and references therein. D. Z. Liu and S. Das Sarma, Phys. Rev. B 49,2677 (1994). M. Janssen, 0.Viehweger, V. Fastenrath, and J. Hajdu, Introduction to the Theory of the Integer Quantum Hall Effect (VCH, New York, 1994). The basic phenomena of the quantum Hall effect and the development of the subject up to about 1985 are discussed in the book The Quantum Hall Effect, edited by R. E. Prange and S. M. Girvin (Springer-Verlag, New York, 1987). The chapter by Pruisken in that book discusses the localization problem from a field-theoretic perspective. For a specialized(and nongeneric) counterexample, see M. Ya Azbel, Phys. Rev. Lett. 67, 1787 (1991). D. Belitz and T. R. Kirkpatrick, Rev. Mod. Phys. 66,261 (1994) and references therein. F. Wegner, Nucl. Phys. B 316,663 (1989). S. Hikami, Prog. Theor. Phys. Suppl. 107,213 (1992) and references therein. C. Jiang, D. C. Tsui, and G. Weimann, Appl. Phys. Lett. 53, 1533 (1988); A. L. Efros, Solid State Commun. 70,253 (1989).See also B. J. Lin, M. A. Paalanen, A. C. Gossard, and D. C. Tsui, Phys. Rev. B 29,927 (1984)for an early weak localization study of two-dimensional GaAs structures. The numerical value for the localization exponent x is 1.5k0.2 for the threedimensional orthogonal ensemble [B. Kramer, K. Broderix, A. MacKinnon, and M. Schreiber, Physica A 167,163 (1990);E. Hofstetter and M. Schreiber, Europhys. Lett. 21, 933 (1933); P. Lambrianides and H. B. Shore, Phys. Rev. B. 50, 7268 (1994)], whereas the nonlinear sigma model based theory gives 0.73 [27]. For the corresponding two-dimensional symplectic case, numerical simulations give x 1.5, whereas theory predicts a very different exponent [27]. Remarkably, numerical simulations essentially given the same localization exponent, x = 1.35 k0.15, for a disordered three-dimensional Anderson model in the presence of a strong external magnetic field [M. Henneke, B. Kramer, andT. Ohtsuki, Europhys. Lett. 27,389 (1994)], which from a nonlinear sigma model viewpoint should be in a completely different universality class! Thus, at the present time there is major qualitative and quantitative disagreement between nonlinear sigma model results and numerical simulations. Whether this arises from crossover and finite-size effects in the simulations or from the inherent problem of an &-expansionin the nonlinear sigma model (e.g. E = 1 in the three-dimensional orthogonal ensemble) is not known. For a very recent serious attempt at constructing a field theory for the localization transition in quantum Hall effect, see the ambitiouseffort by A. W. W. Ludwig, M. P. A. Fisher, R.Shankar, and G. Grinstein, Phys. Rev. B 50, 7526 (1994),where many earlier references on the field-theoretic approach can be found. This paper, however, concludes with the pessimistic remark: “Unfortunately, we have been unable so far to access this generic quantum Hall fixed point analytically.” The problem, as in other field theories, is that strong-coupling fixed points (which are by definition nonpertur-

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33

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LOCALIZATION,METAL-INSULATOR TRANSITIONS

bative) are notoriously difficult to handle analytically unless one somehow finds an exact solution. The quantum Hall effect is a paradigm for the nonperturbative strong-coupling behavior where the effect simply does not exist within the perturbative nonlinear sigma model theories. 31. H. A. Weidenmiiller and M. R. Zirnbauer, Nucl. Phys. B 305, 339 (1988); M. R. Zirnbauer, Ann. Physik. 3, 513 (1994). 32. A. Houghton, J. R. Senna, and S . C. Ying, Phys. Rev. B. 25,2196 (1982); 25,6468 (1982). 33. M. P. A. Fisher, G. Grinstein, and S . M. Girvin, Phys. Rev. Lett. 64,587 (1990);M. P. A. Fisher, Phys. Rev. Lett. 65,923 (1990). 34. Y. Huo, R. E. Hetzel, and R. N. Bhatt, Phys. Rev. Lett. 70,481 (1993). 35. B. Huckestein and B. Kramer, Phys. Rev. Lett. 64, 1437 (1990). 36. S. A. Trugman, Phys. Rev. B 27,7539 (1983). 37. G. V. Mil'nkov and I. M. Sokolov, JETP Lett. 48,536 (1988). 38. H. L. Zhao and S . Feng, Phys. Rev. Lett. 70,4134 (1993). 39. H. Aoki and T. Ando, Phys. Rev. Lett. 54,831 (1985). 40. D. Z. Liu and S . Das Sarma, Mod. Phys. Lett. B 7,449 (1993). 41. B. Huckestein, Europhys. Lett. 20,451 (1992). 42. B. Mieck, Z. Phys. B 90,427 (1993). 43. J. T. Chalker and P. D. Coddington, J. Phys. C 21,2665 (1988). 44. D. H. Lee, Z. Wang, and S . Kivelson, Phys. Rev. Lett. 70,4130 (1993). 45. There has also been a Chern number calculation-based numerical simulation of the critical exponent x in the lowest Landau level for finite-size systems [Y. Huo and R. N. Bhatt, Phys. Rev. Lett. 68, 1375 (1992)], with the results x = 2.4 f 0.1, which is completely consistent with other numerical results. 46. 1. Chang, Z. Lev, A. B. Harris, J. Adler, and A. Aharony, Phys. Rev. Lett. 14, 2094 (1995).Quantum percolation in this paper means something different from that in the quantum Hall context, as in [43]. 47. H. W. Jiang, C. E. Johnson, and K. L. Wang, Phys. Rev. B 46,12830 (1992). 48. T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54,437 (1982) and references therein. 49. C. E. Johnson and H. W. Jiang, Phys. Rev. B 48,2823 (1993). 50. H. W. Jiang, C. E. Johnson, K. L. Wang, and S . T. Hannahs, Phys. Rev. Lett. 71,1439 (1993). 51. A. A. Shashkin, G. V. Kravchenko, and V. T. Dolgopolov, J E T P Lett. 58, 220 (1993). 52. T. Wang, K. P. Clark, G. F. Spencer,A. M. Mack,and W. P. Kirk, Phys. Rev. Lett.72, 709 (1994). 53. R. J. Hughes, J. T. Nicholls, J. E. F. Frost, E. H. Linfield, M. Pepper, C. J. B. Ford, D. A. Ritchie, G. A. C. Johnes, E. Kogan, and M. Kaveh, J. Phys. Condens. Matter 6, 4763 (1994). 54. B. W. Alphenaar and D. A. Williams, Phys. Rev. B 50,5795 (1994). 55. I. Glozman,C. E. Johnson,and H. W. Jiang, Phys. Rev. Lett. 74,594(1995);Phys. Rev. B 52, R14348 (1995).

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56. S. V. Kravchenko, W. Mason, J. E. Furneaux, and V. M. Pudalov, Phys. Rev. Lett. 75, 910 (1995). 57. T. Ando, J. Phys. Jpn. 53,3126 (1984); Phys. Rev. B 40,5325 (1989). 58. T. V. Shahbazyan and M. E. Raikh, Phys. Rev. Lett. 75,304 (1995). 59. K. Yang and R. N. Bhatt, Phys. Rev. Lett. 76, 1316 (1996). 60. D. Z. Liu, X. C. Xie, and Q. Niu, Phys. Rev. Lett. 76,975 (1996). 61. A. A. Shashkin, G. V. Kravechenko, V. T. Dolgopolov, S. V. Kravchenko, and J. E. Furneaux, Phys. Rev. Lett. 75,2248 (1995). 62. I. Glozman, C. E. Johnson, and H. W. Jiang, Phys. Rev. Lett. 75,2249 (1995). 63. H. C. Manoharan and M. Shayegan, Phys. Rev. B 50,17662 (1994). 64. D. Shahar, D. C. Tsui, and J. E. Cunningham, Phys. Rev. B 52, R14372 (1995). 65. H. P. Wei, D. C. Tsui, M. A. Paalanen, and A. M. M. Pruisken, Phys. Rev. Lett. 61, 1294 (1988). 66. H. P. Wei, S.Y. Lin, D. C. Tsui, and A. M. M. Pruisken, Phys. Rev. B 45,3926 (1992). 67. S. Koch, R. J. Haug, K. von Klitzing, and K. Ploog, Phys. Rev. B 46,1596 (1992). 68. S. Koch, R. J. Haug, K. von Klitzing, and K. Ploog, Mod. Phys. Lett. B 6, 1 (1992). 69. A. J. Dahm, private communication. 70. J. Wakabayashi, M. Yamane, and S . Kawaji, J. Phys. SOC.Jpn. 58, 1903 (1989); J. Wakabayashi, A. Fukaro, S. Kawaji, T. Koike, and T. Fukose, Surf: Sci. 229, 60 (1990). 71. D. G. Polyakov and B. I. Shklovskii, Phys. Rev. Lett. 70,3796(1993); Phys. Rev. B 48, 11167 (1993); see also V. Kagalovsky, B. Horovitz, and Y. Avishai, Europhys. Lett. 31, 467 (1995). 72. T. Brandes, L. Schweitzer, and B. Kramer, Phys. Rev. Lett. 72,3582 (1994). 73. T. Brandes, Phys. Rev. B 52,8391 (1995). 74. E. Abrahams, P. W. Anderson, P. A. Lee, and T. V. Ramakrishnan, Phys. Rev. B 24, 6783 (1982). 75. S. Koch, R. J. Haug, K. von Klitzing, and K. Ploog, Phys. Rev. Lett. 67,883 (1991). 76. J. T. Chalker and G. J. Daniell, Phys. Rev. Lett. 61, 593 (1988). 77. S. Das Sarma and D. Z . Liu, Phys. Rev. B 48,9166 (1993). 78. L. W. Engel, H. P. Wei, D. C. Tsui, and M. Shayegan, SurJ Sci. 229, 13 (1990). 79. S. R. E. Yang, A. H. MacDonald, and B. Huckestein, Phys. Rev. Lett. 74, 3329 (1995). 80. J. K. Jain, S. A. Kivelson, and N. Trivedi, Phys. Rev. Lett. 64, 1297 (1990). 81. S. W. Hwang, H. P. Wei, L. W. Engel, D. C. Tsui, and A. M. M. Pruisken, Phys. Rev. B 48,11416 (1993). 82. H. P. Wei, S. W. Hwang, D. C. Tsui, and A. M. M. Pruisken, Surf: Sci.229,34 (1990). 83. D. K. K. Lee and J. T. Chalker, Phys. Rev. Lett. 72,1510 (1994); D. G. Polyakov and M. E. Raikh, Phys. Rev. Lett. 75,1368 (1995); see also C. B. Hanna, D. P. Arovas, K. Mullen, and S . M. Girvin, Phys. Rev. B 52,5221 (1995). 84. L. W. Engel, D. Shahar, C. Kurdak, and D. C. Tsui, Phys. Rev. Lett. 71,2638 (1993). 85. H. W. Jiang, R. L. Willett, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer,and K. W. West, Phys. Rev. Lett. 65,633 (1990); Phys. Rev. B 44,8107 (1991).

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86. T. Sajoto,Y. P. Li, L. W. Enge1,D. C. Tsui, and M. Shayegan, Phys. Rev. Lett. 70,2321 (1993). 87. V. J. Goldman, J. K. Wang, B. Su, and M. Shayegan, Phys. Rev. Lett. 70,647 (1993). 88. T. Sajoto, Y. W. Suen, L. W. Engel, M. B. Santos, and M. Shayegan, Phys. Rev. B41, 8449 (1990). 89. D. Shahar, D. C. Tsui, M. Shayegan, R. N. Bhatt, and J. E. Cunningham, Phys. Rev. Lett. 74,451 1 (1995). 90. L. W. Wong, H. W. Jiang, N. Trivedi, and E. Palm, Phys. Rev. B 51, 18033 (1995). 91. L. P. Rokhinson, B. Su, and V. J. Goldman, Phys. Rev. B 52, R11588 (1995);see also P. T. Coleridge, P. Zawadzki, and A. S. Sachrajda, Phys. Rev. B 49,10798 (1994);P. T. Coleridge, Phys. Rev. Lett. 72,3917 (1994). 92. B. Huckestein, Phys. Rev. Lett. 72, 1080 (1994). 93. D. Z. Liu, unpublished. 94. H. P. Wei, L. W. Engel, and D. C. Tsui, Phys. Rev. B 50,14609 (1994);E. Chow and H. P. Wei, Phys. Rev. B 52, 13749 (1995). 95. E. Chow, H. P. Wei, S. M. Girvin, and M. Shayegan, Phys. Rev. Lett. 77,1143 (1996). 96. C. Pryor and A. Zee, Phys. Rev. B 46,3116 (1992). 97. V. Kalmeyer and S. C. Zhang, Phys. Rev. B 46,9889 (1992). 98. Y. Avishai, Y. Hatsugai, and M. Kohmoto, Phys. Rev. B 47,9561 (1993). 99. T. Sugiyama and N. Nagaosa, Phys. Rev. Lett. 70,1980 (1993). 100. V. Kalmeyer, D. Wei, D. P. Arovas, and S. Zhang, Phys. Rev. B 48, 11095 (1993). 101. A. Barelli, R. Fleckinger, and T. Ziman, Phys. Rev. B 49,3340 (1994). 102. S. C. Zhang and D. P. Arovas, Phys. Rev. Lett. 72,1886 (1994). 103. A. G. Aronov, A. D. Mirlin, and P. Wofle, Phys. Rev. B 49,16609 (1994). 104. D. Z. Liu, X. C. Xie, S. Das Sarma, and S. C. Zhang, Phys. Rev. B 52,5858 (1995). 105. D. N. Sheng and Z. Y. Weng, Phys. Rev. Lett. 75,2388 (1995). 106. J. Miller and J. Wang, Phys. Rev. Lett. 76, 1461 (1996). 107. B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47,7312 (1993). 108. Note that the random gauge field fluctuations in Chern-Simons theories of half-filled Landau levels are dynamic fluctuations, and the half-filled Landau level problem may therefore be dominated by annealed random flux disorder. The static random flux disorder problem being considered here is the problem of quenched disorder (localization, in general, is a problem of quenched disorder). Thus any connection between these two problems may be subtle and quite indirect, arising from the presence of static potential disorder in the real system.

PERSPECTIVES IN QUANTUM HALL EFFECTS: Novel Quantum Liquids in Low-DimensionalSemiconductorStructures Edited by Sankar Das Sarma, Aron Pinczuk Copyright0 2004 WILEY-VCH Verlag GmbH & Co. KGaA

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Experimental Studies of Multicomponent Quantum Hall Systems J. P. EISENSTEIN Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, California

2.1. INTRODUCTION The many-body phenomena collectively known as the fractional quantized Hall effect (FQHE) are, at the most basic level, due to Coulomb interactions between electrons confined to a two-dimensional (2D) plane in the presence of a perpendicular magnetic field large enough to effectively freeze out both the kinetic and spin Zeeman energies. Within this sphere a tremendous amount of interesting physics has been and continues to be done. Nevertheless, a substantial enrichment of this problem results when there are available discrete internal degrees of freedom which are not frozen out by the high magnetic field. For example, if the field is not too large, Coulomb interactions can so dominate the Zeeman energy that new, spin-unpolarized quantum Hall states appear at certain Landau level filling fractions. In general, these new states compete in energy with conventional spin-polarized quantum liquids, and interesting phase transitions between the two result. Similarly, an analogous internal degree of freedom is present in doublelayer 2D electron gases where, ignoring tunneling, each single-particle state is doubly degenerate. These bilayer systems are even richer than the single-layer, two-spin case, owing to the loss of symmetry of the Coulomb interaction arising from the spatial separation of the layers. This fact allows bilayer systems to support quantum Hall states which have no analog in the single-layercase. In this chapter I discuss recent experimental work in both of these areas. Section 2.2 covers the two-spin problem. In this section I discuss the existing evidence for spin-unpolarized quantum Hall ground states, their phase transitions to polarized configurations, and the associated data suggesting the existence of spin-reversed quasiparticles. The tilted magnetic field technique, which Perspectives in Quantum Hall Effects, Edited by Sankar Das Sarma and Aron Pinczuk. ISBN 0-471-11216-X 0 1997 John Wiley & Sons, Inc.

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has proven so essential to the study of these phenomena, is described in some detail. I limit my discussion of phase transitions to the relatively well understood v = 8/5 FQHE. At the end of the section I turn to the first suspected example of an unpolarized fractional quantum Hall liquid, the still poorly understood evendenominator state at v = 5/2. The double-layer quantum Hall problem occupies Section 2.3. This subject is introduced through analogy to the spin problem, but the crucial enhancement created by the physical separation of the layers is emphasized. Some discussion of the techniques for creating high-quality bilayer systems is given, and the flexibility provided by having independent control over the two new energy scales afforded by such systems, tunneling and the interlayer Coulomb interaction, is stressed. The striking observations of an even-denominator quantum Hall state, at v = 1/2, are discussed and a brief outline of the current theoretical understanding of this uniquely bilayer state is presented. Following this, I turn to the remarkable physics that is presently being uncovered at v = 1 in double-layer systems. In particular, the observation of an unexpected phase transition, driven by an in-plane magnetic field, in this integer QHE state, is discussed in detail. The chapter is then concluded briefly in Section 2.4. Finally, I should like to emphasize that this chapter is not intended to be a comprehensivereview. While I have attempted to cite all of the relevant experimental literature, the detailed discussions below center on work done at AT&T Bell Laboratories. For a complete discussion of the theory of multicomponent quantum Hall systems, the reader is referred to Chapter 5.

2.2. SPIN AND THE FQHE The beautiful explanation, put forth by Laughlin [l], for the quantized Hall plateau found at pxy= 3h/e2 by Tsui et al. [2], relies solely on Coulomb interactions between spin-polarized electrons in the lowest Landau level. The assumption of complete spin polarization was a natural one since at sufficiently high magnetic field B, the Zeeman energy gp,B will exceed the Coulomb energy e2/&l,, which scales as B’’2. Soon after Laughlin’s publication, Halperin [3] pointed out, in a prescient paper, that the Zeeman energy for electrons in GaAs, the host material of choice, is in fact not very large, and it made sense to reexamine the assumption of complete spin polarization. Halperin went on to propose a family of generalized Laughlin wavefunctions that could incorporate reversed spins. In particular, a completely unpolarized candidate wavefunction was given for the case of Landau level filling v = 2/5, where a relatively weak FQHE had been observed [4]. Whether this unpolarized candidate state was lower in energy than some other polarized configuration was not known at the time. Following Halperin’s suggestion [3], numerical studies [S] of finite-size systems revealed that the ground state at 2/5-filling is in fact unpolarized, at least in the absence of the Zeeman energy. This conclusion will remain valid even with the Zeeman energy included,provided that the magnetic field (and thus the 2DES

SPIN AND THE FQHE

39

density) is sufficiently small. Increasing the magnetic field will destabilize the unpolarized state, and eventually there should be a phase transition to a conventional spin-polarized quantum liquid. The exact value of the critical magnetic field was hard to determine given the small energy difference between the two phases and the inherent uncertainty in extrapolating finite-size system calculations to the thermodynamic limit. Nevertheless, the intriguing possibility of observing phase transitions between distinct FQHE states that occupy the same filling factor had been established. In addition to the ground-state spin configuration, spin should play a role in determining the spectrum of excited states of FQH systems. For example, although the v = 1/3 ground state is presumably spin polarized, it was predicted [6,7], again for sufficientlysmall Zeeman energy, that its lowest-lyingelementary excitations are actually spin-reversed quasiparticles. For higher Zeeman energy, a crossover to spin-polarized quasiparticles should occur. 2.2.1. Tilted Field Technique

Direct measurement of the spin polarization of the 2D electron gas at mK temperatures and high magnetic fields is a formidableexperimentalchallenge [8]. The standard approach used, in transport experiments, to gain access to the spin degree of freedom is the tiltedjield technique, introduced by Fang and Stiles [9] in their study of the g-factor in Si inversion layers and first applied to the FQHE by Syphers and Furneaux [lo] and Haug et al. [111. Fang and Stiles utilized the fact that the Landau level spacing ho, depends only on the perpendicular field component B,, while the spin-flip Zeeman energy is proportional to the total magnetic field B,,,. Thus, by tilting the sample they could vary the two energies independently. More generally, the efficacy of the tilted field technique lies in the fact that for an ideally thin 2DES, the parallel field component B , couples to the system only through the Zeeman energy. Hence, if two possible many-body configurations of the 2DES differ in their net spin polarization, adding an in-plane field component will favor the more highly polarized state. For example, as mentioned above, for v = 2/5 there are both polarized and unpolarized candidate FQHE ground states. If, at some perpendicular field, the unpolarized state is lowest in energy, adding an in-plane field component should drive a phase transition to the polarized configuration. The assumption that the parallel magnetic field affects only the Zeeman energy of the 2DES is valid only if the system is infinitely tbin. In reality, the thickness of the quantum confined 2D layer is typically 100A. Due to this finite thickness, the parallel field couples the in-plane and perpendicular dynamics. In the absence of a perpendicular magnetic field, this leads to a mixing in of higher subbands of the confinement potential, with the result that the wavefunction is slightly compressed. Some of the earliest tilted field studies [lo] of the FQHE were interpreted along these lines. When a perpendicular magnetic field is present as well, subband and Landau level mixing occur simultaneously,and the combined effect on the FQHE is not completely understood [12,13]. In general, however, we

-

40

MULTICOMPONENT QUANTUM HALL SYSTEMS

anticipate the non-Zeeman effects of the parallel field to be fairly weak [141, and certainly smooth and monotonic. It seems unlikely that they play a significant role in the phase transition experiments about to be described, where strongly nonmonotonic behavior of the FQHE is observed. 2.2.2. Phase Transition at u = 815

The first experimental indications of an unusual ground-state spin configuration in the FQHE came with the discovery by Willett et al. [15] of the even denominator v = 5/2 state and the subsequent tilted field study of it by Eisenstein et al. [161. Despite this, I begin the discussion of spin effects with the conventional odddenominator FQHE and return to the 5/2-state at the end of this section. In the odd-denominator case, it was the v = 4/3 FQHE that provided the first indications of a spin-unpolarized ground state. This evidence came from the experiments of Maksym et al. [17] and Furneaux et al. [lS], which showed that the v = 4/3 state was suppressed in tilted magnetic fields. More compelling evidence for novel spin phenomena in the FQHE was subsequently reported by Clark et al. [19] and Eisenstein et al. [20]. In both cases an FQHE state was found (v = 4/3 and 8/5, respectively) which exhibited reentrant behavior with tilt: The state would first weaken, then later revive as the sample was rotated. This unusual behavior was interpreted as a phase transition between an unpolarized quantum liquid at small tilt angles and a polarized phase at larger angles. In addition to these results, a number of other experiments on FQHE spin phenomena have been reported. These include tilted field studies giving evidence of both spinreversed quasiparticles [21] and a ground-state phase transition [22-241 at v = 2/3, observations of unusual behavior at v = 7/5 and 3/5 [19,241,the application of density changes to probe the spin configuration [22,25], the study of spin phenomena in the FQHE in 2D hole gases [26], and even the interpretation of some experimentsin terms of the anticipated dependenceof the GaAs g-factor on applied hydrostatic pressure [27]. In this subsection I concentrate solely on the evidence for a phase transition at v = 8/5. Figure 2.1 shows the transport coefficientspxxand p,, at T = 25 mK in the filling factor range 2 > v > 1, measured [20] with a 2DES sample having a lowtemperature mobility of about 7 x lo6cm'/V.s and density N, = 2.3 x 10" cm- '. For these data the magnetic field is perpendicular to the 2D plane. The location of the v = 8 / 5 state (at B = 5.95 T) is indicated by the dashed lines; numerous other FQHE states are also evident in the data. Upon tilting the sample, the 8/5 state initially weakens. At about 0 = 25", a weak satellite resistivity minimum appears about 1% higher in magneticfield than the main 8/5 p,, minimum. Increasing the angle further, to about 0 = 30", results in two minima of comparable strength whose field positions straddle the location of v = 8/5. At this angle, the resistivity precisely at v = 8/5 exhibitsa local maximum. Increasing the angle further reverses these trends. The high-field component of the doublet becomes dominant and gradually centers on v = 8/5. Beyond about 0 = 37" a single 8/5 minimum exists and becomes steadily stronger as 0 is increased further. In both the low- and

41

SPIN AND THE FQHE

.1000

1

T r

Y

W

800

0

z a

-

c

E

L 0

5/8

>. 600

t >

W

t;; z

;

a J

112

i

400

513

I

I

a 200

6 6.5 ? MAGNETIC FIELD (tesla)

7.5

Figure 2.1. Diagonal and Hall resistivity data, taken at T = 25 mK,in the filling factor range 2 > v > 1. Dashed lines indicate the v = 8/5 FQHE state. (After Ref. [20].)

high-angle regimes, the Hall resistance exhibits a plateau at pxy= 5h/8e2. The qualitative behavior of the 815 resistivity minimum is depicted in the left panels of Fig. 2.2 (where pxx is plotted against filling factor v rather than magnetic field). The qualitative interpretation [20] of these data is that for this sample, the v = 815 state is making a phase transition from a nonfully polarized ground state at small tilt angles to a polarized state at larger angles. This is just the scenario, discussed at the beginning of the section,which was predicted for the v = 2/5 state. In fact, the 815 and 215 states are closely connected since v = 815 215 = 2 represents the fully filled lowest Landau level. Provided that Landau level mixing can be ignored, particle-hole symmetry dictates that states at v and 2 - v are in fact “identical.” To make the case for a spin phase transition more convincing, a well-defined model for the tilted field behavior of both phases needs to be given and compared quantitatively to experiment. For this we turn to measurements of the energy gap of the quantized Hall state, determined from the activated temperature dependence of the resistivity. Figure 2.2 shows representative Arrhenius plots [log(p,,) versus 1/T] at v=8/5 in both the low- and high-angle phases. The conventional interpretation of the linear portion of these plots is that pxx= const. x exp( - A/27‘), with A being the fundamental energy gap of the FQHE state (i.e., the energy required to excite a well-separated quasielectron-quasihole

+

42

MULTICOMPONENT QUANTUM HALL SYSTEMS

1000

1

FILLING FACTOR

TEMPERATURE" ( K-'

1

Figure 2.2. (Left)Expanded views of p,, versus filling factor v in a narrow range around v = 8/5 at T = 30 mK. Note the splitting of the 8/5 minimum at 8 = 30"(Right) Arrhenius plots for 8/5 minimum at various angles. Curve a, 8 = 0";b, 8 = 18.6"; c, 8 = 42.4"; d, 8 = 49.5'. (After Ref. [20].)

pair). Figure 2.3 shows the observed tilted field dependence of A. The perpendicular magnetic field is fixed at B, = 5.95 T for these data; the gap has been plotted against the total magnetic field B,,, in the figure. This is the natural choice since the Zeeman energy is proportional to B,,,. At small tilt angles A falls linearly with Btot,while in the high-angle phase it rises linearly with B,,,. The magnitudes of the slopes I aA/aB,,, I are nearly equal in the two phases. In essence, the tilted field dependence of the excitation gap is determined by the difference in spin AS (in units of h) of the quantum liquid state before and after excitation of a quasielectron-quasihole pair. In the simplest model we write

where A. is the gap in the absence of the Zeeman energy, p B = 0.67 K/T is the Bohr magneton, and lgl- 0.44 is the bulk GaAs g factor [28]. The quantity A. is in general unknown, but in this simple model it is assumed to depend only on the perpendicular magnetic field (i.e., on the filling factor v), and this is kept fixed in a tilt experiment. We assume further that the quasiparticles are fundamentally spin-1/2 objects [29]. For a maximally spin-polarized quantum Hall ground

SPIN AND THE FQHE

0 15

-

c

Y

>.

0.6

-

8 (degrees)

30

43

45

B i t 5.95T

f9 0:

w

Obr,

I

I

7

8

I

9

Btot (teslo)

Figure 2.3. Results of the tilted field study of the energy gap A of the v = 8 /5 FQHE. (The difference between the symbols is relevant only in the regime where the v = 8/5 resistivity minimum is split. Solid and open symbols refer to low- and high-field components of the doublet, respectively.)(After Ref. [20].)

state, there are four cases to be distinguished [6,7]: (1)both the quasielectron and quasihole spins are polarized just like the parent fluid; (2)the quasielectron spin is reversed but the quasihole remains polarized;(3) the reverse of case 2; and (4) both quasiparticle spins are reversed. For case 1, AS = 0 and the energy gap remains constant in a tilt experiment. In both cases 2 and 3 the net spin change is AS = - 1 and the energy gap contains the positive Zeeman term + Ig I pBBtot.The gap will increase in a tilt experiment. Finally, for case 4 we have AS = - 2 and the gap again increases with tilt, but twice as fast as in case 2 or 3. Hence for a polarized quantum Hall liquid, the energy gap A can either remain the same under tilt (spin-aligned quasiparticles)or it can increase (spin-reversedquasiparticles).The observation of afalling energy gap in the tilt experiment is therefore inconsistent with a polarized ground state, unless the added parallel magnetic field couples to the system in some way other than through the Zeeman energy. On the other hand, such an observation is easily explained if the quantum liquid is less than fully spin polarized, since then it is possible for the net spin to increase upon quasiparticle excitation. In particular, for an unpolarized quantized Hall state such as v = 2/5, the lowest-energy gap will obtain when both quasiparticle spins polarize along the field direction. Excitation of such a triplet state [7] will give AS= + 1. In this case the energy gap will have a negative Zeeman term - lglpBBtot and will decrease with tilt.

44

MULTICOMPONENT QUANTUM HALL SYSTEMS

With this model in hand, the data in Fig. 2.3 are readily interpreted. For total magnetic fields less than about 7 T (i.e., tilt angles below 30") the falling energy gap is consistent with an unpolarized ground state and AS = + 1. From the slope aA/aB,,, in this regime, we deduce a g factor of about 0.4, remarkably close to the bulk GaAs value of g 0.44.Similarly, the rising energy gap for B,,, > 7 T suggests a polarized ground state. The measured slope aA/dB,,, again gives g 0.4, assuming that AS = - 1. It seems quite reasonable that on approaching the critical point from either direction, the spin alignment of the lowest-energy quasiparticle excitations favors the polarization of the phase about to be entered. These experimental results concerning the ground-state spin polarization of the v = 8/5 state are in good agreement with theoretical studies [5,7,30-351 of the particle-hole conjugate state at v = 2/5.Recent exact diagonalization studies [33,35] of systems containing up to eight electrons show that the energetic advantage (per electron and neglecting the Zeeman energy) of the unpolarized over the polarized v = 2/5 state extrapolates to about 0.006e2/d, in the thermodynamic limit. Setting this equal to the Zeeman energy per electron lglp,B/2 in the polarized phase allows us to estimate the critical perpendicular magnetic field B,,, of the transition; for GaAs B,,, x 5 T. Provided that particle-hole symmetry is operative, the same critical magnetic field applies to the v = 8/5 state. Thus in theory at least, an ideal 2D sample with density less than N,,, = 8eB,:,/ 5h x 2 x 10'' cm-2 will possess an unpolarized v = 8/5 ground state, while in higher-density samples the same state will be spin polarized. For a sample with N , < N,,, tilting may be used to induce the phase transition, and it will do so at a total magnetic field less than B,,,. This follows since tilting enhances the Zeeman energy while leaving the Coulomb energies unchanged. Starting in the unpolarized phase at some perpendicular field B , B,,,, tilting should induce the transition at B,,,,, = (BlBl,c)''2. For the sample being discussed here, the data in Fig. 2.3 show that B,,,,c x 7 T,while B, = 5.95 T. Thus the experimentally determined B,,, is approximately 8.2 T. Although this is significantly larger than the theoretical estimate ( x 5 T), we find the agreement satisfactory, considering the simplicity of the theoretical model (infinitely thin 2 D systems, no disorder or Landau level mixing,etc.), and the uncertainty in extrapolating finite-sizecalculations to the thermodynamic limit. Theory and experiment are also in agreement on the nature of the energy gap in the two phases. Both find that the spin change AS upon excitation of a quasielectron-quasihole pair is AS = + 1 in the unpolarized phase and - 1 in the polarized state. (Numerical calculations [33] suggest that in the polarized 2/5 state, the relevant excitation at low fields has only the quasielectron spin reversed while the quasihole remains polarized; this was case 2 above. Reversing the quasihole spin yields comparatively little Coulomb energy advantage, and the possibility the both spins are reversed (i.e., AS= -2) simply costs too much Zeeman energy. Similar results were obtained earlier for the v = 1/3 case [6].) Finally, the experimental finding that there is no noticeable discontinuity in the energy gap at the transition has also been corroborated numerically [33,34].

-

-

-=

SPIN AND THE FQHE

45

There are, of course, aspects of this general picture that are not understood. For example, the splitting of the v = 8/5 resistivity minimum in the vicinity of the transition remains mysterious. A similar splitting was subsequently observed in tilted field studies of the v = 2/3 FQHE [23,24]. One speculative possibility is that these splittings reflect an inhomogeneous state of the quantum Hall liquid. Phase separation of the electron liquid into polarized and unpolarized components is quite plausible near the critical point. On approaching the transition from the unpolarized phase, the satellite p,, minimum first appears on the high-field side of the main minimum. This is at least consistent with the satellite being due to condensed “droplets” of polarized fluid. Conversely, above the transition, in the polarized phase, the satellite eventually disappears off the low-field side ofthe p,, minimum;now the satellite might be due to droplets of the unpolarized fluid. Even if the quantum fluid does actually phase separate into stable domains of polarized and unpolarized liquids, it might simply fluctuate between the two phases. The resistivity splitting might be a sign of enhanced dissipation arising from such fluctuations. Another feature of these results that is not understood is why the transition is observed at v = 8/5 but has not yet [22] been detected at v = 2/5. The theoretical picture that we have applied was developed specifically for v = 2/5; its extension to the 8/5 state is made solely on the grounds of particle-hole symmetry. We can only remark that this symmetry principle connects 2/5 and 8/5 states at the same magnetic field. Experimentally, of course, this means comparing two samples whose densities differ by a factor of 4. Such samples would be quite different; for example, the “thickness” of the 2DES, as measured by the extent of the subband wavefunction,would be considerably larger in the low-density sample than in the high-density sample. This thickness softens the short-range part of the Coulomb interaction and therefore affects the critical magnetic field of the spin phase transition. Other things being equal, the 2/5 critical field would be suppressed relative to the 8/5 case.

2.2.3. The u = 512 Enigma As mentioned above, the first evidence for a spin-unpolarized FQHE state came with the discovery [151, at very low temperature, of a fragile quantized Hall effect in the second Landau level, at the even-denominator fraction v = 5/2. This unusual state remains the only even-denominator FQHE ever found in a single-layer 2DES [36]. Figure 2.4 shows the transport coefficientsp,.. and p,,, around v = 5/2 from the original paper of Willett et al. [l5]. Despite earlier suggestions [37,38] that even-denominator states might exist (at v = 3/4, 9/4, 11/4, and 5/2), these definitive new results were greeted with considerable surprise. This was certainly justified given the total adherence to the odd-denominator rule by all previously discovered FQHE states. The restriction to odd-denominator filling fractions for the primitive Laughlin states at v = l/m stems from the requirement of exchange antisymmetry of the many-body wavefunction. Since the spins are assumed to be fully polarized, this

46

MULTICOMPONENT QUANTUM HALL SYSTEMS I

I

I

I

1

0.1

s!

CU-

c

0.' e Q

2

0.3

MAGNETIC FIELD

(t]

Figure 2.4. Diagonal and Hall resistivity data showing the even-denominator FQHE at v = 5/2. Note the low temperatures involved. (After Ref. [lS].)

antisymmetry requirement forces the exponent m in the Jastrow (orbital) part of the wavefunction to be an odd integer. In the early hierarchical generalizationsof the Laughlin theory, the odd-denominator restriction was found to propagate down from the primitive v = l/m states to the daughter states at v = p/q, encompassing all FQHE states known prior to the v = 5/2 discovery. Thus it was natural to speculate that the existence of an even-denominator state might reflect a breakdown of the assumption of full spin polarization. The relatively low magnetic fields (B x 5 T) at which the v = 5/2 state was first observed made this seem quite plausible. This possibility seemed even more likely after Haldane and Rezayi [39] showed that for a certain hypothetical interaction (the hollow-core

SPIN AND THE FQHE r

47

8 = 23 deg.

3

P

I

J

MAGNETIC FIELD (tesla) Figure 25. Collapse of the v = 5/2 state with tilt. Arrows mark the location of the 5/2 filling fraction. These data were all taken at T = 25 mK. (After Ref. [16].) model) a FQHE could exist at half-filling. The wavefunction they proposed for

this state was explicitly unpolarized. Experimental support for these ideas came from the tilted field studies of Eisenstein et al. [16]. Figure 2.5 shows resistivity data in the vicinity of v = 5/2 for a number of different tilt angles 8. The data show clearly that the 5/2-state collapses as the tilt angle is increased. As discussed earlier for the v = 8/5 case, this behavior suggests that the ground state at v = 5/2 is not fully spin polarized. But again, a detailed study of the energy gap of the 5/2 state is required to strengthen the case. While the relatively low quality of the sample used in the original experiments [l5, 161 and the extreme fragility of the 5/2 state precluded such a study, better samples eventually became available and the necessary data were obtained [40]. Figure 2.6 shows the tilted field dependence of the energy gap A, extracted from the activated temperature dependenceof pxxat v = 5/2, from one such better sample. The gap is plotted against total magnetic field B,,,; the perpendicular field at v = 5/2 is fixed at B , = 3.75 T. The figure shows that A falls, roughly linearly, with B,,,. Just as with v = 8/5, this observation is inconsistent with a fully spin polarized ground state, provided that the parallel magnetic field has coupled to the system only through the Zeeman energy. On the other hand, the linearly decreasing gap is suggestive of an unpolarized (or partially polarized) ground state and a net increase in spin polarization upon quasiparticle excitation (i.e., AS > 0). Assuming that AS = + 1, the slope aA/dB,,, gives a g-factor of 191 x 0.56,

48

MULTICOMPONENT QUANTUM HALL SYSTEMS

loo

' 0

0 (degrees) 15'

I

I

A =A0'QPe BTOT slope

a

BIZ 50

=+g.0.56

3.75 T

-

3.75

1

3.80

I

3.85

I

3.90

3 I5

B m (TI Figure 2.6. Tilted field study of the energy gap A of the v = 5/2 state (from a different sample than shown in Fig. 2.4).The slope of the straight line gives a g factor of g z 0.56. (After Ref. [40].)

some 30% larger than the bulk GaAs value [28]. Note that even for the untilted case, the measured gap at v = 5/2 (A x 0.11K) is only about one-tenth of the Zeeman energy at B = 3.75 T. This suggests that if the Zeeman energy were absent, an unpolarized 5/2 state would be robust, exhibiting an energy gap in the 1 K range. Although the evidence that the v = 5/2 ground state is not fully spin polarized is suggestive, it is not as compelling as that obtained at v = 8/5, 4/3, and 2/3. The reentrant tilted field behavior observed at those odd-denominator states is hard to explain via a non-Zeeman effect of the parallel magnetic field (ie., subband/Landau level mixing). At v = 5/2, however, there is only a monotonic gap suppression on tilting (albeit with a magnitude consistent with an unpolarized ground state). There remains a possibility that the ground state is actually spin polarized, and the tilted field behavior is due to some other effect of the parallel field. Interest in this has been renewed by recent experiments [41] showing that the 5/2 state is observable in (perpendicular) magnetic fields as high as B , 9 T. Given the rapid tilted field collapse of the 5/2 state shown in Fig. 2.6, for which B, = 3.75 T, the existence of the state at B, 9 T seems incongruous within the unpolarized ground-state scenario. There is, however, no obvious inconsistency between this observation and the evidence that the ground state is unpolarized. Indeed, one can construct a simple model [42] that includes sample disorder in a rudimentary way and show that the sample used for Fig. 2.36 could support an unpolarized 5/2 state at B, > 9 T if its density were increased (by a gate, for example). Nevertheless, these new findings [41] do not enhance the case for

-

-

-

FQHE IN DOUBLE-LAYER 2D SYSTEMS

49

a spin-reversed ground state at v = 5/2 and suggest that further work is needed. If, in fact, the 5/2 state is spin polarized, a new explanation for the tilted field experiments has to be constructed. While the experimental debate has centered, thus far, on the ground-state spin configuration, on the theoretical front the very existence of a v = 5/2 FQHE is still not well understood. The excitement over the early work of Haldane and Rezayi [39] largely dissipated when it became apparent that the hollow-core model they employed was a poor representative of real Coulomb interactions, even in the second Landau level [43]. Various modifications to the hollow-core model (e.g., inclusion of Landau level mixing [44]) have been tried, but so far without compelling success; Alternative candidate wavefunctions have also been proposed [43,45-481, but none have so far been shown to be close to the ground state for Coulomb interactions. On the numerical side, exact diagonalization calculations [43,49, SO], while suggesting that the ground state at v = 5/2 is actually spin polarized, reveal at best [SO] only a very weak cusp in the total energy. Such a cusp is the essential ingredient for a quantized Hall effect. It seems, therefore, that after several years of study, a key piece of the 5/2 puzzle is still missing. 2.3. FQHE IN DOUBLE-LAYER 2D SYSTEMS Interestingly, while the v = 5/2 FQHE in the single-layer 2D electron system is hard to observe and even harder to explain, even-denominator states in doublelayer 2D systems are experimentally robust and theoretically well understood. What accounts for the difference? Superficially,one might have expected that the two-spin and the two-layer problems would be very similar. In fact, there are two essential differences between the two cases. In a single-layer system the coulombic repulsion between any two electrons is independent of their relative spin orientation and depends only upon their lateral separation in the plane. For the bilayer case this is obviously not true; the physical separation of the layers makes interlayer interactions weaker than intralayer ones. This asymmetry of the Coulomb interaction is the main reason why v = 1/2 states are readily found in double-layer systems. The second critical difference between the two systems is the existence of interlayer tunneling in the bilayer case. Tunneling hybridizes the quantum states of the two layers and creates an additional energy gap not present in a single-layer system. Imagine first a bilayer system consisting of two identical 2DESs which are separated by a potential barrier sufficiently thick and high that both tunneling and interlayer Coulomb interactions are negligible. Assuming that the layers are electrically connected in parallel, such a system will exhibit a spectrum of QHE states that is identical to that of the individual layers, except that the resistance of each Hall plateau will be one-half of its single-layer value. With v representing the total filling factor of the double-layer system, integer Hall states will appear at v = 2,4,6,. . . and fractional ones at v = 2/3, 4/5, 6/7,. . . No Hall plateaus will be observed at odd-integer or odd-numerator fractional values of v.

50

MULTICOMPONENT QUANTUM HALL SYSTEMS

Now adjust the barrier so that tunneling becomes relevant but interlayer Coulomb effects remain unimportant. In the absence of tunneling each singleparticle state is doubly degenerate;turning on this interlayer coupling hybridizes the individual quantum well states into symmetric and antisymmetric components. These new states are split by an energy gap ASAS which depends on the width and height of the separating barrier. The existence of this new energy gap profoundly affectsthe spectrum of observablequantum Hall states. For example, in the uncoupled case, no v = 1 QHE appears because this would imply the existence of a v = 1/2 QHE in each layer separately. With tunneling, however, odd-integer states appear each time the Fermi level becomes pinned in the symmetric-antisymmetric (SAS) gap of some Landau sublevel. A v = 1 state is then a full Landau level ofsymmetric bilayer states, not two distinct v = 1/2 states in the individual layers. (We are assuming, and will continue to throughout this chapter, that the spin splitting of the Landau level exceeds the tunneling gap ASAS ~511.) If the barrier is at once narrow but very high, tunneling may be unimportant, while interlayer Coulomb effects are essential. It was first shown theoretically by Haldane and Rezayi [52] and Chakraborty and Pietilainen [53] that in such a situation new QHE states, which are intrinsically bilayer in nature, can exist. Subsequently, Yoshioka et al. [54] systematically investigated a wide class of bilayer FQHE states using Halperin’s two-component wavefunctions [3]. These bilayer states depend critically on a delicate balance of inter- and intralayer Coulomb interactions. This balance is conveniently parameterized by the ratio d/Io, where d is the interlayer separation and I, is the magnetic length. Since for a given filling factor I, is proportional to the mean separation between electrons in the same layer, the ratio d/l, is proportional to the ratio of intra- to interlayer Coulomb energies. This ratio needs to be of order unity for the new states to be stable (at least for ideal systems with infinitesimally thin 2D layers.) Although it seems obvious that such states cannot survive at large d/I, values, it is interestingly also true that some (e.g., v = 1 /2) collapse in the opposite limit, d/l, +0. For these, the asymmetry of the Coulomb interaction produced by the finite layer separation is essential. In real samples, of course, tunneling and interlayer Coulomb effects are usually simultaneously operative. In general, an interesting phase diagram can be constructed, with interlayer Coulomb energy on one axis and symmetric-antisymmetric tunneling gap on the other. In some cases the two effects are at odds, and in others they act cooperatively.Indeed, their complex interplay has yielded some of the most interesting QHE phenomena observed in recent years. In this section I discuss the present experimental status of the QHE in double-layer systems, concentrating entirely on the v = 1/2 and v = 1 states. 2.3.1. Double-Layer Samples

Two main approaches have been taken to fabricate bilayer 2D electron systems. Both employ GaAs/AlGaAs heterostructures grown by molecular beam epitaxy.

FQHE IN DOUBLE-LAYER 2 D SYSTEMS

51

In a double-quantum-well (DQW)structure, two thin GaAs layers are emkedded in the alloy Al,Ga, -,As. The GaAs quantum wells are usually 150 to 200.A wide, while the (undoped) alloy barrier between them is typically only 30 A thick. Electrons transfer into the quantum wells from Si doping sheets which are placed well above and below the DQW [55]. Alternatively,bilayer 2D electron systems have also been realized in wide single quantum wells (WSQW)[56]. Electrons in the well accumulate at the upper and lower boundaries, being attracted there by the electricfields of the positively charged donor ions which are placed symmetrically above and below the well. If the well is wide enough, the charge distribution in the well has a distinct dumbbell shape. Unlike the DQW case, where the two 2DES layers are separated by a true semiconductor barrier, in the WSQW case the “barrier” results solely from the Hartree potential of the electrons themselves. One of the main advantages of the DQW geometry is the freedom it provides for tailoring the details of the barrier separating the two quantum wells to a specific experiment. Since both the thickness and the height (via the A1 concentration, x) of the barrier can be adjusted, independent control of tunneling and interlayer Coulomb interactions is possible. The 2D electron layers in DQWs are typically “thinner,” more closely spaced, and yet much more weakly tunneling than comparable WSQW structures. On the other hand, the WSQW geometry yields considerably “cleaner” bilayer 2D systems; this evident from the better developed FQHE spectra seen in such samples. DQW samples suffer from the rather “dirty” Al,Ga, - 4 s barrier (often pure AlAs) between the quantum wells. Most bilayer QHE experiments to date have been performed with equal 2D densities in the layers. This condition is established either by careful choice of the doping layer setback distances or by in situ tuning of the densities using gate electrodes on the bottom or top of the heterostructure sample. In addition, it is generally desirable for the two 2D electron systems to be comparably disordered, but this is usually difficult to assess directly.Transport studies on weakly coupled DQW systems, in which separate electrical connections [57] to the individual layers have been made, show that the two layers can be made to have very nearly the same “quality.” Such separate contacts have not yet been achieved with the strongly coupled systems of interest for bilajler QHE studies. (This is more a question of the tunneling resistancethan of layer spacing per se.) Indeed, all the results discussed in this chapter have been obtained from samples having ohmic contacts that connect to the two layers simultaneously.

2.3.2. The u = 112 FQHE At the qualitative level, the most dramatic discovery made so far in the study of double-layer quantum Hall systems was the observation of the even-denominator v = 1/2 state reported simultaneously by Suen et al. [SS] and Eisenstein et al. [59]. In both cases a deep minimum in the diagonal resistivity pxxwas found at total filling factor v = 1/2 along with a well-defined plateau in the Hall resistance at p,,, = 2h/e2. Figure 2.7 shows relevant resistivity data from both groups.

52

MULTICOMPONENT QUANTUM HALL SYSTEMS

Figure 2.7. Observationsof the v = 1/2 FQHE state in double-layer 2D electron systems. The upper panel shows the data of Suen et al. [SS] from a wide single quantum well; the lower panel shows data from Eisenstein et al. [59], who employed a double-quantum-well structure.

FQHE IN DOUBLE-LAYER 2 D SYSTEMS

53

A well-developed theoretical picture of a v = 1/2 FQHE state in ideal doublelayer 2D systems was in existence prior to the experimentsnoted above. After the initial prediction by Haldane and Rezayi [52], Yoshioka et al. [54] discussed a specific Halperin [3] trial wavefunction appropriate to v = 1/2 (the so-called “331” state) and established its large overlap with the exact ground state of a few-electron system when d/l, 1. As expected,Yoshioka et al. [54] found that this intrinsically bilayer FQHE state collapses at large layer separation where interlayer Coulomb correlations become negligible. Subsequently,He et al. [60] performed additional numerical calculations for realistic sample geometries, thereby allowing closer contact with experiment. The variational wavefunction widely believed to capture the essential physics of the v = 1/2 state in double-layer systems is a member of the broad class of generalized Laughlin-Jastrow functions introduced by Halperin [3] in the two-spin context. Letting z and w represent the coordinates of electrons in each of the 2D planes, the orbital part of the “331” wavefunction is (aside from the ubiquitous Gaussian factors)

-

The first and last products reflect the correlations within the z and w layers separately, while the middle product term contains the interlayer correlations. The intralayer terms are the same as that of the Laughlin 1/3 state [l]; as two electrons in the same layer approach one another, the wavefunction vanishes as the cube of their separation. The interlayer term, however, makes the wavefunction vanish when two electrons in different layers are directly opposite one another. In a crude sense, YJ31represents two Laughlin 1/3 states locked together so that the electrons in one layer are opposite correlation holes in the other. In fact, one may view each layer as a 1/3 state plus one extra quasihole for each electron (i.e., each layer is actually at filling factor 1/4). The quasiholes in layer z are “bound” onto the electrons in layer w, and vice versa. The coulombic attraction between these quasiholes and electrons reduces the total energy of the system when the two layers are close together. Thus the \y331 state contains not only a commensurability between the number of flux quanta and electrons in each layer, but also one between the number of quasiholes in one layer and the electrons in the other. It is these special commensurability conditions that produce the cusp in the total energy required for a quantized Hall effect. To show that the v = 1/2 state they observed depended critically upon interlayer correlations, Eisenstein, et al. [59] examined the effect in a quartet of samples having different densities and quantum well separations. Figure 2.8 displays resistivity data, taken at 300 mK, for samples A, B, C, and D. Samples A, B, and C are structurally identical (180-&wide wells separated by a 31-A AlAs barrier) but have different densities (A lowest, C highest). Note that while the v = 2/3 pxxminima are all similar, the v = 1/2 feature weakens monotonically as the density increases. This observation is contrary to the usual enhancement of (spin-polarized)FQHE states with increasingmagnetic field found in single-layer

54

MULTICOMPONENT QUANTUM HALL SYSTEMS

!

8

12 6 MAGNETIC FIELD

10

8

10

12

(Tesla)

Figure 2.8. Collapse of the v = 1/2 state in four samples with increasing values of d/lo. Data taken at T = 300mK. These data demonstrate that interlayer correlations are essential to the existence of the v = 1/2 state in double-layer systems. (After Ref. [SS].)

systems. In the present double-layer case, however, the ratio of intra- to interlayer Coulomb energies, which is just d/lo, is critical. Setting d equal to the quantum well center-to-center distance (210A for samples A, B, and C) the ratio d/l, at v = 1/2 increases from 2.4 for sample A to 2.9 for sample C. Sample D has a thicker barrier (99A) and a ratio d/l, = 3.6. This last sample has a density comparable to that of sample B and is arguably the cleanest of the quartet (as evidenced by it exhibiting the largest v = 2/3 activation energy), yet it shows no v = 1/2 FQHE. Taken together, the data in Fig. 2.8, showing the v = 1/2 state to weaken and eventually collapse as d/l, increases, provide compelling evidence that this even-denominator bilayer FQHE does indeed depend essentially upon interlayer Coulomb interactions. Numerical calculations by He et al. [61] for DQW geometries equivalent to the samples just described show that the origin of the v = 1/2 FQHE in those samples is well understood and consistent with the properties of the ’I-”,,, variational wavefunction.Their calculations reveal that the finite thickness of the quantum wells leads to stability of the v = 1/2 state at larger values of d/l, than in

FQHE IN DOUBLE-LAYER 2D SYSTEMS

0.8

-

0.6

-

55

-

Q Y

0.4

0.2

-

0.00

100

200 tc

(h

300

Figure 2.9. Calculations by He, Das Sarma, and Xie [61] of the energy gap of the bilayer v = 1/2 state versus the magnetic length 1, in double-quantum-well structures consisting of two 180-A-wide GaAs wells separated by an AlAs barrier of width D,= 31 or 99 A. The solid dots represent the coordinates of the four samples shown in Fig. 2.8. The calculated energy gap shows, qualitatively, the same collapse of the v = 1/2 state that was observed experimentally [59] via the pxxminimum.

the ideal, infinitely thin case [54]. The reason for this is simply that the finite thickness softens the short-range part of the intralayer Coulomb interaction. Consequently, the interlayer interactions can be weaker as well and the v = 1/2 state still exist. Figure 2.9 shows the calculated [61] energy gap at v = 1/2 versus magnetic length I, for DQWs with 180-A quantum wells and either a 31- or 99 A-wide AlAs barrier. The solid dots give the calculated gaps for the various samples of Eisenstein et al. [59]. Although the v = 1/2 activation energies were not measured for these samples, the calculated gaps show the same qualitative collapse as that shown by the resistivity minima in Fig. 2.8. The sample used by Suen et al. [SS] to observe the v = 1/2 FQHE was a 680-Awide single quantum well. This sample differs significantly from those used by Eisenstein et al. [59]. Not only are the “individual” 2D systems much thicker in the Suen sample,but the tunnelingstrength is about an order of magnitudelarger. Setting the layer spacing d equal to the distance between the two (self-consistently calculated)maxima of the charge distribution, Suen et al. [58] concluded that the v = 1/2 state in this sample occurs at d/l, N 7.These conditions are sufficiently far

56

MULTICOMPONENT QUANTUM HALL SYSTEMS

from what was originally expected [54,60] that the possibility of a non-Y33 origin for the v = 1/2 state in the Suen sample had to be considered. Fueling this speculation was the existence of another candidate wavefunction, the Pfaffian state, advanced originally by Moore and Read [46]. This singlecomponent state was originally considered as a possible candidate wavefunction for a spin-polarized v = 5/2 FQHE. Its connection to the present v = 1/2 FQHE in wide single quantum wells derives from the work of Greiter et al. [47], wherein it was suggested that if the Coulomb interaction were sufficiently softened at short distances, a Pfaffian-like v = 1/2 FQHE might exist. With this in mind, Suen et al. [62] performed an extensive study of the v = 1/2 state in wide single wells, varying the well width and sheet density. They concluded that the l/Zstate they were observing was fundamentally two-component in nature and that the Pfaffian was not relevant. Numerical work by He et al. [61] showed that in fact the v = 1/2 ground state in Suen’s sample was basically Y331-likein character, despite the large thickness and substantial tunneling strength. Interestingly, Suen et al. [62] showed that wide single quantum wells can support not only twocomponent states v = 1/2, but also thick variants of the familiar single-component states such as v = 1/3. Which “hat” the sample chooses to wear depends on a subtle balance among tunneling, thickness, and density, and this balance changes from one fraction to another.

2.3.3. Collapse of the Odd Integers It was actually the integer quantized Hall effect (IQHE) that first illustrated the importance of interlayer Coulomb interactions in double-layer 2D systems. In the experiments of Boebinger et al. [63], the IQHE in DQW samples with relatively large amounts of tunneling was investigated. As mentioned above, in the absence of tunneling and interlayer interactions, the IQHE is restricted to even filling factors, v = 2,4,6,. . . . These states correspond to the ordinary IQHE in a single-layer system. If there is tunneling and the symmetric-antisymmetric gap ASAscan be resolved, odd-integer quantized Hall states appear. Provided that the spin Zeeman splitting (gIpBBexceeds ASAS, each of these odd integers corresponds to the Fermi level lying in the SAS gap within a particular Landau spin sublevel [Sl]. What Boebinger et al. [63] and later Santos et al. [64] found was that the lowest odd integers were systematically suppressed. Very strongly tunneling DQWs would exhibit all the odd integers; with slightly weaker tunneling the v = 1 QHE would be missing; weaker still and both v = 1 and 3 were gone; and so on. Representative resistivty data from Boebinger et al. [63] are shown in Fig. 2.10. This suppression of the small odd-integer QHE states in DQWs has been attributed [60,65,66] to a collapse of the SAS tunneling gap induced by interlayer Coulomb interactions. To understand this, imagine first a DQW containing just two electrons. Ignoring the Coulomb interaction, each electron resides in the ground symmetric DQW quantum state. Ifwe now turn on the Coulomb repulsion while holding fixed the in-plane separation of the pair, the interaction energy is

FQHE IN DOUBLE-LAYER 2 D SYSTEMS

57

Figure 2.10. Missing integer quantized Hall states in a strongly tunneling double quantum well. While v = 5,7,9,... are plainly visible, v = 1 and 3 are absent. (After Ref. [63].)

reduced if the two electrons hop into opposite quantum wells, since this increases their net separation. This hopping requires a complete mixing of the symmetric and antisymmetric DQW states and thus costs tunneling energy in the amount A S A P If A S A S is not too large, this penalty can be outweighed by the Coulomb advantage provided that the layers are far enough apart. Now consider the v = 1 QHE state, which is simply a full Landau level of (spin-polarized)symmetricstate electrons.The hopping scenario described above can still occur if, crudely speak-

58

MULTICOMPONENTQUANTUM HALL SYSTEMS

ing, nearest-neighbor electrons conspire to occupy opposite quantum wells, creating a charge density wavelike interlayer correlated state. The precise conditions for the transition to this state depend on the tunneling gap ASAS,the layer separation d, and the sheet density N,.For the v = 1 QHEin a DQW with a given layer separation and tunneling strength, there exists a critical density, and thus magnetic field, above which this new correlated state is favorable. According to theory [60,65,66] the Coulomb interactions in the DQW serve to convert the simple single-particle tunneling gap into a dispersive collective excitation that goes soft at a finite wavevector q = 1; at the critical magnetic field. Owing to this soft mode, the quantized Hall effect is destroyed. Analogous arguments apply to the higher odd-integer QHE states (v = 3,5,...). Ignoring for the moment couplings across the cyclotron and spin Zeeman gaps, only those electrons in the uppermost filled Landau level of symmetricstates can mix with unfilled antisymmetric states. As there are exactly N o = eB/h such electrons per unit area, in a DQW sample with fixed total density there will be a maximum odd-integer v above which the interactions are not strong enough to kill the QHE. Thus, as the tunneling strength is reduced, first the v = 1 state dies, then v = 3, and so on. This simple scenario is in qualitative agreement with experiment [63,64]. 2.3.4. Many-Body u= 1 State

In the preceding section the focus was on the v = 1 QHE in the strong tunneling limit where the main contributor to the quasiparticle energy gap was a singleparticle effect. What of the opposite limit, when the tunneling is very weak? Is there still a QHE, and if so, what is its character? These questions, and their analogs in the single-layer,two-spin v = 1 QHE, have been the subject of considerable recent interest. Indeed, it is becoming increasingly apparent that that this quantized Hall state exhibits physics far more interesting and subtle than originally thought. In the simplest view, the v = 1 QHE arises from a mundane single-particle effect: tunneling in double layers, spin splitting in single-layer systems. Electron-electrun interactions are often described as merely “enhancing” the energy gap. It is now recognized that this is a very incomplete picture. The resurgence of interest in the v = 1state derives primarily from the fact that in both the single- and double-layer cases a QHE should exist at v = 1 euen in the complete absence of the single-particle energy gap. In the single-layer case, recent theoretical work [67] has shown that not only does the v = 1 quantum Hall state exist in the limit of vanishing Zeeman energy, but its charged elementary excitations are not merely exchange-enhanced singlespin flips. They are, instead, large vortexlike distortions of the spin field called skyrmions (after their relatives in nuclear physics),which possess a large total spin. A number of recent experiments have uncovered evidence for these unusual excitations [68]. Double-layer systems exhibit an even richer spectrum of phenomena at v = 1 than do their single-layer counterparts, owing to the physical separation of the

FQHE IN DOUBLE-LAYER 2D SYSTEMS

59

two layers. If the layer spacing is sufficiently small, a v = 1 QHE is predicted to exist even in the absence of tunneling [53,54,69]. The low-lying charged excitations of the bilayer v = 1 state are also vortexlike topological defects (called merons); only the real spin is replaced by a layer index or pseudospin [70,71]. On increasing the layer spacing the v = 1 QHE must collapse [60,65,69] as the system becomes, in effect, two uncoupled layers, each at v = 1/2. This phase transition is obviously not present in the single-layer case. Finally, the separation between the two layers offers another possibility. By applying an additional magnetic field component parallel to the layers, a Bohm-Aharonov phase can be injected into the many-body problem. As we shall see below, this can have dramatic consequences. In this section I draw on the analogy between double-layer quantum Hall systems and 2D ferromagnetism,which has been developed most thoroughly by the Indiana University theory group [70,71]. My discussion is limited to qualitative aspects of this analogy and how they relate to experiment. The reader interested in a deeper understanding of this physics is referred to Chapter 5. Before discussing the experiments, it is useful to introduce the pseudospin description of double-layer 2D systems. In this language an electron in the “upper” layer of a DQW is an eigenstate of ozwith eigenvalue + 1. Similarly, an electron in the “lower” layer is an eigenstate of ozwith eigenvalue - 1. Recalling the properties of the Pauli spin matrices,it is clear that an electron in a symmetric DQW state, which is just (lupper > + llower > )/& is an eigenstate of ox with eigenvalue + 1. Similarly, an antisymmetric state electron has ox = - 1. In the presence of tunneling, the symmetric and antisymmetric states are split by the energy gap ASAS. Therefore, tunneling may be viewed as a pseudomagnetic field lying along the x direction and ASASas the associated pseudo-Zeeman energy. Carrying this a bit further, imagine an electron placed at t = 0 into the upper quantum well. For t > 0 this electron will begin to tunnel back and forth coherently between the layers. In the pseudospin picture this is equivalent to the Larmor precession, in the z-y plane, of an electron with spin initially along 2 in a uniform magnetic field along 2. The Larmor frequency is just AsA$h. Tunneling alone is sufficient to produce a QHE at v = 1 in a double-layer system. This occurs when the Fermi level is in the lowest symmetric/antisymmetric gap and the system is just a filled Landau level of symmetricstate electrons. In the pseudospin picture, the ground state of the system is fully pseudospin polarized along 2. But as mentioned above, a v = 1 QHE should exist even in the absence of tunneling, owing to interaction effects. This many-body integer QHE, first predicted on numerical grounds by Chakraborty and Pietilainen [53], is similar to the bilayer v = 1/2 FQHE insofar as it collapses at large layer spacing. In contrast to the v = 1/2 FQHE, the v = 1QHE also exists in the theoretical limit d/l, +O. The question for experiment is: To what extent can one decide if the v = 1 QHE observed in a bilayer sample derives from single-particletunneling or many-body effects.? To clarify this issue, Murphy et al. [72] performed a systematic study of th? v = 1 QHE in a series of DQW samples, most of which consisted of two 180-A

60

MULTICOMPONENT QUANTUM HALL SYSTEMS

BIB (v = 1) Figure2.11. Existence and lack thereof of the v = 1 QHE in two very similar DQW samples. These samples are represented in the phase diagram in Fig. 2.12 by the leftmost open and solid stars. (After Ref. [72].)

GaAs quantum wells separated by a 31-A Al,Ga, -,As barrier. Within these geometrical constraints, the calculated [73] tunneling gap was varied from a minimum of about A,, x 0.5 K to a maximum of roughly 8.1 K, by tailoring the A1 concentration within the barrier (in both magnitude and profile). These samples had total sheet densities ranging from N,, x 0.8 x 10" to 3.2 x 10" equally split between the two layers. Figure 2.1 1 illustrates resiostivity data from two samples that differ only in barrier thickness (31 versus 40 A) and slightly in density (N,,, = 1.26 versus 1.45 x 10" cm-2. Both show clear quantum Hall states at v=2, 213, and even 415. At v = 1, however, the two samples differ qualitatively. The narrower-barrier, lower-density sample exhibits a strong v = 1 QHE, while the other sample shows no such state. Figure 2.12 contains the phase diagram for v = 1 that Murphy et al. [72] constructed. On the horizontal axis is the tunneling gap [73] ASASin units of the basic Coulomb energy e2/d,, while the vertical axis is d/lo, the ratio of the quantum well center-to-center spacing to the magnetic length, this being the ratio of intra- to interlayer Coulomb energies. The solid symbols denote samples that

FQHE IN DOUBLE-LAYER 2D SYSTEMS I

61

6

NO QHE

.05 &&(e2/e psi

.1

Figure 2.12. Phase diagram for the v = 1 QHE in double-layer 2D electron systems. Solid symbols denote samples that exhibit a v = 1 QHE, while open symbols denote those that do not. (After Ref. [72].)

show a quantized Hall effect at v = 1, while the open symbols indicate those that do not. The figure demonstrates that there exists a well-defined boundary between a QHE and a non-QHE phase. The dashed line estimates the location of that boundary. The phase diagram shows that the QHE is destroyed as the layer separation (or, more properly, d/lo)is increased,even if the tunneling gap is held fixed. In the strong tunneling limit, this is just the conclusion reached earlier by Boebinger et al. [63] and discussed in the preceding section. In this limit the phase transition results from a competition between tunneling and Coulomb interaction effects [60,65,66]. Figure 2.12 contains two important additional results. First, the boundary separating the QHE and non-QHE phases appears to intercept the vertical axis at a finite value of d/l, ( w 2). This is compellingevidence that a v = 1 QHE does exist even in the limit of zero tunneling, as predicted [53]. In this case the phase transition as d/lo increases is driven entirely by Coulomb effects [54,65,69]. Indeed, this would appear to be the case for the samples used in Fig. 2.1 given their coordinates in the phase diagram of Fig. 2.12 (leftmost open and solid stars). Finally, the distribution of points in Fig. 2.12 suggests that the v = 1 QHE evolves continuously from a regime dominated by single-particletunneling to one where many-body effects are paramount. No compressible phase appears in between, and the two regimes cannot be cleanly distinguished. The pseudospin language provides an attractive way to view these results. Consider first the limit of strong tunneling and extremely small layer spacing. In

62

MULTICOMPONENT QUANTUM HALL SYSTEMS

this case, each electron is in a symmetric DQW state. The system is fully pseudospin polarized along A, and the quasiparticle activation gap A at v = 1 is just ASAS, ignoring electron-electron interactions. Turning on the Coulomb interaction can be expected to enhance the gap by adding in an exchange term. If the tunneling is now removed, the QHE survives simply because the exchange energy likes to keep the pseudospins of adjacent electrons parallel. The ground state at v = 1 remains fully pseudospin polarized and the activation gap A is the exchange energy. The crucial point is that the net polarization can point in any direction in the three-dimensional pseudospin space. The v = 1 QHE in this ASAS = d/l, = 0 limit is indifferent to whether all the electrons are in symmetric DQW states or all are in just one of the two layers or in any other combination. The ground state at v = 1 is thus highly degenerate. Tunneling merely breaks the symmetry [30,74], forces the pseudospin to lie along A, and increases the net energy gap. This is completely consistent with the experimental observation that the v = 1 QHE evolves continuously between a purely many-body state in the ASAS +0 limit to a basically single-particle effect at large ASAS. This scenario is very similar to that which occurs at v = 1in a single-layer 2DES [67]. In the limit that d/l, +0 the two problems are in fact identical. When the layer separation is not small (as is the case in the experiments)the picture is modified [70,71]. For sufficiently small d/l, (less than about 2 in the aforementioned experiments [72]), an incompressible many-body QHE state still exists in the ASAS -+O limit. In the pseudospin picture, there are two important differences relative to the d/l, = 0 case. First, the total pseudospin is forced to lie near the x-y plane. The reason for this is simple; if ( S , ) is locally finite, then in that region there are more electrons in one layer than in the other. Owing to the finite layer separation, this produces a large capacitive energy and thus is not favorable. The second important consequence of the layer separation is that the total pseudospin is no longer a good quantum number, owing to the reduced symmetry of the Coulomb interaction. This leads to quantum fluctuations of the pseudospin,which if the layer spacingis large enough, are sufficient to destroy the incompressible quantum Hall state. These fluctuations are presumably responsible for the experimentally observed [72] transition to a compressible phase at d/l, x 2 in the small ASAS limit. In their experiments, Murphy et al. [72] also examined the behavior of the v = 1 QHE in tilted magnetic fields. The motivation for this was not to access the real spin degree of freedom (which was assumed to be fully polarized),but rather, to manipulate the tunneling gap ASAS. The basis for this stems from momentum conservation in the tunneling process. As can readily be shown [75], owing to the altered relationship between wave vector and guiding center for lowest Landau level eigenstates that is induced by an in-plane magnetic field B,,, ASAS is exponentially suppressed:

with a = d/21, and tan 8 = B,,/B,.Murphy et al. [72] planned to use this suppres-

63

FQHE IN DOUBLE-LAYER 2D SYSTEMS

121

.

103

a

p:

21j

10' ri

.I.

',

3

\2

4

a-

I

\

01 0

I

1

10

I

20

I

30

Tilt Angle (0)

I

40

1

50

\

. 60

Figure 2.13. Extreme sensitivity of the v = 1 activation gap A (solid dots) to tilting in a weakly tunneling sample (AsASz0.5 K). In contrast, the v = 2/3 state (open triangles) shows little angular dependence.The dashed line is the calculated angular dependence of ASAS,arbitrarily normalized to equal A at 0 = 0. Inset: Typical Arrhenius plot for v = 1 QHE. (After Ref. [72].)

sion to sweep a single sample through the phase diagram from the strong to the weak tunneling regime. What was observed instead was a dramatic sensitivity to tilt at small B,,that was clearly not due to this matrix element effect. Figure 13 shows the measured activation gap A (not to be confused with the tunneling gap ASAS!)at v = 1 versus tilt angle 8. These data were obtained using a sample that is positioned deep in the many-body region of the phase diagram of Fig. 2.12 (the leftmost solid star). For this sample, d/l, = 1.88 at v = 1 and the calculated tunneling gap is only ASAS = 0.5 K.The figure shows that A exceeds A,,, by almost a factor of 20 in the perpendicularfield case, clearly demonstrating the dominance of interaction effects in this integer QHE state. Further evidence for this is provided by the Arrhenius plot in the inset. There it is apparent that the activated behavior collapses for temperatures above about T* w 0.4 K, some 20 times lower than the measured gap itself [76]. As the magnetic field is tilted away from normal to the plane, the measured activation gap A drops rapidly. By 8, w 8" the gap has been roughly halved. At this tilt angle, the applied parallel magnetic field is only B,,x 0.7 T. Interestingly, the gap was found to be rather insensitiveto further increases in the tilt angle. The dashed curve in the figure is proportional to the expected suppression of the single-particle tunneling gap ASAS,due to the in-plane magnetic field. The curve

64

MULTICOMPONENT QUANTUM HALL SYSTEMS

has been normalized so as to match the gap A observed at 8 = 0. Clearly, the observed tilted field behavior of the measured activation gap is inconsistent with the tunneling gap suppression. Arguing against the possibility that their results were due to a crossover of two different branches of the v = 1 excitation spectrum, Murphy et al. [72] suggested instead that a ground-state phase transition was taking place. The tilted field behavior of the v = 1 QHE in samples which are more strongly tunneling than that in Fig. 2.13 was also examined [72]. A similar transition between two distinct incompressible v = 1 states was observed; only the critical tilt angle was found to be larger. Figure 2.14 illustratesthis effect with two samples, one having ASAS = 3.6 and the other ASAS = 8.1 K. Figure 2.15 displays the dependenceof tan 8,upon AsAs/(ez/d0). [For the open symbols the 8 = 0 of ASASis employed,while for the solid symbols the calculated parallel field suppression of the tunneling gap ASAS has been included by evaluating Eq. (3) at 8 = 8,. This correction has a negligibleeffect on the most weakly tunneling sample but is quite significant for the other two.] Figures 2.13 to 2.15 make clear that tunneling plays a fundamental role in the tilted field transition at v = 1. The data suggest a competition between two ground states, one of which, at 8 < 8,,takes advantage of tunneling by forming a many-body condensate out of symmetric state electrons. The competing state apparently ignores tunneling, but in the presence of a sufficiently large in-plane magnetic field, has the lower Coulomb energy.

0.0 1 0

I

I

I

10

20

30

I

40

Tilt Angle (0)

I

50

I

60

70

Figure 2.14. Angular dependence of the v = 1 activation gap A in two more strongly tunneling samples. Dots, A,,, = 3.6 K, A(O = 0) = 5 K; squares, A,,, = 8.1 K, A(O = 0) = 14.6 K. Arrows indicate assigned critical angles. (After Ref. [72].)

FQHE IN DOUBLE-LAYER 2 D SYSTEMS

1.c

65

tE/ ......... ' i I

/

***.A

0

8 5 4-

/

0.5

/

0.0 0.00

'

/

f

/

IT!

0.05

0.10

AsAs/(e2/e.td

Figure 2.15. Comparison of the experimentally determined [72] critical angle 6, for the v = 1 tilted field transition with the theory of Yang et al. [70]. Open and solid symbols connected by dotted lines refer to the same sample, but with the calculated [73] ASAS evaluated at 6 = 0 and 6 = 6, respectively. F h e angular suppression of ASAS is calculated using Eq. (3). For the datum in the lower left corner the suppression is negligible.] Error bars reflect only the uncertainty in determining the critical angle. The dashed line is the theoretical prediction, evaluated at d/l, = 1.85.

Yang et al. [70] have offered a pretty explanation, based on a textural phase transition within the quantum ferromagnetismmodel, for these tilted field results. Adopting a particular gauge, they first note that owing to the in-plane magnetic field, the tunneling matrix element is not only suppressed in magnitude, but also acquires a spatially varying phase: t = toei#.The phase 4 advances linearly across the 2D plane in the direction perpendicular to B,,. The distance over which 4 advances by 2a is L = h/eB,,d,with d the interlayer spacing. This is just the distance required for the in-plane field B,,to thread one flux quantum between the two 2D planes. The consequence of this phase is that the local direction in pseudospin space which defines symmetric DQW eigenstates rotates as one moves across the plane. For B,,= 0 the v = 1 ground state is fully pseudospin polarized, and whatever tunneling there is orients the polarization along A and thereby lowers the total energy by an amount ASAs/2 per electron. If the system is to maintain this energetic advantage, then, for B,,> 0, the pseudospin field must

66

MULTICOMPONENT QUANTUM HALL SYSTEMS

twist in order to track the phase of the tunneling matrix element. This twisted pseudospin texture maintains the energetic advantage of tunneling, but since neighboring pseudospins are no longer parallel, it costs Coulomb (exchange) energy. As B,,is increased, the pseudospin field winds up more and more tightly and this Coulomb penalty rises. At a critical parallel field B,,,,, Yang et al. [70] predict that the system abandons tunneling and makes a transition to a new uniformly polarized state. The B,,, (or tilt angle, tan 8, = B,,,c/Bl)calculated is proportional to &, at least for small d/l, and ASAS. The dashed line in Fig. 2.15 is the prediction of Yang et al. [70], obtained by evaluating their expression for the pseudospin stiffness at d/l, x 1.85 as appropriate to the experiment. The reader is reminded that the values of ASAS used by Murphy et al. [72] are calculated ones. There is room for significant error in ASAS, owing to uncertainties about the structural parameters (band offsets,effectivemasses in the barrier, etc.) of the samples [73]. [The error bars shown in Fig. 2.15 reflect only the uncertainty in assigning the critical angle ( f 2"); this affects both tan 8, and ASASat 8 = 6,.] With these remarks in mind, the agreement between theory and experiment seems quite good. It should be kept in mind, however, that the theory curve in Fig. 2.15 does not account for quantum fluctuations arising from the relatively close proximity of the samples to the compressible phase boundary at large d/lo. Such fluctuations will probably reduce the pseudospin stiffness and thereby increase the critical angle. Aside from determining the critical angle itself, there is also the issue of why the measured activation gap behaves as it does. This has recently been addressed by Read [77] and by the Indiana group [78]. Both offer arguments for why the gap falls rapidly in the low-angle phase and is roughly constant in the high-angle phase, for 8 >> 8,. Read also gives analytic expressions for the shape of the gap near the transition angle and predicts a sharp downward cusp at 8 = 8,. No evidence for such a cusp has yet been found in experiment [79], but Read [77] remarks that the cusp will probably be wiped out by randomness in the tunneling matrix element. such randomness is surely present in actual samples, fluctuations in the quantum well and tunnel barrier widths being one obvious source. There are several other fascinatingpredictions of the quantum ferromagnetism model of the v = 1 QHE in bilayer systems. These include finite-temperature Kosterlitz-Thouless phase transitions [70,71], linearly dispersing collective modes [69,74], superfluidity in the counterllow transport channel [70,71], possible Josephson-liketunneling behavior [80], and others, all of these represent challenging avenues for future experimental work. 2.4. SUMMARY

In this chapter I have tried to convey a sense of the current excitement surrounding quantum Hall phenomena in multicomponent systems. This subject has yielded some of the most interesting results in the field of low-dimensional

REFERENCES

67

electronic systems in recent years. Despite the highly developed state of the field, much of this new physics was simply not appreciated only a few years ago. In the single-layer two-spin case, the excitement over skyrmion excitations illustrates this point well. The very recent theoretical work of Sondhi et al. [67] and others [Sl] appeared just before the experimental finding by Barrett et al. [68] that the spin polarization of the 2DES decreases both above and below v = 1. Similarly,in double-layer 2D systems the unexpected tilted field phase transition at v = 1 discovered by Murphy et al. [72] spurred the development by Yang et al. [70] of a fruitful analogy between bilayer quantum Hall systems and quantum ferromagnets. This recent theoretical work has led to a number of fascinating predictions worthy of experimental investigation. It will be particularly interesting if experimental probes other than conventional magnetotransport can be brought to bear on multicomponent quantum Hall systems. The application of optically pumped NMR [68] to the spin polarization of the 2DES is a good example of the effectiveness of a new approach. In the double-layer case the development [57] of reliable means for establishing separate ohmic contacts to the individual layers has already opened new avenues for the study of weakly coupled bilayer systems. These include measurements of Coulomb drag [82], interlayer tunneling [83], and electronic compressibility [84]. Extension of this contacting scheme to the kind of strongly coupled bilayer systems that support interlayer quantum Hall states should allow testing some of the dramatic recent predictions [70,71, SO] about such systems. Other nontransport probes, such as optical and surface acoustic wave methods, have so far not been applied with force to the multicomponent problem, but there is little doubt that they could be profitably pursued. Multicomponent quantum Hall systems are still relatively young; I think they’re a good bet for the devoted miner of strongly correlated electron systems.

ACKNOWLEDGMENTS I am indebted to a very long list of collaborators and friends for their help in learning the physics discussed in this chapter. I would like to thank in particular Loren Pfeiffer and Ken West, for growing the heterostructure samples; Sheena Murphy, for her excellent work on the bilayer v = 1 problem; Greg Boebinger, Horst Stormer, and Bob Willett, for several fruitful collaborations; and Steve Girvin, Song He, and Allan MacDonald, for many helpful discussions.

REFERENCES 1. R. B. Laughlin, Phys. Rev. Lett. 50, 1395 C1983). 2. D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Reu. Lett. 48, 1559 (1982). 3. B. I. Halperin, Helu. Phys. Acta 56,75 (1983).

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MULTICOMPONENT QUANTUM HALL SYSTEMS

4. For a review of the early experiments on the FQHE, see the chaptar by A. M. Chang in The Quantum Hall Eflect, edited by R. E. Prange and S. M. Girvin, Springer-Verlag, New York, 1987). 5. T. Chakraborty and F. C. Zhang, Phys. Rev. B 29,7032 (1984); F. C. Zhang and T. Chakraborty, Phys. Rev. B 30,7320 (1984). 6. T. Chakraborty, P. Pietililinen, and F. C. Zhang, Phys. Rev. Lett. 57, 130 (1986). 7. E. H. Rezayi, Phys. Rev. B 36,5454 (1987). 8. There are, however, some new techniques emerging. See Ref. [68]. 9. F. F. Fang and P.J. Stiles, Phys. Rev. 174,823 (1968). 10. D. A. Syphers and J. E. Furneaux, Suif Sci. 196,252 (1988);Solid State Commun. 65, 1513 (1988). 11. R. J. Haug et al., Phys. Rev. B 36,4528 (1987). 12. V. Halonen, P. Pietilainen, and T. Chakraborty, Phys. Rev. B 41, 10202 (1990). 13. J. D. Nickila and A. H. MacDonald, to be published. 14. J. Hampton, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Solid State Commun. 94, 559 (1995). 15. R. L. Willett et al., Phys. Rev. Lett. 59, 1776 (1987). 16. J. P. Eisenstein et al., Phys. Rev. Lett. 61,997 (1988). 17. P. Maksym et al., in High Magnetic Fields in Semiconductor Physics 11, edited by G. Landwehr, Springer-Verlag, Berlin, 1989. 18. J. E. Furneaux, D. A. Syphers, and A. G. Swanson, in High Magnetic Fields in Semiconductor Physics 11, edited by G. Landwehr, Springer-Verlag, Berlin, 1989. 19. R. G. Clark et al., Phys. Rev. Lett. 62, 1536 (1989). 20. J. P. Eisenstein,H. L. Stormer, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 62,1540 (1989). 21. J. E. Furneaux, D. A. Syphers, and A. G. Swanson, Phys. Rev. Lett. 63,1098 (1989). 22. J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer,and K. W. West,Phys. Rev. B41,7910(1990). 23. R. G. Clark et al., Sut$ Sci. 229, 25 (1990); and in Localization and Conjinement of Electrons, edited by F. Kuchar, H. Heinrich, and G. Bauer, Springer-Verlag, Berlin, 1990. 24. L. W. Engel, et al., Phys. Rev. B 45, 3418 (1992). 25. A. Buckthought et al., Solid State Commun. 78, 191 (1991). 26. A. G. Davies et al., Phys. Rev. B 44, 3128 (1991); Sut$ Sci. 263, 81 (1992). 27. N. G. Morawicz et al., Semicond. Sci. Technol. 8, 333 (1993). 28. M. Dobers, K. von Klitzing, and G. Weimann, Phys. Rev. B 38,5453 (1988). 29. We are thus ignoring the possibility of skyrmion-like large-spin excitations. See Section 2.3.4. 30. M. Rasolt, F. Perrot, and A. H. MacDonald, Phys. Rev. Lett. 55,433 (1985);M. Rasolt and A. H. MacDonald, Phys. Rev. B 34,5530 (1986). 31. D. Yoshioka, J. Phys. SOC.Jpn. 55,3960(1986). 32. P. Maksym, J. Phys. C 1,6299 (1989). 33. X. C. Xie, Y. Guo, and F. C. Zhang, Phys. Rev. B 40,3487 (1989). 34. T. Chakraborty, Sut$ Sci. 229, 16 (1990).

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69

35. P. Beran and R. Morf, Phys. Rev. B 43,12654 (1991). 36. Studies of the 3 state have also been reported by T.Sajoto et al., Phys. Rev. B 41,8449 ( 1990). 37. G. Ebert et al., J . Phys. C 17,1775 (1984). 38. R. G.Clark et al., Surf. Sci. 170,141 (1986). 39. F. D. M. Haldane and E. H. Rezayi, Phys. Rev. Lett. 60,956(1988). 40. J. P. Eisenstein et al., Surf: Sci. 229,31 (1990). 41. R. Du et al., to be published. 42. The effect of disorder can be roughly approximated simply by reducing the energy gap by a constant amount r. For an unpolarized ground state with AS = + 1 and no tilt, we write A =A, - J g J p B B r. After choosing a specific value for r, the proportionality constant between A, and e’/do can be determined by fitting the formula to the measured gap value for the sample’s as-grown density. This fixes the overall density dependence, provided that r remains constant. (For gate-induced density changes, this is a good assumption; see Ref. [84].) For state-of-the-art heterostructures r x 2K.Combining this value with the data in Fig. 2.6leads to the conclusion that the v = 4 gap would increase with density out to about B = 6 T and remains positive to beyond B = 9T. 43. A. H. MacDonald, D. Yoshioka, and S. M. Girvin, Phys. Rev. B 39,8044(1989). 44. E. H.Rezayi and F. D. M. Haldane, Phys. Rev. B 42,4532(1990). 45. X. C. Xie and F. C. Zhang, Mod. Phys. Lett. B 5,471(1991). 46. G. Moore and N. Read, Nucl. Phys. B 360,362(1991). 47. M. Greiter, X. G. Wen, and F. Wilczek, Phys. Rev. Lett. 66,3205(1991). 48. L.Belkhir and J. Jain, Phys. Rev. Lett. 70,643 (1993). 49. T.Chakraborty and P. Pietilainen, Phys. Rev. B 38,10097 (1988). 50. R. Morf,unpublished. 51. Ignoring interaction and level mixing effects, this assumption obviously fails at low magnetic field. 52. F. D. M. Haldane and E. H. Rezayi, Bull. Am. Phys. SOC.32,892(1987). 53. T.Chakraborty and P. Pietiliiinen, Phys. Rev. Lett. 59,2784(1987). 54. D.Yoshioka, A. H. MacDonald, and S . M . Girvin, Phys. Rev. B 39,1932(1989). 55. L.N.Pfeiffer, E. F. Schubert, K. W. West, and C. Magee, Appl. Phys. Lett. 58,2258 (1991). 56. Y. W. Suen et al., Phys. Rev. B 44,5947 (1991). 57. J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Appl. Phys. Lett. 57,2324 (1990). 58. Y. W. Suen et al., Phys. Rev. Lett. 68, 1379 (1992). 59. J. P. Eisenstein et al., Phys. Rev. Lett. 68, 1383 (1992). 60.S.He, X. C. Xie, S. Das Sarma, and F. C. Zhang, Phys. Rev. B 43,9339(1991);S . He, X. C. Xie, and S. Das Sarma, Surf. Sci. 263,87(1992). 61. S.He, S.Das Sarma, and X . C. Xie, Phys. Rev. B 47,4394(1993). 62. Y. W. Suen et al., Phys. Rev. Lett. 72,3405 (1994). 63. G. S.Boebinger, H. W. Jiang, L. N. Pfeifer, and K. W. West, Phys. Rev. Lett. 64, 1793 (1990);Phys. Rev. B 45,11391 (1992).

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64. M. B. Santos, L. W. Engel, S. W. Hwang, and M. Shayegan, Phys. Rev. B 44,5947 (1991). 65. A. H. MacDonald, P. M.Platzman, and G. S . Boebinger, Phys. Rev. Lett. 65, 775 (1990). 66. L. Brey, Phys. Rev. Lett. 65,903 (1990). 67. S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. Rezayi, Phys. Rev. B 47, 16419 (1993). 68. S.E. Barrett,et al., Phys. Rev. Lett. 74,5112(1995);A.Schmeller,et al.,Phys. Rev. Lett. 75,4290 (1995); and E. Aifer, et al. Phys. Rev. Lett. 76,680 (1996). 69. H. A. Fertig, Phys. Rev. B 40,1087 (1989). 70. Kun Yang et al., Phys. Rev. Lett. 72, 732 (1994). 71. K. Moon et al., Phys. Rev. B 51, 5138 (1995). 72. S. Q. Murphy et al., Phys. Rev. Lett. 72,728 (1994). 73. The tunneling gap AsAsis calculated self-consistently, at zero magnetic field, in the Hartree approximation. The values given here differ slightly from those given by Murphy et al. [72], owing to the use of a revised r-point conduction band offset between GaAs and AlAs (1.05 eV instead of 0.83 eV). These calculations represent a significant source of possible systematicerror. In Ref. [72], for only the most strongly tunneling sample (AsAsx 8.1 K) could this calculation be verified experimentally (via Shubnikov-de Haas analysis). 74. X.G. Wen and A. Zee, Phys. Rev. Lett. 69, 1811 (1992); Phys. Rev. B 47,2265 (1993). 75. J. Hu and A. H. MacDonald, Phys. Rev. B 46, 12554 (1992). 76. The unusual temperature dependence of the bilayer v = 1 QHE has been studied more thoroughly by T. S . Lay, Phys. Rev. B 50,17725 (1994). 77. N. Read, Phys. Rev. B 52,1926 (1995). 78. See Chapter 5, this volume. 79. S. Q. Murphy et al., unpublished. 80. Z . F. Ezawa, Phys. Rev. B 51, 11152 (1995) and the references therein. 81. H. A. Fertig, L. Brey, R. C&b, and A. H. MacDonald, Phys. Rev. B 50, 11018 (1994). 82. T. J. Gramila et al., Phys. Rev. Lett. 66, 1216 (1991). 83. 3. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 69,3804 (1992), Surf: Sci.305, 393 (1994); Phys. Rev. Lett. 74, 1419 (1995). 84. J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Reu. Lett. 68,674 (1992); Phys. Rev. B 50, 1760 (1994).

PERSPECTIVES IN QUANTUM HALL EFFECTS: Novel Quantum Liquids in Low-DimensionalSemiconductorStructures Edited by Sankar Das Sarma, Aron Pinczuk Copyright0 2004 WILEY-VCH Verlag GmbH & Co. KGaA

3

Properties of the Electron Solid H. A. FERTIG Department of Physics and Astronomy and Center for Computational Sciences, University of Kentucky, Lexington, Kentucky

3.1. INTRODUCTION

The ground state of a collection of noninteracting electrons in a structureless background is well known to be a degenerate Fermi gas. Remarkably, this simple model accounts for much of our understanding of the properties of electrons in metals. One reason this simple model works is that the effect of the periodic potential due to the ionic lattice in which the electrons move can be accounted for approximately by using a renormalized electron mass. More complex questions arise when the electron-electron interaction is taken into account; however, in metallic systems many system properties still behave like those of noninteracting electrons, because of screening effects. The cohesive energy of the degenerate electron gas Egasin a jellium model (i.e., electrons moving in a uniform neutralizing charged background) has a wellknown expansion that is valid at high densities,whose first two terms are just the result of the Hartree-Fock approximation. In three dimensions this takes the form [1J 2.21 0.916 E,,, z -- rf

rs

RY

In this expression, r, is a unitless measure of the average interelectron distance, defined as r, = (rne2/h2)(3/47rp)1/3, where p is the electron density. The energy unit is defined in the usual fashion as 1 Ry = rne4/2h2 = 13.6 eV. This model can also be used for the cohesive energy of electrons in solids by replacing m with the electron effective mass m* and the electron charge with e/&, where E is the dielectric constant of the host crystal. For semiconductors, this typically lowers the effective rydberg value by two to three orders of magnitude. The first and second terms in this expansion represent, respectively, the kinetic energy and the

Perspectives in Quantum Hall ENects, Edited by Sankar Das Sarma and Aron Pinczuk.

ISBN 0-471-11216-X 0 1997 John Wiley & Sons,Inc.

71

72

PROPERTIES OF THE ELECTRON SOLID

Hartree-Fock approximation of the potential energy of the system. Higher-order corrections in r, may be computed systematically as well [2]. Long ago, it was pointed out by Wigner [3] that a degenerate Fermi gas is really not the only possible state of electrons in the jellium model. Wigner hypothesized that at low densities, the electrons might crystallize; the energy of such a state has been estimated [4] to have the form E,,x

1.792 2.26

--

rS

b

+pT+z s

Ry

for a BCC lattice. The first term represents the Coulomb energy and is negative because of the interaction of the electrons with the uniform neutralizing background; the second term represents the energy due to zero-point motion of the phonons in this Wigner crystal(WC),and the third term is the first correction due to anharmonicities in the crystal.The constant b has been estimated to be slightly less than unity [4]. Clearly, for very low densities, Eq. (2) will give an energy below that of Eq. (1), indicating a phase transition from the gas to a crystal phase. Physically, the transition may be understood as being driven by quantum fluctuations: As the density of the crystal is increased, the electrons are confined to ever-smaller regions (i.e., the unit cell of the crystal decreases in volume), and the uncertainty principle requires the admixing of higher momentum states to do this. This contribution to the WC energy is contained in the second and third terms of Eq. (2), and it is clear that for small enough r, (high density),these terms will overwhelm the first term. Thus, due to the uncertainty principle, one pays a quantum-mechanical kinetic energy cost to form the crystal. For small enough r,, it becomes favorable for the system to go over to the gas state, which in fact has the minimum possible kinetic energy for the system. The precise value of r, at which the transition takes place is difficult to estimate, as it requires an accurate knowledge of the higher-order terms in both Eqs. (1) and (2) [S]. 3.1.1. Realizations of the Wigner Crystal

Because of the necessity of obtaining low-density electrons to realize a WC, metals are usually not good candidates for observing it. The first convincing evidence of an experimentally realized electron crystal came much after its first prediction, and was two-dimensional rather than three-dimensional.Looking at a system of electrons adsorbed on a helium surface, Grimes and Adams [6] measured the density response function using a radio-frequency technique. They found a series of resonances that could be identified with the predicted dispersion of phonons of the WC [7,8] of electrons adsorbed on a helium surface. This was regarded as strong evidence that the electrons had indeed crystallized in this system. The areal density of electrons in this system is generally extremely small (typically,n, lo7cm-’), so that for all intents and purposes the system may be regarded as classical (i.e., for such large distances between the electrons, the

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INTRODUCTION

73

energy due to the zero-point motion of the phonons is negligiblecompared to the potential energy of the lattice). While there are a number of interesting properties associated with the WC in this classical limit, particularly relating to its behavior in a disorder potential (which we discuss in detail below),one disadvantage of the helium system is that the densities that may be realized practically are far too low to allow an observation of the transition from the crystal to the gas state. A more favorable system to study in this regard is electrons in a semiconductor environment. Semiconductors offer a particular advantage because one can control the density of electrons over several orders of magnitude by doping the system at an appropriate level. However, for bulk doped semiconductors, an immediate problem arises: The dopants that provide the electrons, once ionized, become strong scattering centers, completely changing the character of the ground state. At zero temperature, the electrons fall back on their donors rather than localizing at crystal sites. The resulting ground state is highly disordered and not collective in nature. Clearly, the problem with bulk semiconductors is that the electrons are free to move in the region of the donors, which inevitably imposes a strong disorder potential on them. A major achievement in the last 15 years that allowed this problem to be circumvented was the invention of modulation doping [9, lo]. These systems use high-quality interfaces between different semiconductors, typically GaAs and AlGaAs, to trap a two-dimensional layer of electrons, provided by dopants that are setback some distance d from the interface. By separating the electrons from the dopants, the effect of the resulting disorder potential is substantially smaller, and the prospects of observing the WC are greatly enhanced. The strength of the disorder can be characterized by a setback ratio d/a,, where a, = J.s/.is the typical interelectron spacing in two dimensions. One wishes to achieve very large values of this ratio to minimize the effects of disorder. One way to do this is to work with high-density samples, for which ratios of d/a, 4 to 6 can in practice be achieved. However, the price one pays for such favorable setback ratios is that the resulting densities are typically above the Wigner transition, so that one is in the gaseous rather than the crystal regime.

-

3.1.2. Wigner Crystal in a Magnetic Field It was recognized early on that the prospect of achieving crystallization in these systems can be greatly enhanced by applying a strong perpendicular magnetic field.The reason is that magnetic fields quench the kinetic energy of the system, so that zero-point motion effects-which tend to favor the gaseous state-can be suppressed. In the presence of a magnetic field, the kinetic energy of the electrons has the form [ll] En = (n 1/2)Aw,, where n is an integer, o,= eB/mc is the cyclotron frequency, B is the magnetic field, and m the effective mass of the electrons. For each value of n, there is a vast degeneracy: The number of states is proportional to the area of the system. These energy states are known as Landau levels, and because of the huge degeneracy within a Landau level, it is possible to form linear combinations of states that represent localized Gaussian orbitals. If

+

74

PROPERTIES OF THE ELECTRON SOLID

one uses just states in the lowest Landau level, working in the symmetric gauge vector potential [l 11, these orbitals have the form

where 1; = hc/eB is the square of the magnetic length and R represents a site near which the electron is localized. As can be explicitly seen, in the limit of very large magnetic fields, I , *0, so that the electron becomes highly localized while maintaining the minimum possible kinetic energy. Thus in the limit of large fields, the energy of the system can be minimized by putting the electrons in states of the form in Eq. ( 3 ) and then finding the set of R s that is optimal for the potential energy. In the high-field limit, this becomes completely equivalent to finding the ground state of a set of classical charged particles [ l 2 ] , so that one expects a crystalline ground state for the system. Thus, for any density of two-dimensional electrons, a large enough magnetic field should allow the electrons to form a crystalline ground state.

3.2. SOME INTRIGUING EXPERIMENTS In this section we describe some of experimental results that have led to renewed interest in the Wigner crystal in the past few years. This is not meant to represent a complete review of the experimental situation. Rather, it is a sampling of results which may or may not indicate that the W C has been realized in semiconductor systems, but definitely indicates that something interesting is happening to the electrons in these two-dimensional systems in very strong magnetic fields.

3.2.1. Early Experiments: Fractional Quantum Hall Effect The earliest experiments on modulation doped semiconductorsfocused on their transport properties. Placing the two-dimensional electron gas (2DEG)perpendicular to an applied magnetic field in the 2 direction and imposing a current density jxA, one may measure the resulting electric field parallel to the current (Ex), yielding the diagonal resistivity pxx= Ex/jx, as well as the electric field perpendicular to the current (Ey),yielding the Hall resistivity pyx= Ey/jx.According to the simplest semiclassicalmodels of electrons [ 1 3 ] , which ignore electronelectron interactions as well as many quantum effects,the expected values of these quantities are pxx= m/ne2z,pxy= - B/nec, where T is a phenomenological scattering time and n is the two-dimensional density of electrons. It was already known [ 1 1 ] from experiments on related MOSFET systems that the lowtemperature behavior of these transport coefficients was dramatically different than the semiclassical analysis suggested. Near filling fractions v E 27tl;n in the vicinity of an integer, the pxyturns out to be quantized in units of h/e2,and pxx

SOME INTRIGUING EXPERIMENTS

75

nearly vanishes for low-enough temperatures. This important discovery is generally known as the integral quantum Hall efect (IQHE). The filling fraction v may be thought of as a measure of how full the Landau levels are for a given density of electrons; it is equivalent to the ratio of the total number of electrons in the system to the number of states available in a Landau level. Thus, if one wishes the electron to crystallize, one needs to reach the limit v << 1, most ideally by applying the strongest magnetic fields available. (Alternatively, one may fabricate a sample with small electron densities; however, such samples tend to be more strongly disordered.) Experimentslooking for the WC in precisely this way yielded still another dramatic discovery: the fractional quantum Hall effect (FQHE) [141.The phenomenologyof the FQHE is essentially the same as that of the IQ H E One finds quantized values of pxyand a vanishing pxx, except now in the vicinity of filling factors v = p/q, with p and q integers. (Usually, q is an odd integer,although it can be even for layered systems or systems in which the electron spin is not fully polarized by the magnetic field.) However, the origin of the FQHE is quite different from that of the IQHE. The latter may be understood purely in terms of noninteracting electrons Ell], whereas in our current understanding, electron-electron interactions are crucial in the former. Interestingly,it was initially thought that the FQHE might be explained by the electrons forming a WC [14]. However, it was quickly realized that the phenomenology of the transport properties could not be understood in terms of a crystalline ground state. It can be shown [151 that the FQHE results from cusps in the total energy per particle of the 2DEG as a function of v; however, Hartree-Fock calculations [16] of the energy of the WC show that its energy is a smooth function of v. Indeed, the transport properties that one expects from a WC, which are generally understood in analogy with charge density wave (CDW) systems [171, are quite different from those of the FQHE. In particular, any random potential, no matter how weak, is expected to pin the crystal, due to its highly collective nature. This means that at zero temperature, the electrons are unable to respond to an arbitrarily weak electric field. Thus a WC is expected to be insulating, characterized by a divergent pxxin the limit T +0. The FQHE has precisely the opposite behavior (pxx+O as T +0).Indeed, the current understanding of the FQHE is that it results from a correlated liquid ground state, which cannot be pinned [ll]. To get transport from a WC at zero temperature, as in CDW systems, one expects that there is a minimum threshold electric field that must be applied to depin the crystal from the impurity potential; for CDWs, this is referred to as a sliding state.

3.2.2. Insulating State at Low Filling Factors: A Wigner Crystal? In 1990, behavior very reminiscent of the phenomenology described above was observed in modulation doped systems, generating great excitement among people thinking about the WC. Working with extremely high mobility (i.e., very high purity) samples, it was found [18] that in the vicinity of v = 1/5 there is

0.21

I

6 2 7

,

-

0.22

I

0.20

,

I

,

,

0.16 FILLING FACTOR Y

(XI0

,

, 0.14

MAGNETIC FIELD ( T I

Figure 3.1. Reentrant behavior of diagonal resistance R,, a p x x near Y = 1/5 in a highmobility heterojunction. Inset: Schematicdrawing of solid-state energy EkC and fractional quantum Hall liquid-state energy EL versus v. The solid line for EL is an interpolation of computed energies for filling factors v = l/m; the dashed line denotes the actual expected dependence of liquid-state energy. (From Ref. [18].)

a transition from FQHE to an insulating state, both below v = 1/5 and for a small range just above it. This behavior is illustrated in Fig. 3.1. The regions of large px, diverge very rapidly as a function of temperature in these data, pxxcc eAIT,with A typically [19] of order 1K. Such activated behavior is precisely what one expectsfor an insulating ground state. More recently, experimentshave exhibited very similar behavior in the vicinity of v = 1/3 for hole systems [20] and v = 1/2 for double-layer electron systems [21]. If the insulating state may be appropriately described as a crystalline state, this transition is highly analogous to the one first proposed by Wigner, for which the crystallineground state is destabilized by quantum fluctuations. Around the same time, experimentson similar samples [22] showed that these systems exhibit a nonlinear current response to an applied voltage, as illustrated

SOME INTRIGUING EXPERIMENTS Y

m

dV

= 0.170 137 mK

0

1

0

dV

0.5

T=44mK

u=0.190

m

77

6% I

I

I

I

1

1

1

1

1

I

in Fig. 3.2. One can clearly see a threshold voltage above which the resistance begins to decrease, which is highly reminiscent of depinning behavior in CDWs. Furthermore,it was found that broadband noise is generated by the system in the sliding state [22,23], again in close analogy with known phenomenology for CDWs. It is also interesting to note that very analogous results have been found [24] for electrons on thin helium films, where the source of disorder is a glass substrate on which the helium film is adsorbed. The remarkable similarity in the helium and semiconductor data is particularly significant because there is broad agreement that in the former system,the electrons form a crystal in the absence of disorder. However, it is unclear to what degree the electronic state becomes disordered by the glass substrate. Indeed, this isue of how the external random potential deforms the crystal-and whether the system can still be described as a crystal at all-has only recently been addressed for the WC [25,26]. The reentrance phenomenon between the FQHE and the insulating state may be understood by noting that the energy of the correlated liquid state must have a cusp at v = 1/5 in order for pxy to be quantized. The energy of the crystal, by contrast, is a smooth function of the filling fraction.This means that if the energies of the WC and the liquid ground states are competitivein the vicinity of this filling fraction, the liquid-state energy might be expected to dip below the crystal energy for a finite range of filling fractions, as illustrated in the inset of Fig. 3.1, leading to the small island of insulating behavior for v > 1/5. To test this idea properly, careful calculations of both the liquid-state energy and the WC energy are necessary. The basic form for the liquid ground state is the well-known Laughlin

78

PROPERTIES OF THE ELECTRON SOLID

wavefunction[11,27], which describes the liquid precisely at v = l/m for rn an odd integer, whose energy may be evaluated quite accurately [11,27-291. As suggested by Eq. (2), the first correction to the Hartree-Fock energy of the WC ground state would be due to zero-point motion of the phonons. This was computed using a trial wavefunction method [30], with the resulting estimate for the transition between FQHE and the WC taking place near v = 1/7. An important improvement to both the liquid-state energy [31] and the WC energy [32] is inclusion of Landau level mixing. This not only shifts the transition to near v = 1/5 for the electron systems, but also predicts a transition in the vicinity of v = 1/3 for hole systems. This impressive agreement with experiment strongly favors the interpretation of the insulating state as a WC. Many other experimental probes have been employed to probe the insulating state. Some early experiments [33] used radio-frequency techniques similar to those employed for electrons on helium, looking for the shear modes associated with the crystal. The interpretation of these data proved to be controversial [34], however, as it was unclear whether the electric field probe coupled to phonon modes of the underlying semiconductor structure in addition to, or instead of, the phonon modes of the WC. However, this early data did show the reentrant behavior near v = 1/5 that was found in subsequent dc transport experiments. There are also a number of more recent experimentsprobing the collectivemodes of the 2DEG in its insulating phase by measurements of the frequency-dependent conductivity via surface acoustic wave methods 135,361, and the dielectric response using capacitive coupling methods [37]. The former measurements reveal broad resonances at low temperatures and low filling factors near 1 GHz, and the latter a remarkably large response at very low frequencies. Both results may be qualitatively understood as a collective response of a pinned WC, although the correct quantitative analysis of the data-in particular, what they say about the positional correlation length of the crystal-remains a matter of some controversy [38-401. Some of the transport data in the insulating phase are not clearly consistent with a WC interpretation. Narrowband noise in the sliding state [22,23], which arises in analogous CDW systems [17], has not been observed in the insulating state. Narrowband noise arises in CDW systems because the lattice should deform in a periodic way as it slides across a disorder potential. This leads to a noise spectrum with a peak at a frequency proportional to the velocity of the sliding electrons. (Very recent experimental work on the WC has found some nontrivial coupling between ac and dc electric fields, which may be a precursor of narrowband noise [37].) However, the absence of narrowband noise in these systems should not be construed as evidence that the electron state is extremely disordered, as one expects to observe this phenomenon only if the positional correlation length is on the order of the system size [17]. Computer simulations of the classical electron crystal [26] suggest that such long correlation lengths are unlikely to be achieved for presently available samples. A perhaps more troubling experimental observation has been that the Hall resistivity pxyof these systems appears to be finite in the low-temperature limit,

SOME INTRIGUING EXPERIMENTS

79

and equal to its classical value to within a few percent [41,42]. This is in contrast to what one might naively think For example,if the current at finite temperatures is carried by point defects (as occurs for bulk ionic solids) such as vacancies and interstitials, one expects the effective number of carriers neff to vanish exponentially with temperature. It has already been remarked that the diagonal resistivity, which classically is proportional to the inverse of the carrier density, behaves precisely as if this is occurring. However, classically pxycc he;,', so that this model predicts activated behavior. The distinction between the behaviors of pxx and p x yis so dramatic that one group [43] has postulated that the ground state of these systems is not a WC at all, but rather is a disorder-dominated state dubbed the Hall insulator. However, as described below, when quantum fluctuations are accounted for, one can understand how a model with finite temperature conduction due to point defects would lead to a classical Hall resistivity [44,45].

3.2.3. PhotoluminescenceExperiments Another promising line of experimental research is photoluminescence (PL). These experiments use holes to probe the electron gas and are generally introduced into the system in one of two ways. In localized hole experiments [46-491, a layer of acceptors is intentionally grown into the crystal relatively close to the electron plane. In their ground state, the acceptors are negatively charged; however, by shining light on the sample at appropriate wavelengths an electron can by ionized out of the core of the acceptor, leaving behind a neutral object which presumably has little effect on the state of the 2DEG. By observing recombination of electrons in the 2DEG with the core hole, a characteristic spectrum consisting of a doublet is typically observed.This doublet structure has interesting temperature and magnetic field dependence: at low temperatures and most fields, the greatest portion of the oscillator strength is found in the lower portion of the doublet. At temperatures of order 1 to 2 K, the oscillator strength is transferred to the upper PL line (Fig. 3.3). This behavior has been interpreted as a signal of melting for the WC, and is supported by the observation of very similar behavior at very low temperature when the magnetic field is tuned such that the filling factor is 1/5, where it is expected the system forms a correlated quantum liquid (i.e., the Laughlin state). Another interesting experimental observation is that similar behavior occurs near v = 1/7, 1/9, where no FQHE is observed in transport; it is thus an open question precisely what is happening at these very small filling fractions. The second class of PL experiments uses valence band holes, generally in wide quantum well structures [SO-SZ]. The phenomenology of these data is in many respects quite similar to that of the localized hole experiments as illustrated in Fig. 3.4. The advantage of such geometries is that the holes may in principle by itinerant, allowing them to be sensitive to global properties of the lattice. Localized holes, by contrast, can only probe local properties. The disadvantage to valence band hole experimentsis that not much is currently understood about the initial quantum states of the holes [52], making interpretation of currently

80

PROPERTIES OF THE ELECTRON SOLID I

I

I

0 s = 6xlOlocm-2

H = 26T

l2=0 I

-L I

1 2 Temperature (k)

I

3

Figure3.3. Intensity ratio for lines in photoluminescence doublet as a function of temperature in an insulating state, as measured in localized hole experiments. (From Ref. [46].)

1.8 c, .-

1.41

v)

c

W 4-

0

W

a

-c

m

1.0

\

1.2

0

0.4

?c2

E 2.2

.3

0

c W V

8 W .-c

- 1.8 4-

C

5

-I

0 1.521

0.4

1.4

1.523 Enegy (eV)

0

1

2 T (K)

3

4

Figure 3.4. Photoluminescence spectra observed in an insulating state as measured in valence hole experiments:(a)temperature dependence of the spectrum; (b) intensity ratio of lines labeled S and B' in part (a)as a function of temperature. The ratio drops markedly in the range of temperatures between T,, and Tc2.(From Ref. [Sl].)

DISORDER EFFECTS ON THE ELECTRON SOLID

81

available data difficult. There is also a class of related experiments using recombination of electrons in higher electric subbands of a heterostructure with valence band holes to probe the Wigner crystal structure [53]. One exciting possibility of itinerant hole PL experiments is that one might observe some features of the Hofstadter spectrum [54,.55]. This is the energy spectrum of a charged particle moving in a uniform magnetic field and a periodic potential and is very sensitive to the precise value of the magnetic field. Specifically, for a rational filling fraction v = p / q , the spectrum is expected to have q bands with q - 1 gaps. An observation of this behavior would constitute strong evidence that the electrons have indeed crystallized. The PL spectrum expected from real samples in the WC states requires a detailed treatment of the coupling between the hole and the WC, and some features of the Hofstadter spectrumparticularly the smallest gaps-are in practice dificult to observe [56]. This problem is compounded by the effects of disorder. Nevertheless, the possibility of seeing some of the basic structure of the Hofstadter spectrum in PL experiments remains a tantalizing possibility. 3.3. DISORDER EFFECTS ON THE ELECTRON SOLID: CLASSICAL STUDIES From the discussion in Section 3.2 it is clear that interpretation of almost any experimental data on the WC in semiconductor environments requires some understanding of the sources of disorder in these systems, as well as what effect they have on the ground state of the electrons. The importance of disorder is further emphasized by very general arguments due to Imry and Ma [57] that one cannot have long-range positional order at zero temperature in a classical system in an arbitrarily weak random potential in two dimensions. It thus becomes relevant to study whether crystallization of electrons in the heterostructure systems is in any sense possible even if quantum fluctuations are ignored. Such studies are directly relevant to electrons at the very highest magnetic fields (very low filling fractions),where quantum exchange effects are believed to be small [ 5 8 ] , as well as to electrons on helium surfaces. In principle, classical states may be used as a starting point from which quantum effects and corrections may be studied. 3.3.1. Defects and the State of the Solid Since two-dimensional solids in a random potential cannot exhibit long-range positional order, they cannot be classified as crystals in the sense of classical crystallography. Nevertheless, it is known that if the disorder is not strong enough to induce isolated topological defects of the crystal, specifically dislocations, the system can possess long-range orientational order [59,60]. Furthermore, the positional correlations in this state fall off as a power law with distance rather than exponentially,so that the state may be characterized as having quasilong-range order. Thus, one usually thinks of this as a crystal state.

82

PROPERTIES OF THE ELECTRON SOLID

In the absence of disorder, it is known that temperature can destroy the crystal above the Kosterlitz-Thouless transition temperature [61,62] via the unbinding of dislocation pairs with equal and opposite Burgers vectors. (In two-dimensional crystals, a dislocation may be thought of as a line of particles that comes to an abrupt end somewhere inside the crystal. The Burgers vector, which characterizes the strength of the dislocation, is the amount by which a circuit of steps that should close on itself in a perfect lattice fails to do so when the circuit surrounds a dislocation core.) The resulting state has short-range (exponential) order and quasi-long-range orientational order; because of the latter it is not a true liquid and is known as a hexatic state [63]. This state is destroyed at a still higher temperature by disclination defects. Disclinations are defect sites that have the wrong number of nearest neighbors relative to the number expected for a perfect lattice; for triangular lattices, dislinations typically have five or seven nearest neighbors. Such defects may be said to have disclination charge k 1. (However, this should not be confused with electric charge.) When bound together, a net neutral pair of disclinations separated by a single lattice constant forms the core of a dislocation. The hexatic state is destroyed when the temperature is high enough that these disclinations can unbind and be found in isolation in the resulting liquid state.

3.3.2. Molecular Dynamics Simulations It is natural to look for an analog of this phenomenology at zero temperature as a function of disorder strength. Simulations [25,26] of classical electrons moving in a random potential modeled after the heterostructure systems reveal that only half this phenomenology is actually reproduced. The model used contains N electrons in a plane setback from a plane of N positively charged ions by a distance d (Fig. 3 . 9 , and low-energy states are generated by a moleculardynamics simulated annealing method, in which the electrons are assumed to follow Newton’s equations, given an initial temperature well above the melting

d 0

0

Q 0

o

0

0

Figure3.5. Model system of N electrons and N quenched positively charged ions, in planes separated by a distance d. (From Ref. [26].)

DISORDER EFFECTS ON THE ELECTRON SOLID

83

transition, and then are slowly cooled by removing kinetic energy from the electrons in small discrete amounts. Details of these simulations may be found in Ref. [26]. Figure 3.6 shows some typical low-temperatureconfigurations,well below the freezing transition, for different values of the parameter d/u,, where here a, is the lattice constant of a perfect WC at the electron densities used in the simulation. The electric forces in these simulations were screened dielectrically with E = 13 (the dielectric constant of GaAs), and the electron densities shown in the figures were taken to be 5.7 x 10'0cm-2. Periodic boundary conditions were assumed for these systems. For all values of d/a,, disclinations appear in the lowtemperature state. These are denoted in the figures as + for sites with five nearest neighbors, and as x for sites with seven nearest neighbors. For large values of d/a, (Fig. 3 . 6 ~isolated )~ dislocations (denoted as bound pairs of + and x in the figures) may clearly be seen in the configuration. As d is increased further, the density of dislocations decreases until the average spacing between them exceeds the sample size. It is interesting to note that grain boundaries do not appear in these configurations.Some of the present literature on the WC has assumed that the effect of disorder is to introduce well-ordered microdomains, separated by sharp boundaries. This would be reflected in these configuration if the dislocations collected together to form grain boundaries [62]. Apparently, this is not energetically favorable for this type of disorder. However, one cannot rule out that other disorder potentials could introduce these defects. As the ratio d/a, is decreased (Fig 3.6b and c), the ground state of the WC undergoes a zero-temperature phase transition, from a hexatic state to an isotropic state. Both states must be characterized as glasses, because there is only short-range order in the positional correlation functions. In Fig. 3.6b, one may see disclination pairs that are separated by more than a single lattice constant but are still clearly bound together. Above a critical setback distance (Fig. 3.6c), isolated disclinations may be found. Figure 7 illustrates the orientational correlation functionsfor several values of d/a, for low-temperature states generated by the MD method. They can be well fitted by exponential forms in the strong disorder limit (small d/a,), and the resulting correlation length is found to be a slowly varying function for d/a, < 1. In the interval 1.1 < d/a, < 1.2, the correlation length rises rapidly, suggesting a possible divergence (and an associated phase transition). However, once the correlation length exceeds the system size, fits to either an exponential form or a power law become possible, and it becomes difficult to identify precisely what value of d/a, would be the critical one in an infinite system.One can estimate from Fig. 3.7, however, that the transition between the states occurs near d,la,.= 1.15. Several comments are in order. It must be noted that these simulations include neither the finite thickness of the layer nor the finite value of the magnetic length I, when a magnetic field is present. Both these effects tend to soften the electronelectron interaction relative to the electron-ion interaction, so the actually value of d, is expected to be somewhat higher than the simulation value in real heterojunction systems. Experimentally, the best such samples have d/a, x 6, so

<

84

PROPERTIES OF THE ELECTRON SOLID

Figure 3.6. Sample ground-stateconfigurations, for different levels of disorder. Locations of disclinations are marked by x for a sevenfold site and by + for a fivefold site. Bound pairs of these defects are equivalent to dislocations. Pictures contain = 1600 particles; actual simulations samples contained 3200 particles. (a) d/a, = 1.5; (b) d/a, = 1.3; (c) d/ao= 1.0. (From Ref. [26].)

DISORDER EFFECTS ON THE ELECTRON SOLID

0

5

10 rlao

15

85

20

Figure 3.7. Orientational correlation functions for various setback distances. Different symbols represent data for samples with different setback distances. From top to bottom, d/a, = 2.0, 1.7, 1.5, 1.3, 1.2, 1.1, 1.0,0.9,and 0.8. (From Ref. [25].)

that these systems are likely to be well inside the hexatic phase. In principle, one cannot have a crystal phase for this type of disorder: isolated dislocations are always present, and their density simply decreases with increasing setback distance. In practice, however, the distance between dislocations turns out to increase very rapidly with setback distance. The conclusion one then reaches based on these simulations is that it is not possible to have a true electron crystal state in the heterojunction systems; the best one can hope for is an hexatic state. However, the positional correlation lengths associated with the large setbacks in real samples may be extremely large, so that the system could for many properties behave very much like a crystal. Electrons on helium surfacesalso present a possible system in which this phase transition might be observed. Since the distance between the electrons and the disorder source(typica1lya glass slide)can be varied by increasing the helium film thickness, such an experiment represents a direct way of probing the transition. Furthermore, because the nature of the disorder in the helium system is considerably different from that in delta-doped heterostructures, there remains the possibility of observinga true crystal phase in these systems. This is discussed in more detail below. A novel way of detecting the transition is by measuring the depinning field. Associated with the appearance of isolated disclinations, the behavior of the threshold electric field is changed as shown in Fig. 3.8. The crossover takes place at approximately d/a, x 1.15 f 0.1, which is consistent with the vanishing of orientational order. To understand this behavior qualitatively, it is necessary to observe the motion of the electrons as they depin [26]. It is found that the

86

PROPERTIES OF THE ELECTRON SOLID

0 : N = 3200 A:N=800

.-L

0.5

1.5

1.0

dlao

2.0

3

i

2.5

Figure 3.8. Depinning threshold electric field, in units of E , = e/Kai. Dotted lines are guides to the eye. (From Ref. [25].)

electrons tend to flow along directions of the local bond orientation (i.e., to flow along local symmetry directions of the crystal). Since the system does not have long-range orientational order, it is necessary for electrons to pass through regions ofgreat strain in the lattice, where the orientation changes.These regions of strain represent bottlenecks in the electron flow. As the disorder strength is tuned and orientational order changes from quasi-long range to short range, the number of bottlenecks proliferates and there is a sharp increase in the threshold field. We also note that the threshold electric field is very sensitiveto the setback distance, which might explain the very disparate values of this quantity in experiments.

3.3.3. Continuum Elasticity Theory Analysis There are two principal results here that we would like to understand: (1) Why are dislocations present for arbitrarily weak disorder strengths? and (2) Why is there an apparently sharp disorder strength above which free disclinations are present? This may be addressed with a continuum elasticity theory model [64] of the two- dimensional crystal, in which the energy to create a strain field uij(r)= + djui],whereu is the displacementfield of the lattice and i,j = x, y, is given by

We have taken our unit of length in the above to be the lattice constant (i.e.,

DISORDER EFFECTS ON THE ELECTRON SOLID

87

a, = l), and p and A are Lamb coefficients.The quantity 6p represents a random field, which for simplicity we assign an uncorrelated Gaussian distribution, P[Gp(r)] = (l/fi)e-'p(r)*'zu. The quantity ,/% here should be identified as po, the density of impurities, which in these units is precisely 1 (since there are as many electrons as impurities in the model). One can see in this model that the coupling of the strain field to the disorder has the effect of forcing in fluctuations in the lattice density, which is proportional to ukt.This model has been studied previously to describe a crystal with random substitutional disorder [65]. To guarantee net charge neutrality at long wavelengths, we consider the A + 03 limit of this model [25,60]. A more realistic model would include a finite correlation length for the disorder of size scale d (the model described here may be easily modified to describe this [25]); however, it turns out that only the long-wavelengthproperties of the disorder determine whether dislocations are present in the ground state, so that such improvementsdo not change the final result. We will see below that one always finds dislocations in the ground state because increasing d/a, only eliminates short-wavelength components of the disorder (i.e., it lengthens its correlation length);it has no effect on the long-wavelength components. To obtain ground states without isolated dislocations, one needs to reduce drastically the number of ions relative to the number of electrons. Such a geometry could be achieved for electrons on liquid helium films, although it is impractical for electrons in semiconductors. The strain field in Eq. (4) may be separated into a smoothly varying part 4ij and a part due to dislocations with cores at sitesij and Burgers vectors'i;i. The energy of the resulting configuration turns out to be separable in these two contributions [65]. The contribution to the energy due to the presence of dislocations has the form

where K' = 4pA/(2p+ A), and Ed,'"is the energy of the dislocations in the absence of the disorder field [63,65]. We begin by considering whether it is energetically favorable to have isolated dislocations in the groundstate of the disordered crystal. Suppose that one attempts to find the ground state for a given disorder realization in a finite size system of area A. We can first minimize the energy with respect to the smooth ~ ~ introducing any dislocations. We now ask Can one displacements c # ~without find a site in the sample for which the introduction of a disloca$on lowers the energy?The energy to create a dislocation with the Burgers vector b in the absence of disorder has the form E , = (Kbz/16a)In A for large A where K = 4 d p + A)/ ( 2 p + A); to be energeticallyfavorable,the interaction energy between the dislocation and the disorder El [i.e., the second term in Eq. ( 5 ) ] must more than balance this energy cost. For a given site, the ensemble of disorder configurations will generate a distribution of interaction energies P(El) that in the limit of large

88

PROPERTIES OF THE ELECTRON SOLID

sample sizes should be independent of the site location. The probability distribution P ( E , ) may be computed exactly by a functional integral, with the result P(E,) = (l/,/%&-E;/2'J, with

The probability that a site is energeticallyfavorable for creation of a dislocation is then given by p = J r ~ P ( E , ) d E ,which , for large A is easily shown to have the form p e-E:/2q= A-(bK/4K')2/''a.Since the number of sites in the sample scales as A, the number of sites for which it is energetically favorable to create the dislocation scales as A1-(bK14K')21"e, so that it will be possible to find sites for the dislocation in the thermodynamic limit only if

-

(r

-(-)

1 K 2 > Isc= n 4K'

where we have set the Burgers vector to its lowest nontrivial value, b = a, = 1. Noting that po = @ and taking the A+ cc limit, one finds that dislocations are favored for ion densities po > 9/n3a; x 0.29/a& which means that in our model one always expects to see dislocations in the ground state. However, systems for which the number of electrons may be much larger than the number of impurities should exhibit this transition. One potential candidate for the realization of this transition is a system of electrons on a thin helium film over a disordered substrate [24], for which the density of electrons can be controlled over a wide range without appreciably affecting the disorder potential. Recent simulations [64] have confirmed that states with no isolated dislocations can be stabilized for systems with much fewer impurities that electrons. Figure 3.9 illustrates the positional correlation in the low-temperature state of the electron system for various ratios of number of electrons to number of impurities; a clear divergencecan be seen around 0.075 in this ratio, strongly suggestingthat there is a phase transition. The disorder-induced transition between the hexatic and isotropic states, in which free disclinations appear above a threshold disorder strength, can be described in a very similar fashion [26]. The energy to create a disclination with a screening cloud of dislocations around it may be shown after an arduous calculation [66] to have energy nE,In A, where E, is the core energy of a dislocation. The interaction of this screened disclination centered at location7, with dislocations already present in he sample turns out to have the form [66]

whereX(7) is the complexion ofdislocations that minimizes the energy for a given

DISORDER EFFECTS ON THE ELECTRON SOLID

89

Figure 3.9. The positional correlation lengths of two-dimensional electron solid in finitesize systems at zero temperature, where N , is the number of electrons and N iis the number of quenched impurities. Small symbols represent actual data, while bigger symbols represent corresponding average values. The correlation lengths rise sharply as the disorder strength approaches a finite critical value from above. Samples with N J N , < n, are not shown, because long-range behavior is present in correlation functions and correlation lengths cannot be defined.

disorder realization, and s is the disclination charge, f 1. If we assume a simple Gaussian distribution for the dislocations (assuming that 0 > o,),P [ x ( 7 ) ]cc e-b'(7)'ns,where n b is the density of dislocations, the calculation goes through as above, and one finds that screened disclinations are energetically favorable if nb > n, = 1/4x x 1/13. The numerical simulations described above give a critical dislocation density of approximately 1/20. Several comments are in order. First, this is clearly the generalization of the original Kosterlitz-Thouless argument [61] for entropy-driven phase transitions in two-dimensional systems; one is balancing the probability due to its energetics of a given lattice site being occupied by a defect with the number of available sites. As in that work, the screening by defect pairs affects the precise values of a, and n,. One also cannot rule out the possibility that this phase transition will be circumvented by a first-order phase transition, for example, to a state with grain boundaries [62].

90

PROPERTIESOF THE ELECTRON SOLID

3.3.4. Effect of Finite Temperatures Previous studies of the model is defined by Eq. (4) [65] have given a rather interesting prediction for the finite-temperature behavior of the crystal state: It was found using a renormalization group approach that there is a reentrant crystal phase, which for fixed disorder strength exists only over some range of temperatures. The phase diagram of Ref. [65] is reproduced in Fig. 3.10. In particular, it was predicted that for arbitrarily weak disorder, the crystal state becomes unstable at zero temperature. Very similar predictions have been made for closely related XY models [67] and Josephson junction arrays [68]. Interestingly, the reentrance phenomenon has not been oberved either in experiment [69] or in Monte Carlo simulations [69,70]. Obviously, the phase diagram in Fig. 3.10 is inconsistentwith the prediction of a zero-temperature disorder-driven phase transition described above. In fact, a finite temperature version of the argument may be constructed by assigning a thermal probability 1/(1 + e(Eo+E')/kBT) that a site will be occupied by a dislocation, even if the energy to do so is positive. (This is the probability distribution expected if we only allow the possibility of a site being occupied or unoccupied by a single dislocation of the lowest nontrivial Burgers vector F = a,. We assume that larger dislocations are so energetically unfavorable that they

\

Non-crystal

Figure 3.10. Schematic phase diagram of the disordered two-dimensional crystal as temperature and disorder strength change. The solid line represents the phase diagram suggested in this work.The dashed line is from Ref. 1651.

QUANTUM EFFECTS ON INTERSTITIAL ELECTRONS

91

may be ignored.) The probability that a given site will have a dislocation then takes the form p = jT03d E , P(E,)/(l e(Eo+El)lkeT). The behavior of this integral for very large areas may be computed by breaking the integral up into two parts: p= d E P(E' - E,)/( 1 + eE'/kET)+ d E P(E - E,)/( 1 + eE'/ksT). We then expand the thermal factor in powers of eE'lkBT in the first term and in powers of e-E'lkeTin the second. The resulting integrals may be expressed as sums over complementary error functions, from which one finds for the asymptotic behavior p e-Ek2sfor /?q> E,,p e-BEo+B's/2 for /?q E,, with /? = l/k,T. The resulting asymptotic behavior for large areas has the form p A-", with

+ jr

-

-

-=

-

B-K,,

2K

Recalling that the number of sites for the dislocation scales as A, the probability of a site being occupied with a dislocation for A + co diverges if a < 1. With Eqs. (7) and (S), this defines the phase boundary illustrated in Fig. 3.10 as a solid line. It is clear in Fig. 3.10 that the previous results [65] coincide precisely with the analysisdescribed above for the high-temperature half of the phase boundary but are markedly different on the low-temperature side. While the analysis in Ref. [65] involves a detailed renormalization group (RG) treatment of this problem, one can identify precisely why our approach obtains different results at low temperatures. The RG analysis relies on an expansion in the fugacity e-Ec/kaT, where E, is the core energy of a dislocation, to derive scaling relations for the system parameters. This is completely equivalent in our approach to expanding in ,-EolkeT. , in particular, one is then led to approximate the thermal factor 1/(1 + e(Eo+El)lksT) x e - ( E o + E l ) / kSubstituting BT. this into our expression for p, one obtains precisely the same result as in Ref. [65]. The failure of the fugacity expansion occurs because of rare but nonnegligible disorder configurations in which El is large and negative, leading to unboundedly large values of e-(Eo+E1)lkBT. However, this thermal probability should never exceed 1 when properly normalized; thus the fugacity expansion breaks down when the fluctuations in the dislocation energy due to disorder are larger than those due to thermal effects. A very similar breakdown in perturbation theory is known to occur for random-field Ising models [71].

3.4. QUANTUM EFFECTS ON INTERSTITIALELECTRONS For electrons in heterostructure materials in strong magnetic fields, it is clear that quantum fluctuations play an important role. This is especially true near v = 1/5, where the reentrance phenomenon occurs: Presuming that the insulating state slightly above and below this filling fraction is indeed a WC, the crystal is

92

PROPERTIES OF THE ELECTRON SOLID

necessarily melted by quantum fluctuations as v + 1/5. At present there is no consensus as to what is the nature of these fluctuations, or indeed if they become critical in the melting process. This last point is important in determining whether the transition between liquid and crystal states is first or second order [72]. It is interesting to note that one of the earliest theories [73] of the fractional quantum Hall effect focused on large ring exchanges of the WC as the important quantum fluctuations that melt the crystal at v = l/m; however, it is unclear whether such fluctuations are important away from these filling factors,in the crystallinestate. Beyond this, the finite value of pxyat low temperatures in the insulating state is likely to be a quantum effect as well. One of the simplest scenarios for charge transport in the WC is that current is carried at finite temperatures by thermally activated point defects, either vacancies or interstitials [74]. Consider a WC with filling fraction close to, but just below, filling fraction v = l/m. It is clear that the introduction of an interstitial electron somewhere in the lattice raises the local density, thereby bringing the local filling factor closer to, and possible above, l/m. One must expect that quantum fluctuations that lead to the fractional quantum Hall effect should become important in the interstitial state in this situation. 3.4.1. Correlation Effects on Interstitials: A Trial Wavefunction

To model these effects, one may consider a trial wavefunction for a WC with a single interstitial in which Laughlin-Jastrow correlations are introduced [MI:

where i,bWc(z1,.. .,zN)is the wavefunction for a perfect WC of N electrons, with electrons localized near sites zi = xi- iy, is an electron coordinate in complex notation, u is the interstitial coordinate, m is an integer chosen such that l/m is larger than the filling fraction v (we show below that the wavefunction no longer represents an interstitial for v > l/m), and d is the antisymmetrization operator. The lattice sites in $wc are chosen such that the origin is an interstitial site, so that the location of maximum probability for the interstitial is its expected location for a classical crystal. We work here in units of the magnetic length I, = (hc/eB)1'2, where B is the applied magnetic field. We note that if one uses a Hartree approximation for t j W c , the case m = O corresponds to an antisymmetrized Hartree approximation for the interstitial. Such approximate forms for wavefunctions of the WC at low enough fillings have been argued to be quite good, since one may show that exchange corrections to the WC energy at low fillings are quite small ~581. To understand the effect of the correlation factor on the interstitial, one may use the plasma analogy introduced by Laughlin [27]. Dropping the antisymmetrization in Eq. (9),and using a Hartree approximation for r(/wc, we write the square

a:,

QUANTUM EFFECTS ON INTERSTITIALELECTRONS

93

modulus in the form 1$12 = e-@, where for f l = 1,

This represents the energy of a classical two-dimensional Coulomb particle with charge - 1, interacting with a neutralizing backgrund of charge density ab= 1/2n, and a lattice of charges - m with charge density a, = - m/u2,where a is the WC lattice constant. On a coarse scale, the interstitial is thus interacting with a system of net charge density a = (I - mv)/2x (v is the filling in the absence of the interstitial),and thus spreads out in the center of the disk to an area of l/a. We note that precisely at the filling v = l/m, the interstitial will spread out uniformly over the disk, and for higher fillings the interstitial electron will have its highest probability at the system edge, essentially being ejected from the system. This is the reason the wavefunctions are physically reasonable only for fillings below v = l/m. This behavior of the interstitial wavefunction can be demonstrated quite directly, by computing the probability p( ii) of finding the interstitial at some position ii.Using the Hartree approximation for $wc in Eq. (9)and neglectingthe antisymmetrization, one has

For simplicity, we choose the lattice sites Rf to form a square lattice. The interstitial electron density distribution

can be evaluated directly from the expression above since the integrations over the lattice electron coordinates may be computed analytically. Figure 11 shows the result of this calculation along a crystal symmetry axis. One can clearly see that p( i7)becomes quite spread out as v + l/m. We are now in a position to understand why the correlation factor might lower the energy of the interstitial, especially close to v = l/m. In essence, the Jastrow factor allows the interstitial to sample a large region of the system and thereby can take advantage of quantum fluctuations in the WC. When there are configurations in t,bWc that have large holes in them, the interstitial has a high probability of being located there, especially when v l/m, for which the interstitial density (Fig. 3.1 1) samples a large region of the crystal. Conversely, when the crystal forms regions of high density, the interstitial has a low probability of approaching that region. In this way, the interstitial electron does an excellent job of minimizing its energy.

-

94

PROPERTIES OF THE ELECTRON SOLID

m=5

0

0

u = 0.19 u = 0.17 u = 0.10

x/a Figure 3.11. The probability distribution for the interstitial electron in a correlated interstitial state described by Eq. (11) with m = 5 and v = 0.19, 0.17, 0.1. The distance is along they y = 0 axis in a Cartesian coordinate reference frame, where the square lattice R t is expressed as ( i - 0.5,j - 0.5).The distance is in unit of the lattice constant a,. (From Ref. [a].)

The energy of this state may also be computed; details may be found in Ref. [MI. The result is displayed in Fig. 3.12. The energy shown here is defined as the differencebetween the energies to add a particle to the lattice as an interstitial and as a regular lattice electron (i.e,, the chemical potential). Alternatively, one may think of this as the energy to remove an electron from the lattice, rearrange the particles into a perfect lattice with a slightly larger lattice constant, and then replace the electron at an interstitial site. As may be seen, the energy is lowered dramatically by the correlation factor (see inset of Fig. 3.12). This is in fact consistent with experiment, for which the activation energies measured [191 are significantly smaller than expected from simple classical [8] or Hartree-Fock calculations [75] of the vacancy-interstitial creation energies. Furthermore, the energy becomes negative when the filling factor closely approaches v = l/m [76]. This indicates that the Hartree (and probably Hartree-Fock) Wigner crystal ground states are unstable with respect to these excitations if one is too close to v = l/m for m = 5,7; similar results were found for m = 9.

;*'I

QUANTUM EFFECTS ON INTERSTITIAL ELECTRONS

95

1

a

0.1

n 0 rl

cu' 23 W

3 23 W

c

m

V

Figure 3.12. The energy of the correlated interstitial as a function of the Landau level filling factor v for m = 5 and m = 7.The inset shows the energy E at v = 0.19 as a function of the exponent m of the Jastrow factor (i.e., as a function of the strength of the correlation). (From Ref. [MI.)

Although the prediction of an instability of the crystal near v = 1/5 is consistent with experiment,one must not jump to the conclusion that the resulting state is a liquid. This is especially highlighted by the fact that similar results are found for m = 7,9 near fillingfractions 1/7,1/9, where the fractional quantum Hall effect is not observed experimentally.There are indications that something interesting happens near these small filling fractions in photoluminescence experiments [46,47], and there are unexplained anomalies in transport data [7nnear v = 1/7. Since the results above indicate clearly that correlations become important for v = l/m even for large m,the nature of the insulating state at these fillings remains an interesting open question. 3.4.2. Interstitials and the Hall Effect

Another indication that Laughlin-Jastrow correlations are important for interstitial electrons is that they dramatically change their transport properties. In particular, if the current at finite temperatures is indeed carried by correlated interstitials, one finds that the Hall resistivity pxyis equal to its classical value for a noninsulating state, B/nec, where n is the full two-dimensionalelectron density.

96

PROPERTIES OF THE ELECTRON SOLID

The key to seeing this is to consider a generalization of Eq. (9) of the form

which allows the correlation factor between the interstitial and the other electrons to be site dependent. This new degree offreedom is useful if we consider systems with weak disorder, such that the WC density is not perfectly uniform throughout the sample (e.g., Fig. 3.6a).By adjusting the values of mi to avoid the regions of higher energy for the interstitial, one can create a state with particularly low energy. Furthermore, by choosing values of the exponents such that (mi)= l/v, where (...) denotes an average over all sites, the interstitial can sample the entire system and take best advantage of the low-energy regions. We thus expect such choices for Eq. (12) would give particularly low energies; this is supported by the observation that the energy in the inset of Fig. 3.12 is a monotonically decreasing function of m. To understand the effect of the correlation factors on the transport properties of the interstitial,it is useful to think of them as being due to magnetic flux tubes of magnitude m,40 attached to the electrons on each lattice site, where 4' is the magnetic flux quantum [78]. This flux is directed to oppose the actual applied magnetic field Be"' = n&,/v. In a mean-field description, the localized electrons will generate a fictitious magnetic field BEL = nL40/v, where n, is the density of lattice electrons, which partially cancels Be"'. The excited conduction electrons are then traveling in a net field of B""' = Be"' - BCL= nC4dv, where n, is the density of interstitial (i.e., conduction) electrons. To the lowest order in the disorder potential, the motion of the conduction electrons can then be described in the Drude model. Suppose that there is a current j , = n,eu with px, = l/n,ep, where p is the effective mobility. A Hall field E, must be present to balance the Lorentz force, so that the measured Hall resistivity would be pxy= Ey/jx= (vB/c)/ n,eu = B/n,ec. For the strongly correlated systems B = B""' = (nc/n)Bex',so that pxy= B/nec, regardless of the number of electrons that are localized. Remarkably, this implies that pxyremains finite and equal to its classical value even at very low temperatures, for which the number of electrons actually conducting becomes vanishingly small. This behavior is precisely what is observed in experiment for the insulating phase [41,42]. One final detail of this model is that it assumes that when lattice electrons are thermally ionized to become interstitials, the vacancy that is left does not carry a flux tube, so that some magnetic field is left behind to produce a small Lorentz force on the interstials when they move. This is sensible since if the flux tubes remained, the interstitials would be avoiding sites that are actually particularly low in energy. This means that the effectivemagnetic field seen by the electrons is highly nonuniform; it essentially looks vanishingly small except at vacancy sites, where it is quite large. However, it may be shown using a Boltzmann approximation that even this very strong inhomogeneity in the field does not alter the Hall resistivity [79].

PHOTOLUMINESCENCE OF THE WIGNER CRYSTAL

97

3.5. PHOTOLUMINESCENCE AS A PROBE OF THE WIGNER CRYSTAL As discussed in the introduction, one probe of the electronic insulating state in semiconductor systems in strong magnetic fields that has generated considerable excitement over the last few years is photoluminescence (PL). In such experiments, either a valence band hole [SO] or a hole bound to an acceptor [46-481 recombines with an electron in the 2DEG, producing a characteristic photon spectrum. We will see that a mean-field analysis of the PL spectrum has, in principle, characteristic signatures of the W C a Hofstadter butterjy [54] spectrum for the case of weak interactions between the electrons and the hole, and a characteristic shift in the PL spectrum upon melting of the crystal. When the possibility of exciting phonons and other collective excitations of the WC in the recombination process-shakeup effects-is included, the Hofstadter spectrum is lost in localized hole experiments and is replaced by a spectrum with several collective mode satellites. For itinerant holes, coupling between the hole and the WC phonons has important effects, but some of the features of the Hofstadter spectrum are likely to survive. 3.5.1. Formalism

We begin by describingcalculations for the localized hole geometries. To find the PL spectrum expected from a WC, it is convenient to employ a supercell method, in which one approximates the initial state of the system as an electron lattice that is commensurate with a superlattice of holes. The supercells contain a single hole per unit cell, and one tries to accommodate as many electrons in them as possible. This approach is sensible since the actual density of holes in real systems is extremely small, so that correlations among the holes are not expected to be important. The advantage of this approach is that it allows one to take advantage of the symmetriesof the electron lattice. Working in the lowest Landau level (as is appropriate in strong magnetic fields),the PL intensity may be written in the form [SS] P(o)K Im [ R ( o id)], where

+

R(G,o)e-G21i/4 R(o)= 7 nh 2x10

c

and we have approximated the core-hole wavefunctions as delta functions, nh is the density of holes, R the volume of the system, and the vectors G are the reciprocal lattice vectors of the superlattice. The magnetic length I , = (hc/eB)’/’ will be set to unity in the remainder of this chapter. R(G, o)is defined as

where g is the Landau level degeneracy, Xis the guiding center quantum number,

98

PROPERTIES OF THE ELECTRON SOLID

and

with a& creating an electron in state X and c i creating a hole in the unit cell i, and fi is the inverse temperature. The quantity R, should be interpreted as a Green’s function for the electron lattice, and formally has poles at the quasihole excitations of the WC. We will see that these poles merge together into narrow bands and form well-defined peaks in the PL spectrum. The equation of motion for Rij(G,o)may be written as

- nh

c V(G)eic”G/2-G”/4R.(G - G’,z) 0

C’

-

JOB

d z ’ c (G, G’; z - z’)Rij(G,z’)

where the self-energy is defined implicitly by

In Eqs. (13) and (14), v(q) and V(q) are the Fourier transforms of the electronelectron and electron-hole interactions, respectively, t o is the energy of the localized hole, and the sum over G’ is only over reciprocal lattice vectors, while the sums over q are over all wavevectors.Ri specifiesthe position of the hole in the ith unit cell, and (p(G))e-”14 is the expectation value of a Fourier component of the electron density [SS, 801.Finally, the correlation function Cij(pl,pz)isjust the Fourier transformation of

Equations (13) and (14) represent the first in an infinite series of equations relating an n-particle Green’s function to an (n + 1)-particleGreen’s function [81]. To make practical the computation of the PLYone must find a sensible approximation for the self-energy in Eq. (14). We discuss next two schemesfor doing this.

PHOTOLUMINESCENCE OF THE WIGNER CRYSTAL

99

3.5.2. Mean-Field Theory

The simplest approximation possible for the self-energy in Eq. (14) is to use a Hartree-Fock (HF) decomposition of Eq. (15). This yields the result C (G, G , T - z’) = CHF(G,G)6(z - z’), with CHF(G,G ) = W(G - G’)( p(G G ))eiC c”z + (1/2x)C,Y(q)(p( - q))e-2’4-q‘L6,,G., where W is the sum of the direct and exchange Coulomb potentials [55,80]. To understand the results qualitatively, two limiting cases are of interest: (1) no electron-hole interaction (which is appropriate for a hole very far from the electron plane), and (2) with a Coulomb electron-hole interaction. For case 1 one finds a characteristic spectrum for rational filling fractions v = p / q , where p and q are integers, of p narrow peaks in the PL. An example of this is illustrated in Fig. 3.13. This may be understood in terms of the single particle density of states for the WC in the mean-field approximation (MFA): in this case, an individual electron moves in the periodic potential of the other electrons and a magnetic field with q / p flux quanta penetrating each unit cell. The spectrum for this situation is the wellknown Hofstadter butterfly of q bands, and in the MFA, p of these are filled. The

600

01

Y

(a) T = O.o045T,,,,~t

- 13.0

= 217,

= OmeV I

I

I

- 12.0 E

- w(meV)

-11.0

- ...

-10.0 .

Figure 3.13. Example of a photoluminescencespectrum for v = 2/7 with no electron-hole interaction,with T below the melting temperature Tme,,. The PL spectrum has two peaks because the numerator of the rational filling factor is 2. Inset: Same, for T above melting temperature. (From Ref. [SS].)

100

PROPERTIES OF THE ELECTRON SOLID

PL spectrum just reflects the recombination of electrons occupying these bands. Thus PL is a weighted measure of the electron density of states. One unfortunate characteristic is that at the lowest filling fractions, for which the WC might be realized in real systems(v < 1/5), the gaps between the bands are extremely narrow (on the order 0.01 MeV); thus in practice one is likely only to resolve a single PL peak. In any case, these gaps (for the localized hole case) do not survive either electron-hole interactions or shakeup effects, so that one should not expect to observe the Hofstadter spectrum in this way. For case 2 (electron-hole interaction), only a single PL line is present, which turns out to be the contribution from a single electron in a bound state with the hole. PL from recombination of other electrons in the crystal is in principle present, but its magnitude is very small. For both cases 1 and 2, there is an interesting finite-temperature effect: When the crystal is brought to melting, there is a sudden shift in the spectrum to higher energies, as may be seen in the inset of Fig. 3.13. The interesting property of this effect is its suddenness,which is a result of the first-order nature of the transition in the Hartree-Fock approximation. Thus, in real, weakly disordered samples, one might expect to see two peaks over a range of temperatures, one each from regions of melted and crystalline electrons, which coexist for some temperature range due to disorder effects [25]. This is in qualitative agreement with experiment [46,47,50], where a shift in oscillator strength between two peaks is observed over a temperature range of about 1 K.

3.5.3. Beyond Mean-Field Theory: Shakeup Effects This general formalism allows one to go beyond mean-field theory, to include the excitation of phonons and other collective modes in the electron-hole recombination process. The strength of this approach is that it treats both the tunneling electron and the lattice electrons on the same footing (i.e., the formalism does not choose out a single electron and specify it as the one that will recombine with the hole). This is contrast to (albeit simpler)independent boson models, in which one electron is specified to recombine with the hole. This acts as a suddenly switched potential for the remaining electrons, whose motion is treated in the harmonic approximation [82]. Such methods do not allow a description of the PL spectrum when the temperature approaches the melting temperature. Furthermore, quantum treatments of the density response [80] of the WC have indicated that there are sharp collective modes with large wavevectors; these cannot be described within a harmonic approximation. The equation-of-motion formalism allows both these effects to be included in a very natural way. The general approach is to carry the hierarchical set of equations described above to one more level. Specifically,rather that write down a H F decomposition of C ,, one derives an equation of motion for it that includes the collective excitations. The result is then inserted into Eqs. (13) and (14). The actual calculation is quite arduous [83], but the result contains the collective modes of the system in a very clear way. One finds that Cij may be approximately

PHOTOLUMINESCENCEOF THE WIGNER CRYSTAL

101

Photoluminescence Power I

0

- 15.5 E

,

- 13.5 - E, - Eh (mew

I

-11.5

Figure 3.14. PL spectra including shakeup effects for (a) v = 1/5, T = 0, no electron-hole interaction; (b) v = 1/5, T = 0, with electron-hole interaction; (c) v = 2/7, T = 0, no electron-hole interaction. Inset: PL spectra for (a) and (b), with Tjust above the melting temperature. To distinguish different spectra, they are separated by 250 units. (From Ref. [84].)

expressed in terms of the density response function [84] x(G, + q, G , + q; t)E x&q; T) = - g ( Tp(G, + q, t)p( - G , - q; 0)). The Fourier transform of this response function contains poles at the collectivemode frequenciesof the system; it is thus through x that shakeup effects are introduced. Examples of the PL spectra calculated in this way are shown in Fig. 3.14 for filling fractions v = 1/5 and v = 2/7. For the case of no electron-hole interaction, at low temperature, a well-defined shakeup peak may be seen approximately 1 MeV below the main PL peak; a second, very weak satellite is observed approximately 1.6 MeV below the main peak. The origins of these peaks may be understood in terms of the collectivemode density of states, which is illustrated in Fig. 3.15. A van Hove singularity, arising from zone-edge phonons, appears as a strong double peak near 0.4 MeV. Two other peaks may be seen above this. There are weak sidebands associatedwith each of these peaks in the PL spectrum. The precise interpretation of these peaks is unclear; however, it has been speculated that these represent vacancy-interstitial excitations [80]. It is crucial to use a fully quantum mechanical treatment of the collective excitations of the

102

PROPERTIES OF THE ELECTRON SOLID

100

Phonon Density of States

F

Ee . f!50 3 G 5

0 C Figure 3.15. Collective-mode DOS for v = 1/5, 7' = 0. (From Ref. [84].)

lattice to observe these higher-order satellites; classicaltreatments of the phonons do not produce these unusual excitations. As in the mean-field calculation, there is a sudden shift in the PL spectrum upon melting. The inset to Fig. 3.14 illustrates the PL spectrum in the melted state both without and with an electron-hole interaction. In the former case, there is no phonon sideband present. This is necessarily so, because in the melted phase, the density is uniform and there are no collective modes in the LLL [80]. By contrast, when an electron-hole interaction is present, there is a nonuniform electron density near the hole, allowing some local collective modes to persist even above the melting temperature. With further increase in temperature, or increased setback between the hole and the 2DEG, the oscillator strength of this mode will decrease significantly. Figure 3 . 1 4 ~illustrates results for v = 2/7 in the absence of electron-hole interactions. As can be seen, except for a change in energy scale caused by changing the magnetic field, the lineshape is essentially identical to the case of v = 1/5. This contrasts sharply with the results found in mean-field theory, where without electron-hole interactions, a filling v = p / q generally yields p distinct lines for a localized hole. While the splittings are so small in that situation that they are difficult in practice to resolve, evidently shakeup effects wipe out this structure even in principle.

PHOTOLUMINESCENCE OF THE WIGNER CRYSTAL

103

3.5.4. Hofstadter Spectrum: Can It Be Seen?

Is it possible that any remnant of the Hofstadter spectrum is observable in photoluminescence?A possible avenue for this may lie in itinerant hole experiments, in which the hole can hop between sites and is not forced to be in a single initial quantum state before the recombination process. In particular, if the hole is mobile, it then has a density of states of a positively charged particle moving in a periodic potential-which itself will have a Hofstadter spectrum. The higher Hofstadter states will in general be occupied by holes at finite temperatures, so that considerable structure could arise due to the hole density of states. Figure 3.16 illustrates a spectrum for itinerant holes as evaluated using the mean-field approximoation, under the assumption that the holes are in a narrow quantum well 250A from the electron layer, at low and intermediate temperatures. The new structure on the high-energy side of the main PL line comes from thermal excitations to the higher Hofstadter levels in the hole density of states. Observation of a Hofstadter structure in the hole spectrum would indeed give dramatic evidence that crystalline order is present in the electron system at low filling factors. However, as in the case of localized holes, one must consider which features of the Hofstadter spectrum survive the coupling of the hole with distortions in the electron lattice. There are two important considerations that arise in this context: 1. The hole must not be too close to the electron layer, or it will become

energetically favorable for an intersitial to be introduced in the lattice, to

i?

8 1 600

h

E

m

a,

400

-

200 -

-

Y

0 E-E,-Eh-A(mev) Figure 3.16. Photoluminescence for itinerant hole at electron filling v = 2/11, for T z

0.005Tm,,, (solid line) and T z O.OST,,,, (dotted line). Electron and hole cyclotron frequen-

cies given here by o,and oh, respectively,and A is the conduction band-valence band gap. (From [55].)

104

PROPERTIESOF THE ELECTRON SOLID

which the hole will become tightly bound. Hartree-Fock calculations [56,85], as well as classical treatments, give an estimate of the distance below which this occurs to be approximately 0 . 5 ~ ~ . 2. Above the critical setback distance between the hole and the electrons,there is still a polaron effect, in which the lattice is distorted as the hole moves through the system. This will certainly narrow the Hofstadter bands, and can close up the smaller gaps in the spectrum. However, preliminary work [56] has shown that at least some of the bands do remain well separated, with energy gaps very close to those found in the mean-field approximation. Further theoretical work is necessary to sort out this complicated issue. 3.6. CONCLUSION SOME OPEN QUESTIONS Many interesting open questions remain regarding the Wigner crystal. In both the helium system and the semiconductor systems, it is found in the sliding state that the noise spectrum is broadband, and large at low frequencies (possibly following a llf”law [23]). Since the frequency range of this noise lies well below the characteristic phonon frequencies, it is unclear precisely what the source of this noise is and what it might say about the motion of the electrons in the sliding state. One possibility is that the system undergoes a collective creeping motion, which may involve a distribution of waiting times for different patches of the crystal to move. Such possibilities have been discussed in the context of charge density wave [86] depinning, and it has been pointed out that such motion may be described as a self-organized critical phenomenon [87]. There is some indication that this sort of creep motion is relevant to the sliding state in molecular dynamics simulations [26], although it is difficult to glean information about the noise spectrum at very low frequencies from finite time simulations. For the heterostructure systems in a strong magnetic field, where the electrons must choose between a liquid (fractional quantum Hall) state and some other insulating state, the question of the nature of the phase transition between these two states is still unresolved. Are there quantum fluctuations in the crystal state that hint at the liquid states that are nearby in energy, and if so, how do they affect the crystal properties and the transitions between states? The study of correlated interstitials above suggests that such fluctuations are indeed important, particularly for filling fractions in the vicinity of v = l/m for large m, but it remains unclear how they should be incorporated in the crystalline ground state. Nevertheless, photoluminescenceexperiments suggest that something interesting does occur in the insulating ground state near v = 1/7, 1/9,. .. . For heterostructure systems, the most important development one could hope for would be an experiment that leaves a “smoking gun”: data that clearly distinguish between a disordered insulating state and a crystal state with some large correlation length. The observation of a Hofstadter spectrum would certainly give such dramatic evidence, and there is some possibility that itinerant hole photoluminescencecould observe this. However, current experiments [SO]

REFERENCES

105

of this sort are still limited by a lack of knowledge of the initial state of the hole, and further refinements of samples will be necessary before such effects become clearly accessible. On the theoretical side, one must also account for the effects of lattice distortions on the initial states of the hole, which is likely to make only the largest gaps in the Hofstadter spectrum accessible [SS] (at least for low filling factors), thus blowing away some of the smoke in such experiments before it is observed. Nevertheless, the observation of even the broadest features of the Hoftstadter spectrum could yield valuable information about the degree of order in these systems. ACKNOWLEDGMENTS The author has benefited greatly from numerous conversations and collaborations related to this subject. Luis Brey, Min-Chul Cha, Renk Cbte, Sankar Das Sarma, E. Kolomeisky, Dongzi Liu, Allan MacDonald, J. P. Straley, D. C. Tsui, and Lian Zheng have had particular influence in the development of this perspective. Some of the work described was supported by the NSF through Grant Nos. DMR-92-02255and DMR95-038 14, as well as by the Sloan Foundation, the Research Corporation, and NATO. REFERENCES 1. 2. 3. 4. 5.

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106

PROPERTIES OF THE ELECTRON SOLID

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39. D. C. Glattli, G. Deville, V. Duburcq, F. I. B. Williams, E. Paris, B. Etienne, and E. Y. Andrei, Surf Sci. 229,344 (1990). 40. M. Ferconi and G. Vignale, Phys. Rev. B 48,2831 (1993). 41. V.J. Goldman, J. K. Wang, B. Su, and M. Shayegan, Phys. Rev. Lett. 70,647 (1993). 42. T. Sajoto, Y.P. Li, L. W. Engel, D. C. Tsui, and M. Shayegan,Phys. Rev. Lett.70,2321 (1993). 43. S. Kivelson, D. H. Lee, and S. C. Zhang, Phys. Rev. B 46,2223 (1992). 44. L. Zheng and H. A. Fertig, Phys. Rev. Lett. 73,878 (1994). 45. L. Zheng and H. A. Fertig, Phys. Rev. B 50,4984 (1994). 46. H. Buhmann, W. Joss, K. von Klitzing, I. V. Kukushkin, G. Martinez, A. S. Plaut, K. Ploog, and V.B. Timofeev, Phys. Rev. Lett. 66,926 (1991). 47. I. V. Kukushkin, N. J. Pulsford, K. von Klitzing, K. Ploog, R. J. Haug, S. Koch, and V. B. Timofeev, Phys. Rev. B 45,4532 (1992). 48. I. V. Kukushkin, R. J. Haug, K. von Klitzing, K. Eberl, and K. Ploog, Surf Sci. 305,55 (1994). 49. I. V.Kukushkin, V. I. Fal’ko,R. J. Haug, K. von Klitzing,K. Eberl, and K. Totemayer, Phys. Rev. Lett. 72, 3594 (1994). 50. R. G. Clark, Phys. Scr. T39,45 (1991) and references therein. 51. E. M. Goldys, S. A. Brown, R. B. Dunford, A. G. Davies, R. Newbury, R. G. Clark, P. E. Simmonds,J. J. Harris, and C. T. Foxon, Phys. Rev. B 46,7957 (1992). 52. S.A. Brown, A.G. Davies, A.C. Lindsay, R.B. Dunford, R.G. Clark, P.E. Simmonds, H. H. Voss, J. J. Harris, and C. T. Foxon, Surf Sci. 305,42 (1994). 53. I. N. Harris, H. D. M. Davies, R. A. Ford, J. F. Ryan, A. J. Turbefield, C. T. Foxon, and J. J. Harris, Surf Sci. 305,61 (1994). 54. D. Hofstadter, Phys. Rev. B 14,2239 (1976). 55. H. A. Fertig, D. Z. Liu, and S. Das Sarma, Phys. Rev. Lett. 70, 1545 (1993); Surf Sci. 30!5,67(1994);D. Z. Liu, H. A. Fertig, and S. Das Sarma, Phys. Rev. B 48,11184(1993). 56. L. Zheng and H. A. Fertig, Phys. Rev. B 52, R2321(1995). 57. Y.Imry and S. K. Ma, Phys. Rev. Lett. 35, 1399 (1975). 58. K. Maki and X.Zotos, Phys. Rev. B 28,4349 (1983). 59. N. D. Mermin, Phys. Rev. 176,250 (1968). 60. A. B. Dzyubenko and Y.E. Lozovik, Sur$ Sci. 263,680 (1992). 61. J. M. Kosterlitz and D. J. Thouless, J . Phys. C 6, 1181 (1973). 62. For a review of two-dimensionalmelting, see K. Strandburg, Rev. Mod. Phys. 60,161 (1988). 63. B. 1. Halperin and D. R. Nelson, Phys. Rev. Lett. 41,121 (1978);D. R. Nelson and B. I. Halperin, Phys. Rev. B 19, 1855 (1979); A. P. Young, Phys. Rev. B 19, 1855 (1979). 64. M. C. Cha and H. A. Fertig, Phys. Rev. Lett. 74,4867 (1995). 65. D. R. Nelson, Phys. Rev. B 27,2902 (1983). 66. H. A. Fertig and M. C. Cha, unpublished. 67. D. R. Nelson, M. Rubinstein, and F. Spaepen, Phil. Mag. A 46,105 (1982). 68. E. Granato and J. M. Kosterlitz, Phys. Rev. B 33,6533 (1986). 69. M. G. Forrester, H. J. Lee, M. Tinkham, and C. J. Lobb, Phys. Rev. B 37,5966 (1988).

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70. A. Chakrabarti and C. Dasgupta, Phys. Rev. B 37,7557 (1988);M. G. Forrester, S. B. Benz, and C. J. Lobb, Phys. Rev. B 41,8749 (1990). 71. V. Dotsenko, J . Phys. A 27,3397 (1994);R. B. Griffith, Phys. Rev. Lett. 23, 17 (1969). 72. Some of the present literature assumes that the transition is first order; see Refs [31] and [32]. An interesting result that follows if the transition is indeed first order is that the liquid-solid phase boundary in the temperature-density plane at v = l/m for m odd integral is reentrant in nature: For a range of densities the system is liquid at zero temperature, but due to entropic effects solidifies at moderate temperatures and melts once again at high temperatures. See R. Price, P. M. Platzman, S. He, and X. Zhu, Sutf Sci. 305, 126 (1994). 73. S. Kivelson, C. Kallin, D. P. Arovas, and J. R.Schrieffer,Phys. Rev. Lett. 56,873 (1986); Phys. Rev. B 36,1620 (1987)D. H. Lee, G. Baskaran, and S . Kivelson, Phys. Rev. Lett. 59,2467 (1987). 74. Such transport mechanisms arise quite naturally in Hartree-Fock calculations of the conductivity. See R. CBtB and A. H. MacDonald, Sutf Sci. 263, 187 (1992). 75. D. Z. Liu, H. A. Fertig, and S. Das Sarma, unpublished. 76. One may note that for m = 1/5 the energy is slightly negative over a broad range of filling factors; however, the amount by which it is negative is smaller than the estimated errors in the energy calculation and should not be taken seriously. 77. V. J. Goldman, M. Shayegan, and D. C. Tsui, Phys. Rev. Lett. 61,881 (1988). 78. J. K. Jain, Phys. Rev. Lett. 63,199 (1989);B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47,73 12 (1993). 79. L. Brey and H. A. Fertig, Phys. Rev. B 47, 15961 (1993). 80. R. CBte and A. H. MacDonald, Phys. Rev. Lett. 65,2662 (1990);Phys. Rev. B 44,8759 (1991). 81. L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics, W. A. Benjamin, Reading, Mass., 1981. 82. G. Mahan, Many-Particle Physics, Plenum Press, New York, 1981. For a recent related application, see P. Johansson and J. M. Kinaret, Phys. Rev. Lett. 71, 1435 (1993). 83. D. Z. Liu, H. A. Fertig, and S. Das Sarma, unpublished. 84. H. A. Fertig, in Proc 1 1 th international Con5 on the Physics of Semiconductors in High Magnetic Fields, Cambridge, Mass., 1994. 85. I. M. Ruzin, S. Marianer, and B. I. Shklovskii, Phys. Rev. B 46,3999 (1992). 86. S. N. Coppersmith, Phys. Rev. Lett. 65, 1044 (1990). 87. P. Bak, C. Tang, and K.Wiesenfeld, Phys. Rev. Lett. 59,381 (1987);J. M. Carlson and J. S. Langer, Phys. Rev. Lett. 62, 2632 (1989).

PERSPECTIVES IN QUANTUM HALL EFFECTS: Novel Quantum Liquids in Low-DimensionalSemiconductorStructures Edited by Sankar Das Sarma, Aron Pinczuk Copyright0 2004 WILEY-VCH Verlag GmbH & Co. KGaA

4

Edge-State Transport C. L. KANE Department of Physics, University of Pennsylvania,Philadelphia,Pennsylvania

MATTHEW P. A. FISHER Institute for Theoretical Physics, University of California, Santa Barbara, California

4.1. INTRODUCTION

The most striking feature of the quantum Hall effect is the remarkably precise quantization of the Hall conductance [l]. It was in 1980 that von Klitzing [2] observed unanticipated plateaus in the Hall conductance of a two-dimensional electron gas, quantized precisely at integer multiples of the fundamental unit, e2/h.This spectacular result was so surprising that a Nobel prize was awarded to von Klitzing for his discovery of the integer quantum Hall effect (IQHE) within five years of the discovery. Only two years later, in 1982, the fractional quantum Hall effect was discovered [3]. In 1981 Laughlin proposed an appealing and general explanation of the precise quantization in the IQHE [4]. Laughlin’s argument, as elaborated on by Halperin [5], focused on an annulus that was threaded by a time-dependent magnetic flux-a Corbino disk geometry. Each time the flux was increased by one flux quantum, an electron was argued to be adiabatically transferred from the inside to the outside edge of the annulus. This resulted in a flow of electrical current proportional to the electronmotive driving force, with a precisely quantized coefficient of proportionality, the Hall conductance. Although theoretically appealing, the connection with experimental transport measurements was somewhat unclear. In particular, the experimental samples were not multiconnected, but rather, had a single outer edge to which multiple current and voltage probes were attached. The important current-carrying edge states, elucidated by Halperin [S], could then carry the transport current between adjacent contacts along the edge. In contrast to the Corbino-disk geometry, a bulk current was, at least in principle, unnecessary. Laughlin’s wavefunction [6] for the FQHE was the critical step in identifying the origin of plateaus with fraction Hall conductance. At filling v = l/m, for odd Perspectiuesin Quantum Hall Eflects, Edited by Sankar Das Sarma and Aron Pinczuk. ISBN 0-471-11216-X 0 1997 John Wiley & Sons, Inc.

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integer m, the wavefunction described a featureless fluid with a gap to all excitations. The lowest-lying excitations were shown to be fractionally charged quasiparticles, with charge ve. Within a Corbino-disk geometry, the quantized Hall conductance could be understood as a discrete transfer of such quasiparticles between the inner and outer edges, one for each magnetic flux quantum threading the bore of the disk. But again, there were questions of relevance to experimentalgeometries. Since there was an energy gap for quasiparticlecreation, should one really expect the current to be carried by them? Also, as in the IQHE, there were nagging questions about the role of disorder. Impurity scattering was believed to be important for giving the Hall plateaus an observable width, but was also expected to destroy the incompressibility,and lead to low-lying,possibly localized bulk excitations. These, however, had better not destroy the precise quantization! The difficulties in relating the microscopic Laughlin wavefunction to the measured quantized conductance were reminiscent of the situation in superconductivity soon after BCS theory. While leading to many verifiable microscopic predictions, the connection of the BCS wavefunction to the macroscopic behavior of superconductors-the Meissner effect and zero resistance even with impurity scattering-was not altogether apparent. In fact, the macroscopic behavior of superconductors follows more directly from effectivetheories,such as Ginzburg-Landan theory. Guided by this, a number of theorists sought to develop a more phenomenological approach to the FQHE. The key step was undertaken by Girvin and MacDonald [7], who emphasized a striking analogy between Laughlin’s wavefunction and superfluidity. As emphasized earlier by Halperin [S], in Laughlin’s v = 1/3 wavefunction three vortices, or zeros in the wavefunction, sit on each electron. Such binding of electrons to vortices was believed to be a universal feature of the incompressible Hall fluid. Girvin and MacDonald pointed out that the electron-vortex composite, when viewed as a new “particle,” is not a fermion, but rather, has bosonic statistics. Moreover, these composite boson particles were shown to be Bose condensed in Laughlin’s state, exhibiting (algebraic)off-diagonal long-range order, much as in a conventional superfluid. This appealing picture was placed on a firm theoretical foundation by Zhang et al. [9] and Read [lo], who developed a full-blown Ginzburg-Landau type of description of the quantum Hall fluid. The superfluidity accounted naturally for the dissipationless flow in the QHE, and the bound vortices dragged along by the current generated a quantized Hall voltage. However, initially, these effective theories did not address the question of realistic finite geometries, so the relation with the quantized conductance measured in transport experiments remained unclear. During this period of intense interest in effective theories of the FQHE, Buttiker was reconsideringthe role of edge states in the IQHE [l I]. He emphasized that experiments with complicated geometries involving multiple current and voltage probes could easily be interpreted in terms of transmission and reflection of edge states. In Landauer transport theory [12], generalized by Biittiker [13] to multiple contacts, the conductance is expressed in terms of the

INTRODUCTION

111

transmission of electron waves incident from the leads at the Fermi energy. Within this framework, for a fluid of noninteracting electrons within an IQHE state, all of the transport current is confined to flow along the edges of the sample. This approach offered a simple and unifying picture of numerous transport experiments in the IQHE. But being a theory for noninteracting electrons,it was unclear how to generalize the edge-state approach to the FQHE. Following early work by Chang [14], Beenakker [lS], and MacDonald [16], a pioneering step was made by Wen, who developed a general theory for the edge excitations in the FQHE [171. This theory rested firmly on the Ginzburg-Landau description of the bulk FQHE. In this paper we offer an overview of the edge-state approach to transport in the FQHE. While providing a simple and direct understanding of the quantized Hall plateaus, the edge excitations of a fractional Hall fluid are extremelyinteresting in their own right. As emphasized by Wen [17], the FQHE edge excitations cannot be described in terms of noninteracting electrons. Rather, they must be thought of as a fluid of fractionally charged quasiparticle excitations. For the Laughlin sequence of fractions, v = l/m, the edge can be viewed as a gas of Laughlin quasiparticles, which is liberated and free to move along the sample edge. Wen emphasized the close analogy between these edge excitations and the low-energy excitations in models of interacting one-dimensional electron gases. It was over a quarter of a century earlier that pioneering theoretical work first revealed the profound effects that electron interactions can have on a onedimensional electron gas [18-201. Even weak interactions were found to destabilize a Fermi-liquid description of the one-dimensional gas. Instead, the one-dimensional gas exhibited a novel phase, termed a Luttinger liquid some years later by Haldane [21]. In a Luttinger liquid the low-energy excitations are not weakly dressed electrons but are collectivedensity waves, moving to the right and left without scattering.Wen emphasized that the edge excitationsin a FQHE state at v = l/m, which move only in one direction (say, right), are formally equivalent to the right-moving half of a Luttinger liquid. He coined the term chiral Luttinger liquid to describe such edge excitations [171. The Luttinger liquid has remained primarily a theoretical curiosity, since even with modern lithographic techniques it is very difficult to fabricate clean one-channel quantum wires. However, the current-carrying FQHE edge states, which are unaffected by disorder (see below), provide a unique laboratory for the study of “ideal” Luttinger liquids. For the Laughlin sequence v = l/m,the FQHE edge is predicted to consist of a single-branch chiral Luttinger liquid [17]. In this case the IQHE edge-state transport theory can readily be generalized.The quantization of the conductance followsreadily, even in the presence of disorder. In the hierarchical FQHE states, multiple edge excitations are predicted, in some cases with edge modes moving in both directions. For example, for v = 2/3 two oppositely moving edge modes are predicted [16,17,22]. This disagrees, however, with a recent time-domain experiment [23] in which only a single propagating mode was observed. Moreover, in these cases, edge-state theory predicts a Hall conductance that is neither univer-

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EDGE-STATE TRANSPORT

Figure 4.1. Schematic portrait of the edge ofa quantum Hall state with two channels. The

solid lines with arrows represent the edge states. The presence of random impurities, denoted by the small circles, allows for momentum nonconserving scattering between the different channels. (a) When the channels move in the same direction (e.g.v = 2), interchannel scattering does not effect the net transmission of the edge. (6) When the channels move in opposite directions,as in v = 2/3, the backscatteringof charge plays a crucial role.

sal nor quantized [24]. The conductance depends on microscopic details, such as the strength of the electron interaction between the two edge modes. This is clearly an unsatisfactory state of affairs. For such hierarchical states it is absolutely essential to incorporate impurity scattering at the edge. Edge impurity-scattering transfers charge between the channels, as depicted in Fig. 4.1. When the channels move in the same direction (Fig. 4 . 1 ~interchannel ~)~ charge transfer does not affect the net transmission. In contrast, charge transfer between oppositely moving edge channels (Fig. 4.lb) modifies the net transmitted edge current and conductance. As shown below, for v = 2/3 such impurity scattering drives an edge phase transition which separates a weakly and strongly disordered edge phase. An exact solution in the disorderdominated phase [24] reveals the existence of two modes: a charge mode which gives an appropriately quantized Hall conductance, (2/3)e2/h, and a neutral mode, propagating in the opposite direction. Since an electron is a superposition of the charge and neutral modes, it should be possible to excite the neutral mode by tunneling an electron into the edge. With suitable time-domain experiments it might thus be possible to detect the neutral mode directly. At finite temperatures the neutral mode is predicted to decay, with a rate varying as T2, behavior reminiscent of a quasiparticle in a Fermi liquid. Exact solution for the disorderdominated phase of the v = 2/3 edge can readily be generalized to a broad class of FQHE fluids [25], at filling v = n/(np + l), with integer n and even integer p. Unfortunately, the rich low-energy structure at the edge of FQHE fluids is not readily exposed via bulk transport measurements. Much more revealing are

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113

Figure 4.2. Schematic portrait of a point contact, in which the top and bottom edges of a Hall fluid are brought together by an electrostaticallycontrolled gate (G),allowing for the tunneling of charge between the two edges. Here S and D denote source and drain,

respectively.

laterally confined samples, in which different edges of a given sample are brought into close contact, allowing for interedge tunneling. The simplest case is that of a single point contact or constriction, in an otherwise bulk Hall fluid, as depicted in Fig. 4.2. In this case, backscattering between edge states on the top and bottom edges can occur via tunneling at the point contact. The magnitude of this backscattering can be inferred simply by measuring the conductance, or transmission, through the point contact. For a v = l/m Hall fluid, in which each edge has only one mode, a point contact is in fact isomorphic to a point scatterer in a one-channel one-dimensional electron gas. Moreover, for v = 1 the electron gas is a noninteracting Fermi liquid, whereas for fractional v the gas is a Luttinger liquid. Point contacts in Hall fluids thus provide an ideal arena to study experimentally the differences between Fermi and Luttinger liquids. In particular, since the transport current through the point contact depends on the tunneling density of states (DOS)for each edge mode, by varying the temperature one can effectively use transport to directly extract the energy-dependent DOS of a Luttinger liquid. In striking contrast to the IQHE, where the conductance through the point contact should go to a constant at low temperatures, the conductance for a FQHE fluid with v = l/m is predicted [26,27] to vary strongly with temperature, vanishing algebraically at . can be attributed to the suppressed low temperatures, G(T ) T ( 2 / v ) - 2This tunneling density of states in a Luttinger liquid [28]. Preliminary evidence for this signature of a Luttinger liquid has been seen in recent experiments by Milliken et al. [29]. In the presence of two adjacent defects, one has the possibility for resonant tunneling through an isolated localized state. In contrast to resonances in the IQHE, which are expected to have temperature-independent Lorentzian linewidths at low temperatures, for v = 1/3 the resonance widths are predicted [27,30] to vanish with temperature as T2l3.Moreover, at low temperatures the resonance is predicted to be strongly non-Lorentzian, with a completely universal lineshape. This lineshape has been computed by Monte Carlo simulations [27] and more recently by Bethe ansatz methods [31], and is in reasonable agreement with measurements by Milliken et al. [29].

-

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The point contact geometry in Fig. 4.2 should also be useful as a probe of the composite edge structure of hierarchical Hall fluids. Indeed, the presence of the predicted neutral modes, at fillings such as v = 2/3, leads to a modification in the temperature dependence of the conductance through the point contact [24]. Although indirect, such an experiment could thus provide evidence of the exotic neutral edge excitations. This chapter is organized as follows. Section 4.2 is devoted to a brief overview of the chiral Luttinger-liquid edge-statetheory for clean IQHE and FQHEedges. In Section 4.3 we consider the effects of impurity scattering of hierarchical states which have multiple edge modes, focusing on the intermode equilibration. Transport through a point contact in the FQHE is discussed in Section4.4 as a probe of Luttinger liquids. Section 4.5 is devoted to a brief summary and discussion, emphasizing experimental implications.

4.2. EDGE STATES 4.2.1. IQHE Consider first a noninteracting electron gas moving in two dimensions and confined by a potential V(y),which is constant (say zero) for l y ( < W/2, where W is the width of the system, and rises rapidly for larger values of lyl. A magnetic field of strength B is taken perpendicular to the x-y plane. In the Landau gauge with vector potential Ax = By the Hamiltonian takes the form

'(

H=G

e & + - Bc y

)

2

1 + -2m p;+

V(y)

with momentum p p = - iha,. Being separable, eigenstates can be written

where O(y)satisfies

Here the cyclotron frequency is o,= eB/mc, and y o = k12, with the magnetic length 1= If the confining potential is taken as slowly varying, d,,V(y)<< hwJl, the potential V ( y )in (3) can be approximated by the constant V(yo),and the eigenfunctions are then simply harmonic oscillator wavefunctions,Oncentered at yo. Thus wavefunctions that satisfy the time-dependent Schrodinger equation take the form

JG.

EDGE STATES

115

Figure4.3. Dispersion of energy levels in a quantum Hall bar as a function of the one-dimensional momentum k. Here p is the Fermi energy at bulk filling v = 1.

with energies

ha, = (n -k f)hU, -/- V(k1’) The eigenstates are plane waves in the x direction with dispersion wk, sketched in Fig. 4.3. When the Fermi energy, p, lies between bulk Landau levels, the only low-lying excitations are at the edges, where hw, = p. For one full bulk Landau level, there are then only two Fermi points, a right-moving one that is confined to the top edge, and a left mover confined to the bottom edge. The low-energy physics is isomorphic to a one-dimensionalnoninteracting electron gas, which also has two Fermi points, at & k,. At low energies it is legitimate to linearize the spectrum so that the wavefunctions take the form I) eik(x-u‘) with velocity v = awJdk. The quantization of the Hall conductance in the IQHE can be understood very easily in terms of edge states, as emphasized by Biittiker [ll]. Imagine raising the chemical potential (or Fermi energy) of the “source” electrode by an amount eY while keeping the “drain” electrode at p. The top edge state, being injected from the source electrode, will be at higher chemical potential and carry more current. The additional current can be expressed as

-

where dn = dk/27t is the change in the (one-dimensional)electron density on the top edge. Since the velocity v = dw/dk, this can be rewritten as

with a conductance G = e’/h. Notice that the conductance is independent of the velocity u, depending only on fundamental constants. More generally, for n-full Landau levels there will be n edge modes, each contributing a quantized value,

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EDGE-STATETRANSPORT

e2/h, to the conductance. The edge-state approach to quantum transport can be suitably generalized to more complicated geometries, with multiple current and voltage probes [ll]. An appealing feature of the edge theory of the quantized conductance is the insensitivity to disorder. Impurity scattering at the edge cannot degrade the source-to-drain current, since all of the n edge modes move in the same direction. The only effect of scattering is an unimportant forward-scattering phase shift. Backscattering is possibly only when the opposite edges are brought into close proximity, so that interedge tunneling becomes feasible. The quantized conductance is also insensitive to electron-electron interactions, as we argue below. For some hierarchical FQHE states, multiple edge modes that move in both directions on each edge are predicted. In these cases impurity scattering, rather than being unimportant, is in fact essential to explain the observed quantized conductance. Before generalizing the FQHE edge states, it will be useful to discuss a secondquantized formulation of the IQHE edge. For simplicity, consider an edge with only one mode, corresponding to one full Landau level. The low-energy states may be described by linearizing ( 5 ) about the Fermi wavevector to obtain

sr i,

H = 0 - k$+(k)$(k) Here we have taken the Fermi energy to be the zero of energy and k , to be the zero of momentum. It is also useful to transform to real space, which leads to

s

H = u dx @(x)ia,$(x)

(9)

where $ ( x ) is a one-dimensional fermion field operator, which satisfies the usual anticommutation relations, [$(x), $+(x’)] - = 6(x - x’). It will often be convenient to consider a Grassman path integral for the associated partition function. The appropriate euclidean action is

where z is imaginary time. As discussed,each free fermion edge channel contributes a conductance of 1,in units of e2/h.However, in the FQHE the conductance is fractional, G = ve2/h.For this reason a free fermion description of FQHE edge states is not possible. Rather, an appropriate description is in terms of a bosonic field, roughly analogous to the displacement field for phonons in a solid. Before discussing this, we first show how the IQHE edge can be “bosonized” [32-341. Although unnecessary for the IQHE edge, the bosonized description is useful since it can easily be generalized to describe FQHE edges.

EDGE STATES

117

By way of motivation, consider the nature of the low-energy edge excitations of a quantum Hall fluid. The incompressible Hall fluid can support long-wavelength fluctuations at its edge, analogous to water waves on the surface of the sea. However, unlike water waves the edge waves can propagate only in one direction, which is determined by the sign of the magnetic field. Classically, this can be understood as an E x B drift of the electrons caused by the edge confining potential. A quantum mechanical description may be developed in terms of the one-dimensionaledge density operator, defined as

where the dots denote a normal ordering with respect to the filled Fermi sea of the Hamiltonian (8).It is convenient to work with a new field $ defined by

From the continuity equation for charge conservation the electron edge current is then e .

I=-$ 2n

By carefully accounting for the normal ordering with an appropriate point splitting, one can show that the operator &(x) obeys the Kac-Moody commutation relation [32-341 [$(x),

&’)I

= in sgn(x - x’)

(14)

Notice that the momentum n(x) conjugate to $(x) is not independent, but rather is given by n ( x ) = ( 1/2)ljx4. The electron operator at the edge may be constructed by noting that removing a charge is equivalent to creating a 27z instanton in the field configuration $(x). This may be accomplished via

Using the Kac-Moody algebra, one can readily show that the fermion anticom= - $(x’)$(x). mutation relations are obeyed, @(x)@(x’) After normal ordering the Hamiltonian (8)-subtracting an appropriate constant from the energy so that the filled Fermi sea has zero energy-it can be reexpressed in terms of the density as [32-341

f

H=nu dxp2(x)

(17)

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Figure 4.4. Schematic diagram of a two-terminal conductance measurement for a v = 1 quantum Hall state. The shaded regions denote the reservoirs, which are assumed to be in equilibrium at different chemical potentials.

Upon using the conjugate momentum to obtain the Lagrangian, one arrives at the appropriate bosonized action, which in imaginary time is

The first term reflects the Kac-Moody commutation relations. Clearly, this action describes modes that propagate in one direction at a velocity u. It is instructive to rederive the quantized conductance for the IQHE using the bosonized action (18). To this end, consider an edge state that flows between two reservoirs which are in equilibrium at different chemical potentials (see Fig. 4.4). We model the reservoirs by considering an infinite edge, in which the “sample” resides between x L and xR.The left and right reservoirsare then defined for x c x L and x > xR, respectively. We suppose that the system is driven from equilibrium by an electrostatic potential eV(x),which couples to the edge charge density p(x) and is a constant eVL(R,in the left (right) reservoir. The underlying physical assumption of this approach is that the edge states that emanate from a given reservoir are in equilibrium at the chemical potential of that reservoir. Since the edge current operator is linear in the boson field, the edge current at x which flows in linear response to V(x’)may be computed directly. Specifically,

where the retarded response function is given by

This may be computed using (18) by analytically continuing the imaginary time

EDGE STATES

119

response function

to real frequencies, ion-+ o + k. Here q = k 1 determines the direction of edge propagation. We find that

The 8function reflects the chiral nature of the edge propagation, showing that the current at x depends only on the voltages at positions x’ “upstream” of x. In the limit w +O, the integral in (19)will be dominated by values of x’ that are deep into the “upstream” reservoir. Thus, for q = 1, which corresponds to an edge that propagates from left to right, the current is

+

The two-terminal conductance of a Hall bar follows upon adding a contribution from the opposite edge which emanates from the right reservoir and contributes a current - (e2/h)VR’,. The net current is thus Z = G(V , - VR),with an appropriately quantized two-terminal conductance: G = e2/h. It is straightforward to generalize the foregoing approach, based on the right/ left conductances, to compute the conductance measured in a four-terminal geometry. One thereby reproduces the multiterminal Buttiker-Landauer [131 transport formula for noninteracting electrons. An advantage of bosonization even for the IQHE edge is the ease in which electron interactions can be incorporated. Consider specifically short-range electron-electron interactions acting between the electrons along the edge. (In this chapter we ignore the long-ranged piece of the Coulomb interaction, assuming it to be screened by a ground plane.) The appropriate term to add to the Hamiltonian is uintpZ, with p the electron density. This term is quartic in fermion fields but can simply be absorbed into the velocity in the bozonized Hamiltonian (17). Thus electron interactions simply shift the edge velocity. They do not alter the quantized conductance, which is independent of u. 4.2.2. FQHE

In the fractional quantum Hall effect one typically has a partially filled Landau level. In the absence of electron-electron interactions there would then be an enormous ground-state degeneracy, but this degeneracy is lifted by the interactions. At special filling factors such as v = 1/3, the system is expected to condense into a correlated liquid state. This liquid state is incompressible and has a gap for

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all excitations. In the presence of edges one anticipates low-lying edge excitations, as in he IQHE. There are several routes to the appropriate description of the FQHE edge states. We describe two briefly. The first, heuristic in nature, involves generalizing the bosonized action of the IQHE edge so that the resulting conductance is fractional. The second, discussed before, generates the equivalent edge description starting from a Landau-Ginsburg theory for the bulk Hall fluid. The latter approach can easily be generalized to hierarchical FQHE states, as described later. Heuristic Motivation. In the derivation of the conductance for the IQHE edge from the bosonized action, the quantization can be traced to the prefactor of the first term in (18),which is a fixed dimensionless number (1/4n).This prefactor also determines the coefficient on the right side of the Kac-Moody commutation relation (14).By simply replacing the prefactor 1/4n by 1/4nv, where v is the filling factor, one obtains an edge action,

which has the required fractional conductance: e2 G=vh This innocuous-looking change has most striking consequences. The most dramatic is that the charge e excitation will generally have fractional statistics. To see this, note first that the boson field now satisfies a modified commutation relation [+(x), &x‘)] = inv sgn(x - x’)

(26)

The conjugate momentum is thus no longer the edge density, but rather, ll = (l/v)p, with p = (1/2n)a,+ as before. The operator for a charge e edge excitation then becomes $,(x) = ei2rrf”dx‘n(x’)+ei$/v

(27)

+

Upon using the commutation relation (26) for the field, one deduces that the charge e operator generally has fractional statistics:

Notice that for the special case of l/v an odd integer, the charge e excitation is fermionic and can then be associated with the electron. For more general v, the absence of a charge e fermionic edge excitation is rather worrisome. Since the bulk FQHE state is built from physical electrons, one

EDGE STATES

121

would expect electrons to be present at the edge also. This reasoning suggests that the effectiveaction (24) is only a valid description of the Hall edge, when v = l/m, with m an odd integer [17]. This conclusion will be confirmed below, where it is shown that for Hall states with v # l/m,there are multiple edge modes. Besides the operator #(x) = ei9/’, which creates a charge e fermion at the edge of the v = l/m fluid, one can consider other operators, such as ei9. This operator creates an edge excitation with fractional charge, ye. The statistics is also fractional, with a phase factor of exp( k ivn) under exchange. It is apparent that ei+ creates a Laughlin quasiparticle at the edge. This will be confirmed below starting from a Ginsburg-Landau description of the Hall fluid. From Ginzburg-Landarr Theory. The physical idea behind the Ginzburg-Landau theory of the FQHE [9, lo] is that vortices in the electron wavefunction bind to the electrons. For the Laughlin sequenceof states at filling v = l/m,each electron binds with m vortices. This can be seen explicitly in Laughlin’s celebrated wavefunction, in which m vortices sit right on top of each electron but is expected to be a general feature of the incompressible Hall fluid. Due to the 2n phase accumulated upon encircling each vortex, the statistics of the electron-vortex composites are bosonic, since m is an odd integer. At the magic filling v = l/m, all of the field-induced vortices are accommodated by binding to electrons, and the composite bosons do not “see” any residual vortices. They can then Bose condense, forming a superfluid. This condensed fluid describes dissipationless flow and a (bulk)quantized Hall conductance, the two hallmarks of a quantum Hall fluid. The original Ginzburg-Landau theory focused on the bosonic wavefunction of the electron-vortex composite. There is an alternative dual description, though, which consists of a bosonic field c ’ ~which creates a vortex in the GinzburgLandau complex boson field [35,36]. This field is minimally coupled to a gauge field, a,. The gauge field is related directly to the electron three-current via

Thus when a vortex moves, it sees the electrons as a source of “fictitious” flux. In a quantum Hall fluid at magic filling, there are no free vortices in the GinzburgLandau field. Vortex antivortex pairs can be created but cost a finite energy and are unimportant at low temperatures. (This corresponds to a Laughlin quasiparticle and quasihole pair.) The low-energy description thus reduces to that of the gauge field alone. Keeping the most important terms, the resulting Euclidean action for this gauge field is 1

r

This effective action describes the long-wavelength density fluctuations of the condensed fluid. The energy gap for the bulk quasiparticle excitations is not

I22

EDGE-STATE TRANSPORT

specified. Provided that the temperature is well below this gap, the effectiveaction (30) provides an adequate description. However, at filling factors away from v = l/m, there will be some residual vortices, and the electron-vortex composites can condense only if these residual vortices are pinned and localized by bulk impurities. In this case there will be many low-energy, but spatially localized, excitations involving rearranging the positions of these vortices. The effective action (1) can presumably still be used to extract transport properties, though, since at low temperatures the localized vortices will not contribute significantlyto the transport. (This is not the case for other physical properties, such as the electronic specific heat.) Although the vortex (quasiparticle)excitations are not important at low Tin the bulk, they play a crucial role at the edge. At the edge their gap vanishes and they form the edge states. The edge excitations can naturally be expressed in terms of the phase 4 of the vortex creation operator, ei4, which is minimally coupled to the gauge field, 8,,4 - a,,. Spccifically,as shown by Wen [17], the bulk degrees of freedom can be eliminated from (30) by integration over a,, which imposes an incompressibility constraint on bulk density fluctuations: x i? = 0, with the vector referring to the two spatial components. A scalar field can then be introduced to solve this constant, ii = 94, where 4 can be interpreted as the phase of the vortex. After an integration by parts, the final Euclidean action for the edge states is given by (14), the form we arrived at by heuristic argument above. As discussed by Wen [171, the magnitude of the edge velocity u depends on the precise boundary conditions assumed for the field a,. Since u will depend on the edge confining potential and the edge Coulomb interaction, it cannot be determined from the bulk action (30). It is thus appropriate to take the velocity u as a phenomenological parameter. It follows from (29) that the one-dimensionalcharge density along the edge is given by p = 8,4/27t, precisely as in (12).Our identification of ei4as the creation operator for a quasiparticle (or vortex) at the edge was also appropriate. We have thereby arrived a description of the v = l/m edge, identical to that obtained heuristically above. While neither “derivation” is rigorous, the final effective action (24) is undoubtedly correct. The Ginzburg-Landau approach is advantageous, though, since it can readily be generalized to hierarchical quantum Hall fluids, as we describe next. HierurchicalStutes. Soon after Laughlin’s wavefunction, Haldane and Halperin

[37] suggested a way to account for Hall plateaus at other rational fillings, besides v = l/m. Their basic idea was that the vortices introduced upon moving away from filling v = l/m would themselves condense into a Laughlin fluid. By successively iterating this procedure, incompressible Hall fluids at arbitrary rational v, with odd denominator, could be generated. An alternative hierarchical construction, suggested by Jain, consists of attaching an even number, p, of flux tubes (or vortices) to each electron, and then putting the resulting fermionic electron-vortex composites into n full Landau levels. This describes states at

EDGE STATES

123

+

filling v = n/(np l), a robust sequence of fractions. Although the wavefunctions in these different constructions certainly differ in detail, it was initially unclear as to whether they describe the “same phase.” This was clarified by Wen and Zee [36] and Read [38], who emphasized a “hidden” topological order in the Hall fluids and introduced a scheme to characterize and classify them. Specifically, at the nth level of the HaldaneHalperin hierarchy,the fluid can be characterized by a symmetric n x n matrix K. The appropriate effective action that generalizes(30) can be expressed in terms of n gauge fields and takes the form

The electron three-current is given by

where ti are basic dependent integers or “charges.” This is the multicomponent generalization of (29). The filling factor is given by v=

1tiK,G I t j ij

(33)

The K matrix also characterizesthe charge and statistics of the bulk quasiparticle excitations. Specifically,the quasiparticles are labeled by a set of integers mj, with j = 1,2,. ..,n, with charge (in units of the electron charge) Q=

and statistics angle

1miK,; ‘ti ij

- = 1mi& ‘mj 0

ij

(34)

(35)

In this approach, all of the universal properties of the bulk quantum Hall state follow directly from the K matrix. The explicit form of the K matrix for a given quantum Hall state depends on the choice of basis, as do the integers ti [36,38]. Under a basis transformation

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EDGE-STATE TRANSPORT

where W is an n x n matrix with integer matrix elements and unit determinant. This transformation leaves the filling v as well as the quasiparticle charge and statistics invariant. The basis in which ti = 1 for all i, is called the “symmetric” basis by Wen and Zee. In the hierarchical basis, t = 1 and ti = 0 for i = 2,3,...,n. The form of the K matrix obtained for a FQHE fluid at filling v from the Haldane-Halperin hierarchy scheme is identical to that obtained via Jain’s construction [41]. Thus the two states are topologically equivalent. In the symmetric basis, for filling v = n/(np + 1) with integer n and even integer p, the K matrix is given explicitly by

,

K .U. = 6lJ. .+ p

(39)

At filling v = l/(p, - l/p2) where p , and p 2 are odd and even integers,respectively, the K matrix in the hierarchical basis takes the simple form -1

-1) P2

The edge excitations may be described by eliminating the bulk degrees of freedom, as described above. Upon integration over at, a constraint on the density fluctuations in the bulk is imposed: x i f i = 0, for all i ==+l,2,. ..,n. Scalar fields can then be introduced to solve these constraints, iii= V $ J ione , for each gauge field. As above, the edge excitations are described in terms of these scalar fields. The appropriate effective action at the edge can then be written as S = So + S , with

s, =

J

f

1

dxdz--l-C 4n i j

vijax$Jiax4j

The first term is solely determined from bulk physics of the Hall fluid. This term determines the commutation relations for the Bose fields: [$Ji(x),4j(x’)] = inK; sgn(x - x‘)

(43)

the generalization of (26). In addition, we also have interaction terms of the form 8x$Ji8x$J? These interaction strengths are nonuniversal, depending on the edge confining potential and edge electron interactions. It follows from (32) that the edge charge density is given by

Operators that create charge at the edge can be deduced from the conjugate

EDGE STATES

125

Figure 4.5. Schematic diagram of a two-terminal conductance measurement for a quantum Hall state such as v = 2/3 in which two edge channels move in opposite directions.

momentum: Hi = ( 1 / 2 ~ ) K ~ , dAn ~ 4 ~operator . of the form e~p[i4~(x)] creates instantons in the boson fields 4j at position x. These instantons carry charge, as see from (44).The general operator Qx) = eixy=i m f i i x )

(45)

for integer mj creates an edge excitation at x with charge Q given by (34). Since the effective action (41)-(42) is quadratic in the boson fields, all physical quantities can readily be computed. Unfortunately,when the eigenvaluesof K are not all of the same sign, the computed conductance is not given [24,25] by the quantized value Ivle2/h. To see why, consider for simplicity a two-channel case such as v = 2/3. Since the sign of the eigenvalues of the K matrix determine the direction of propagation, the two modes will be moving in opposite directions, as sketched in Fig. 4.5. Consider a two-terminal transport situation in which the source and drain are held at different chemical potentials. Since an edge mode is, by assumption, in equilibrium with the reservoir from which it emanates, the current on the top edge will depend on the voltages in both reservoirs. Specifically, the current at the top edge can be written

where g + (g -) is the dimensionless conductance for the right (left)moving channel. This form can be derived explicitly by generalizing (19) to include multiple edge modes. The two-terminal conductance follows by considering the other edge, which carries a current g - VL- g + V,, giving

Notice that the conductances add, even though the channels are moving in opposite directions. Using (19)and the action (41)-(42),one can compute g+ and g -

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EDGE-STATE TRANSPORT

explicitly. Their difference is found to be universal and quantized, g+ - g- = v. However, their sum, which enters in the conductances, is nonuniversal, depending on the interaction strengths Vij in (42). It is straightforward to generalize this approach to compute the conductance measured in a four-terminal geometry. In particular, we find that the fourterminal Hall conductance is given by

It is only when all channels propagate in the same direction that G, is universal and equal to ve2/h. The nonquantized conductance for hierarchical states with multiple modes that move in both directions is in glaring contradiction with experiment.Clearly, some important physics must be absent from the simp1 effective action (41)-(42). A clue can be seen from Fig. 4.5, where it is clear that in a transport situation, right-moving edge modes are in equilibrium with the left reservoir, and left movers are in equilibrium with the right reservoir. Thus in the presence of a nonzero source-to-drain voltage, opposite-movingedge modes on a given edge will be out of equilibrium with one another. But since these modes are in close proximity, that stops then from equilibrating? In the effective action (41)-(42) there are no terms that transfer charge between the different edge modes, to allow for possible equilibration. But surely in real experimental systems there will be equilibration processes present. A stringent constraint is that charge transfer must conserve energy and momentum. Generally, different edge modes have different momenta-the gauge-invariant momentum difference between two modes being proportional to the magnetic flux threading the space between them. Since the edge modes are all at the same energy (in equilibrium),charge-transfer processes with the emission of phonons or photons to take up the momentum will not conserve overall energy. These processes are thus forbidden. However, if there are impurities near the edge, momentum of the edge modes need not be conserved. Momentum can be transferred to the center of mass of the sample, through the impurities.Thus impurity scattering at the edge will mix the different modes and tend to equilibrate them. In Section 4.4 we study this effect in detail. Such equilibration can also occur at the contacts to the sample. This has been discussed in Ref. [39].

4.3. RANDOMNESS AND HIERARCHICAL EDGE STATES In Section 4.2 we argued that for hierarchical states with multiple edge branches moving in both directions, the conductance is predicted to be nonuniversal, due to an absenceof intermode equilibration. Edge impurity scattering is then critical, to allow for equilibration, and must be incorporated into a realistic edge-state

RANDOMNESS AND HIERARCHICAL EDGE STATES

127

theory. We now consider the general problem of random impurity scattering at the edge of a quantum Hall state. For a broad class of hierarchical states, we will show that the low-temperature physics is described by a new random fixed point. In contrast to the hierarchical states, for Laughlin states that have a single edge, channel disorder is unimportant. To see this, consider first a v = 1 edge with a spatially dependent random edge potential p(x). Treating the electrons as noninteracting, we may add this random potential to the fermion action (10):

s

S = d x d z $*(a,

+ id,)$ + Ax)$*+

(49)

The random term can be eliminated by performing a spatially dependent U(1) gauge transformation, $ = $ exp[id(x)], with

Thus the only effect of the random potential is to introduce an unimportant forward-scattering phase shift, d(x). This result may be extended to interacting systems and to Laughlin states by using the chiral boson representation (24). In this representation, the random potential p(x) couples to p(x) = 8,4/2a and may be eliminated via the transformation &XI = 4(x) + vd(x). Clearly, any nontrivial effects of edge disorder must arise from tunneling between different channels. To study such effects, we consider first the IQHE with v = 2 in detail. In doing so, we develop the necessary machinery to describe hierarchical FQHE states. We then focus on three specific cases, v = 213, v = 215, and v = 415. As emphasized in Section 4.2, the case v = 213 is particularly important, since the model without disorder predicts a nonquantized conductance. 4.3.1. The v = 2 Random Edge

Fermion Representufwn. Consider first the simplest-possible two-channel model noninteracting electrons with v = 2, where the two edge modes are identical. In terms of a two-component chiral fermion field, Y = $J, the appropriate action generalizing (10)is

Note that So is invariant under a global U(2)transformation. This symmetry is a product of a U(1) symmetry, arising from conservation of charge, and an SU(2) symmetry,arising from the conservation of "spin." As we shall see below, this high symmetry, which appears rather artificial since it requires the channels to be

128

EDGE-STATE TRANSPORT

identical, is actually a generic property of the ground state when impurity scattering is present. Consider now the effect of randomness. In addition to random potentials coupling to the charge densities in each channel, randomness will also introduce tunneling between the two channels. In general, we may write

where 5 are Pauli matrices. Here p(x) (,(x) are random potentials coupling separately to the two channels. The spatially random coefficients, 5 , = t, k it,, multiply and and correspond to interchannel tunneling. As in the single-channel case, p(x) may be eliminated by the U(1) gauge transformation (50). Similarly, ?(x) may be eliminated by an SU(2)rotation. We thus write

where 6 is given by (3.2) and U(x) is an SU(2)rotation, given by

where T, is an x-ordering operator. The resulting random problem is then given by

J

P

So

+ Srandom = dx dz ‘Ft(a,+ i d x ) $

(55)

In terms of the new field ‘F, the random problem still has an exact global U(2) symmetry. The exact solution (55) gives a complete description of the noninteracting problem. The eigenstates are simply plane waves in which the channel index (or “spin”) is rotated by V(x). The strength of the random interchannel tunneling introduces a mean free path, I, which is the length scale over which U(x)varies. On length scales longer than I, eidU(x)will be an uncorrelated random U(2)matrix. Wen has generalized the solution above to allow for different velocities of the two channels [40]. In this case, the U(2)symmetry of the clean edge is broken. Nevertheless, on length scales longer than the mean free path 1, all excitations move at a single velocity: V = 2u,u2/(u, + u2). This is physically reasonable, since upon averaging over long times an electron spends equal time in each channel, regardless of its initial channel. The long-distance behavior is thus argued by Wen to be the same as that of the U(2)symmetric model (51) in which the velocities are equal.

RANDOMNESS AND HIERARCHICAL EDGE STATES

129

Boson Representation. The noninteracting electron theory above cannot easily be generalized to include the effects of interactions, and moreover, is not suited to the FQHE. It is therefore desirable to reformulate the random edge in a chiral boson representation. In doing so, we establish that the low-energy physics of the random v = 2 edge is described by a stable random fixed point. However, in contrast to the noninteracting problem ( 5 9 , in which both modes move at the same velocity, the interactions play an important role by giving rise to “spincharge” separation, in which the charged and neutral modes propagate at different velocities. The final fixed point has a U(1)x SU(2)symmetry rather than the full U(2) symmetry present in the noninteracting case (55). Using (18),the bosonized action for a v = 2 edge can be written as

’

= SO+ ‘random

with

Electron interactions between the two channels are accounted for by u12. By using (16),which expressesthe electron operator as eib,the interchannel tunneling may be written

As in (52) <+(x)are spatially random complex tunneling amplitudes.We omit the random potentials p(x) and tZ(x),which couple to & &,), since they may be eliminated via a suitable redefinition of q51,2. Since the operators in (58)are nonlinear in the boson fields, the random model appears rather intractable. One approach is to expand for small {Jx) about the free theory, So. This is problematic, however, because the perturbation theory is divergent at low energies. One can, nevertheless,define a perturbative renormalization group (RG)transformation [24,25] in powers of the variance, W, defined via [t +(x)(-(0)Iens= Wd(x). Here the brackets denote an ensemble average over realizations of the disorder. As outlined in Appendix A, the leading-order RG flow equations take the form

aw

-= (3 - 2A)W ai

(59)

where A is the scaling dimension of the operator 6 = expCi(4, - 4,)] evaluated in the free theory, So. One finds A = 1, which also follows from the fermion representation (51),since 6 is the product of two fermion operators. From (59)it is clear that the random potential is relevant, so that nonperturbative methods

130

EDGE-STATETRANSPORT

are necessary. Fortunately, the low-energy physics is controlled by a stable fixed point that is accessible nonperturbatively, as we now show. The proceed it is convenient to introduce charge and spin variables [24],

The net charge density on the edge is dx4,/2n.The field 4,, which commutes with $, represents the neutral degree of freedom associated with tunneling between the channels. In terms of these new variables, we may rewrite (56) as

s = s, + s, + S i n l

(62)

with charge and neutral pieces r

coupled together via

C

s

Vint Sinl= d x d z ax$,dx4a 4n

The velocities up, u,, and uintdepend on the original velocities in (57). Note that v, is not necessarily equal to v,. Indeed, a short-range interaction that couples to the total charge affects only up. Consider first the U(2) symmetric model, as considered in (51), with u1 = u2 = u, u12 = 0. Upon transforming to the charge-spin variables, we then find up = u, = u and uinI= 0. The system thus decouples into a charge sector described by S, and a spin sector S,. Note that the interchannel tunneling terms affect only the spin sector. Despite the presence of complicated nonlinear random terms in S,, we know that this model is equivalent to the exactly soluble fermion model (55). The hidden simplicity in S, is a consequence of the fact that the operators cos 4a,sin 4a,and 8,4, transform as S,, S,, and S , in an SU(2) algebra, known as a level-one SU(2) current algebra. It is this connection that will allow us to formulate a general solution of the interacting problem. Consider now the case in which vint= 0 but up # u,. This no longer corresponds to noninteracting electrons. The charge and neutral sectors are still decoupled, but now the charge and neutral modes propagate at different velocities. Again, the only nontrivial part of the problem is the neutral sector, which involves nonlinear random terms. But the neutral sector is identical to the neutral sector of

RANDOMNESS A N D HIERARCHICAL EDGE STATES

131

the exactly soluble problem (51) provided that we identify u = u,. Since the soluble fermion problem includes a charge sector, in order to take advantage of this connection we must introduce an auxiliary field $p with an action which is the same as S,, except that up is replaced by u,. The field $p does not enter physical observables but allows a convenient representation of the SU(2)symmetry in the neutral sector. The combined action S, may be “fermionized,” by letting $, = exp[i($p + 4,)/2] and $2 = exp[i(J, - 4,)/2]. The operator corresponding to dx4,is YtozY.The resulting fermion problem is identical to (52) and may be solved by the space-dependenttransformation 9 = U(x)Y with U(x)given in (54). The action is then simply by (55) with u = u,. This form exposes the hidden but exact global SU(2) symmetry of the neutral sector (3.16). Thus we see that when uin1= 0, the problem is exactly soluble even in the presence of interactions, and the ground state has an exact U(1) x SU(2)symmetry described by (63) and (55). However, in contrast to the noninteracting case the charge and neutral modes need not move at the same velocity; there is “chargespin separation.” We must now study the effects of uint, which couples the charge and neutral sectors and hence breaks the SU(2) symmetry. Using the exact solution above, we will show that in the presence of a random potential, uinl is irreleuant. Under an RG transformation, the system scales to the charge-neutral decoupled fixed point. It is convenient to reexpress Sintin terms of the fermion field 9.We find that

sp

sp+

The relevancy of this term may be deduced from the scaling dimension of the which we denote 6,. Using (65) and (63), one readily operator Gij = ~3,+,9$’~ obtain 6, = 2. Since this operator has a random x-dependent coefficient v(x) zuintUo,Ut,which is uncorrelated on spatial scales large compared to the mean free path l, it is most useful to consider the scaling behavior of the meansquare average, W , = [ U ( X ) ~ ] where ~ ~ ~ , the brackets denote an ensemble average. As outlined in the appendix to this chapter, to leading order in W , the RG flow equation is

-dwu-(3-26”)w, dl

(67)

Since 6, = 2, the perturbation is irrelevant. It should be emphasized that in the absence of randomness, the dimension 2 operators in Sintare marginal and do not renormalize to zero! Thus disorder is seen to be absolutely critical in the stability of the decoupled U(1) x SU(2) fixed point. In summary, impurity scattering has played a crucial role in driving the charge-neutral coupling to zero. The final fixed-point theory has a global U(1) x SU(2) symmetry, a much higher symmetry than the underlying random Hamiltonian. Although it is difficult to compute the fully renormalized velocities

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EDGE-STATE TRANSPORT

up and u,, it is clear that they will be equal only in the absence of interactions. Thus, generically, we expect charge-spin separation on the v = 2 edge.

4.3.2. Fractional Quantum Hall Random Edge Using the results of the preceding section for v = 2, we are now in a position to analyze the effects f impurity scattering on hierarchical FQHE edges. We will focus initially on Hall states at the second level of the Haldane-Halperin hierarchy, with v = l/(pl - 1/p2), where p1 and p2 are odd and even integers, respectively. We will consider three specific examples that display different generic behavior: v = 2/3, 2/5, and 4/5. We will then discuss generalizations to other fractions, including those at higher levels in the hierarchy. At filling v - = p1 - l/pz, the edge consists of two modes, as described in Section 11. Without impurity scattering the edge can be described by the action (41)-(42), with K an appropriate 2 x 2 matrix. In the hierarchy basis the explicit form for K is given in (40).As in the preceding section,it will be useful to represent the problem in terms of charge and neutral (or spin) variables. These variables are related to the hierarchy basis variables by

The charge and neutral fields commute with one another, as can be seen using (43). The action (41)-(42) may then be expressed in the form

s = s, + s, + sint with charge and neutral pieces

coupled together via

Again, the velocities up, u,, and uin, depend on the original velocities Vijin (42). The most important terms generated by a random edge potential will be tunneling terms, as in (58), which transfer charge between the two channels. Charge conservation dictates that such processes do not create a net charge, so

RANDOMNESS AND HIERARCHICAL EDGE STATES

Q = 0 in (34). The most relevant such term is given by exp[ k i(& exp(i4,). We are thus lead to consider

f

Srandom = d x dz[<+(x)e'+O

+ c.c.]

- p2&)]

133 =

(74)

Note that the random tunneling operators involve only the neutral field, 40. We now consider three specificexamples that display different types of generic behavior. 1. v = 2/3. The v = 213 edge is described by p1 = 1 and p 2 = - 2. Consider first an impurity free edge with Srandom = 0. In this case, when uint = 0, the charge and neutral modes decouple, and the neutral mode propagates in the direction opposite to the charge mode. Since the conductance depends only on the charge mode, it may be seen using (25) that it is appropriately quantized, G = (2/3)ez/h. However, when uint# 0, the conductance will be nonquantized and nonuniversal, depending on uint.This can be seen explicitly using (19).The two modes continue to propagate in different directions, but both contribute to the conductance. In general we may write G = (2/3)Aez/h,where A > 1. Then A = 1only when uint= 0. We shall now argue that as the v = 2 case, the presence of a random edge potential can drive the system to the decoupled fixed point with uint = 0. Thus the conductance is quantized even when channels move in opposite directions. Consider again the decoupled point, uinl = 0. Since p 2 = - 2, the neutral sector (72)and (74) is identical to the neutral sector (64) for v = 2 studied in the preceding section (up to a sign that determines the direction of propagation). Thus the exact solution obtained there by refermionizing the problem may be applied. Moreover, we may use exactly the same arguments to show that uint is irrelevant. We thus establish that the decoupled fixed line is stable. The fixed point is characterized by an exact SU(2) x U(1) symmetry. Having established the stability of the decoupled fixed line, we must also consider the possibility of other stable fixed points that could describe different phases. Consider treating the randomness (74) perturbatively. As in (59), we may analyze the relevancy of weak disorder by considering the leading-order RG for the variance, W, of 5, :

dW dl

-= (3 - 2A)W

(75)

Here A is the scaling dimension of the operator exp(i4,) and may be computed explicitly using the action (71)-(73). When uint= 0, then A = 1, but A > 1 is nonuniversal when uin1# 0. Indeed, A is the same quantity that entered into the conductance above. As seen from (75), when uint is tuned so that A exceeds 312, there will be an edge phase transition to a phase in which (weak) disorder is irrelevant. For filling v = 2/3, this transition was analyzed in Ref. [23]. There it was shown that the transition is of Kosterlitz-Thouless type, with the RG flows shown in Fig. 4.6.

134

EDGE-STATETRANSPORT

0

1

312

A

Figure 4.6. Renormalization group flow diagram for a v = 2/3 random edge as a function of disorder strength Wand the scaling dimension A of the tunneling operator. For A c 3/2 all flows end up at the exactly soluble fixed line A = 1. For A > 3/2 there is a KosterlitzThouless-likeseparatrix separating the disorder-dominated phase from a phase in which disorder is irrelevant.

We thus conclude that there are two possible phases for the random v = 2/3 edge. For sufficientlylarge uinl there is a phase in which disorder is irrelevant. At zero temperature, this phase is characterized by a nonuniversal conductance and nonuniversal tunneling exponents as shown in Section 4.4. The more generic phase, favored by electron interactions, is characterized by charge-spin separation, an exact U(1) x SU(2) symmetry,and a quantized conductance. Unlike the case v = 2, however, the charge ad neutral (or spin) modes propagate in opposite directions. 2. v = 215. The v = 215 edge is described by p1 = 3 and p2 = 2. In this case, sincep, > 0, both modes propagate in the same direction, as seen from (72).Thus, as shown in Section 4.2, even in the absence of edge impurity scattering, the conductance is quantized: G = (2/5)ez/h.Nonetheless,when impurity scattering is present, the edge restructures and exhibits spin-charge separation. To see this, simply repeat the argument for v = 2/3, which shows that the charge and neutral sectors decouple at low energies, uint-+0.The fixed point with U(1) x SU(2) symmetry is stable. Both the charge and neutral modes move in the same direction, generally with different velocities. Since the scaling dimension of the tunneling operator in (74) is A = 1 for any interaction strength, the disorder free edge is always perturbatively unstable to impurity scattering, in contrast to the v = 213 case. 3. v = 4/5. The Hall fluid with v = 415 is described by p1 = 1 and p , = - 4, and the two edge modes move in opposite directions. In this case, since 1 p , I # 2, the neutral sector is no longer identical to the neutral sector of v = 2, so the exact solution employed there can no longer be used. Weak impurity scattering can be analyzed perturbatively, however, by computing the scaling dimension of the tunneling operators (74) in the clean theory. When uin, = 0 it can be shown that A = 2. Including uint, we find that A > 2. Thus A > 312, and from (75) weak

RANDOMNESS AND HIERARCHICAL EDGE STATES

135

impurity scattering is always irrelevant. Thus the only low-energy fixed point is the clean edge, (71)-(73). Since the channels move in opposite directions, the zero-temperature conductance is thus predicted to be nonuniversal. At finite temperatures, however, as we see in Section 4.3.3, quantization is restored. SU(n) Generalizations. We have shown that the edges of disordered v = 2/3 and 2/5 fluids are described by a stable T = 0 fixed point, with an exact U(1) x SU(2) symmetry. This will also be the case for any state with Ip21 = 2. These filling factors can be written as v = 2/(2p k l), with p an even integer. Within Jain’s construction [40], these are precisely the states that can be obtained by attaching p flux quanta to each electron and filling two Landau levels. This suggests that the foregoing results can be generalized to the quantum Hall states derivable from n full Landau levels, which occur at filling factors v = n/(np + 1). This case was studied in detail in Ref. [25], where it was shown that a random potential drives the edge to a stable fixed point characterized by an exact U(1) x SU(n) symmetry. Again, the stable fixed point is characterized by spincharge separation, but now with a more general SU(n)spin. In this case, there is a single charged mode, and n - 1 neutral modes. The neutral modes are related by the exact SU(n)symmetry, and it follows that they all move at the same velocity. In general, the charge mode moves at a different velocity,and when p < 0 it moves in the opposite direction of the n - 1 neutral modes. Only the charge mode contributes to the conductance, which is appropriately quantized, G = ve2/h.

4.3.3. Finite-Temperature Effects The exact solution of the random edge that describes a stable zero-temperature fixed point can also be used to extract physical properties of the edge at low but nonzero temperatures.These properties will be determined by the structure of the fixed point itself and the leading irrelevant operators, such as qntin (3.18). At low but nonzero temperatures these operators have not had “time” to fully renormalize to zero and can then have an important effect on physical observables. Although one can show that the irrelevant operators do not modify the quantized Hall conductance itself, they do dramatically effect the propagation of the neutral modes at finite temperature. To see why, we first note that the existenceof the propagating neutral modes is tied intimately to the exact SU(n) symmetry in the neutral sector at the fixed point. But at finite temperatures, this symmetry is no longer exact, due to the presence of irrelevant operators, so that the neutral modes should no longer be strictly conserved. Thus one expects that at finite temperatures the neutral modes should decay away at a nonvanishing rate, 1/7,. Equivalently, one expects a finite decay length, or inelastic scattering length, I , = ~ ~ 7On , . scales Lmuch larger than I,, the neutral modes should not propagate. Since the fixed point is approached as T-0, however, the decay length should diverge in this limit. By analyzing the leading irrelevant operators [such as (73)] that control the flows into the zero-temperature fixed point, it was shown in Ref. [25] that the

136

EDGE-STATE TRANSPORT

decay rate vanishes algebraically with temperature: 1

-K

=,

TZ

In contrast, the charge mode cannot decay, even at finite temperature, since electriccharge is always conserved. However,due to irrelevant operators, such as (73), which couple the charge and neutral sectors, the charge mode can scatter off the neutral modes. This leads to a charge mode that propagates with a dispersion o = opq iDq2,with a “diffusion”constant D that is temperature independent at low temperatures. This implies a diffusive spreading of a charge pulse as it propagates along an edge. For v = 4/5 we arrived at the striking conclusion that impurity scattering at the edge was ineffective at equilibrating the two edge modes, leading to a nonquantized conductance at T = 0. But a quantized plateau is seen at v = 4/5, albeit with less vigor than might have been expected given estimates for the bulk energy gap. This apparent conflict is resolved when one considers finite temperature effects at the v = 4/5 edge. Although disorder, W, is formally irrelevant, since A > 2 in ( 7 9 , at finite temperatures W has not had “time” to scale all the way to zero. In fact, by cutting off the RG flows with temperature, it was shown in Ref. [25] that there is a characteristic inelastic scattering length, which diverges at low temperatures as

+

On scales longer than this length, equilibration takes place and charge does not propagate upstream. Provided that this length is shorter than the distance between sample probes, a quantized conductance is recovered, as discussed in Ref. [25]. However, this does raise the interesting possibility of observing deviations from quantization in short Hall bars at v = 4/5 and low temperatures. A very clean sample would be favorable for observing such deviations.

4.4. TUNNELING AS A PROBE OF EDGE-STATE STRUCTURE The rich physics “hidden” at the edge of FQHE fluids is not easily revealed via bulk transport measurements. An ideal way to access this edge physics is via laterally confined samples in which two edges of a given sample are brought into close proximity, allowingfor interedgetunneling. The simplest situation is a point contact in an otherwise bulk quantum Hall fluid, as depicted in Fig. 4.2. The point contact can be controlled electrostatically by a gate voltage. When the constriction is open, the two-terminal conductance is given by its quantized value. However, as the channel is pinched off and the top and bottom edges are brought into close proximity, charge will begin to backscatter between the right and left moving edge channels. Such backscattering reduces the two-terminal

TUNNELING AS A PROBE OF EDGE-STATE STRUCTURE

137

1.0

0.8

;u'

u5l

0.6

Y

C 3 0.4

0.2 0.0

-

-0

Figure 4.7. Two-terminal conductance as a function of gate voltage of a GaAs quantum Hall point contact taken at 42mK. The two curves are taken at magnetic fields that correspond to v = 1 and v = 1/3 plateaus. (From Ref. [29].)

conductance. Ultimately, as the gate voltage is increased, the Hall bar will be pinched off completely, and the two-terminal conductance will be zero. In Fig. 4.7, the two-terminal conductance as a function of gate voltage is shown for a GaAs quantum Hall point contact [29] taken at 42mK. The two curves are taken at magnetic fields that correspond to v = 1 and v = 1/3 plateaus. To the right the point contact is open, and the conductance is quantized, whereas to the left the point contact is pinched off. Between there are many resonant structures resulting from random impurities in the vicinity of the point contact. Note the qualitativedifference between the behavior for v = 1 and v = 1/3. For v = 1/3,the valleys between the resonances are deeper and the resonances are sharper. How are we to understand this qualitative difference? In this section we present a theory of tunneling and resonant tunneling at a point contact that answers this question. We begin in Section 4.4.1 with a discussion of tunneling at a point contact. We show that in contrast to the IQHE, the conductance of a FQHE point contact vanishes in the limit of zero temperature. We study resonant tunneling in Section 4.4.2, where we show that in the fractional quantum Hall effect resonances have a temperature-dependent width and a universal, non-Lorentzian lineshape at low temperatures. Finally, in Section 4.4.3, we discuss low-frequency shot noise at a quantum Hall point contact and suggest a method for the direct observation of the fractional charge of the Laughlin quasiparticle. We confine our attention initially to the Laughlin states, v = l/m, for which the edge states have a single branch. A discussion of tunneling in hierarchical quantum Hall states is deferred to Section 4.4.3.

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EDGE-STATETRANSPORT

4.4.1. Tunneling at a Point Contact

A point contact in an IQHE fluid at v = 1, is isomorphic to a barrier in a onedimensional interacting electron gas. As discussed in Section 4.2, the right- and left-moving edge states, which feed the point contact, are equivalent to the right and left Fermi points of a one-dimensionalnoninteracting electron gas. According to Landauer-Buttiker transport theory [111, the two-terminal conductance through the point contact is proportional to the transmission probability, 1 t 12, for an incident wave-the edge state-to propagate through the point contact: e2 G=-lt12

h

For an IQHE state at v = n, the transmission will involve the transmission probabilities of all n of the edge channels. For a FQHE fluid at v = l/m point contact is isomorphic to a barrier in a onedimensionalinteracting electron gas-a Luttinger liquid. How is the relation (78) for the point-contact conductance modified in this case? To answer this question we use the chiral boson description of FQHE edges described in Section 4.2. The effects of the point contact can be analyzed perturbatively in the two limits depicted schematically in Fig. 4.8: (1) a pinched-off channel with weak tunneling, and (2) an open channel with weak backscattering. These limits are discussed next, after which we piece these two limits together into a unified description. Weak Tunneling Limit. Consider a point contact that is pinched off almost completely. As shown in Fig. 4.8~1,this may be described by quantum Hall fluids

Figure 4.8. Quantum Hall point contact in the (a)weak tunneling limit and (b)weak backscattering limit. The shaded regions represent the quantum Hall fluid, with edge states depicted as lines with arrows. The dashed line represents a weak tunneling matrix element connecting the two edges.

TUNNELING AS A PROBE OF EDGE-STATE STRUCTURE

139

on the left- and right-hand sides, which are coupled by a weak perturbation that tunnels electrons between them. For v = l/m the low-energy physics will be described by an edge-state model of the form

s = s; + s::+ s,,,

(79)

where the left and right edges are described by 4n

+ a&,

dx, dr aX4,(iar

with a = L, R. The tunneling between these two edge states at the point contact can be expressed in terms of the edge creation and annihilation operators and has the form

s

+

S,,, = dr teim(4L-4R)C.C.

where 4, is evaluated at the point contact, x, = 0. Here t is the amplitude for the tunneling process [not to be confused with real time, which appears in (85) below]. The two-terminal conductance through the point contact can now be computed perturbatively for small tunneling amplitude t. In the presence of a voltage V across the junction, the tunneling rate to leading order can be obtained from Fermi's golden rule:

Here HI,, is the tunneling Hamiltonian corresponding to (81). The sum on n is over many-body states in which an electron has been transferred across the junction in the s, = f 1 direction. It is straightforward to reexpress this as 2nh dECGL(E)G;(E- eV) - G,'(E

- eV)G,'(E)]

(83)

where G,' and G: are (local)tunneling-in and tunneling-out densities of states for the edge modes, related by G c ( E )= G'( - E). These can be expressed as G:(E) = 2n

=s

n

1 (nleim@EIO) I26(E, - E ,

dt eiEt (eim4&)e

where

- imnC(0)

)

-E )

(84) (85)

+,, is evaluated at x = 0. The tunneling density of states is related to the

140

EDGE-STATE TRANSPORT

imaginary-time Green’s function

via analytic continuation, G’(t) = G(T-,it). Since the Euclidean action (80) is quadratic, G(z) may readily be computed, giving G(z) =

(F) + m

TI

zc

(87)

where z, is a short-time cutoff. Upon analytic continuation and Fourier transformation, the tunneling density of states is thereby obtained:

Form = 1, corresponding to the IQHE at v = 1, the tunnelingdensity of states is a constant at zero energy (the Fermi energy).From (83) this gives an ohmic I-V characteristic, with a tunneling conductance, Z/V, proportional to t2.This is the expected result, consistent with Landauer transport theory (78). However, for m > 1 the tunneling density of states uanishes at zero energy, giving rise to a nonohmic Z- V characteristic [17,28]:

The linear conductance is strictly zero! At finite temperatures the density of states is sampled at E x kT, and a nonzero (linear) conductance is expected. Generalizing Fermi’s golden rule to T # 0 gives the expected result for the conductance [17,28]: G oc t2TZrn-’

(90)

For v = 1/3 the predicted conductance vanishes with a large power of temperature, G cc T 4 . In striking contrast to the IQHE, the FQHE point-contact conductance, vanishes identically at zero temperature. What is the physical origin of this difference?At the edge of an IQHE fluid the electrons behave at low energies as if they were not interacting-the edge state is a Fermi liquid. Thus an electron can be added or removed from the edge without appreciably disturbing the other electrons. In contrast, the electrons at the edge of a Laughlin FQHE fluid are in a highly correlated state. Indeed, after removal of an electron from the edge, the remaining electrons are not left in the ground state of the edge with one less electron. Rather, the resulting state contains a “shakeup” spectrum of many low-energy edge excitations and is orthogonal to the ground state. It thus follows that the many-body matrix element for tunneling in (82) vanishes. This is a nice example of a general phenomenon known as the orthogonality catastrophe [41], which arises in such contexts as the Kondo problem and the x-ray edge problem.

TUNNELING AS A PROBE OF EDGE-STATE STRUCTURE

0.20

I

141

1

I

V’c=-O.853 & -0.899 V

T (mK) (0)

% 3 040%. Y

0

0.00 0.0

0.2

0.4

0.6

T4 (K4)

0.8

1.0

(b) Figure 4.9. Conductanceofa quantum Hall point contact as a functionof temperature for (a) v = 1 and (b) v = 1/3. (From Ref. [29].)

This orthogonality catastrophe is directly accessible to measurement. Figure 4.9 shows data for the conductance as a function of temperature through a point contact in an IQHE fluid at v = 1 and a FQHE fluid at v = 1/3. The difference in behavior is striking. For v = 1 the conductance approaches a constant at low temperatures, whereas for v = 1/3 the conductance continues to decrease upon cooling. Moreover, the low-temperature behavior for v = I/3 is consistent with the T4 dependence predicted in (90). In our view, these data provide the first compelling experimental evidence for the Luttinger liquid, a phase discussed theoretically over 30 years earlier. It is instructive to recast the result (90)in the language of the renormalization group (RG) [28]. Specifically,the vanishing conductance in the FQHE indicates

142

EDGE-STATE TRANSPORT

that the tunneling perturbation, t, is irrelevant. As shown in the appendix, the lowest-order RG flow equation can be obtained from the scaling dimension, A, of From the power-law behavior of te Green's the tunneling operator eimn(+L-4R). function in (87), we may deduce that the scaling dimension of ei"4LR is m/2. It then follows that A = m. The RG flow equation is then simply

dt dl

- = (1 - m)t

For FQHE states with m > 1, t is irrelevant as expected. The perturbative results (89) and (90) can be ontained by integrating this RG flow equationa until the cutoff is of order kT (or eV), giving tcff tm-' and G tZff.

-

N

Weak Backscattering Limit. Having established that the conductance of a FQHE point contact vanishes at T = 0 for weak tunneling, we now turn to the opposite limit, in which the point contact is almost completely open. Consider a bulk quantum Hall fluid in which the top and bottom edges are weakly coupled together, as depicted in Fig. 4.8b. As before, the low-energyphysics is at the edges and can be described by the action

s = so,+ s: + S,", Here SOT and S: describe the top and bottom edge modes, respectively, and are given by the chiral boson action (80). In contrast to the weak tunneling limit, the charge that tunnels between the two edges is now tunneling through the quantum Hall fluid. It is therefore possible that a single Laughlin quasiparticle, with fractional charge e/m, could tunnel between the edges. This process can be described by a term of the form

where u is the tunneling amplitude. In addition, higher-order processes involving tunneling of multiple quasiparticles (or electrons) are also possible. However, as shown below, such processes are less "relevant" at low energiesand temperatures. Consider now applying a voltage V between the source and drain electrodes. In the absence of any coupling, the top and bottom edges would then be in equilibrium at chemical potentials differing by the voltage I/. This results in the flow of a net edge current, I = (l/rn)(e'/h)V. Quasiparticle tunneling between the top and bottom edges backscatters charge and will tend to reduce this current. The reduction may be computed perturbatively in u, also using the golden rule. In fact, the backscattering current will be given by the golden rule expression (82), with two differences. First, the charge e in (82) must be replaced by the quasiparticle charge, e* = e/m. Second, the electron tunneling operator (81) must be replaced by the quasiparticle tunneling term in (93). This difference is crucial,

TUNNELING AS A PROBE OF EDGE-STATE STRUCTURE

143

repiacing the electron tunneling DOS (88) with the density of states for the addition of a quasiparticle. This follows from the quasiparticle Green's function

which has the same form as the electron Green's function (87) with rn -+ l/m. The backscattering current can thus be obtained from (89) by replacing m with l/m, giving at zero temperature c17.281

Similarly, at temperature T, the backscattering contribution to the (linear) conductance is given by

Once again, for the IQHE with rn = 1, a temperature-independent correction to the two-terminal conductance is obtained, as expected from Landauer transport theory. However, for the FQHE the perturbation theory (97) is divergent at low temperatures. The quasiparticle tunneling rate grows at low energies, in contrast to electron tunneling. Rather than an orthogonality catastrophe, quasiparticle tunneling causes quite the opposite-an overlap catastrophe. The divergent perturbation theory indicates that the quasiparticle tunneling operator is a relevant perturbation. This may be seen directly by noting from (95) that the scaling dimension of the quasiparticle creation operator ei4 is equal to 1/2rn. The leading-order RG tlow equation for the quasiparticle tunneling amplitude is then simply

dv = (1 dl

);

-

v

For the FQHE, u grows upon scaling to lower energies, flowing out of the perturbative regime where (98) is valid. The behavior in this limit is discussed in the next section. It is instructive to consider, in addition, backscattering processes involving multiple quasiparticles. The operator that tunnels n-quasiparticles is of the form u,e'n(or-4B). It is straightforward to show that the leading-order RG flow equation for u, is

dl

(99)

144

EDGE-STATE TRANSPORT

Notice that for v = 1/3 (rn = 3), the single-quasiparticle backscattering process is the only relevant perturbation. In contrast, for m = 5 7 , . . .more than one operator is relevant. In all cases, however, the single-quasiparticleterm is the most relevant. Crossover Between the Two Limits. The preceding results can now be pieced together to form a global picture of the behavior of a point contact in the FQHE. The perturbative results above describe the stability of two renormalization group fixed points. The “perfectly insulating” fixed point, with zero electron tunneling t =0, is stable, whereas the “perfectly conducting” fixed point, with zero quasiparticle tunneling v = 0, is unstable. Provided that these are the only two fixed points, it follows that the RG flows out of the conducting fixed point eventually make their way to the insulating fixed point. This is a very striking conclusion, since it implies that an arbitrarily weak quasiparticle backscattering amplitude v will cause the conductance to vanish completely at zero temperature. Of course, for u very small, very low temperatures would be necessary to see this. In this scenario, the conductance as a function of temperature will behave as shown in Fig. 4.10. At high temperatures, the system does not have “time”to flow out of the perturbative regime, so the conductance is given by G z ( l/m)(e2/h) v2T2/mn-2 . A s the temperature is lowered below a scale T* a ’), perturbation theory breaks down. Eventually, the system crosses over into a low-temperature regime in which the conductance vanishes as T2’”-’. The validity of this scenario rests on the assumption that no other fixed points intervene. This assumption has been verified both by quantum Monte Carlo simulations [27], and more recently by exact nonperturbative methods based on the thermodynamic Bethe ansatz [31]. As seen from (99),for v = 1/3 the single-quasiparticlebackscattering operator, with amplitude v = vlr is the only relevant perturbation about the conducting fixed point. All higher-order processes are irrelevant and hence less important at

T*

Figure 4.10. Schematic plot of the crossover from the weak backscattering limit to the weak tunneling limit as the temperature is lowered. At high temperatures,weak backscattering leads to a small correction to the quantized conductance. As the temperature is lowered below T* cc urn/(”’-’), the system crosses over to the insulating limit.

TUNNELING AS A PROBE OF EDGE-STATE STRUCTURE

145

low temperatures. Indeed, for small v and T, the conductance will depend on these parameters only in the combination u / T ' / ~In. this limit the conductance can be expressed in terms of a universal crossover scaling function [27,30,43]

where c is a nonuniversal dimensionful constant. The limiting behavior of the scaling function, G X ) , may be deduced from the perturbative limits. For smallargument X, the perturbation theory result (97) implies that

G ( X )= 1 - x 2

(101)

For large argument, corresponding to the limit T-0, the scaling function must match onto the low-temperature regime (90), which gives a T4 dependence for nu = 1/3. This implies that for X -,a,

G(x) a x - ~

(102)

The exact scaling function, computed by Fendley et al. [31], indeed reduces to (100) and (101) for small and large arguments, respectively. The universal crossover scaling function G is of particular interest because it determines the experimentally accessible lineshape for resonant tunneling, as we describe next.

4.4.2. Resonant Tunneling We now consider the phenomenon of resonant tunneling through a point contact in the FQHE. Resonancesin the conductance are expected when the energy of the incident edge mode coincides with a localized state in the vicinity of the point contact. As a point of reference, we first review resonant tunneling theory for a noninteracting electron gas, which should be applicable to a point contact in the IQHE. As the chemical potential p of the incident edge mode sweeps through the energy of the localized state, E ~ the , conductance will exhibit a peak described by

Here Tr.and r R are tunneling rates from the resonant (localized)state to the left and right leads and r = (r,+ rR)/2.The Fermi function is denoted f (E). At high temperatures, T > r, the resonance has an amplitude T / T and a width T. At low temperatures the lineshape is Lorentzian, with a temperature-independent width. Moreover, when the left and right barriers are identical, the on-resonance transmission at zero temperature is perfect, G = c2/h. How is this modified for tunneling through a FQHE point contact? Since arbitrarily weak quasiparticle backscattering causes the zero-temperature con-

146

EDGE-STATE TRANSPORT

ductance through the point contact to vanish, one might expect that resonances simply are not present at T = 0. As we now show, this is not the case. Rather, perfect resonances are possible, but in striking contrast to (103)for noninteracting electrons, they become infinitely sharp in the zero-temperature limit. As before, it is useful to consider perturbatively two limits, the weak tunneling limit and then the opposite limit of weak backscattering. Weak Tunneling Limit. Consider tunneling through a localized state separating two FQHE fluids. Focusing once again on the FQHE edge modes, we consider the model

s = s; + s:: + s,,, + s,,,

(104)

where S; and S;, given in (80), describe the edge modes in the two FQHE fluids, and r S,,, = dz E,dtd

J

describes the localized state with energy c0. The edge modes are coupled to the localized state via a tunneling term,

s

+

S,,, = dz t(ei4L+ eibR)d h.c.

with the tunneling amplitude, t, taken to be the same for left and right edge modes. Once again, the boson fields c$L,R are evaluated at x = 0. Consider first computing the rate, r, for an electron to tunnel from the localized state into the edge modes, perturbatively in t. From Fermi's golden rule, this will depend on the density of states for tunneling into the edge, which is given in (88). We thus find that

At finite temperatures and p x c0, we thus have

Once again at zero temperature there is an orthogonalitycatastrophe that prevents tunneling to FQHE edge states, rn > 1. However, it is only half as severe as that for tunneling between two edge modes (go), since only a single mode is being disturbed. For FQHE states (m2 3), (108) implies that r << T at low temperatures. In Refs. [44]and [45] it was argued that in this limit the conductance is well approximated by the form (103) with a temperature-dependent tunneling rate r in (108).This implies resonances with a width varying as T and a height T(T)/T,

TUNNELING AS A PROBE OF EDGE-STATE STRUCTURE

147

which gives an on-resonance conductance varying as

Thus, in the limit of small tunneling t, resonances are indeed suppressed at zero temperature. However, at finite temperatures, peak heights are predicted to vanish more slowly (as T for v = 1/3) than the tails ( T 4 for v = 1/3). What happens when the tunneling, t, to the localized state is increased?In Ref. [43] a renormalization group calculation was performed to higher order in t, which revealed that the exponent in (109) is renormalized when t is finite. Specifically, to O(t2),the following RG flow equations were obtained:

"=[

1 - i (1m - u ) ] t

dl

dU

The RG flow is shown in Fig. 4.1 1. Initially, u = 0, however, when t is finite it becomes positive. For small t , the RG flows to a fixed line with t = 0, and u = u*. The on-resonance conductance then decays with a modified exponent,

For v = 1/3 (m = 3) when t is larger than a critical value t,, for which u: = m - 2, the flows cross a Kosterlitz-Thouless like separatrix and scale toward large tunneling t (see Fig. 4.1 1). In this case it is extremely plausible that the flows take the system all the way to the perfectly conducting fixed point, as argued in Ref. [43]. This certainly happens for the IQHE m = 1, since (103) shows a perfect

t*

m-2 Figure 4.1 1. Renormalization group flow diagram describing the on-resonance transmission. For weak tunneling, small t, the system flows to the fixed line with t = 0. The peak conductancethen vanishes at low temperature as Tm-2-rr.. As the coupling t is increased, the system crosses a Kosterlitz-Thouless separatrix and flows to the D = 0 fixed point described in Section 4.4.2. 0

148

EDGE-STATETRANSPORT

on-resonance conductance when rL= rR.[In this case the renormalization of a is inconsequential,however, since from (1lo), t is always relevant.] For the FQHE the equality of the left and right tunneling amplitudes assumed in (106) is critical. Indeed, when they are unequal, the RG flows are modified, and the system crosses over to off-resonance behavior (90). For m 2 5 the RG flows reveal that the tunneling amplitude always scales to zero. Thus for v < 1/5 perfect resonances are not readily attainable. In summary, we conclude that for a localized state coupled symmetrically to two v = 1/3 FQHE fluids, robust resonances are indeed possible. Since the conductance on resonance is expected to be large, the perturbative analysis in t above cannot be used to calculate the resonance lineshape. The behavior near the resonance peak, however, can be obtained easily in the opposite limit of weak backscattering, as we now describe. Weak Backscattering: Theory of Perfect Resonance. Resonant tunneling is not normally studied for weak backscattering, since in this limit the transmission is large even off resonance, which tends to obscure the resonance. However, for FQHE states, the off-resonance conductance vanishes at zero temperature, leaving an unobscured resonance peak. Consider a point contact that has two nearby parallel tunneling paths for the backscattering of quasiparticles, as depicted in Fig. 4.12. These tunneling paths may be due to a random impurity potential or an intentionally created quantum dot. By varying the gate voltage and magnetic field, it should be possible to achieve a destructive interference,

which shuts off the interedge quasiparticle tunneling. This is the condition for a resonance. Provided that all higher-order tunneling processes are irrelevant, there will be perfect transmission on-resonance at T = 0, with a conductance G = ve2/h. Upon tuning away from the resonance, so that ueff # 0, theconductance will vanish at zero temperature, as shown in Section 4.4.1. Thus at zero temperature there will be an infinitely sharp perfect resonance [30]. How easy is it to achieve such a perfect resonance? The criterion is that the renormalized value of ueff and all other relevant u’s equal zero. In general, the net

Figure 4.12. Quantum Hall point contact with two parallel backscattering paths, u1 and ut. A perfect resonance occurs when the backscattering amplitudes interfere destructively.

TUNNELING AS A PROBE OF EDGE-STATE STRUCTURE

149

quasiparticle backscattering amplitude ueff is a complex number, so that the resonance condition requires the simultaneous tuning of two parameters. If barriers (or quasiparticle tunneling paths) are symmetric, however, ueff may be chosen real, so that only a single parameter, such as a gate voltage, need be tuned. For v = 1/3, all higher-order backscattering processes are indeed irrelevant, so that tuning ueff = 0 is sufficient to achieve resonance. For v = 1/5, however, the parameter u2 in (99)is also relevant, so a perfect resonance requires the tuning of four parameters. The situation gets even worse for rn > 5. For this reason we focus on resonances for v = 1/3,which should be the easiest to find. Consider tuning through such a perfect resonance by varying a parameter, such as the gate voltage. It is convenient to denote by 6 the distance from the peak position in the control parameter. Close enough to the resonance, one has ueff cc 6 . For very small 6 the RG flows will thus pass very near the perfectly conducting fixed point, since all of the other irrelevant operators will scale to zero before ueff has time to grow large. Eventually, ueff does grow large and the flows cross over to the insulating fixed point, as depicted in Fig. 4.13.Temperature serves as a cutoff to the RG flows, as usual. This reasoning reveals that for both 6 and temperature small, the conductance will depend only on the universal cross over trajectory, which joins the two fixed points. The uniqueness of the RG trajectory implies that the conductance will be described by a universal cross over scaling function. Thus, for small T and 6, the resonance lineshape is given by a universal scaling function,

where G is the scaling function introduced in (100). The scaling form (114)shows that the resonance width scales as T2I3,at low temperatures. Moreover, rescaled data from different temperatures should col-

Figure 4.13. Schematic renormalizationgroup flow diagram showing the universal trajectory connecting the perfectly conducting v = 0 fixed point to the insulating t = 0 fixed point. On resonance, the system flows into the unstable v = 0 fixed point. Slightly off-resonance, the system flows past the v = 0 fixed point and flows along the universal trajectory to t = 0.

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EDGE-STATE TRANSPORT

Conductance

1

0.1

Q

0.01

-

exactcurve MonteCarlo experimental data x

0.001

0.01

0.1

1

10

X=.74313(T-BK)"(Z3) Figure 4.14. Log-log scaling plot of the lineshape of resonances at different temperatures from Ref. [29]. The x axis is rescaled by T2I3.The crosses represent experimental data at temperatures between 40 and 140 mK. The squares are the results of the Monte Carlo simulation, and the solid line is the exact solution from Ref. [31].

lapse onto the same universal curve. As seen from (102),the resonance lineshapeis predicted to be non-lorentzian,with a tail falling off as Figure 4.14shows a scaling plot of one of the resonances for v = 1/3 in Fig. 4.7. The widths of the resonances at several different temperatures have been rescaled by T2I3, as suggested by (114).Since the peak heights were also weakly temperature dependent [and roughly one-third of the quantized value (1/3)e2/h],the amplitudes have also been normalized to have unit height at the peak. The temperature scaling of the peak widths is indeed very well fit by T2/3. Also shown in Fig. 4.14are quantum Monte Carlo data and an exact computation from Bethe ansatz for the universal scaling function in (1 14).The agreement is striking.

TUNNELING AS A PROBE OF EDGE-STATE STRUCTURE

151

Although the experimental lineshape does drop somewhat faster in the fails, the shape is distinctly non-Lorentzian, with a tail decaying with a power close to that predicted by theory. It should be emphasized that the experimental data do not represent perfect resonance, since the peak amplitude is dropping (slowly) upon cooling rather than approaching the quantized value, ( 1/3)ez/h. By varying an additional parameter in addition to the gate voltage (such as the magnetic field), though, it should be possible to find a perfect resonance for v = 1/3.

4.4.3. Generalizationto Hierarchical States The theory of tunneling through a point contact for FQHE fluids at filling v = l/m described above can be generalized to hierarchical FQHE states [25]. The additional complication is that the hierarchical states have composite edges, with multiple branches, as described in detail in Section 4.2. Moreover, for those states with edge branches moving in both directions,such as v = 2/3, the conductance is nonuniversalunless edge impurity scattering is present. Similarly,it can be shown that without impurity scattering, the conductance through a point contact in a v = 213 fluid varies as

with a nonuniversal exponent a. However, impurity scattering drives an edge phase transition, as shown in Section 4.3 and the system flows to a fixed point that exhibits an appropriately quantized conductance. At this fixed point, the exponent a is likewise universal. Thus, measuring a gives critical information about the low-energy properties of the composite edge. Generally, the exponent a can be determined from the density of states to tunnel an electron into the composite edge. Specifically, one must consider all tunneling operators of the form (45),which create an edge excitation with charge e, as determined from (34). For filling factors v = n/(np + l), with integer n and even integer p, the edge fixed point with impurity scattering is known (see Section 4.3.2), so that the tunneling DOS can readily be computed by generalizing (86). For v = 2/3 one finds that

This exponent depends on the presence of the neutral mode at the v = 2/3 edge. The electron edge creation operator is a combination of the charge and neutral modes, so that tunneling an electron into the edge also shakes up the neutral mode. Indeed, the second contribution in (116) is due to shakeup of the neutral mode. The charge mode gives the first term, which has the same form as (91) for filling v = l/m. Thus an observation of a = 2 for a v = 2/3 point contact would constitute a measurement of the neutral mode.

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EDGE-STATE TRANSPORT

TABLE 4.1. Tunneling Exponent a for the Temperature Dependent Conductance Through a Point Contact Separating Two Quantum Hall Fluids at Filling Factor u

More generally, for v = n/np + 1 it can be shown that

I

for p a 0

Notice that a depends on the sign of p, which determines the direction of propagation of the neutral modes relative to the charge mode. For the p = - 2 sequence, the predicted exponents are displayed in Table 4.1. The exponents approach a = 4 as v approaches 112. For filling v = 4/5 the conductance at T = 0 was argued in Section 4.3.2 to be nonuniversal even in the presence of edge impurity scattering, although at T # 0 a universal quantized conductance is restored for samples larger than the edge equilibration length (77).On contrast, the exponent a is predicted to be nonuniuersal even for samples much longer than the edge equilibration length. For a point contact that is only very weakly pinched off, the conductance can be computed perturbatively for small backscattering. The backscattering will reduce the conductance from its quantized value, as in the single-channel case (97). Generally, for v = n/(np + l), we find a temperature-dependent suppression given by

where u is the amplitude of the most relevant backscattering operator of the general form (45). If there were no other relevant backscattering operators, as for v = 1/3, this would imply that resonances narrow with temperature as T'' -Iv1)the generalization of (114).However, for hierarchical states there will generically be several relevant backscattering processes, so that resonances will not be very robust, and tend to vanish at very low temperatures. Nevertheless, one expects there should be a range of temperature over which the resonance width narrows 1 as ~ ( -1~1)4.4.4. Shot Noise

In addition to measuring the conductance of a point contact, it is also possible to measure time-dependent fluctuations, or noise, in the transmitted current. Most

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153

interesting is the nonequilibrium noise present at finite bias voltage rather than the equilibrium Nyquist noise. At frequencies comparable to the bias voltage, there may be Josephson oscillations, as discussed by Chamon et al. [46]. At low frequencies one expects shot noise, arising from the discreteness of the electron. As we describe briefly, shot noise might enable a rather direct measurement of the fractional charge of the Laughlin quasiparticle [47]. Consider first a very high resistance point contact in a QHE fluid. In the presence of a bias voltage, electrons will occasionally tunnel. This will give rise to temporal fluctuations in the current. At low enough currents, successive tunneling events will be uncorrelated. Assuming Poisson statistics for the tunneling events, the low-frequency current noise will be given by the classic expression

Note that the amplitude of the low-frequency noise depends on the absolute magnitude of the electric charge, e. With increasing current, correlations between tunneling events are expected, and the expression (119) must break down. Lesovik [47-501 has analyzed shot noise for a one-dimensional noninteracting electron gas, which is relevant to a v = 1 point contact. For a single barrier with transmission probability ltI2, he finds that

(lm412)a-o

e2

=$tI2(1

- bI2)eY

( 120)

where Y is the bias voltage. In the limit I t I 2 -+ 0, this reduces to the classic expression (119). But when (t('+ 1, the noise is greatly suppressed. Indeed, in the absence of any backscattering (Ill2 = l), the noise vanishes altogether. For 1 - Iti2 small, (120) may be rewritten as

The noise arises from the discrete, uncorrelated backscattering of electrons at the point contact. How are these results modified for a point contact in the FQHE? In Ref. [47] we developed a detailed theory of nonequilibrium shot noise at FQHE point contacts for filling v = l/m. The results may be understood very simply. In the weak tunneling limit the transport is dominated by the discrete tunneling of electrons through the point contact. As for noninteracting electrons, the tunneling events satisfy Poisson statistics and the noise is given by (119). In the opposite limit of weak backscattering, however, there are qualitative differences, because the dominant backscattering processes in the FQHE are fractionally charged quasiparticles. When the conductance is just slightly less than ve2/h(attained by adjusting the gate on the point contact), these backscattering processes are

IS4

EDGE-STATE TRANSPORT

infrequent and should be uncorrelated. Indeed, in this limit we find a lowfrequency noise given by

with e* the quasiparticle charge: e* = e/m. This form is identical to the noninteracting result (121), except with the electron charge replaced by the quasiparticle charge. A measurement of the current noise and mean current, (I), in this regime should enable direct measure of the quasiparticle charge. 4.5. SUMMARY

In this chapter we have presented a theory ofedge-state transport in the fractional quantum Hall effect based on the chiral Luttinger liquid model. For the Laughlin states at filling v = l/m with odd na, this model provides a very simple description of the low-energy edge excitations, which consist of a single propagating mode corresponding to charge-densityfluctuations.This provides a simple framework for understanding the quantization of the Hall conductance in an edge-state picture, analogous to the Landauer-Biittiker theory for the integer quantum Hall effect. In addition, this theory makes specific, experimentally testable predictions for the behavior of tunneling and resonant tunneling through a point contact in a Hall fluid. The behavior for a FQHE fluid is predicted to be qualitatively different than that in the integer effect. Specifically,for v = 1/3 the conductance through a point contact is predicted to vanish at low temperatures at T4,in contrast to the temperature-independent result expected for v = 1. Moreover, resonances in the tunneling between two v = 1/3 states are predicted to have a temperature-dependent linewidth, which vanishes at T2I3at low temperatures. The shape of the resonances are universal and described by a scaling function that has been computed exactly. These predictions agree qualitatively-if not quantitatively-with recent measurements of edge-state transport through a point contact at v = 1/3 [29]. It is worth emphasizing that a point contact in a QHE fluid provides a simple and experimentallyaccessibleexample of a broad class of quantum impurity problems [Sl], which consist of an “impurity” that is coupled to an extensive set of low-energy degrees of freedom. The classic example is in the Kondo effect [52], where an impurity spin is coupled to the particle-hole excitations of a metallic host. In the quantum Hall effect, the point contact is an impurity, coupled to the low-energy edge excitations. The powerful methods of boundary conformal field theory [52] and boundary integrability are ideal for analyzing this class of problems. In contrast to the Laughlin sequence, v = l/m, the edge excitations of hierarchical quantum Hall states cannot be described by a single mode. Rather, multiple Luttinger liquid edge modes are expected, which in general can propa-

SUMMARY

155

gate in different directions. This can lead to a breakdown of conductance quantization, due to a lack ofequilibration between opposite-moving modes. It is thus essential to incorporate edge impurity scattering, which has a profound effect on the low-energy edge-state structure. Specifically, for a broad class of hierarchical quantum Hall states, v = n/(np + 1)with integer n and even integer p, impurity scattering is perturbatively relevant, and drives the edge to a new disorder-dominated low-energy fixed point, where quantization is restored. At this fixed point the charge is carried in a single mode. The remaining n - 1neutral modes propagate at a different velocity and are related by an exact SU(n) symmetry. In the specific case v = 2/3, there is a single neutral mode, which propagates in the direction opposite to the charge mode. Despite carrying no charge, the upstream propagating neutral mode can be detected in at least two ways. The first, which is less direct, involves tunneling through a constricted point contact in a v = 2/3 Hall fluid. For filling v = l/m, where there is only a single edge mode, the point-contact conductance is predicted . v = 2/3 one might therefore to vanish with temperature as G(t ) T 2 / v - 2 For expect a power law, G(T) T. However, the presence of the neutral mode at the v = 2/3 edge increases this power by one, giving the prediction G(T) T 2 . Timedomain transport experiments at filling v = 2/3, might enable a much more direct measurement of the neutral mode. Imagine two leads coupled via tunnel junctions to the opposite sides of a large Hall droplet at filling v = 2/3. A short current pulse incident in one lead, upon tunneling into the droplet edge, would excite both the charge and neutral edge modes. These excitations,after propagating in opposite directions and with different speeds along the droplet edge, would, upon arrival at the far tunnel junction, excite two current pulses into the outgoing lead. By tailoring the placement of the leads, a measurement of the direction of propagation and decay length of the neutral mode should also be possible. The disorder-dominated fixed point that describes the edge structure at filling v = n/(np + 1)has higher symmetry-an exact U(1) x SU(n)symmetry-than the underlying Hamiltonian. This is reminiscent of Fermi-liquid theory, where the attractive zero-temperature fixed point also has higher symmetry than the underlying Hamiltonian. In addition to conserved electric charge, the (T = 0) Fermiliquid fixed point has an infinity of conserved charges [and hence an infinity of U(1)symmetries] associated with each point on the Fermi surface. This is the symmetry responsiblefor the quasiparticle excitations. At finite temperatures this symmetry is broken, leading to a finite scattering lifetime for the quasiparticles, proportional to T - '. Since total electriccharge is always conserved,propagating zero sound in a Fermi liquid does not decay even at T # 0. At the FQHE edge, it is the SU(n)symmetry that is responsible for the existence of the neutral modes. But as in Fermi liquid theory, this symmetry is exact only at T = 0, so that the neutral edge excitations are expected to decay at finite temperatures. It is amusing that the scattering rate predicted for the neutral modes vanishes with the same power of temperature- T2-as for quasiparticles in a Fermi liquid. In conclusion, edge states of fractional Hall fluids provide an ideal arena for the study of correlations in low-dimensionalquantum transport. The remarkable

-

-

-

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EDGE-STATE TRANSPORT

richness of the edge-state structure in both the Laughlin states and the hierarchical quantum Hall states is directly accessible to experimentalstudy. We hope that this overview will stimulate further experimental and theoretical work in this exciting area. ACKNOWLEDGMENTS We gratefully acknowledge our collaboratorsin this work Steve Girvin, Kyungsun Moon, Joe Polchinski, and Hangmo Yi. We also thank A. H. MacDonald, N. Read, M. Stone, R. A. Webb, X. G. Wen, and A. Zee for fruitful discussions. M.P.A.F. has been supported by the National Science Foundation under Grants PHY89-04035 and DMR-9400142. APPENDIX RENORMALIZATION GROUP ANALYSIS The renormalization group (RG) provides a powerful framework for understanding the global behavior of the models in this chapter and for piecing together results obtained in perturbative limits. Here we outline the procedure for deriving the lowest-order RG flow equations which are referred to in the text. Consider first a point contact in a v = l/mfluid in the limit of weak backscattering, discussed in Section 4.4.1. The action in (92)-(93) may be written S = So

+u

s

dz ei4(*)+ C.C.

(All

where S,=SO,+S: is the quadratic edge action given in (80) and &)= &(x = 0,T) - 4B(x = 0, z). The perturbation u acts at a single space point, x = 0. The RG of two steps:

1. Integrate out degrees of freedom a)that lie in a momentum shell A/b < k c A. First split the field into “slow” and “fast” modes, below and inside the momentum shell, respectively: 4 = 4s+ 4J.To lowest order in u one must average over the fast modes:

Here A is the scaling dimension of the operator ei@. The scaling dimension is most easily deduced from the two-point correlation function

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157

2. The RG transformation is completed by rescaling space and time, r’ = r/b, x’ = x/b. The resulting action is then equivalent to the original one, with u replaced by u’ = ubl-A. Upon setting b = e’, one thereby obtains the leading-order differential RG flow equation, du -=(l dl

-A)u

Consider now a spatially random perturbation, as in Section 4.3, S =S,

+

s

dx dr ((x)ei4(x*r) + C.C.

where ( ( x )is a Gaussian random variable satisfying [ ~ * ( X ) ( ( X ’ = ) ] W6(x ~ ~ ~ - x’) (the brackets denote an ensemble average over the quenched disorder). Our analysis follows closely that of Giamarchi and Schultz [53], who studied the effects of randomness on a (nonchiral) Luttinger liquid. To lowest order in W, ensemble-averaged quantities may be computed from the ensemble average of the partition function. (At higher order the introduction of replicas would be useful.) Performing the average over t gives the effective action,

The leading-order RG flow equation for Wmay now be derived by applying steps (1) and (2) to (A7). This gives dW -=(3-2A)W dl

where again A is the scaling dimension of ei4. The “2” arises from step (1)because eibappears twice in (A7).The “3” arises from rescaling, step (2), because there are three space-time integrals.

REFERENCES 1. See, for example, R. Prange and S.M. Girvin, eds., The Quantum Hall Effect, 2nd ed., Springer-Verlag, New York, 1990. 2. K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45,494 (1980). 3. D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982). 4. R. B. Laughlin, Phys. Reu. 23,5632 (1981). 5. B. I. Halperin, Phys. Rev. 25,2185 (1982). 6. R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).

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7. S. M. Girvin and A. H. MacDonald, Phys. Rev. Lett. 58,1252 (1987) 8. B. I. Halperin, Helv. Phys. Acta 56,75 (1983). 9. S. C. Zhang, T. H. Hansson, and S. Kivelson, Phys. Rev. Lett. 62,82 1989). 10. N . Read, Phys. Rev. Lett. 62,86 (1989). 11. M. Buttiker, Phys. Rev. B 38,9375 (1988). 12. R. Landaner, Phil. Mag. 21,863 (1970). 13. M. Buttiker, Phys. Rev. Lett. 57, 1761(1986). 14. A. M. Chang, Solid State Commun.74,871 (1990). 15. C. W. J. Beenakker, Phys. Rev. Lett. 64,216 (1990). 16. A. H. MacDonald, Phys. Rev. Lett. 64,222 (1990). 17. X. G. Wen, Phys. Rev. B 43,11025 (1991); Phys. Rev. Lett. 64,2206 (1990). 18. J. M. Luttinger, J. Math. Phys. 15,609 (1963). 19. A. Luther and L. J. Peschel, Phys. Rev. B 9,291 1(1974); Phys. Rev. Lett. 32,992 (1974); A. Luther and V. J. Emery, Phys. Rev. Lett. 33,589 (1974). 20. J. Solyom, Adv. Phys. 28, 201 (1970); V. J. Emergy in Highly Conducting OneDimensional Solids, edited by J . T. Devreese, Plenum Press, New York 1979. 21. F. D. M. Haldane, J . Phys. C 14,2585 (1981); Phys. Rev. Lett. 47, 1840 (1981). 22. M. D. Johnson and A. H. MacDonald, Phys. Rev. Lett. 67,2060 (1991). 23. R. C. Ashoori, H. Stormer, L.PfeiRer, K. Baldwin, and K. West. Phys. Rev. B 45,3894 ( 1992). 24. C. L. Kane, M. P. A. Fisher, and J. Polchinski, Phys. Rev. Lett. 72,4129 (1994). 25. C. L. Kane and M. P. A. Fisher, Phys. Rev. B 51,13449 (1995). 26. X.G. Wen, Phys. Rev. B 44,5708 (1991). 27. K. Moon et al., Phys. Rev. Lett. 71,4381 (1993). 28. C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. 68,1220 (1992). 29. F. Milliken, C. Umbach, and R. Webb, Solid State Commun. M, 309 (1996). 30. C. L. Kane and M. P. A. Fisher, Phys. Rev. B 46,7268 (1992). 31. P. Fendley, A. W. W. Ludwig, and H. Saleur, Phys. Rev. Lett. 74,3005 (1995). 32. For a nice discussion of abelian bosonization, see Chapter 4 in E. Fradkin, Field Theories of Condensed Matter Systems, Addison-Wesley, Reading, Mass., 1991. 33. R.Shankar, in Current Topics in Condensed Matter and Particle Physics, edited by J . Pati, Q. Shafi, and Y. Lu, World Scientific, Singapore, 1993. 34. A. Ludwig, in Low-Dimensional Quantum Field Theories for Condensed Matter Physicists, edited by S. Lundqvist, G. Morandi, and Y. Lu, World scientific, Singapore 1995. 35. D. H. Lee and M. P. A. Fisher, fnt. J. Mod. Phys. 5,2675 (1991). 36. X. G. Wen and Z. Zee, Phys. Rev. B 46,2290 (1992). 37. F. D. M. Haldane, Phys. Rev. Lett. 51,605 (1983); B. I. Halperin, Phys. Rev. Lett. 52, 1583 (1984). 38. N. Read, Phys. Rev. Lett. 65, 1502 (1990). 39. C. L. Kane and M. P. A. Fisher, Phys. Rev. B 52, 17393 (1995). 40. X. G. Wen, Phys. Rev. B 50,5420 (1994).

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159

41. J. K. Jain, Phys. Rev. Lett. 63,199 (1989). 42. P.W. Anderson, Phys. Rev. 164,352(1967). 43. C. L. Kane and M. P. A. Fisher, Phys. Rev. B 46,15233(1992). 44. A. Furusaki and N. Nagaosa, Phys. Rev. B 47,3827(1993). 45. C. de Chamon and X. G. Wen, Phys. Rev. Lett. 70,2605(1993). 46. C.de C. Chamon, D. E. Freed, and X. G. Wen, Phys. Rev. B 51,2363 (1995). 47. C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. 72,724(1994). 48. G. B. Lesovik, Pis’ma Zh. Eksp. Teor. Fix. 49,513(1989)[JETP Lett. 49,592(1989)l. 49. M.Biittiker, Phys. Rev. Lett. 65,2901 (1990);Phys. Rev. B 46, 12485 (1992). 50. R. Landauer, Physica (Amsterdam)38D, 226 (1989). 51. I. Affleck and A. W. W. Ludwig, Nucl. Phys. B 360,641 (1991);Phys. Rev. B 48,7297 (1993). 52. J. Kondo, Prog. Theor. Phys. 32,37 (1964). 53. T.Giamarchi and H. J. Schultz, Phys. Rev. B 37,325(1988).

PERSPECTIVES IN QUANTUM HALL EFFECTS: Novel Quantum Liquids in Low-DimensionalSemiconductorStructures Edited by Sankar Das Sarma, Aron Pinczuk Copyright0 2004 WILEY-VCH Verlag GmbH & Co. KGaA

3

Multicomponent Quantum Hall Systems: The Sum of Their Parts and More S. M. GIRVIN and A. H. MAcDONALD Department of Physics, Indiana University, Bloomington, Indiana

5.1. INTRODUCTION

The fractional quantum Hall effect [l-41 is a remarkable example of strong correlations in a two-dimensional electron gas (2DEG).In a zero magnetic field, a dimensionlessmeasure of the strength of the Coulomb interaction for a system with dielectric constant E, Fermi energy E ~ and , Fermi wavevector k , is

which is small in the limit of high density. In this limit one can frequently treat the effects of the Coulomb interaction perturbatively. Physically, this can be visualized as being due to the electrons moving rapidly past each other at the Fermi velocity and thus not scattering as strongly as they would at lower densities. A strong magnetic field changes this situation completely. Semiclassically, the rapid motion of the electrons is converted into circular cyclotron orbits. The particles now scatter strongly from each other and, in fact, semiclassicallydo not move except under the E x B drift induced by their mutual interactions. A full quantum treatment of the motion shows that the kinetic energy is quenched and now occurs only in discrete values (n + 1/2)ho,, where n is the Landau level index. Since the kinetic energy within a given Landau level is completely degenerate,the Coulomb interaction inevitably induces highly nonperturbative effects. (The Landau level degeneracy is N, = BA/cP,, where a, = hc/e is the magnetic flux quantum and A is the area of the system.) The essential feature of the fractional quantum Hall effect is a condensation of the electrons into special highly correlated states [5] that minimize the Coulomb Perspectives in Quantum Hall Effects, Edited by Sankar Das Sarma and Aron Pinnuk. ISBN 0-471-11216-X 0 1997 John Wiley & Sons, Inc.

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MULTICOMPONENT QUANTUM HALL SYSTEMS

energy by having the electrons avoid each other as much as possible. These states are characterized by an unusual topological order [6-91 that costs a finite amount of energy to break. Hence the fluid is incompressible and has an excitation gap for both its charged [5] and neutral [lo, 111 excitations. Anderson has characterized this state as a Mott insulator induced by the magnetic field c121. The essence of this phenomenon is captured in a remarkable class of wavefunctions first constructed by R. B. Laughlin:

Here, since we are in two dimensions, we are using the dimensionless complex number Z = ( x + i y ) / l to represent the position vector ( x , y ) in units of the This wavefunctiondescribes spinless fermions in the magnetic length 1 = lowest Landau level (in the symmetric gauge). To satisfy the analyticity requirement placed on the wavefunction by the constraint of being in the lowest Landau level [13], the parameter m must be an integer. To satisfy the antisymmetry requirement for fermions, m must be odd. Laughlin's plasma analogy [5] shows that the parameter m fixes the Landau level filling factor to be v = N/N4= l/m. Experiment [14] has indeed observed gapped quantum Hall states at filling factors v = 1,1/3, and 1/5. It is clear that Laughlin's wavefunction builds in good correlations because it vanishes as Iri - Ti("' when any two particles i and j approach each other. Thus there is only a small amplitude for the particles to be near each other, and the Coulomb energy is lowered. Note that no pair of particles ever has relative angular momentum less than m. Hence the Laughlin function is a zero-energy exact eigenfunction for Hamiltonians with the appropriate finite number of nonzero Haldane pseudopotentials [151, V,,:

m.

where P,,[i,j] is the lowest Landau level (LLL) projection operator for the relative angular momentum state n of particles i and j. In the lowest Landau level, relative angular momentum is proportional to the square of the separation [131. Hence the Laughlin wavefunction is very nearly an exact ground state for any sufficiently short-range repulsive interaction. In addition to the primary filling fractions v = l/m, numerous other fractions have been observed, all of which (for single-component systems) have odd denominators (again because of the Pauli principle). These phases have been explained in terms of a hierarchical picture using bosonic [151, anyonic [16], and fennionic [5,17] representations. More recently, Jain [lS, 191 has discussed an appealing composite fermion picture. Read has argued that all of these representations are mathematically equivalent [20] and contain the same physics. Which

INTRODUCTION

163

representation is most convenient depends on circumstances.Jain’s approach has inspired several new and important experiments [191. Our purpose here is to consider the nature of the various phases that can occur in multicomponent systems. There are several physically differnt realizations of systems with extra degrees of freedom that require a multicomponent representation, and important early work on this problem was done by Halperin [21,22]. The first and simplest example is that of ordinary electron spin. In free space for electrons with g factor 2 (i.e,, neglecting QED corrections) the Zeeman splitting gpJ3 is precisely equal to the Landau level splitting ho,. If this were true in quantum Hall samples, even a noninteracting system with filling factor v = 1 would have a large excitation gap for flipping spins, and the ground state would be fully polarized at low temperatures. In this case spin excitations are frozen out and we can treat the electrons as being effectivelyspinless. However in the solidstate environment of the 2DEG, two factors conspire to make the effective g factor much smaller in many semiconductors, particularly in GaAs samples, in which almost all fractional quantum Hall studies have been done. The first is that the small effective mass (m* 0.068 in GaAs) increases the cyclotron energy by a factor of approximately 15. Second, spin-orbit coupling causes the spins to tumble and reduces their coupling to the external magnetic field by roughly a factor of 4. Thus the ratio of the Zeeman splitting to cyclotron splitting is reduced from unity to about 0.02in GaAs. The spin-orbit contribution is pressure dependent and may be reduced further as a result of size quantization effects in narrow quantum wells [23]. For a small enough g factor and weak enough magnetic fields, spin fluctuations become an important dynamical degree of freedom and we must use a twocomponent wavefunction to describe.the system. While the Coulomb force is spin independent, we shall see below that exchange effects can lead to spontaneous ferromagnetism as well as to gapped “local singlet” spin liquids, which are, loosely speaking, itinerant antiferromagnets.(Unlike antiferromagnets, however, they do not break translation symmetry nor do they have gapless Goldstone modes.) Ferromagnetism turns out to be important at filling factor v = 1, where Coulomb exchange effects are much more important than originally realized [24-271. A different class of quantum Hall states with gaps has been observed experimentally [28-311 at filling fractions with even denominators such as v = 5/2. These have been theorized [9,32-351 to be either manifestations of spin liquid states or special p-wave pairing spin-polarized states known as Pfaffian states. One useful signature of spin effects in 2DEGs is that they are sensitive to tilts of the magnetic field. To a first approximation, the orbital degrees of freedom are sensitive only to the perpendicular component of the field because of the 2D confinement, while the Zeeman splitting is proportional to the total magnetic field. (However, the coupling of orbital degrees of freedom to the parallel component of the magnetic field is not completely negligible in typical quantum wells [36], and this can complicate the situation considerably.)

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MULTICOMPONENT QUANTUM HALL SYSTEMS

A second example of a multicomponent system is found in silicon, where the conduction-band minimum occurs not at the r point (the zone center) but rather at six symmetry equivalent points lying near the zone boundary along the principal cubic directions. Thus electrons doped into the conduction band of Si must be described by a six-component wavefunction (if we ignore spin). The presence of the oxide barrier in a Si MOSFET device and the enormous electric field perpendicular to it (which is used to confine the electrons into the inversion layer) breaks the cubic symmetry and lowers the energy of two of the valleys. For typical electron densities, only these two valleys are occupied, thus yielding a system that is effectively two-component and has SU(2)symmetryjust like that of a spin-1/2 system [37]. A third example, which is discussed extensively here and by Eisenstein [38], occurs in double-quantum-wellstructures [39]. With modern MBE techniquesit is possible to grow GaAs heterostructures containing two 2DEGs separated by a distance comparable to the spacing between electrons within each layer. Remarkably, it is also possible to make separate electrical contact to each layer. A closely related system is a single wide quantum well in which the two lowest electric subbands are nearly degenerate [40]. We will make the (only approximately correct) assumption throughout our discussion that the low-energy physics of a single wide well can be mapped onto that of a double well with appropriately chosen parameters. In all two-component systems it is useful to define a pseudospin representation in which spin-up and spin-down refer to the two possible values of the layer index (or subband index) for each electron [41]. We will frequently frame our discussion in a spin or pseudospin language; the reader should be aware that the discussion applies equally well to double-layer systems. When the distinction between the U(1) and SU(2) symmetries of the interaction term in the Hamiltonian is important, we will say so. Because the intra- and interlayer Coulomb matrix elements are different, double-well systems do not have full SU(2) symmetry but only U(1) symmetry associated with the conservation of the charge difference between the two layers (assumingthere is no interlayer tunneling)[41-471. These systemsexhibit gapped quantum Hall states with both even and odd denominators. In addition, there are gapless XY ordered phases. These phases are destroyed above some critical temperature by a Kosterlitz-Thouless phase transition. This phase transition (not yet observed experimentally) is the first example of a finite-temperature phase transition in a quantum Hall system. This chapter is organized as follows. In Section 5.2 we introduce the physics of incompressible states in multicomponent cases by presenting the appropriate generalizations of Laughlin’s many-particle wavefunctions. Section 5.3 directs the reader briefly to references on the Chern-Simons effective field-theoretic approach to these problems. In Section 5.4 we discuss fractional charges in multicomponent systems and Section 5.5 covers collective modes from the point of view of the single-mode approximation. In both these sections we will see that some of the multicomponent wavefunctions have correlation functions whose qualitative properties deviate from the norm. These differences lead to changes in

MULTICOMPONENTWAVEFUNCTIONS

165

the nature of the fractionally charged excitations and in the collectivemodes. The qualitative difference is associated with a broken symmetry that occurs in some cases, as we discuss in Section 5.6. In our view, it is in the properties of these broken-symmetry states that some of the most interesting new physics in multicomponent quantum Hall states is revealed. In Section 5.7 we discuss the field-theoretic gradient expansion approach to the broken-symmetry case. In double-layer systems, the broken-symmetry state has spontaneous phase coherence between the electrons in different layers even when these layers are isolated apart from interlayer Coulomb interactions. Section 5.9 deals with the external symmetry breaking introduced by interlayer tunneling. Spontaneous coherence in double-layer systems leads to remarkable effectsupon tilting the magnetic field away from the normal in a double-layer system. Some of these effects are discussed in Section 5.10. Finally in Section 5.11 we present a summary of the central ideas.

5.2. MULTICOMPONENT WAVEFUNCTIONS In this section we discuss wavefunctions for spin or pseudospin-1/2 particles that can be written in the symmetric gauge in the form

where A is the antisymmetrization operator, [i] = N,+ i, and ak and P k are the spinors for the kth electron aligned, respectively, parallel and antiparallel to the Zeeman field (which we take to be in the 4 direction). Thus @[Z] is the orbital wavefunction for the spin configuration in which the first N, electrons have spin-up and the remaining electrons have spin-down [48]. The two-component orbital wavefunctions originally proposed by Halperin [21,22] have a form analogous to that of the Laughlin functions:

where 2, = (xk + iyk)/lis the two-dimensionallayer coordinate of the kth electron expressed as a complex number, and rn and m' are odd integers. @,,,.,,[Z] excludes relative angular momenta less than rn between up-spins, less than m' between down-spins, and less than n between an up-spin and a down-spin. Hence the same arguments presented above for the Laughlin wavefunction can be used to motivate the idea that the Halperin wavefunctins are good approximations to the ground state for short-range repulsive potentials. In the single-component case, Laughlin wavefunctions are expected to approximate the ground state accurately for rn = 3 and rn = 5 (v = 1/3 and v = 1/5);

166

MULTICOMPONENT QUANTUM HALL SYSTEMS

at smaller filling factors the Wigner crystal state takes over. In the two-component case, the realization of a Laughlin state at a particular filling factor may be dependent on other parameters of the system, such as the layer separation in double-layer systems. One slight complication is presented by the SU(2)symmetric case [l5, 491. Here the total spin S, commutes with the Hamiltonian and one may, without loss of generality, require that the energy eigenstatessimultaneously be eigenstatesof S;. This requirement is not satisfied in general by the Halperin wavefunctions. It is easy to pick out two special cases (among others) that do work, however. The choice m = m’= n yields a fully antisymmetric spatial wave functin and hence implies a fully symmetricspin function. This immediately tells us that we have a fully aligned ferromagnetic state with total spin quantum number S = N/2. On the other hand, the choice {m,m‘,n} = (1, l,O} corresponds to a simple Slater determinant with both spin states of the lowest Landau level fully occupied, giving v = 2. Hence it is automatically a spin singlet. We can generalize this to {m, m’, n} = {m,m, (m - l ) } since this corresponds simply to multiplying the filled Landau level function by a fully symmetric spatial polynomial. Thus these states are also spin singlets. Using the fact that every extended single-particleorbital in the lowest Landau level involves a polynomial with a fixed (average) density of zeros [21,22] given by B/a0(where a0= hc/e is the flux quantum), we may derive a pair of equations for the density p of each component:

From this we obtain for the filling factors

Partial and total filling factors for some of these two-component Laughiin states are listed in Table 5.1. At these partial filling factors, @m,mp,n[Z] is unique in the sense that it is the only wavefunction that excludes its corresponding low relative angular momentum channels, and just as in the case of Laughlin states in one-component systems, we may expect that these wavefunctions will be nearly exact ground states for any sufficiently short-ranged repulsive interaction. In the SU(2) invariant case it seems that we should [50], however, require that the wavefunctions be eigenstates of the total spin operator S, and that their Zeeman energy be not too unfavorable. There is numerical evidence, for instance, that the {3,3,2} state, which has v 3 vt + v, = 2/5, has a lower energy than the usual

MULTICOMPONENT WAVEFUNCTIONS

167

TABLE 5.1. GeneralizedLaughlin States for Two-ComponentSystems“ m

m‘

n

1 1 1 1

1 1 3 5 3 3 3 3 5 5 5

0 1 0 0 0 1 2 3 0 1 2 0 1 2 3 4 5

3 3 3 3 3 3 3

5 5

5 5 5 5

5 5 5 5 5 5

vi

Vt

V

S

Source: After Ref. [22]. “Sis the total spin quantum and * denotes a state that is not an eigenstate of S:. The nominal filling factors v, and v, are shown in parentheses for the ferromagnetic {m, m,m} states because these are not unique (only their sum v is fixed).

hierarchical state if (and for typical field strengths, only if) one ignores the Zeeman energy. For further references and detailed discussion on this point and other topics related to spin in the FQHE, the reader is directed to the book by Chakraborty and Pietilainen [3]. We note that Eq. (7) is ill-defined ifmm’ - n2 vanishes, as it does, for example, in the fully ferromagnetic{m, m, m} states. In this case, however, one can compute the filling factor by simply noting that the fully spin aligned system has an orbital wavefunction equivalent to the Laughlin function at total filling factor v = l/m. The relative filling factors of the two components is necessarily ill-defined because of the SU(2) rotational symmetry. There are 2s + 1 = N + 1 orthogonal but macroscopically degenerate states differing only by their S’ quantum number. Thus we have 1 v t + v -‘-m vt-V

2s’ -l-Nm

The degeneracy of these states leads to a broken symmetry, which we discuss in greater detail in Section 5.6.

168

MULTICOMPONENT QUANTUM HALL SYSTEMS

Generalizations of the Halperin states can be made to the case of an arbitrary number of components [51-531. This would have application, for example, to a superlattice of closely spaced quantum wells, should these become technologically feasible to produce at some point in the future. It is known experimentally [28-311 that there exists an incompressible (but unusually delicate) Hall state at filling factor v = 5/2. This state has also been observed numerically for various artificially chosen interaction models but not for a pure Coulomb interaction [3,33,34,54]. It has been argued that the ground state at this filling factor is not fully spin polarized because the Hall state is easily destroyed by tilting the magnetic field at constant filling fraction [31]. In principle, the same state should be observed at filling factor v = 112 because v = 5/2 = 2 + 1/2 has both spin states of the LLL filled, and these electrons are essentially inert, leaving an effective filling factor veff = 1/2 in the next Landau level. However, the analogous state is not observed at v = 1/2 because, to reach this lower filling factor (in the same GaAs sample), it is necessary to increase the perpendicular component of the magnetic field by a factor of 5. This increases the Zeeman energy advantage of the fully spin-polarized state (which is gapless and does not exhibit a Hall plateau) and makes it the ground state. Thus nonobservation of a quantum Hall state at v = 1/2 provides additional evidence that the v = 5/2 state is not fully spin polarized. As described by Eisenstein [38], tilted field experimentscan also help identify spin-reversed quasiparticle excitations above polarized ground states [3,551. Haldane and Rezayi [32] have proposed a two-component spin-singlet wavefunction to explain the existence of the v = 5/2 quantum Hall plateau. This wavefunction is an exact ground state for the hollow-core model [32] and may be written in two different ways by modifying two different Halperin states, each of which has filling factor 1/2 [56]. The first uses the Halperin fermionic {3,3, l} function @HC = a331 P r l M

I

(9)

where M is an N / 2 x N/2 matrix whose ij element is given by

Mij=(Zi-Z,)-'

(10)

The permanent of the matrix, per(M1,is by definition just like the determinant except that there are no minus signs for odd permutations.The subtle effect of this permanent on the wavefunction is to cause it to be an eignefunction of total spin (with S = 0) without changing the density (at least in the thermodynamic limit). Note that the permanent causes some spin-up and spin-down particles to have a finite probability of having relative angular momentum zero. It turns out that no particles ever have relative angular momentum 1 in this state. Hence this wavefunction is an exact zero-energy singlet ground state for the hollow-core potential model V m = Vdm.1

(11)

FRACTIONAL CHARGES IN DOUBLE-LAYER SYSTEMS

169

Despite the unphysical appearance of this model, it has been argued that it might capture the correct physics when the form of the effectivepseudopotentials in the second Landau level is taken into account [32]. A second way to write the same state uses the (222) bosonic Halperin wavefunction @HC=@222detlGl where

(12)

is an N/2 x N/2 matrix whose element is given by

Gij= (Zi -2 ),

-

(13)

A curious mathematical identity [32] allows one to show that these two representations are precisely equivalent. It is possible that orbital effects [36] confuse the tilted field test for spinunpolarized states and the v = 512 is actually spin polarized. A competing spin-polarized Pfufian state developed by Read [9] and also studied by Greiter et al. [35] is a kind of p-wave paired state. It is not known for certain at this point what the true nature of the very delicate 5/2 state is [38,54]. It may be one of the proposed states [9,32-341 or something completely unknown. Jason Ho has considered interesting connections between the internal order in these types of wavefunctions and analogous order in superfluid 3He [57]. In particular, he has discussed the incompressibledeformation of the various states into each other, connecting, for example, the {3,3, l} state and the Pfaffian state [35,57,58]. 5.3. CHERN-SIMONS EFFECTIVE FIELD THEORY One interesting approach to the quantum Hall effect in general, and multicomponent systems in particular, is the Chern-Simons effective field theory. Unfortunately space limitations prevent us from discussing this approach in any detail. The reader is directed to the work by Halperin [59], the references therein, and to the many references in what is now a vast literature [24,42-44,46,60-731. A succinct and introductory summary of the bosonic representation for doublelayer systems is given in Ref. [46]. We present a brief discussion of the collectivemode predictions of the Chern-Simons approach for double-layer systems in Section 5.5. 5.4. FRACTIONAL CHARGES IN DOUBLE-LAYER SYSTEMS The fractional quantum Hall effect occurs because, at particular filling factors, electrons in a partially filled Landau level are able to organize themselves into such strongly correlated states that the energy cost of making decoupled particles and holes remains finite, even in the thermodynamic limit (i.e., there is a charge gap). It is unusual to have a charge gap due entirely to electron-electron inter-

170

MULTICOMPONENT QUANTUM HALL SYSTEMS

actions (i.e., in a translationally invariant continuum system), although the example of superconductivity is familiar. In the fractional quantum Hall effect, not only do interactions produce a charge gap, but the free charges responsible for the thermally activated dissipation measured experimentally contain only a fraction of the charge of an electron. The fact that sharply defined fractinal charges occur in the fractional quantum Hall effect can be understood as a necessary consequenceof the quantization of the Hall conductance [5, 741 (see the related discussion in Section 5.7). It can also be understood in terms of the variational wavefunctions introduced by Laughlin in his pioneering work on the theory of the fractional quantum Hall effect [ 5 ] . In this section we discuss the fractionally charged excitations of the Y,,,m,,n states by generalizing the plasma arguments made by Laughlin for single-layer systems. The Ym,m,,n wavefunctions have the property that pairs of electrons are excluded from certain relative angular momentum states. A low-energy charged state must have a localized excess or deficiency of charge without destroying the energetically favorable correlations associated with the relative angular momentum state exclusions. For a single-component system, Laughlin suggested that an accurate approximation to the many-body wavefunction for a state with a charged hole at the origin could be obtained simply by multiplyinghis wavefunctions for incompressiblestates by the factor n i Z i .For two-component systems this argument has an obvious generalization.We can produce two, in general different, charged excitations centered on the origin, by multiplying the orbital manyparticle wavefunction by the product of Z i for all electrons in one pseudospin state. This operation is the variational wavefunction equivalent of introducing an unattached flux tube in Chern-Simons theories, and the conclusionswe reach below can equally well be obtained by using an algebraically equivalent argument in that language. The plasma analogy results from writing the quantum distribution function, the square of the many-body wavefunction, as a classical statistical mechanics distribution function for interacting particles in an external potential

1$12

=e-"

(14)

The classical systems that result are generalized two-dimensional Coulomb plasmas [75] and it is convenient in discussing them to adopt the convention of using Roman indices for one component of the plasma and Greek indices for the other component. With this notation, the trial wavefunctions we consider for charged excitations are

These trial wavefunctions clearly reduce the density near the origin without At first sight it might appear that only the ruining the good correlations in Ym,rn,,n.

FRACTIONAL CHARGES IN DOUBLE-LAYER SYSTEMS

171

t,,,,,

Roman particle density is reduced in Y and only the Greek particle density is reduced in Y “,.,,,n, but this is not the case in generalbecause of the correlationfactors. The classical potential energy corresponding to Y is

t,,.,,

This potential is a generalized two-dimensional Coulomb plasma [75) in the sensethat the coupling constants outside the sums in the interaction terms are not constrained to be the products of charges for the two species [i.e., we allow n # (mm‘)1’2]. This difference results in long-range interactions in the plasma which depend on the density of each species separately rather than just on the total charge. U:,m,,nis the potential energy function for a system consisting of Roman and Greek particles. All particles have repulsive mutual two-dimensional Coulomb interactions with coupling constant rn between two Roman particles, coupling constant m‘ between two Greek particles, and coupling constant n between a Roman particle and a Greek particle. All particles are attracted to a neutralizing background which can be considered to have resulted from interaction with unit coupling constant with nonresponding particles of uniform charge density (27d2)-’. For UA, only Roman particles interact with a unit coupling constant with an impurity particle located at the origin. The charge densities induced in each species of particles by the impurity can be calculated using the perfect screening properties that result from the long-range interactions of the plasma. Far enough from the impurity the direct long-range interaction must vanish for each speciesof particle; that is, the sum of the impurity charge times its coupling constant plus the induced charges in each plasma component times the coupling strength for that plasma component must vanish so that

In Eq. (17) e{ is the contribution, in units of the magnitude of the electron charge, to the quasiparticle charge from Roman particles and e i is the contribution from Greek particles. Equation (17) can be solved for e i and 4and the total quasiparticle charge e$ = e i + e;:

f4=

m’ mmt

- ,2

9

-n eg = mm’ - n2 ’

m’ - n

4 = mm’ - n2

(18)

The fractional chargesfor Y :,,.,n differ only through the interchange of m and m’: ei =

-n mm’ - n2 ’

e: =

m-n

m

mm’-

n2’

G = mm’ - n2

Fractional charges calculated from these expressions are listed in Table 5.2.

(19)

172

MULTICOMPONENT QUANTUM HALL SYSTEMS

TABLE 5.2. Fractional Charges for Some Two-Component Fractional Quantum Hall Effect States“ 1 1

1

1 3 3 3 3 3 3 5

5 5 5

5

5

1 1 3 5 3 3 3 3 5 5

5 5 5

5 5 5

0 1 0 0 0 1 2 3 1 2 0 1

2

3 4 5

1 ? 1 1 113 318

0 ? 0 0 0

? 5/14 5111 115 5/24 5121 5/16 519 ?

?

315

- 1/8 - 215 - 1/14

- 2111

0

- 1/24

- 2/21

- 3/16 - 419

?

1 1 1 1 113 114 115 113 217 3111 115 116 117 118 119 1/10

0 ? 0 0 0 - 118 - 215 ? - 1/14 -2111 0 - 1/24 - 2/21 - 3/16 - 419 ?

1 ? 113 115 113 318

315

? 3/14 3111 115 5/24 512 1 5/16 519 ?

1 1 113 115 113 114 115 113 117 1111 115 116 117 118 119 1/10

“4, gives the contribution to the charge from the X component in quasihole state X. We note that the total charge ofwhat is presumably the lowest-energycharged excitation at each filling factor has a value e/q, where q is the denominator of the fractional total filling factor. When the two components of the incompressible Hall fluid are correlated, a reduction of charge density in one component leads to an increase of charge density in the other component. The total charge thus tends to consist of partially canceling contributions from the two layers. This cancellation reaches its extreme limit for the case where m = m’= n, for which the total charge of the excitation is well defined but its separation into contributions from separate components cannot be fixed by the perfect screening requirement on the plasma. We will see later that what is behind this behavior is the existence of long-range order in the Y m,m,m wave-function. This long-range order has delivered a bonanza of new physics in two-component systems, which will be the focus of much of this chapter.

5.5. COLLECTIVE MODES IN DOUBLE-LAYER QUANTUM HALL SYSTEMS In this section we discuss both intra-Landau-level and inter-Landau-level(cyclotron) collective (neutral) excitations of the incompressible ground states whose origin we have explained in previous sections. Our discussion is based on the projected single-mode approximation, which has proved extremely useful [10,76,77] in understanding the nature of the collective-modestructure in single-

COLLECTIVE MODES IN DOUBLE-LAYER QUANTUM HALL SYSTEMS

173

layer systems. The projected single-mode approximation can be appropriate in the strong magnetic field limit where there is little Landau level mixing in either the ground state or the low-lying excited states. Many-body eigenstates of the system can then be distinguished by the quantized integer number of units of h a , by which the kinetic energy exceeds the minimum value NfiwJ2 (where N is the number of particles). The single-mode approximation for the collective energy spectrum follows from the assumption that there is a unique many-body state, I";), with energy E; within each quantized kinetic energy manifold, which is coupled to the ground state by the one-body density operator:

[We employ complex number notation (k = k , + iky) for two-dimensional vectors when convenient.] To use the single-mode approximation it is necessary to evaluate separately contributions to moments of the dynamic structure factor from transitions involving different numbers of quantized kinetic energy units. We write for the dynamic structure factor

where

and A is the area of the system. Here lYi,n)is an exact eigenstate of the Hamiltonian involving n excess quantized kinetic energy units. In the singlemode approximation it is assumed that only a single eigenstate contributes to the sum in Eq. (22),

Both the matrix element and the energy that appear in this expression have physical significance. The matrix element determines how strongly one-body external probes (e.g., far infrared or microwave radiation) couple to the collective excitation. The energies of the collective modes can be measured in transmission experiments or in inelastic light-scattering experiments [78]. The thermodynamics and linear response functions of the system depend qualitatively on the presence or absence of collective modes whose energies vanish in the limit of long wavelengths. Two moments of sn(k,E ) are relatively easy to evaluate, and we will use these two moments to determine both the matrix element and the collective-mode

174

MULTICOMPONENTQUANTUM HALL SYSTEMS

energy:

:1

s,(k) =

d&s,(k,E )

Given s,(k) and f,,(k), we have for the matrix element

and for the excitation energy

Since P k = O is a constant, and the ground and excited states must be orthogonal, it follows that at long wavelength, s,(k) must vanish at least as fast as k2. Long-wavelength probes, like far-infrared radiation, produce [78] observable coupling to a long-wavelength collective mode only if sn(k)ock2 at long wavelengths.We refer to modes for which sn(k)oc k2 as dipole active and those for which s,(k) vanishes with a higher power of k as dipole inactive. The quantity f,(k) is proportional to the product of the square of the matrix element and the excitation energy, and in analogy with atomic physics we refer to this quantity as the projected oscillator strength. We refer to s,(k) as the projected static structure factor. The usualf-sum rule, valid with or without a magnetic field, states that for parabolic bands with effective mass m*,

To evaluate projectedf-sum rules we exploit the property that the single-particle Hilbert space of a charged particle in a magnetic field can [74] be considered as the product space of a factor space in which states are distinguished by the number of quantized kinetic energy units in the cyclotron orbit and a factor space for the cyclotron-orbit-center degree of freedom that exists within each Landau level and is responsible for the macroscopic Landau level degeneracy. To separate the dynamic structure factor into contributions associated with different quantized kinetic energies, we write the kinetic energy operator in the form

where

n'.n

COLLECTIVE MODES IN DOUBLE-LAYER QUANTUM HALL SYSTEMS

175

The G"'*"(k)are related to Laguerre polynomials [74], the sum over i is over particle labels, and B,(k)is a factor coming from the projection of pk onto a single Landau level and operates on the intra-Landau-level degree of freedom of particle i. The commutators that appear below are evaluated by using Eq. (30) and the identity

(k:2k')

+

Bi(kl)Bi(k2)= exp - Bi(k, k,) At strong magnetic fields, only p;" contributes to s,(k) and f,,(k). We restrict our attention to cases of intra-Landau-level (n = 0) and magnetoplasmon (n= 1) collective excitations. (Excitation modes out of the LLL with n > 1 are never dipole active.) Using completeness relations of the many-body eigenstates, it follows from the definitions above that 0.0

s O ( k ) = ~1( y O l p - k p k

0.0

Iy 0 )

(32)

and that

where P is the electron-electron interaction term in the Hamiltonian. From general properties of (translation invariant) many-body eigenstates within the lowest Landau levels, it can be shown [lo] that for small k, so(k) cc k4, so that the intra-Landau-levelcollectivemode is dipole inactive. After a somewhat laborious calculation, involving repeated application of Eq. (31), the n = 0 projected oscillator strength can be expressed in terms of so(k),and one finds the result that fo(k) cc k4, independent of any details of the electron-electron interaction or the ground-state wavefunction, so that the magnetoroton intra-Landau-level collective modes have a gap at long wavelengths: lim A; # 0

k-0

(34)

On the other hand, an elementary calculation based on Eq. (31) and the strong-field-limit assumption that the ground state lies entirely within the lowest-Landau-level subspace of the full Hilbert space implies that

Similarly [76],

176

MULTICOMPONENT QUANTUM HALL SYSTEMS

and the second term on the right-hand side of Eq. (36)can be shown to vanish as k3 for small k. The magnetoplasmon mode is dipole active and in the longwavelength limit completely exhausts the full f-sum rule. For a single layer, the magnetoplasmon mode is the only dipole-active mode and its energy is not shifted from hw, by electron-electron interactions. These behaviors result from the conservation of particle number and invariance under translation: A longwavelength electromagnetic field couples only to the cyclotron motion of the center of mass of the system [79]. We are now prepared to discuss how these results are altered in double-layer systems. We restrict our attention to the case where the two 2DEGs are identical and tunneling between them may be neglected. In this case the number of electrons in each layer is a good quantum number and collective modes corresponding to the sum and difference of the density oscillations in the two layers decouple.To generalize the projected single-mode approximation to the doublelayer case, we evaluate separately projected oscillator strengths for both sum (in-phase) and difference(out-of-phase) modes. We will find that the behavior of the sum modes for double-layer systems is similar to the behavior of the modes of a single-layer system, while the behavior of the difference modes departs from this pattern. For the difference modes, both n = 0 and n = 1 modes are dipole active. The n = 1 mode is shifted from ha,and the n = 0 mode usually has a finite energy as k -+ 0. An exception occurs for those ground-state wavefunctions that have a type of long-range order which we have not yet made explicit. This long-range order is associated with a broken-symmetry ground state. These broken-symmetry ground states are, arguably, responsible for the most surprising and appealing new physics, which is introduced on going from one-component to two-component fractional quantum Hall systems, and we will have much more to say about them later in the chapter. The difference in behavior between sum and difference modes is due to the fact that the Hamiltonian is not invariant under relative translations of the two layers. The Hamiltonian of the double-layer system in the absence of interlayer tunneling may be written in the following form, which is convenient for calculations:

H = hocC[af(L)a,(L)+ a!(R)a,(R)] i

where a,(L)is the Landau level lowering [74] operator for particle i in the left (L) layer, Vf = V f R is the intralayer Coulomb interaction, is the interlayer Coulomb interaction, and p , ( X ) is the density operator for layer X.[For explicit calculation we ignore the finite thickness of the two-dimensional layers so that V r = 2ne2/&qand V y= exp( - 4 d ) V y . l In Eq. (37) the Hamiltonian includes infinite constant terms corresponding to the self-interaction of each electron in the system. It is convenient to retain these terms so that the interaction terms in the Hamiltonian can be expressed in terms of density operators. Since, in all

"5"

177

COLLECTIVE MODES IN DOUBLE-LAYER QUANTUM HALL SYSTEMS

subsequent calculations, the Hamiltonian enters only in commutators, these nonphysical constant terms never contribute. The operators that generate the sum and difference collective modes are Qi* = [p;O(L) Ifr p ; O ( R ) ] / f i With . the Hamiltonian expressed in terms of density operators, a laborious but directcalculation makes it possible to express fj:* and s;* in terms ors,"* using Eq. (31). Explicit expressions for the collective-mode energies and the coupling matrix elements are given elsewhere [65, SO]. Here we comment only on some of the physically interesting conclusions of these calculations. As mentioned above, the energy of the n = 1 sum mode approaches the cyclotron energy hw, in the k +O limit, in agreement with Kohn's theorem. (Since interactions are invariant under simultaneous translations in both layers, the proof for a single-layer system [79] trivially generalizes to the case of the two-layer in-phase mode.) However, the energy of the n = 1 difference mode is shifted from the cyclotron energy in the long-wavelength limit, where it is given by the following expression:

A'-(k = 0)= hw, - J$$q2VyhLR(q) Here hLR(q)is the Fourier transform of the interlayer pair correlation:

where N is the number of particles per layer. If the layers are uncorrelated, hLR(q)= hLL(q)= 0 and A' -(k = 0) = hw,. For correlated layers h&) tends to be negative, at least at small q, since the density in the left layer will tend to be reduced when the density in the right layer is increased, and we can expect that A'-@ = 0) > ho,.In fact, it is possible to prove that interactions (of either sign) always increase the frequency of this mode, as we note below. By expanding the expressionfor f,"' at small k it can be shown that f,"' k4, whereas N

c

N k2 f,"- = - - A 2 4

q 2 V?hLR(q)

+ O(k4)

The n = 0 difference mode is dipole active. It is interesting that for the difference mode, the interaction contributions to the dipole (aIkI2) portions of the n = 0 and n = 1 oscillator strengths are identical. By definition,f,"-is positive definite, so that A-(k = 0) - hw, must also be positive. In the single-mode approximation, the n = 1 (cyclotron) difference mode is always shifted to higher energy by

178

MULTICOMPONENT QUANTUM HALL SYSTEMS

electron-electron interactions. The situation is similar to that for the effect of disorder on the vibration modes of the Wigner crystal at strong magnetic fields, where pinning of the crystal shifts both intra-Landau-level and inter-landaulevel modes upward by the same amount [Sl]. To determine whether or not the n = 0 mode is gapped, it is necessary to determine how *:s behaves at small k. From general properties of wavefunctions in the lowest Landau level, it is possible [65] to conclude that [82]':s k4, whereas the behavior of the n = 0 difference-mode structure factor depends on the difference between interlayer and intralayer correlation functions.It is possible to prove [65] that s,"- k2 at small k, provided that intralayer and interlayer correlation functions separately vanish at large spatial separations. In general it is easy to show from the plasma arguments outlined in Section 5.4 that this is a property of the Ym,m,n wavefunctions.However, an exceptinon occurs for n = m. In this case there is no distinction between the effective plasma interactions of particles in the same layer and particles in different layers. The weighting of particle configurations depends only on the total charge densities of the two layers, and only correlations in the total charge densities of the two layers go to ,,' it is possible [65] zero at large distances. For the broken-symmetry state Y to show explicitly that -:s = exp (- Ikl 12/2), which goes to a constant for k +0. The SMA collectivemode then goes like k2 for this special case ifthe ground state is approximated by Y,. (The italics above are pregnant, as we will see later.) For the {m, m, rn} states, the n = 0 difference mode (intra-Landau level) is gapless at long wavelengths. It is often the case that gapless collective modes can be identified as Goldstone modes associated with a broken symmetry in the ground state, and that is indeed the case here, as we shall see. We emphasize that the results discussed above follow from general sum rules and are independent of the approximate many-body wavefunction (Ym,m,,n) in terms of which we have framed our discussion so far. Since f,"' k4 and f,"- k2 independent of the long-wavelength behavior of S:*, it follows quite generally that the n = 0 sum mode has a gap and that the n = 0 difference mode has a gap except in the case where long-range order is present, which results in correlation functions that do not vanish at large distances. In Fig. 5.1 we show results obtained for the collective-mode energies of a double-layer system with a total Landau level filling factor vT = 1/2 and a layer separation d = 1.51, close to the effective layer separation value for which novel double-layer fractional Hall effects have recently been observed [83]. Numerical calculations [84,85] have established that the ground state at this value of d/l is accurately approximated by the {m,m,n> = {3,3, l } Halperin [21] wavefunction, and we have used the correlation functions [86,87] of that wavefunction to evaluate the oscillator strengths and structure factors. Fork -rO the n = 1 sum mode (the Kohn mode) is unshifted by interactions while the n = 1 difference mode is shifted to higher energies as discussed above. The shift, which is a direct measre of interlayer correlations, should [88] be observable in cyclotron resonance experiments in double-layer systems. Note also that both sum and difference n = 0 modes have a finite gap, as expected from the discussion above.

-

-

-

-

COLLECTIVE MODES IN DOUBLE-LAYER QUANTUM HALL SYSTEMS

179

0.20

0.15

0.10

0.05

'

0.00 0.0

I

8

1.o

4

2.0

3.0

Figure 5.1. Collective-modedispersionfor a double-layersystem at vT = 112 and d/l = 1.5. The energies of the inter-Landau-levelmodes are measured from ha,. The ground state is approximated by the (3,3,1) Halperin state. The plotting symbols refer to the following modes: triangles (n = 1 sum mode),circle (n = 1 difference mode),square (n = 0 sum mode), diamond (n = 0 difference mode).

We are now able to compare our results for the collective modes with the Chern-Simons Landau-Ginzburg (CSLG) theory of the double-layer system C42-441. In the CSLG random-phase approximation, the sum and difference density response functions are given by

where the collective-mode frequencies are given by a+= o,= eB/m*c, 0-= w,(m-n)/(m+n), and m* is the effective band mass of the electrons. For the single-layer case the CSLG random-phase-approximation predictions are correct for the dipole active mode and we might have expected the same to be true in double-layer systems. The sum mode for double-layer systems can clearly be identified with the Kohn mode [64]. However, there are difficulties in identifying the density difference modes in this theory. From Eq. (41) we see that the n = 0 difference mode is dipole active and should have a dipole oscillator strength

180

MULTICOMPONENT QUANTUM HALL SYSTEMS

proportional to VLRhLR. One might therefore be tempted to identify w - with the n = O difference mode. Then, for the case of the {m,m,m}-state random-phase approximation, w - = 0, and it would then be tempting to identify this mode with the gapless n = 0 Goldstone mode. Unfortunately, the single density-difference mode calculated within the double-layer CSLG theory saturates the full dipole oscillator strength Nq2/m*A.This is not acceptable since an excitation within the lowest Landau level cannot contain explicit dependence on the band mass m*. The second possibility is to interpret the difference mode obtained in Eq. (42) as the n = 1 difference mode. In this case one is faced with the difficulty that the mode energy is shifted downward from the cyclotron energy by an amount proportional to w,,whereas the SMA calculations show that it should be shifted upward by an amount proportional to the interlayer Coulomb energy. It has been suggested [89] that these difficulties can be resolved by including the mixing of the vortex excitations with the Gaussian fluctuations in the CSLG theory. Similar difficulties arise in the ferminon-Chern-Simons theory of the single-component v = 1/2 state and can be resolved in that case by taking sufficient care with the Landau parameters of the composite-fermion Fermi liquid [59].

5.6. BROKEN SYMMETRIES

Ferromagnetic states break spin rotation symmetry since they are defined by an order parameter (S), which gives the magnitude and orientation of the magnetization. For the SU(2) symmetric case with no Zeeman term, this orientation is arbitrary. As we will see below, the case of a double-layer system is described by a pseudospin with easy-plane anisotropy [U(l) or X Y symmetry]. Here the magnetization vector is forced to lie in the X Y plane in the ground state. The origin of ferromagnetismin all these systemsis the Coulomb interaction,just as it is for itinerant ferromagnetssuch as iron. Exchangeeffectsare particularly crucial in a 2DEG in a large magnetic field because the kinetic energy is quenched into highly degenerate Landau levels. It is advantageous to follows Hund’s rule and maximize the spin in order to make the spatial wavefunction fully antisymmetric, thereby lowering the Coulomb energy. Since the Landau level is degenerate, this spin alignment can in some cases be complete since it costs no kinetic energy as it does in iron [90]. Before considering the physical consequences of this broken symmetry, let us return to Table 5.1 to consider how the total spin quantum number S for a state can be determined. A portion of our discussion hare follows that of Ref. [22]. As already mentioned, states of the form {m, m, m} are fully ferromagnetically aligned and have total spin S = N/2. To derive the spin quantum numbers for the other states in Table 5.1, we write

BROKEN SYMMETRIES

181

and use the fact that N k.1

We see that

where e(i,k)@(...,Zir...,z k , . ..) = @(. . .,z k , . . .,Zi,...) is a label exchange operator. In Eq. ("I) S* , = X k S t and S,+ and S; are spin raising and lowering operators, and the prime on the sum in Eq. (44) indicates that k = 1 is excluded. In Eq. (45) S is the total-spin quantum number. To identify Halperin {m, m', n} states that are total spin eigenstates, we start with states where m = 1, n = 0, and m' has any odd value. Since j(Zi - Z j )is a Vandermonde determinant, these orbital wavefunctions have the up-spin Landau level full (see Table 5.1). They are eigenstates of S' with eigenvalue

ni,

( N ,-N,) - Nl(m' - 1) - N(m'- 1) = S,. 2 2 2(m' 1)

+

(46)

Moreover, S+Yl,m,,o = 0 since the up-spin Landau level is already full and there are no wavefunctions with a larger value of N,. It follows that @l,m,OIZ] satisfies Eq. (45) with S = (N, - N1)/2,and hence that

(This result can also be established by an explicit algebraic proof.) Since e(i, [j])Q[ZJ@[Z] = Q[Z]e(i, [j])@[Z] for any symmetric polynomial Q[Z], we have from Eqs. (47) and (45) that

+ Zp,m*+ z p , z p C z : ~ I= Srn*(Srn,+ ')@I + z p . m * + z p . z p C z : ~ I

+

(48)

In addition, it follows from Eq. (45) that S2,Y[Z:x] = N ( N / 2 1)/2@[Z:x] for any completely antisymmetric function @[Z], and in particular for generalized Laughlin states with m = m' = n. These states are merely the S' = 0 members of the set of the (N + 1) fully polarized Laughlin states, which are degenerate in the absence of the Zeeman term. We return now to the question of the physical consequences of the spontaneously broken symmetry of ferromagneticstates. We focus initially on the SU(2)

182

MULTICOMPONENT QUANTUM HALL SYSTEMS

tt tttttt tttttttttttttttttttttt ttt Figure 5.2. (a) Simple spin-flip excitation which creates a widely separated particle hole pair; (b) skyrmion spin configuration (shown in cross section). The spins gradually and smoothly rotate from up at the perimeter to down at the origin in a circularly symmetric spin textural defect. For the case of Coulomb interactions,this object costs only 1/2 the energy of the simple spimple spin flip.

invariant case of “real” spins with zero g factor. Consider a state with v = 1and all spins up. Because the up Landau level is maximally filled, the Pauli principle forces us to flip a spin if we are to move an electron to create a pair of charged excitations. This is illustrated in Fig. 5.2a and shows that spin and charge are intimately connected in this case. Transport dissipation measures the thermal activation of charged excitations. In the absence of interactions, the energy cost of charged excitations is zero and there will be no v = 1 quantum Hall plateau because there is no gap. In the presence of Coulomb interactions a flipped spin particle-hole pair causes a loss of exchange energy of [46]

where E is the bulk dielectric constant. This occurs because the electron spatial wavefunction is no longer perfectly antisymmetric.This energy gap is quite large (- 150K at B = 10T) and is vastly larger than the bare Zeeman splitting. Hence Coulomb interactions and the associated ferromagnetism play a dramatic role in producing the charge gap at v = 1 (and because of the spontaneous magnetic ordering will continue to do so, even if g is strictly zero). The exchange energy cost of particle-hole pair excitations is so large that it is worth searching for a modified form of the excitation that is less costly. A prescient analysis of smooth spin textures by Sondhi et al. [24] yielded the exciting idea that “skyrmion” [60]-like spin textures (shown in cross section in Fig. 5.2b) can have relatively low energy and carry a fermion number proportional to their topological charge (Pontryagin index) [24,46]: d2reNvm(r)* [d,m(r) x d,m(r) J

(50)

where m(r) is the unit vector field representing the local spin orientation. The

BROKEN SYMMETRIES

183

fermion number AN is an integer multiple of v because it is the number of times the unit sphere is wrapped around by the order parameter. That is, it is the winding number of the spin texture [91]. For the Laughlin parent states v = l/m, elementary spin textures carry the same fractional charge as the quasiparticles discovered by Laughlin [S] for spinless electrons. As we discuss below, the fact that the charges are the same follows from very general considerations. Actually, the spin texture states we have defined must contain precisely the same number of particles as I$o)since the spin-rotation operator does not change the total electron number. However, the spin density may contain a number of well-separated textures with well-defined nonzero topological charge densities and hence well-localized charges; only the net charge in the spin-texture states defined above will be zero. The system clearly has states with locally nonzero net charge in the spin textures. A simple variational wave function for a skyrmion of size A centered on the origin and carrying p units of topological charge is given by

where Qmmm is defined in Eq. (5), (.) refers to the spinor for the jth particle, and the variational parameter II is a fixed length scale. This is a skyrmion because the 2-j component of the spin has a vortex centered on the origin and the &component is purely down at the origin (whereZ j = 0) and purely up at infinity(where lZjl >>A), as shown in Fig. 5.2. The parameter A is simply the size scale of the skyrmion [24,92]. Notice that in the limit A - 0 [where the continuum effectivefield theory is invalid (see Section 5.7), but this microscopic wavefunction is still sensible], we recover a fully spin-polarized filled Landau level with p Laughlin quasiholes at the origin. Hence the number of flipped spins associated with the presence of the skyrmion interpolates continuously from zero to infinity as A increases. To analyze the skyrmion wavefunction in Eq. (51),we use the Laughlin plasma analogy [S]. In this analogy the norm of $A, Tr(,,JD[zJ IY [z] 12, is viewed as the partition function of a Coulomb gas. To compute the density distribution, we simply need to take a trace over the spin (we specialize heie to the case p = 1 for simplicity)

z=

s

~ [ z ~ e ( 2 / m ) I ~ , , , l o ~ ~ z ,i -+z( m , /2)~~10g(iZk~2+~z)-(m/4)~~iZt~21

(52)

This partition function describes the usual logarithmically interacting charge m Coulomb gas with uniform background charge plus a spatially varying impurity background charge Apb(r), 1 A2 Apb(r)3 ---V2V(r) = 2n n(r2+ 22)2

(53)

184

MULTICOMPONENTQUANTUM HALL SYSTEMS

For large enough scale size 1 >> 1, local neutrality of the plasma [69] implies that the excess electron number density is precisely ( l/m)Apb(r), so that Eq. (54) is in agreement with the standard result for the topological density [92], and the skyrmion carries electron number l/m (for p = 1) and p / m in general. These objects are roughly analogous to the Laughlin quasihole. Explicit wavefunctions for the corresponding quasielectron objects (“anti-skyrmions”)are more difficult to write down, just as they are for the Laughlin quasielectron, due to the analyticity constraint [S]. Sondhi et al. [24] have shown that for the case of pure Coulomb interactions (i.e., with no finite inversion-layer thickness corrections), the optimal skyrmion configuration costs precisely half the energy of the simple spin flip [93]. The reason for this is simply that the skyrmion keeps the orientation of spins close to that of their neighbors and so loses less exchange energy.This is discussed in more detail from a field-theoreticpoint of view in Section 5.7. Optical [94] and standard transport [95] experiments show that the charge excitation gap is indeed much larger than would be expected if interactions were neglected and has approximately the correct Coulomb scale, although there is not yet precise agreement between the observed gap and the best estimates, including the finite g factor [24,25]. The main source of error is probably neglect of finite thickness effects. Calculations including finite thickness corrections do not exist at present. Quantum fluctuation effects (i.e., corrections to HartreeFock) may also be important. It should be noted that the uniform ground state of the v = 1 ferromagnet does not have quantum fluctuations (Hartree-Fock is exact here). However, for finite g factor, the length scales associated with the skyrmion are small and quantum corrections could well become important. The idea that skyrmions are the lowest-energy excitations has received very strong and unequivocal support from numerical simulations, which show that quite remarkably, adding a single electron to a v = 1 system (with g = 0)suddenly changes it from a fully aligned ferromagnet (S = N/2) to a spin singlet (S = 0) due to the formation of a skyrmion [24,96]. (It should be noted that this occurs in the spherical geometry. Things are slightly more complicated on the torus [46].) The notion of charges being carried by skyrmion textures has received additional dramatic experimental confirmation in recent optically pumped NMR measurements by Barrett et al. [26,27] Their Knight shift measurements (see Fig. 5.3) indicate that the electron gas spin polarization has a maximum at filling factor v = 1 and falls off sharply on each side. The rate of fall-off indicates that each charge added or removed from the Landau level turns over about four spins. This is consistent with the charge being carried by skyrmions of finite size. The size of the skyrmion is determined by a competition between the Zeeman coupling,which wants to minimize the number of flipped spins, and the Coulomb self-energy,which wants to expand the skyrmion to spread out the excess charge density over the largest possible area. As mentioned above, Rezayi found numerically that a single charge destroys the spin polarization of a system entirely (if g = 0).Using an effective field-theoreticapproach, Sondhi et al. [24] have estimated the skyrmion size in the regime of small g factor. Microscopic

FIELD-THEORETIC APPROACH

I

0 ’

0.6

185

i i \

0.8

1.0

1.2

v

1.4

1.6

I

1.8

Figure 53. Knight shift measurements of the electron spin polarization of a ZDEG in the vicinity of filling factor v = 1. (After Ref. [26].)

Hartree-Fock calculations expected to be more accurate in the physically accessible Zeeman energy regime have been performed by Fertig et al. [25]. These estimates are roughly consistent with the experimental value of x 3.5 spins per unit of charge. It should be noted that for finite g factor, the energetic advantage of the skyrmion over the simple flipped spin is considerably reduced [25]. Wu and Sondhi have shown that in higher Landau levels, skyrmions have higher energy than other charged excitations 1973. For the SU(2)symmetriccase discussed in this section,the existence of a vector order parameter (S) is in some sense trivial because the magnetization commutes with the Hamiltonian. For the case of double-layer systems we will see that the pseudospin Hamiltonian has only U(1) symmetry and the fact that a nonzero expectation value of (S) appears is highly nontrivial. We will discuss this in more detail in Section 5.7. 5.7. FIELD-THEORETZC APPROACH

In this section we study the ferromagneticbroken-symmetry ground state and its excitations from the point of view of (quantum) Ginzburg-Landau effective field theory. We will begin with the SU(2)invariant case of real spin and then move on to the pseudospin analogy in double-layer systems. We give here an introductory qualitative discussion of the physics. Part of our discussion follows fairly closely the presentation in Ref. [46]. The technical details of the calculations can be found there and elsewhere [24,25,47].

186

MULTICOMPONENT QUANTUM HALL SYSTEMS

The standard first step in this procedure is always to identify the slowly fluctuating order parameter field, which in this case we have reason to believe is the local magnetization. We believe that on long length scales the (coarsegrained) magnetization fluctuates very slowly, because we know that the zerowavevector component of the spin operator (i.e., the total spin) commutes with the Hamiltonian and so is a constant of the motion. We focus on slow tilts of the spin orientation, ignoring variations in the magnitude of the coarse-grained magnetization and so define the order in terms of a local unit vector field m(r). General symmetry arguments can now be used to deduce the form of the Lagrangian. We cannot have any terms that break spin rotational symmetry, and thus the leading term that is an allowed scalar is I

f

where ps is a phenomenologicalspin stiffnesscoefficient and the energy is relative to the ground-state energy. It expressesthe cost due to loss of Coulomb exchange energy when the spin orientation varies with position. For the SU(2) invariant v = 1 case, this stiffnessmay be computed exactly [46]. For Coulomb interactions (with no finite thickness corrections) this calculation yields PS=-

e2/d 1 6 6

In two dimensions the stiffness has units of energy and is approximately 4 K at a field of 10T. Numerical estimates for v = 1/3 yield a value that is about 25 times smaller [46]. As usual, there is a linear time derivative term in the Lagrangian that can be deduced from the fact that each spin processes under the influence of its local exchange field. Equivalently, we may note that when the orientation of a spin is moved around a closed loop, the quantum system picks up a Berry’s phase [98] proportional to the solid angle R enclosed by the path w of the tip of the spin on the unit sphere,as shown in Fig. 5.4.Noting that a charged particle moving on the surface of a unit sphere with a magnetic monopole at the origin also picks up a Berry’s phase proportional to the solid angle subtended by the path [98], we may express the Berry’s phase for a spin S as eiy

= eiSIS

or equivalently, eiy

= ,i

-e

s

i

fa

dm.A(m)

dl(dnt/at).A(m)

(57)

(58)

where A(m) is the vector potential of a unit monopole [24,46,91] at the center of

FIELD-THEORETICAPPROACH

187

Figure 5.4. Path o of a spin Sm on a unit sphere. When viewed from the origin of spin space, the path subtends a solid angle Q, so the path contributes a Berry's phase SR.

the sphere evaluated at point m. That is, V, x A = m. This phase is correctly reproduced in the quantum action by adding the following total derivative term to the Lagrangian for the spin

am L, = S--.A(m) at

(59)

Using the fact that the electronic density is v/2d2, the analog for the present problem for a large collection of electrons with S = 1/2 at filling factor v may be written simply

which yields the Lagrangian V

d2r[VmN(r)].[VmN(r)]

at

(61)

We shall see shortly that higher gradient terms can be unexpectedly significant, but this Lagrangian is adequate to recover the correct spin-wavecollective mode. Taking the spins to be aligned in the 2 direction and looking at small transverse oscillations at wavevector q, we obtain from this Lagrangian the following equation of motion:

dm 4np,q2 A=- 2 x m, dt

hv

188

MULTICOMPONENT QUANTUM HALL SYSTEMS

This yields the dispersion relation hw=-q 471Ps

2

V

which agrees with the long-wavelength limit of exact results obtained by a variety of means [99,100]. At this point we have expanded the Lagrangian to lowest order in gradients and we have correctly found the neutral collective spin-wave modes. Their dispersion is quadratic in wavevector just as it is for the Heisenberg ferromagnet on a lattice. However, here we have an itinerant magnet and we have so far seen no sign of the charge degrees of freedom. It turns out that we have to go to higher-order gradient terms in the action to see charged objects. We have already seen in the discussion of Fig. 5.2 in Section 5.6 that for a filled Landau level, the Pauli principle forces there to be a connection between charge excitations and flipped spins. It turns out that the existence of a finite Hall conductivity in this itinerant magnet causes smooth spin textures to carry charge proportional to their topological density. One can derive this result from a Chern-Simons effective field theory [24,26], or from microscopic considerations involving the fact that the spin density and charge density operators do not commute when projected onto the lowest Landau level [46], or from macroscopic considerations connecting the Berry phase term to the Hall conductivity [46,101]. The latter is the least technical and the most instructive, so we shall pursue it here. Imagine that the order parameter of the ferromagneticsystem is distorted into a smooth texture, as illustrated in Fig. 5.5. As an electron travels around in real space along a path ar which is the boundary of the region r,the spin is assumed to follow the orientation of the local exchange field b(r) and hence traces out a path in spin space labeled w in the (schematic) illustration in Fig. 5.4. That is, given any sufficientlysmooth spin texture, we can write a Hartree-Fock-like Hamiltonian for the electrons which will reproduce this texture self-consistently:

H=

N

-

C b(r).Sj

j= 1

If we drag one electron around in real space along X a n d its spin follows the local

Figure 5.5. Smooth spin texture. An electron moving along the boundary of the region r in real space has its spin follow the path in spin space along the boundary of the region labeled w in Fig. 5.4.

FIELD-THEORETICAPPROACH

189

b(r) adiabatically (which we expect since the exchange energy is so large), the electron will acquire a Berry’s phase Q/2, where Q is the solid angle subtended by the region w shown in Fig. 5.4. In addition to the Berry’s phase from the spin, the electron will acquire a Bohm-Aharonov phase from the magnetic flux enclosed in the region r.At least in the adiabatic limit, the electron cannot distinguish the different sources of the two phases [loll. The electron would acquire the same total phase in the absence of the spin texture if, instead, an additional amount of flux

n

A@=-@o 47l

(where a0is the flux quantum) were added to the region r. We know, however, that a