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 adding flux to a region in a system with a finite Hall conductivity changes the total charge in that region [102]. To see this, let @(t) be the time-dependent flux inside r. Then the electric field along the perimeter obeys (from Faraday’s law)

$arE*dr =

1 dQ c dt

Because of the Hall conductivity (and the fact that axx=O), the field at the perimeter induces a current obeying

2.J x dr = axyE-dr

(67)

Integrating this expression along the boundary and using the continuity equation, we have that the total charge inside r obeys

dQ -= dt

+--a,d@ c dt

or

AQ AQ = - evQO

where we have used the fact that the Hall conductivity is quantized (and is negative for B = IBI2): bXY =

ye2 h

--

Thus v electrons flow into the region for each quantum of flux added to the region. This makes sense when we recall that there is one state in each Landau level per quantum of flux penetrating the sample.

190

MULTICOMPONENTQUANTUM HALL SYSTEMS

Figure 5.6. Infinitesimal circuit in spin space associated with an infinitesimalcircuit in real space.

From Eq. (65) we see that the spin texture thus induces an extra charge of

A Q = --v-

n 4R

The solid angle R is, of course, a functional of the spin texture in the region r.For simplicity of analysis of this functional let us consider making up r out of a set of infinitesimal square loop circuits in real space of the form

The corresponding circuit in spin space illustrated in Fig. 5.6 is

+

m(x,y ) m(x dx, y ) -+ m(x + dx, y -+

+dy)

-+

m(x,Y + d y ) -,m(x,y )

(73)

Approximating this circuit as a parallelogramas shown in Fig. 5.6, the solid angle subtended is (to a sufficient approximation)

This may be rewritten in a suggestive form, which tells us the curl of the Berry “connection” [98]:

dw = $&,,m.a,,mx a,mdxdy We can now add up all the infinitesimal contributions to obtain =

dxdyf~,,,m.a,mx a,m

(75)

FIELD-THEORETIC APPROACH

191

which yields a total charge of

or a local-charge-densitydeviation of ev

Sp = --&,,,,m.~,,m x d,m 871

The expression on the right-hand side of Eq. (78) is simply the Pontryagin topological charge density of the spin texture. Its integral over all space is an integer and is a topologically invariant winding number known as the Pontryagin index. The spin textures, which have nonzero Pontryagin index, are the skyrmion configurations illustrated in Fig. 5.2b. A microscopic variational wavefunction for these spin textures was discussed in Section 5.6. The charge density in Eq. (78) can be viewed as the timelike component of a conserved (divergenceless) topological three-current, which results in the following beautiful formula: V

j a = - -&a~Y&,bcm“(r)~smb(r)a,mc(r) 871

(79)

Using the fact that m is a unit vector, it is straightforward to verify that a,,jr = 0. We note that the fact that the expression for the topological current is not parity invariant is a direct reflection of the lack of parity symmetry in the presence of the external magnetic field. The mechanism we have seen here that associatescharge with flux is the reason that quantum Hall fluids are describedby Chern-Simons theories [6,60,63,64,66] and is the same mechanism that causes Laughlin quasiparticles (which are topological vortices) to carry quantized fractional charge proportional to the quantized value of bxY[S]. Having established that the electron charge density is proportional to the topological density of the spin order parameter field, we must now return to our Lagrangian to see what modifications this implies. We have already taken into account the long-range Coulomb force, but it led only to the local spin gradient term, whose coefficient is the spin-wave stiffness. There are, however, additional effects of the charge (topological) density fluctuations which we must take into account: 4711’

- Ap 2 sId’r[Vm*(r)].[Vm’’(r)]

192

MULTICOMPONENT QUANTUM HALL SYSTEMS

Here 6 p , is the Fourier transform of the charge density in Eq. (78).Note that it is second order in spin gradients. The first of the new terms in Eq. (80)representsthe coupling of the charge fluctuations to the external and random disorder potentials V(-q) and is second order in spin gradients. The second new term represents the mutual interaction of the charge fluctuations via the Coulomb potential. Note that this is fourth order in gradients (and so is not a duplicate of the p, term, which also comes from the Coulomb interaction). In general, there will be additional fourth-order terms allowed by symmetry, but we do not bother to write them down since they will not have the divergent Coulomb interaction coefficient 2n/~q,which makes the term we have kept effectively third order in q. We can immediately conclude several interesting things from the rather peculiar nature of our itinerant ferromagnet. First, unlike the case of a regular ferromagnet,a scalar potential can induce the formation of charged skyrmions in the ground state. Thus sufficientlystrong disorder would have the effect of greatly reducing the net spin polarization of the ground state, something that should be directly observable experimentally. Second,we note that (at the classical level)the energy of a skyrmion due to the gradient term is scale invariant:

because we have two spatial integrations and two derivatives. [The quantity E , is defined in Eq. (49).] Now, however, the Coulomb self-energywill want to expand the size of the skyrmion. In real spin systems this effect competes against the small but (usually) nonzero Zeeman coupling, which wants to minimize the number of flipped spins. This competition has been studied in some detail and appears to be essential to explain the experiments of Barrett et al. [24-271.

5.8. INTERLAYER COHERENCE IN DOUBLE-LAYER SYSTEMS The details of the double-layer experiments of Murphy et al. [39] are described by Eisenstein [38]. Here we introduce the main ideas briefly. Double-layer quantum Hall systems (and wide single-well systems [40]) exhibit a variety of nontrivial collective states at different filling factors. Here we focus on the case of total filling factor v = 1(i.e., 1/2 in each layer), which is most closely analogous to the fully ferromagnetic broken-symmetry state for v = 1 with real spins that we have discussed in previous sections. There are many other interesting states that we do not have room to discuss here. One example is the state at total filling factor v = 1/2 (i.e., 1/4 in each layer), which is believed to be described by Halperin’s {3,3, l } wavefunction [3,21,22,84,85]. This state is more nearly like a gapped spin liquid state, although, as we have already seen, it does not satisfy the Fock cyclic condition and so is not a true spin singlet.

INTERLAYER COHERENCE IN DOUBLE-LAYER SYSTEMS

n

Figure 5.7. Schematic conduction-band edge profile for a double-layer two-dimensional electron gas system. Typical widths and separations are W d - lOOA and are comparable to the spacing between electrons within each inversion layer.

193

I

..._._____. ,_____.._..

N

-d-

The schematic energy-level diagram for the growth direction degree of freedom in the double-layer system is shown in Fig. 5.7. For simplicity we assume that electrons can only occupy the lowest electric subband in each quantum well. If the barrier between the wells is not too strong, tunneling from one side to the other is allowed. The lowest energy eigenstates split into symmetricand antisymmetric combinations separated by an energy gap ASASwhich can, depending on the sample, vary from essentially zero to hundreds of kelvin. The splitting can therefore be much less than or even greater than the interlayer interaction energy scale, E , = e 2 / d The analogy with the spin systems studied in previous sections is that within the approximations just mentioned, the electrons in a two-layer system have a double-valued internal quantum number-the layer index. The tunnel splitting in the double well plays the role of the Zeeman splitting for spins. In addition to double quantum wells, there are wide single quantum wells in which the two lowest electric subband states are strongly mixed by Coulomb interactions [40]. These systems exhibit very similar physics and can also be modeled approximately as a double-layer system. Throughout our discussion we assume that the real spins are aligned and their dynamics frozen out by the small Zeeman energy. This is not necessarily a good approximation in experimentally relevant cases but simplifies matters greatly. Dynamics of real spins in double layers is currently a topic of investigation [103,104].

5.8.1. Experimental Indications of Interlayer Phase Coherence Here we review very briefly the main experimental indications that double-well and wide single-well systems at v = 1 can show coherent pseudospin phase order over long length scales and exhibit excitations that are highly collective in nature. When the layers are widely separated, there will be no correlations between them

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MULTICOMPONENT QUANTUM HALL SYSTEMS

4.0 1

I

Figure 5.8. Phase diagram for the double-layer QHE system. Only samples whose parameters lie below the dashed line exhibit a quantized Hall plateau and excitation gap. (After Ref. [39].)

and we expect no dissipationless quantum Hall state [1051 since each layer has v = 1/2. For smaller separations, it was predicted theoretically [33,85,106] and subsequently observed experimentallythat there is an excitation gap and a quantized Hall plateau [39,40,107]. The resulting phase diagram is shown in Fig. 5.8 and discussed in more detail by Eisenstein [38]. The existence of a gap has either a trivial or a highly nontrivial explanation, depending on the ratio ASAdEc.For largeAsAsthe electronstunnel back and forth so rapidly that it is as if there is only a single quantum well. The tunnel splitting ASASis then analogous to the electric subband splittingin a (wide)single well. All symmetric states are occupied and all antisymmetric states are empty and we simply have the ordinary v = 1 integer Hall effect. Correlations are irrelevant in this limit and the excitation gap is close to the single-particle gap ASAS (or ho,, whichever is smaller). What is highly nontrivial about this system is the fact that the v = 1 quantum Hall plateau survives even when ASAS << E, (see Fig. 5.8). In this limit the excitation gap has clearly changed to become highly collective in nature since the observed [39,40) gap can be on the scale of 10 to 20K even when ASAS" 1 K. Because of the spontaneous broken symmetry [42-45,1083, the excitation gap actually survives the limit AsAs -,0. This crossover from single-particle to purely collectivegap is quite analogous to the result we discussed earlier-that for spin-polarizedsingle layers, the excitation gap survives the limit of zero Zeeman splitting. Hence, to borrow a delightful phrase from Sondhi et al. [1097. "Y = 1 is a fraction too." A second indication of the highly collective nature of the excitations can be seen in the Arrhenius plots showing thermally activated dissipation [38,39]. The low-temperature activation energy A is, as already noted, much larger than Asas. If A were, nevertheless, somehow a single-particle gap, one would expect the

195

INTERLAYER COHERENCE IN DOUBLE-LAYER SYSTEMS

Figure 5.9. Process in double-layer two-dimensional electron gas systems that encloses flux from the parallel component of the magnetic field. The quantum amplitude for such paths is sensitive to the parallel component of the field.

’i.1

1-

W n

b

c -

m L-

Arrhenius law to be valid up to temperatures of order A. Instead, one observes a fairly sharp leveling off in the dissipation as the temperature increases past values as low as about 0.1A. This is consistent with the notion of a thermally induced collapse of the order that had been producing the collective gap. This behavior is very similar to that seen in real spins [26,27]. The third significant feature of the experimental data pointing to a highlyordered collective state is the strong response of the system to relatively weak magnetic fields B , applied in the plane of the two-dimensional electron gases. Within a model that neglects higher electric subbands, we can treat the electron gases as strictly two-dimensional. This is important since B,, can then affect the system only if there are processes that carry electrons around closed loops containing flux. A prototypical such process is illustrated in Fig. 5.9. An electron tunnels from one layer to the other at one point, then travels to a second point. Then it (or another indistinguishable electron) tunnels back and returns to the starting point. The parallel field contributes to the quantum amplitude for this process (in the two-dimensional gas limit) a gauge-invariant Aharonov-Bohm phase factor exp(2ni@/@,),where 0 is the enclosed flux and is the quantum of flux. Such loop paths evidently contribute significantly to correlations in the system since the activation energy gap is observed to decrease very rapidly with B,, falling by factors of order 2 or more until a critical field, B t 0.8 T,is reached at which the gap essentially ceases changing [39]. To understand how remarkably small Br is, consider the following. We can define a length L,,from the size of the loop noeeded to enclose one quantum of flu?: L,,Bfd= (Do. (LIIIA] = 4.137 x 105/d[A]B~[T).) For B t = 0.8 T and d = 210 A, L,,= 2460 A, which is approximately six times the spacing between electrons in a given layer and more than 20 times larger than the quantized cyclotron orbit radius 1 = (hc/eB,)’’’ within an individual layer. Significant drops in the excitation gap are already seen at fields of 0.1 T, implying that enormous phase-coherent correlation lengths must exist. Again, this shows the highly collective nature of the ordering in this system. The fourth indication about the nature of the coherent ordering is the fact that the gapped quantum Hall state at v = 1 can survive a finite amount of imbalance in the layer charge densities [38]. Charge imbalance can be controlled by applying a gate voltage, which increasesthe equilibrium charge density in one layer and decreases it in the other. In some cases, such as the gapped state at v = 1/2, layer imbalance immediately destroys the quantum Hall state. For v = 1 it does not.

-

1%

MULTICOMPONENT QUANTUM HALL SYSTEMS

This is a strong hint about the different natures of the ordering in the two cases. We defer discussion of what this hint means to Section 5.8.3.

5.8.2. Effective Action for Double-Layer Systems Having established the correct form of the effective low-energy Lagrangian for the SU(2) invariant case, let us now turn to the U(1)symmetric case in doublelayer systems. The spin analogy is relativelystraightforward but can be confusing until one gets used to it. The idea is simply that each electron can be in either the upper layer or the lower layer, and we refer to these two states as pseudospin up and down, respectively. For the moment we assume that the layers are identical and also neglect the possibility of tunneling between the two layers. The confusing point is that quantum mechanics neverthelessforces us to consider the possibility that an electron can be in a coherent superposition of the two pseudospin states, so that its layer index is uncertain. Portions of our discussion here and in the following subsectionsfollows that of Ref. [46]. Further technical details can be found therein. The formal mapping that we use to define the pseudospin density operators is the following. The z component of the pseudospin density represents the local charge density difference between the layers:

The x and y components of the pseudospin density are off-diagonal and can be combined to form tunneling operators:

If for instance, tunneling is present, the Hamiltonian contains a term

s

T = - t d2r[S+(r)

+ S-(r)]

= - 2t

s

dzrS”(r)

(84)

We know that ferromagnetism in real spin systems is not the result of truly spin-dependent forces but rather a by-product of Coulomb exchange forces. Hence the effectivespin Hamiltonian must contain only spin scalars such as Sj*S,. Here, however, the Coulomb forces are explicitlypseudospin dependent, since the intra- and interlayer Coulomb interactions are not identical. Following our previous discussion [41], we define

v,o= f(V: + V,”)

(85)

V’= Ir

(86)

-m - m 1

where V t is the Fourier transform with respect to the planar coordinate of the

INTERLAYER COHERENCE IN DOUBLE-LAYER SYSTEMS

197

interaction potential between a pair of electrons in the same layer and t ': is the Fourier transform of the interaction potential between a pair of electrons in opposite layers. If we neglect the finite thickness [41] of the layers, t ': = 2ne2/eq and V:=exp(-qd)t',A, where d is the layer separation. The interaction Hamiltonian can then be separated into a pseudospin-independent part with interaction t" and a pseudospin-dependent part. The pseudospin-dependent term in the Hamiltonian is

Here is the Fourier transform of the spin density at wavevector k and the over:'t Vf is bar indicates projection onto the lowest Landau level [46]. Since V t > , positive and this term produces an easy-plane, as opposed to Ising, pseudospin anisotropy. That is, this term prefers for the spin to lie in the XY plane. If the spin orientation moves up out of the XY plane so that ( S ' ) # 0, the energy increases. We can view this energy cost as simply the charging energy of the capacitor formed by the two layers since the pseudospin component S' measures the charge difference between the two layers. The pseudospin symmetry of the Hamiltonian is reduced from SU(2) to U(1) by this term. In addition, this term increases the quantum fluctuations in the system since it does not commute with the order parameter

where p = x, y. Thus total spin is no longer a sharp quantum number. However, for small layer separations, we expect the quantum fluctuations to be small. At very large layer separations we expect the quantum fluctuations to become dominant and produce the disordering phase transition, which uncouples the two layers. In the absence of the symmetry breaking term (d = 0), we know the exact quantum ground state for v = 1 since it is simply a fully occupied pseudospinpolarized Landau level. This state is 2S + 1 = (N+ 1)-folddegenerate (sincethere is no charging energy in this unphysical limit). As a first approximation for finite d, we assume that the eigenstates remain exactly the same, but the charging energy lifts the degeneracy among these states by favoring the ones with S' 0. That is, we assume that the spin vector is still fully polarized and nonfluctuating, but it now lies in the X Y plane. We argue that the form of the energy functional that we derive must remain valid even when quantum fluctuations due to the pseudospin-dependent terms in the Hamiltonian are present. However, the coefficientsthat appear in the energy functional will be altered by quantum fluctuations, and the explicit expressions we derive below are accurate only when the pseudospin-dependent interactions are weak (i.e., only when the layers are close together). Estimates have been obtained for the quantum fluctuation corrections to these coefficients from

-

198

MULTICOMPONENT QUANTUM HALL SYSTEMS

finite-size exact diagonalization and many-body perturbation theory calculations [46,47]. To better understand what we are assuming, consider the specific example of a state that has S = N / 2 and has its spin oriented in the I direction so that it is an eigenstate of total S":

Here X is a LLL orbital label and 10) is the electron vacuum. It is clear from the construction that this state has every orbital in the LLL filled, so that it has v = 1. However, each electron is in a linear combination of up and down pseudospin, which puts it in an eigenstate of S". This particular combination is familiar because it is the symmetric state that is the exact (noninteracting)ground state in the presence of tunneling between the layers. In the absence of VSs,this would be an exact eigenstate of the system (with or without tunneling) even in the presence of the symmetricinteraction term V , because it is a filled Landau level. This state has zero net charge on the capacitor on the average (( S') = 0, so first-order perturbation theory in V,, favors it for the ground state. The confusion that many people have at this point is over the fact that this is a coherent state with uncertain S'. It makes perfect sense if the Hamiltonian includes tunneling that the charge in each layer will be uncertain. However, we are arguing that this state is a good approximation to the ground state even in the absence of tunneling because it has favorable Coulomb energy. That is, we are assuming that the system spontaneously breaks the U(1) symmetry associated with conservation of layer charge difference [42-44,1061 and is acting much like a superfluid [41] which breaks V(1) symmetry. We refer to this state as having spontaneous interlayer phase coherence. It is quite analogous to a BCS state that has an uncertain total number of Cooper pairs. In the absence of tunneling, the electron energy cannot be sensitivet o the actual relative phase between the two layers, so any coherent state of the form

will have equally good energy. The global U(1) phase rp simply determines the orientation of the total spin in the XY plane:

N

(YlS+[Y) E ( Y ( S X + i S y l Y ) = - e i q

2

(91)

This is exactly analogous to the situation in a BCS superconductor where the energy is independent of the phase rp, which determines the coherence between states of different numbers of Cooper pairs:

INTERLAYER COHERENCE IN DOUBLE-LAYER SYSTEMS

199

The complex order parameter in a double-layer system given by Eq. (91) has an amplitude and phase like that of a superconductor but in many ways is more reminiscent of an excitonic insulator since it is charge neutral (i.e., it contains I#~, not and physically represents a particle in one layer bound to a hole in the other layer (but we do not know which layer contains the particle and which contains the hole). Some years ago, Datta [l lo] considered similar states in double-layer systems in a zero43 field. At first, it may seem counterintuitive that in the absence of tunneling, the system energy could depend on the relative phase rp for finding the particle in the upper or lower layer. Indeed, the energy is unchanged when a constant is added to rp. However, the energy does depend on gradients of p. The spin stiffness ps is nonzero because of the loss of exchange energy that occurs when rp varies with position. Imagine that two particles approach each other. They are in a linear superposition of states in each layer (even though there is no tunneling). If these superpositions are characterized by the same phase, the wavefunction is symmetric under pseudospin exchange, so the spatial wavefunction is antisymmetricand must vanish as the particles approach each other. This lowers the Coulomb energy. If a phase gradient exists, there is a greater amplitude for the particles to be near each other, and hence the energy is higher. This loss of exchange energy is the source of the finite spin stiffness and is what causes the system to “magnetize” spontaneously. We may describe this by a term in the effective action of the form

$;#I)

‘1

E =-p, 2

d2rlVrp12

(93)

We see immediately that at finite temperatures this system will be described by a classical XY model and will undergo a Kosterlitz-Thouless phase transition. We cover this in more detail below. 5.8.3. Superfluid Dynamics

We consider now the dynamics of a two-layer system with easy-plane pseudospin anisotropy but no tunneling. Let us back up and note that implicit in Eq. (93) is the assumption that the spin lies exactly in the XY plane. To be more complete we should derive the effective Lagrangian in the presence of the easy-plane anisotropy. On general symmetry grounds this must be of the form (neglectinghigherorder gradients needed to describe charge fluctuations) V

at

fi(rnZ)2+ ~ l V m z l +$[IVrnXlz z

+ lVrnYIZ]

Within the Hartree-Fock approximation, which is presumably valid only for small layer separations, the coefficients fi, pA, and pEcan be evaluated in terms of

200

MULTICOMPONENT QUANTUM HALL SYSTEMS

the density-density correlation function of the ground state [46]. [The gradient expansion in m' is not strictly valid since it turns out [46] that the long range of the Coulomb interaction leads to a nonlocal m'm' interaction term not included above. This term vanishes in the SU(2)invariant case but here is more important than the IVmz12 term and less important than the /31rnr12 term at long wavelengths. We retain the (Vm'(' term in the following expressions only to remind us of its importance in the SU(2) invariant limit.] Since mzis massive, the equations of motion derived from this Lagrangian lead to a linear rather than quadratic collective-mode dispersion [41-44,46,69, 1061 like that of the Goldstone mode in a superfluid or an antiferromagnet. We take the pseudospin of the system to be polarized in the ft direction and consider the linear response to a time- and space-dependent pseudospin Zeeman field in the $4plane. Using the equations of motion determined from the Lagrangian in Eq. (94) and Fourier transforming with respect to both time and space, we find that

where h, and h, are the Fourier coefficients of the pseudospin magnetic field at frequency o and wavevector q. Physically h, corrsponds to a time- and spacedependent bias potential between the two wells, while h,, could arise from a spaceand time-dependent interlayer tunneling amplitude. We see immediately that the response has a singularity at the collective-modefrequency

For the dfl = 0 case, p= 0, p A = p, = p:, and the collective-mode frequency reduces to the result obtained previously for the spin-wave collective mode of isotropic ferromagnets (oq = 4nq2p:/v). The collective mode corresponds to a spin precession whose ellipticity increases as the long-wavelength limit is approached. The presence of the mass term (/3 # 0) changes the collective-mode dispersion at long wavelengths from quadratic to linear [41]. In the limit of small q,

Thus we see that the system is acting very much like a bosonic superfluid of weakly repulsively interacting particles which has a Goldstone mode (if and only if there is nonzero repulsion) [41-46,62,69]. However, we must again emphasize that we have a charge-neutral order parameter. The singular response occurs for electric fields of opposite signs in the two layers. Another way to say this is that

INTERLAYER COHERENCE IN DOUBLE-LAYER SYSTEMS

201

the charge conjugate to the U(1) phase field cp is S'. The current associated with phase gradients,

is the flow of pseudospin density which is the difference between the electrical currents in the two layers [43,44,46,69]. This simply reflects the fact that the object which condenses is a neutral excitonic composite of a particle and hole. These analogies with superfluidity naturally raise the question of analogs of other phenomena known to occur in superfluids and superconductors. It is believed that there is no analog of the Meissner effect in these systems [44-461. However, it has been suggested that Josephson or Josephson-like effects should occur in the tunneling transport between the two layers C42-441. In this picture one views each layer as analogous to one side of a Josephson junction. This appealing analogy appears to imply measurable consequences only in the case where superconducting contacts are made to each two-dimensional electron layer; the distinction between the system and an ordinary Josephson junction system then seems to be artificial. More interesting, in our view, is the recent sideways-tunnelingproposal of Wen and Zee [11 11, which appears to provide a quite precise analog of the Josephson effect, but in this case for pseudospin superfluidity.Here one views the pair of layers as constituting a single superfluid on one side of a Josephson junction. A separate pair of layers is imagined to be coupled to the first by weak sideways tunneling. In each layer there is a phase variable cp and a pseudospin supercurrent given by Eq. (98). When this current reaches the junction, it can coherently jump across the weak link and continue onward. Physically, this coherent tunneling of pseudospin involves an electron tunneling one way between the upper layers and another electron tunneling in the opposite direction between the lower layers. It should be possible in principle to see ac Josephson oscillations of the current at a frequency of 2eV/h, where V is an appropriately applied bias voltage (of opposite sign in the two layers). It is unlikely, however, that a sample with no tunneling within each pair of layers, but finite tunneling across the junction, can ever be produced. One could imagine using a parallel B field to shut off the tunneling effectively within each pair, but even weak disorder will pin the discommensurations and probably ruin the effect. As an aside to the question of spin channel superfluidity, we point out that double-layer systems are ideal for probing electron-electron interactions via mutual drag effects in transport at zero field [112]. More details on some of the mutual transport effectsassociated with spontaneous interlayer phase coherence are presented in Ref. [46]. In addition, it has been predicted that for double-layer fractional quantum Hall states without broken symmetries but with interlayer correlations of the type described by the Halperin {m,m',a} wavefunctions, anomalously large (and quantized) mutual drag effects will occur [6162].

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MULTICOMPONENTQUANTUM HALL SYSTEMS

We are now in a position to return to a discussion of the fourth experimental indication of interlayer phase coherence that was briefly alluded to at the end of Section 5.8.1, namely the fact that unlike the v = 1/2 state, the v = 1 state survives a finite amount of charge imbalance in the two layers. The rigidity of the v = 1/2 state can be understood macroscopically from the Chern-Simons effective fieldtheoretic point of view [58,113], or microscopically from the fact that the Halperin {3,3,1} state is believed to give a good description of the state. The plasma analogy shows that this state has a sharply defined total filling v = 1/2 and that the density in each layer must be exactly equal (see Section 5.4). Another way to say this is that the (3,3,1} state is almost a singlet spin liquid which must have S' = 0 (although strictly speaking, it is not an eigenstate of total spin); that is, it is pseudospin incompressible. In contrast to this, the v = 1 state is pseudospin compressible.This is clear from the existence of pseudospin superfluidity with its gapless collective Goldstone mode. The state is ferromagneticallyordered and there is a whole family of states differing only in S' which would be degeneratewere it not for the charging energy. The charging energy lifts this degeneracy and picks out the S' = 0 state. Application of a bias voltage to unbalance the layers simply picks a different state having nonzero S" out of the manifold. We can supplement this picture by thinking macroscopically of the XY model describing the ordering. Layer imbalance corresponds to tilting the order parameter slightly up out of the XY plane. This changes no symmetries in the problem, and simply renormalizes the spin stiffness slightly because the projection of the spin onto the XY plane is reduced. Thus the ordering that produces the charge gap is weakened but not (immediately)destroyed by layer imbalance. One can also describe this microscopically by modifyingthe variational wave function in Eq. (90) to the form

which obeys

We close by noting that a class of experiments that would be very useful in probing the pseudospin superfluidity would be inelastic (Raman) light scattering to try to detect the gapless collective excitation mode. To see this at finite frequency requires finite wavevectors, which in turn would require some sort of grating coupler. These are difficult to fabricate on sufficientlyshort length scales. Here, however, it turns out that it is possible to have the system act as its own grating by tilting the applied magnetic field, which will induce a twisting of the order parameter [47,114] as we discuss in Section 5.10.

INTERLAYER COHERENCE IN DOUBLE-LAYER SYSTEMS

203

5.8.4. Merons: Charged Vortex Excitations Continuing the superfluid analogy,we study vortex excitations in this section and discuss the Kosterlitz-Thouless phase transition induced by the unbinding of these topological defects [42,43,45,46]. The order parameter of the system in the presence of a vortex at the origin has the following approximate form:

Here the f refers to right- and left-handed vortices, respectively, and 8 is the azimuthal angle made by the position vector r. At asymptotically large radii, mz vanishes to minimize the charging energy. However, in the vortex core we must have mz+ 1 (and mx,mY-,O) to prevent a singularity in the gradient energy. Thus there are four flavors of topologically stable objects which we refer to as merons since it turns out that they are essentially half skyrmions. These are illustrated in Fig. 5.10. The local topological charge density calculated from dp = - ( l/8n)eij(dim x djm).m can be expressed in the form 1 dm' dp(r) = -4nr dr

Figure 5.10. Four flavors of merons. These are vortices that are right or left handed and have topological charge k 1/2.

204

MULTICOMPONENTQUANTUM HALL SYSTEMS

and the total charge is

s

Q = d 2 r dp(r) = $ EmZ(00) - mz(0)]

(103)

For a meron, the spin points up or down at the core center and tilts away from the 2 direction as the distance from the core center increases. At asymptoticallylarge distances from the origin, the spins point purely radially in the fi-i plane. Thus the topological charge is If: 1/2 depending on the polarity of core spin. The general result for the topological charge of the four meron flavors may be summarized by the following formula:

Q = 3[rn'(oo)

- mz(0)]n,

(104)

where n, is the vortex winding number. The formulas derived above for the meron charge do not rely for their validity on the variational ansatz assumed in Eq. (101). They are quite general and follow from the fact that a meron topologically has half the spin winding of a skyrmion. The meron charge of f 1/2 is a topological invariant and implies that the electrical charge is f ve/2. The fact that merons carry fractional charge f ve/2 can be deduced from a Berry phase argument similar to the one used to find the skyrmion charge. We simply note that an electron moving at a large distance around a meron will have its spin rotated through 2nin the R-9 plane due to the vorticity. We know that the Berry's phase for rotating a spin one-half object in such a way is exp(i2nS) = - 1. Thus the meron produces the same Berry's phase as half a flux quantum. From Eq. (69) we then obtain A Q = k ve/2. [The ambiguity of the sign of the charge associated with half a flux quantum can only be resolved by examining the behavior of rnzin the meron core. It depends on whether the midgap state induced by the topological defect (discussed below) is empty or occupied.] It is instructive to write down explicit microscopic variational wavefunctions for vortices (merons).We start with the simplest example: a meron with vorticity 1 and charge - 1/2 that has the smallest possible core size:

+

Here 10) is the fermion vacuum, cLtfi) creates an electron in the upper (lower) layer in the angular momentum rn state in the LLL, and M is the angular momentum quantum number corresponding to the edge. The vorticity is + 1 because far away the spin wavefunction is essentially

where 8 is the polar angle. The charge is - 1/2 because we have created a hole in

INTERLAYER COHERENCE IN DOUBLE-LAYER SYSTEMS

205

the center of the lower layer (every state has occupancy 1/2, except for m = 01, which is unoccupied). Since the spin is pointing up at the center, this agrees with the spin-charge relation derived earlier. From the spin-charge relation we know we can change the sign of the charge of a meron by changing the direction of spins in the core region without changing the vorticity. This can be seen explicitly from the wavefunction:

This state has charge + 1/2 because we have put in an electron in the m = 0 state in the lower layer. Obviously, what we did (in terms of the spin texture language) is to flip the spins in the core region to the down direction without changing the vorticity of the meron at long distances. In this construction, one sees that in a sense, the merons are like fractionally charged midgap states in polyacetylene, which can be empty or occupied [1151. A meron with vorticity - 1 is readily obtained simply by interchanging the labels m and m + 1 in the subscripts in Eqs. (105) and (107). Invariance under pseudospin reversal guarantees the equality of the energies of corresponding f vorticity merons. However, the two charge states for a given vorticity are not necessarily degenerate, just as Laughlin quasiholes and quasiparticles are in general nondegenerate. (For the special case v = 1, particle-hole symmetry guarantees degeneracy.) It is instructive to attempt to find a variational wavefunction for a pair of merons to show that a meron is half a skyrmion. Consider the situation where we have a pair of merons of opposite vorticity but the same charge, located at points Zl and Z2.To achieve this, one meron must have mz= + 1 in its core and the other must have mZ= - 1in its core. The following wavefunction seems to do the job:

where Qmmm is defined in Eq. (5), cp is an arbitrary constant phase angle, and refers to the spinor for thejth particle. Note that at very large distances away from Zl and z2,the spinor for each particle asymptoticaily approaches

and so corresponds to a fixed spin orientation in the xy plane at an angle cp from the x axis. Hence the net vortex content of the pair of excitations is zero. It is clear from the construction that the spin orientation is purely up for an electron located at Z, and purely down at 2,. Furthermore, the net charge must be + v e

206

MULTICOMPONENT QUANTUM HALL SYSTEMS

since asymptoticallythe factor of Zj is the same one as for the Laughlin quasihole in a spin polarized state. By symmetry, it seems that there must be half that amount, + ve/2 associated with each of the two objects located at and (It turns out, however, that the charge is not split in half and localized near 2, and z, as we would really want for a meron pair.) We can easily show that this particular wavefunction actually is just another way to represent a skyrmion. Let 2, = I and 2, = - I and, for simplicity,assume that the spins are asymptotically oriented in the f direction so that cp = 0. Now perform a global rotation of all the spins about the 9 axis by an angle of - 4 2 . Using

z,

z,.

we see that we have recovered the variational skyrmion wavefunction studied in Eq. (51). The form in Eq. (108) is the appropriate one in the U(1) symmetriccase since it keeps the spins primarily in the X Y plane. We expect that a pair of merons can be deformed into a skyrmion, and each meron can be properly viewed as carrying half the topological and electrical charge of a skyrmion. However, there must exist variational wavefunctions which are better for the U(1)case than the one discussed here in the sense that the present one has a smooth charge distribution centered in between 2, and 2, rather than being more closely associated with the two points defining the meron positions. Finally, we note that merons carry fractional statistics 1/4; they are quarterons [46]. This can be seen from the Chern-Simons theory [46] or from the fact that two of them together make a skyrmion that is a fermion. 5.8.5. Kosteriitz-Thouless Phase Transition

It is well known that the presence of vortex topological defects can disorder the XY phase of the ground state. This will occur even at zero temperature, due to quantum fluctuations, if the layer separation exceeds a critical value d*. One could also conveniently tune through this transition in a single sample with fixed d by varying a gate voltage to induce layer imbalance which will renormalize the spin stiffness. We focus here, however, on the effect of finite temperature and thermally induced vortices in samples that are otherwise ordered at zero temperature (because d < d*). Integrating out the massive mzfluctuationsand going to finite temperature, we are led to a classical XY model of the form shown in Eq. (93). Hartree-Fock estimates of the spin stiffness at finite layer separation give values in the range 0.1 to 0.5 K for typical experimental sample parameters. The Kosterlitz- Thouless phase transition is an unbinding of vortex defects in the XY model and occurs at a temperature given approximately by the value of the bare spin stiffness. These unbound vortices will cause the long-wavelengtheffectivespin stiffness in Eq. (93) to renormalize discontinuously to zero at the Kosterlitz-Thouless critical point.

INTERLAYER COHERENCE IN DOUBLE-LAYER SYSTEMS

207

in the usual way, the scale-invariant classical action of Eq. (93) leads to a logarithmic interaction among vortices. A gas consisting of M merons will have an energy of the form

Here E,,,, is the meron core energy [116], R,,,, is the meron core size, R, is the separation between the i th andjth merons, and ni is the vortex charge (winding number, f 1) of the ith meron. The last term is new and is unique to the present problem. it represents the Coulomb interaction among the fractional charges bound to the merons. qi = 1 is the sign of the electrical charge (& 42) of the ith meron. The origin of the logarithmic interaction in a superconducting or superfluid film is the kinetic energy stored in the supercurrents circulating around the vortices due to order-parameter phase gradients. Here the logarithmic interaction arises from the loss of Coulomb exchange potential energy in the presence of phase gradients produced by the vortices. The Coulomb interaction in Eq. (111) falls off more rapidly than the log interaction (it is effectively one order higher in derivatives than the log interaction) and so is perturbatively irrelevant at the Kosterlitz-Thouless (KT) critical point. That is, the KT temperature may be shifted somewhat, but the phase transition itself is unaffected. However, in the limit of strong Coulomb effects (due to small E or large vortex fugacity) the global phase diagram becomes extremely rich. Among other things there is a phase transition to a chiral state with nonzero order parameter ( niqi) in which vortex charge and electrical charge are no longer independent. The rich physics and the novel phase diagram of this model has been elucidated in an interesting paper by Tupitsyn et al. [1 17J. if found experimentally, this KT transition would be the first finite-temperature phase transition in a quantum Hall system. All other transitions between plateaus, for example, are zero-temperature transitions because the vortices there (Laughlin quasiparticles) do not interact logarithmically and are unconfined by an analog of the Anderson-Higgs mechanism [6,7,63,64]. As noted above, the characteristic energy scale for p s is 0.1 to 0.5 K. This is not a problem from the point of view of experimentally achievable temperatures. However, to have the needed U(1) symmetry it is essential (as we discuss in Section 5.9) that the tunneling amplitude between the layers be much smaller than this scale. Such samples have not yet been constructed, but could be in principle because the tunneling amplitude falls off exponentially with layer separation, whereas the Coulomb interactions that control the stiffness ps fall off only as a power law. Nevertheless, it will not be easy since present samples are already close to the critical separation d */f. The latter could perhaps be circumvented by going to lower-density samples and correspondingly lower magnetic fields. The standard experimental signatures of the KT transition are zero linear response dissipation below TKT, but zero critical current, and a characteristic

208

MULTICOMPONENT QUANTUM HALL SYSTEMS

jump in the exponent associated with the nonlinear response

v

-

IP

from p = 1 above the transition to p = 3 just below the transition [118). Dissipation is caused by phase slips associated with the motion of unconfined vortices. A transport current exerts a magnus force whose direction depends on the sign of the vorticity and which moves the vortices at right angles to the current. Below the transition the vortices are confined into (vorticity) neutral pairs by their logarithmic attraction and no longer couple to the supercurrent. In a superconductor or superfluid at finite temperature there is always some sort of “normal” fluid present (thermally excited Bogoljubov quasiparticles, for example), which can produce dissipation. However, there are no electric fields to couple to these particles since the supercurrent shorts them out. Thus there is no dissipation. Analogous normal fluid excitations exist in the quantum Hall systems we are studying here. If we define normal fluid as anything that is charged but has no net vorticity, we see that a pair of opposite vorticity but like-charged merons constitutes normal fluid that can be thermally activated. The validity of the meron pair picture has been confirmed by numerical calculations [1191. In the pseudospin superfluid channel (i.e., opposite electrical currents in each layer) this normal fluid will not couple to the supercurrent since it is vortex neutral. The linear response dissipation will therefore drop to zero below TKT, and all the other usual signatures of the Kosterlitz-Thouless transition will be present . Doing transport experiments in the pseudospin channel requires separately contacting the two layers and having extremely high tunneling resistance between the layers without moving them so far apart that they decouple. It would be easier to do the experiment in the ordinary charge channel in which both layers are contacted simultaneously and current flows in the same direction in both layers. Unfortunately, in this channel there is a net electric field induced by the v = 1quantized Hall resistivity and the normal fluid will couple to this producing thermally activated dissipation 1461. While we would expect to see the dissiit would not show any disconpation decrease rapidly toward zero below TKT, tinuity as it would in the pseudospin channel. This is the same mechanism that causes the small but nonzero thermally activated dissipation in ordinary quantum Hall plateaus. The energy for an excitation of normal fluid consisting of a pair of likecharged but opposite vorticity merons given by Eq. (111) is minimized at a separation of

For typical values of ps, R , 2 101, and so is large enough to justify the fieldtheoretic continuum approximations used in deriving Eq. (1 1 1).

209

TUNNELING BETWEEN THE LAYERS

5.9. TUNNELING BETWEEN THE LAYERS A finite tunneling amplitude t between the layers breaks the U(1) symmetry, t

--cosq 2d2 by giving a preference to symmetric tunneling states. This can be seen from the tunneling Hamiltonian

s

H, = - t d2rC$p)$l(r) + $](r)$,(r)l

(115)

which can be written in the pseudospin representation as

s

H , = - 2t d2rSx(r)

(116)

(Recall that the eigenstates of S" are symmetric and antisymmetric combinations of up and down.) We can shed some more light on the spontaneous symmetry breaking by considering the tunneling Hamiltonian H , in Eq. (1 15) as a weak perturbation. Naively, since particle number is separately conserved in each layer for t = 0, one might expect 1 lim-(i,bIHTI$) = O 1-0

t

That is, one might expect that the first-order term in the perturbation series for the energy shift due to t to vanish. Instead, however, we find that the energy shifts linearly in t, lim lim r+o

A+W

tA

where A is the system area, and mXis, by definition, the magnetization, which is the system's order parameter [l20]. If the interlayer spacingd is taken to be zero, one can readily show [46] that the variational wavefunction in Eq. (89)is exact, hence limI,,mx = 1, and t = ASAS/2.For finite d, Eq. (89) is no longer exact and quantum fluctuations will [46] reduce the magnitude of mx and we must renormalize the hopping parameter t appropriately. As the layer separation d increases, a critical point d* will be reached at which the magnetization vanishes and the ordered phase is destroyed by quantum fluctuations [41,45,46]. This is illustrated in Fig. 5.9 and discussed in more detail

210

MULTICOMPONENT QUANTUM HALL SYSTEMS

from the experimental point of view by Eisenstein [38]. Forfinite tunneling t, the collective mode becomes massive and quantum fluctuations will be less severe [47]. Hence the phase boundary in Fig. 5.8 curves upward with increasing ASAS. For ASAS = 0 the destruction of long-range order and the charge excitation gap are intimately related and occur simultaneouslyat d* and zero temperature. For finite ASAS the system always has nonzero mx even in the phase with zero charge gap. We have already seen that finite layer separation reduces the pseudospin symmetry from SU(2) to V (1). The introduction of finite tunneling amplitude destroys the U(1) symmetry and makes the simple vortex-pair configuration extremely expensive,thereby destroying the KT transition. To lower the energy, the system distorts the spin deviations into a domain wall or string connecting the vortex cores as shown in Fig. 5.11. The spins are oriented in the f direction everywhere except in the shaded domain-line region, where they tumble rapidly through 271. The domain line has a fixed energy per unit length, so the vortices are now confined by a linear string tension rather than logarithmically. We can estimate the string tension by examining the energy of a domain line of infinite length. The optimal form for a domain line lying along the y axis is given by p(r)= 2 arcsin (tanh

;)

where the characteristic width of the string is

The resulting string tension is El211

Provided that the string is long enough (R >> t),the total energy of a segment of length R will be well approximated by the expression e2 Ebair= 2Ek + -+ TOR 4tR The prime on Em,in Eq. (122)indicates that the meron core energy can depend on ASAS. ELairis minimized at R = Rb =.-/, Note that apart from the core energies, the charge gap at fixed layer separation (and hence fixed pa)is proportional to T:I2 cc t’/4 A b L ‘ which contrasts with the case of free electrons, for which the charge gap is proportional to ASAS.Note that because the exponent 1/4 is so small, there is an extremely rapid initial increase in the charge gap when

-

212

MULTICOMPONENT QUANTUM HALL SYSTEMS

not play an important role in determining the nature of the lowest-energy charged pseudospin texture. As t increases, Rb a t - 'I4decreases and will eventually reach R,, which is, of course, independent oft. Since

the characteristic width of the domain line becomes comparable to Rb in the same range oft values where Rb and R, become comparable. We may concludethat the nature of the charged pseudospin texture crosses over directly from the meronpair form to the finite-length domain-line string form for ps/(e2/c<) 1/25, or equivalently for t t,,, where

-

N

-- - 3.9 x I,'

e2/d

103(

AT e2/d

The crossover tunneling amplitude is thus typically smaller than 5 x 10-4(e2/d). Typical tunneling amplitudes in double-layer systems are smaller than abort 10- '(e2/d) and can be made quite small by adjusting the barrier material or by making the barrier wider. Nevertheless,it seems likely that t will be larger than t,, except for samples that are carefully prepared to make t as small as possible. As t increases beyond t,,, Rb will continue to decrease. When Rb becomes comparable to the microscopic length, 1, the description given here will become invalid, and the lowest-energy charged excitations will have single-particle character. However, the domain-wall string picture of the charged pseudospin texture has a very large range of validity since Rbat-'14 decreases very slowly with increasingt at large t. Writing Rb (e'/~87tp,)(t,,/t)''~, we find that Rb lonly for t 10-2[(e2/~l)2/ps].Using typical values of p s we see that the charged excitation crosses over to single-particlecharacter only when the hopping energy t becomes comparable to the microscopic interaction energy scale. The various regimes for the charge excitations of double-layer systems are summarized in Table 5.3.

-

-

N

TABLE 5.3. Charged Spin Texture Energies at v, = 1 for Double-Layer Systems with Tunneling" Regime

Nature of charged excitations Excitation size Excitation energy

T S ~ X lo3&

Meron pairs N-

e2

8XPS

2nPs

4 x 1 0 3 f i , 3 ~ ~ ~ 1 0 - 2 / f i10-2/;dt ,

Finite-length domain-line strings

-

Single-particle excitation

K a t 1 1 4

I

m at'14

t

"3= p,/(e2/0 and f at/(e2/f),where ps is the pseudospin stiffness, t the renormalized tunneling amplitude, C the magnetic length, To = domain wall width.

PARALLEL MAGNETIC FIELD IN DOUBLE-LAYER SYSTEMS

213

Almost all typical double-layer systems lie within the regime of the domain-wallstring pseudospin texture charge excitation, and hence will not exhibit a true Kosterlitz-Thouless phase transition.

5.10. PARALLEL MAGNETIC FIELD IN DOUBLE-LAYER SYSTEMS Murphy et al. [39] have shown that the charge gap in double-layer systems is remarkably sensitive to the application of relatively weak magnetic fields B,,, oriented in the plane of the two-dimensional electron gas. Experimentally, this field component is generated by slightly tilting the sample relative to the magnetic field orientation. Tilting the field (or sample) has traditionally been an effective method for identifyingeffects due to (real)spins because orbital motion in a singlelayer 2DEG system is primarily [36] sensitiveto B,, while the (real)spin Zeeman splitting is proportional to the full magnitude of 8. Adding a parallel field component will tend to favor more strongly spin-polarizedstates. For the case of the double-layer v = 1 systems studied by Murphy et al. [39], the ground state is known to be an isotropic ferromagnetic state of the true spins, and the addition of a parallel field would not, at first glance, be expected to influence the low-energy states since they are already fully spin-polarized.(At a fixed Landau level filling factor, B, is fixed, so the total B and the corresponding Zeeman energy increase with tilt.) Nevertheless,experiments [39] have shown that these systems are very sensitive to B,,.The activation energy drops rapidly (by factors varying from 2 up to an order of magnitude in different samples) with increasing B,,.At B,,= Be: there appears to be a phase transition to a new state whose activation gap is approximately independent of further increases in The effect of B,, on the pseudospin system can be visualized in two different pictures, one microscopic, the other macroscopic. We present the latter here. The technical details of the former are described elsewhere [45,47]. Recent work presents a discussion of higher Landau levels [122]. We use a gauge in which B,,=.V x A,,,where A,,= B,,(O,O,x). In this gauge the vector potential points in the 2 direction (perpendicular to the layers) and varies with position x as one moves parallel to the layers. In this gauge, the only change in the Hamiltonian caused by the parallel field is in the term that describes tunneling between layers. As an electron tunnels from one layer to the other, it moves along the direction in which the vector potential points, and so the tunneling matrix element acquires a position-dependent phase t + t eiQx,where Q = 2n/L,, and L,,= Q&d is the length associated with one flux quantum Q,, between the layers (defined in Fig. 5.9). This modifies the tunneling Hamiltonian to H, = - d 2 r h(r).S(r), where h(r) “tumbles” [i.e., h(r) = 2t(cos Qx, sin Qx,~)].The effective XYmodel now becomes

214

MULTICOMPONENT QUANTUM HALL SYSTEMS

which is precisely the Pokrovsky-Talapov (PT) model [123] and has a very rich phase diagram. For small Q and/or small ps, the phase obeys (at low temperatures) q(r) E Qx; the order parameter rotates commensurately with the pseudospin Zeeman field. However, as B,,is increased, the local field tumbles too rapidly, and a continuous phase transition to an incommensurate state with broken translation symmetry occurs. This is because at large B,,, it costs too much exchange energy to remain commensurate and the system rapidly gives up the tunneling energy in order to return to a uniform state V q x 0, which becomes independent of B,,.As explained in further detail below, we find that the phase transition occurs at zero temperature for [45,47]

Using the parameters for the sample of Murphy et al. [39] with weakest tunneling [124] (ASAS= 0.45 K) and neglecting quantum fluctuation renormalizations of both t and ps(i.e., using Hartree-Fock), we find that the critical field for the transition is x 1.3 T, which is slightly larger than theobserved value [39,124] of 0.8 T but still corresponds to a very large length Ll,.A graphical comparison showing the qualitative agreement between the predicted and observed values of the critical tilt angle for several samples is shown in Fig. 2.15 in this volume. In addition to the Hartree-Fock calculations described here, we have made numerical exact diagonalization studies on small systems to find the critical value of the parallel field. Although it is difficult to extrapolate the results to the thermodynamic limit, they do confirm that at finite layer separation, quantum fluctuations can reduce the predicted critical parallel field [47]. As mentioned previously, the observed value Br = 0.8 T corresponds in these samples to a large value for L,,:Lll/l 20, indicating that the transition is highly collective in nature. We emphasize again that these very large length scales are possible in a magnetic field only because of the interlayer phase coherence in the system associated with condensation of a neutral object. Having argued for the existence of the commensurate-incommensurate transition, we must now connect it to the experimentally observed transport properties. In the commensurate phase, the order parameter tumbles more and more rapidly as B,,increases. As we shall see below, it is this tumbling that causes the charge gap to drop rapidly. In the incommensurate phase, the state of the system is approximately independent of B,,and this causes the charge excitation gap to saturate at a fixed value. Recall that in the presence of tunneling, the cheapest charged excitation was found to be a pair of vortices of opposite vorticity and like charge (each having charge f 1/2) connected by a domain line with a constant string tension. In the absence of B,,the energy is independent of the orientation of the string. The effect of B,,is most easily studied by changing variables to

-

O(r) = q(r) - Qx

PARALLEL MAGNETIC FIELD IN DOUBLE-LAYER SYSTEMS

215

This variable is a constant in the commensurate phase but not in the incommensurate phase. In terms of this new variable, the PT model energy is

We see that B,, defines a preferred direction in the problem. Domain walls will want to line up in they direction and contain a phase slip of a preferred sign ( - 2~ for Q > 0) in terms of the field 8. Since the extra term induced by Q represents a total derivative, the optimal form of the soliton solution is unchanged. However, the energy per unit length of the soliton, which is the domain-line string tension, decreases linearly with Q and hence B,,: T = To( 1-2) where To is the tension in the absence of parallel B field given by Eq. (121), and B: is the critical parallel field at which the string tension goes to zero [125]. We thus see that by tuning B,,one can conveniently control the chemical potential of the domain lines. The domain lines condense and the phase transition occurs (in mean field theory) when the string tension becomes negative. Recall that the charge excitation gap is given by the energy of a vortex pair From Eq. (122)we have that the separated by the optimal distance R, = energy gap far on the commensurate side of the phase transition is given by

Jm

A = 2Em,+ = 2Em

+

(

p[( :)‘I2

1-

$)]lI2

As B,,increases the reduced string tension allows the Coulomb repulsion of the two vortices to stretch the string and lower the energy. Far on the incommensurate side of the phase transition, the possibility of interlayer tunneling becomes irrelevant. From the discussion of the preceding section one can argue that the ratio of the charge gap at B,,= 0 to the charge gap at B,,-+ oc) should be given (very roughly) by

-

Putting in typical values o f t and ps gives gap ratios sin the range 1.5 to 7 in qualitative agreement with experiment. According to the discussion of the preceding section, gap ratios as large as (tmax/tcr)1/40.07(e2/d)/p, can be expected in the regime where the pseudospin texture picture applies. Here t,,, is

-

-

216

MULTICOMPONENT QUANTUM HALL SYSTEMS

the hopping parameter at which the crossover to single-particle excitations occurs. Thus gap ratios as large as an order of magnitude are easily possible. Of course, all the discussion here neglects orbital effects (electric subband mixing) within each of the electron gas layers, and these will always become important at sufficiently strong parallel fields. (Note also that for d near d*, the system will have enhanced sensitivity to renormalization of parameters by electric subband mixing in tilted fields.) It should be emphasized that only this highly collectivepicture involving large length-scale distortions of topological defects can possibly explain the extreme sensitivity of the charge gap to small tilts of the B field. Recall that at Bf, the tumbling length L,, is much larger than the particle spacing and the magnetic length. Simple estimates of the cost to make a local one-body type of excitation (a spin-flip pair, for example)shows that the energy decrease due to B,,is extremely small since l/L,,is so small. Numerical exact-diagonalization calculations on small systems confirm the existence of this phase transition and show that the fermionic excitation gap drops to a much smaller value in the incommensurate phase [47]. The collective excitation modes of the system in the commensurate and incommensurate phases appear to be quite interesting [114]. As mentioned previously, the tumbling of the order parameter may produce a self-grating effect that will allow one to tune the wavevector which couples to optical probes. All of our discussion of the phase transition in a parallel field has been based on mean-field theory. Close to the phase transition, thermal fluctuations will be important. At finite temperatures there is no strict phase transition at Bf in the PT model. However, there is a finite-temperature KT phase transition at a nearby B,, > B f . At finite temperatures translation symmetry is restored [123] in the incommensurate phase by means of dislocations in the domain string structure. Thus there are two separate KT transitions in this system, one for t = 0, the other fort # 0 and B,,> Bf. Read [126] has studied this model at finite temperatures in some detail and has shown that just at the critical value of B,,there should be a square-root singularity in the charge gap. The existing data does not have the resolution to show this however. Fisher [127] and Read [126] have pointed out that at zero temperature the commensurate-incommensurate phase transition must be treated quantum mechanically. It is necessary to take account of the world sheets traced out by the time evolution of the strings which fluctuate into existencedue to quantum zero-point motion. They have also pointed out that the inevitable random variations in the tunneling amplitude with position, which we have not considered at all here, cause a relevant perturbation. 5.11. SUMMARY

We have discussed the origin of spontaneous ordering in multicomponent fractional quantum Hall effect systems. For real spin at filling factors v = 1 this spontaneous ferromagnetism induces a very large charge excitation gap even in

SUMMARY

217

the absence of a Zeeman gap. The charged excitations are interesting topological skyrmion-like objects. Magnetic resonance experiments [26,27] have confirmed this remarkable picture which was developed analytically by Sondhi et al. [24] to explain the numerical results of Rezayi [96]. We have discussed in detail a pseudospin analogy which shows how spontaneous interlayer coherence in double-layer quantum Hall systems arises. This coherent XY phase order occurring over long length scales is essential to explain the experimental observations of Murphy et al. [39] described in Chapter 2. The essential physics is condensation of a charge-neutral bosonic order parameter field (pseudospin magnetization). This condensation controls the charge excitation gap and is very sensitive to interlayer tunneling and parallel magnetic field. We summarize a portion of this rich set of phenomena in the schematic zero-temperature phase diagram of double-layer systems shown in Fig. 5.12. First consider the plane with ASAS= 0 (zero tunneling). We have argued that the system develops spontaneous interlayer phase coherence despite the fact that the tunneling amplitude is zero. However, if the layer spacing d exceeds a critical value d*, the system is unable to support a state with strong interlayer correlations and the spontaneous U(1) symmetry breaking is destroyed by quantum fluctuations [41]. At this same point we expect the fermionic gap Ap to collapse. Little is known about the nature of this quantum transition, which can be viewed as arising from proliferation of quantum-induced vortices (merons). This is similar to the quantum XY model, but in the present case, the merons are fractionalstatistics anyons which will presumably change the universality class of the transition [46]. At finite tunneling, the U(1) symmetry is destroyed and the quantum fluctuations are gapped and hence stabilized. This causes the critical layer spacing to increase, as shown in Fig. 5.12. The third axis in the figure is the tilt of the magnetic field. Magnetic flux between the layers causes the order parameter to want to tumble. For small tilts, the system is in a commensurate phase, with the

Fig. 5.12. Schematic zero-temperature phase diagram (with d/l increasing downward). The lower surface is d*, below which d > d* and the interlayer correlations are too weak to support a fermionic gap, Ap. The upper surface gives I??, the commensurate- incommensurate phase boundary. As d approaches d*, quantum fluctuations soften the spin stiffness and therefore increase B;.

218

MULTICOMPONENTQUANTUM HALL SYSTEMS

order parameter tumbling smoothly. However, above a critical value of the parallel field, this tumbling costs too much exchange energy, and the system goes into an incommensurate phase which spontaneously breaks translation symmetry (in the absence of disorder). This phase transition has been observed by Murphy et al. [38,39] through the rapid drop in the charge gap as the field is tilted. We have presented arguments that the charge gap is determined by the cost of creating a highly collective object: a pair of fractionally charged vortices connected by a string. It is the decrease of the string tension with tilt that causes the extremesensitivity to small tilts. In addition to all this, there is (forzero tunneling) a finite-temperature Kosterlitz-Thouless phase transition. If observed experimentally, this would represent the first finite-temperature phase transition in a quantum Hall system. We have tried to keep the present discussion as qualitative as possible. The reader is directed to the many references for more detailed discussion of technical points. We close by noting that there are still many open questions in this very rich field concerning such things as edge states in multicomponent systems, proper treatment of quantum fluctuations for the highly collective excitations that appear to exist in these systems, and the nature of the phase transition at the critical value of the layer separation. ACKNOWLEDGMENTS The work described here is the result of an active and ongoing collaboration with our colleagues L. Brey, R. Cat&, H. Fertig, K. Moon, H. Mori, K. Yang, L. Belkhir, L. Zheng, D. Yoshioka, and S.-C. Zhang. It is a pleasure to acknowledge numerous useful conversations with D. Arovas, S. Barrett, G. Boebinger, J. Eisenstein, Z. F. Ezawa, M. P. A. Fisher, I. Iwazaki, T.-L. Ho, J. Hu, D. Huse, D.-H. Lee, S . Q. Murphy, A. Pinczuk, M. Rasolt, N. Read, S. Renn, M. Shayegan, S. Sondhi, M. Wallin, X.-G. Wen, Y.-S. Wu, and A. Zee. The work at Indiana University was supported by NSF Grant DMR-9416906. REFERENCES 1. R. E. Prange and S. M. Girvin, eds., The Quantum Hall Effect, 2nd ed., SpringerVerlag, New York, 1990. 2. A. H. MacDonald, ed., Quantum Hall Effect: A Perspective, Kluwer, Boston, 1989. 3. T. Chakraborty and P. Pietiliiinen, The Fractional Quantum Hall Effect: Properties of an incompressible quantumjuid, Springer Series in Solid State Sciences 85, SpringerVerlag New York, 1988, and references therein. 4. M. Stone, ed., The Quantum Hall Effect, World Scientific, Singapore, 1992. 5. R. B. Laughlin, Chapter 7 in The Quantum Hall Effect, 2nd ed., edited by R. E. Prange and S. M. Girvin, Springer-Verlag, New York, 1990.

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29. R. L. Willett, J. P. Eisenstein, H. L. Stonner, D. C. Tsui, A. C. Gossard, and J. H. English, Phys. Rev. Lett. 59, 1776 (1987). 30. P. L.Gammel, D. 1. Bishop, J. P. Eisenstein, J. H. English, A. C. Gossard, R. Ruel, and H. L. Stormer, Phys. Rev. B 38,10128 (1988). 31. J. P. Eisenstein, R. L. Willett, H. L. Stonner, D. C. Tsui, A. C. Gossard, and J. H. English, Phys. Rev. Lett. 61,997 (1988). 32. F. D. M. Haldane and E. H. Rezayi, Phys. Rev. Lett. 60,956 (1988);60,1886: (1988); E. H. Rezayi and F. D. M. Haldane, Phys. Rev. B 42,4352 (1990). 33. D. Yoshioka, A. H. MacDonald, and S. M. Girvin, Phys. Rev. B 39,1932 (1989). 34. L. Belkhir and J. K. Jain, Phys. Rev. Lett. 70,643 (1993); L. Belkhir, X. G. Wu, and J. K. Jain, Phys. Rev. B 48,15245 (1993);X. C. Xie and F. C. Zhang, Mod. Phys. Lett. B 5,471 (1991). 35. M. Greiter and F. Wilczek, Nucl. Phys. B 370,577 (1992);M. Greiter, X.-G. Wen, and F. Wilczek, Phys. Rev. Lert. 66,3205 (1991);Nucl. Phys. B 374,567 (1992). 36. Tapash Chakraborty and P. Pietilainen, Phys. Rev. B 39,7971 (1989); V. Halonen, P. Pietilgnen, and T. Chakraborty, Phys. Rev. B 41, 10202 (1990);J. D. Nickila, Ph.D. thesis, Indiana University, 1991. 37. See for example M. Rasolt, Solid State Phys. 43,94 (1990) and references therein. 38. J. P. Eisenstein, Chapter 2, this volume. 39. S. Q. Murphy, J. P. Eisenstein, G. S. Boebinger, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 72,728 (1994). 40. M. B. Santos, L. W. Engel, S. W. Hwang, and M. Shayegan, Phys. Rev. B 44, 5947 (1991); T. S. Lay, Y. W. Suen, H. C. Manoharan, X. Ying, M. B. Santos, and M. Shayegan, Phys. Rev. B 50,17725 (1994). 41. A. H. MacDonald, P. M. Platzman, and G. S . Boebinger, Phys. Rev. Lett. 65, 775 (1990). 42. X. G. Wen and A. Zee, Phys. Rev. Lett. 69,1811 (1992);X . G. Wen and A. Zee, Phys. Rev. B 47,2265 (1993). 43. Z. F. Ezawa and A. Iwazaki, Int. J. Mod. Phys. B 6,3205 (1992);8,2111 (1994);Phys. Rev. B47,7295 (l993);48,15189(1993);Phys.Rev. Lett.70,3119(1993);Z. F. Ezawa, A. Iwazaki, and Y. S. Wu, Mod. Phys. Lett. B 7 , 1223 (1993);7, 1825 (1993). 44. Z. F. Ezawa, Phys. Rev. B 51, 1 1 152 (1995). 45. Kun Yang, K. Moon, L. Zheng, A. H. MacDonald, S. M. Girvin, D. Yoshioka, and S. C. Zhang, Phys. Rev. Lett. 72. 732 (1994). 46. K. Moon, H. Mori, K. Yang, S. M. Girvin, A.H. MacDonald, L. Zheng, D. Yoshioka, and S.-C. Zhang, Phys. Rev. B 51,5138 (1995). 47. K. Moon, H. Mori, K. Yang, L. Belkhir, S. M. Girvin, A. H. MacDonald, L. Zheng, D. Yoshioka, and S.-C. Zhang, Phys. Rev. B, submitted (1996). 48. It is important to realize that if @ [ Z ]is antisymmetric among spin-up electrons and among spin-down electrons, the expectation value of an operator that does not flip spins can be calculated as if the “up”- and “down”- spin electrons were distinguishable, thus avoiding all the complications associated with the antisymmetrization. 49. See, for example, D. Yoshioka, A. H. MacDonald and S.M. Girvin, Phys. Rev. B 38, 3636 (1988). 50. See, however, G. Baskaran and E. Tosatti, Europhys. Lett. 25,369 (1994).

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51. A. H. MacDonald, Science 267,977 (1995). 52. Xiu Qiu, R. Joynt, and A. H. MacDonald, Phys. Rev. B 40, 11943 (1989). 53. C. B. Hanna and A. H. MacDonald, Bull. Am. Phys. SOC.40,645 (1995); Phys. Rev. B, to appear (1996). 54. Numerical tests and further discussion of the competition between the hollow-core

state and the Pfaffian state can be found in L. Belkhir, Ph.D. thesis, SUNY Stonybrook, 1993, unpublished. 55. J. P. Eisenstein, H. L. Stormer, L. Pfeiffer, and K. W. West, Phys. Rev. Lett. 62,1540 (1989).

56. For simplicity we write the wavefunction for the essentially equivalent v = 1/2 case 57. 58. 59. 60. 61. 62. 63.

rather than dealing with the complications of the higher Landau level wavefunctions needed to describe v = 512. T.-L. Ho, Phys. Rev. Lett. 75, 1186 (1995). B. I. Halperin, Surf. Sci. 305, 1 (1994). Chapter 6, this volume. D.-H. Lee and C. L. Kane, Phys. Rev. Lett. 64,1313 (1990). S. R.Renn has recently considered mutual drag effects in mmm states of double well systems: Phys. Reu. Lett. 68,658 (1958). J.-M. Duan, Eur. Phys. Lett. 29,489 (1995). S. C. Zhang, H. Hansson, and S. Kivelson, Phys. Rev. Lett. 62, 82 (1989); 62, 980 (1989).

64. D. H. Lee and S. C. Zhang, Phys. Rev. Lett. 66,1220 (1991). 65. A. H. MacDonald and S.-C. Zhang, Phys. Rev. B 49,17208 (1994). 66. A. Lopez and E. Fradkin, Phys. Rev. Lett. 69, 2126 (1992); Phys. Rev. B 47, 7080 (1993); Phys. Rev. B 51,4347 (1995).

67. 68. 69. 70. 71. 72. 73. 74.

75. 76.

77. 78.

J. Zang, D. Schmeltzer, and J. L. Birman, Phys. Rev. Lett. 71,773 (1993). N. E. Bonesteel, Phys. Rev. B 48, 11484 (1993); Bull. Am. Phys. SOC.40,645 (1995). T.-L. Ho, Phys. Rev. Lett. 73,874 (1994). J. Dziarmaga, Vortices and hierarchy of states in double-layer fractional Hall effect, preprint (cond-mat/9407085). A. Zee, in Field Theory, Topology and Condensed Matter Physics, edited by H. B. Geyer, Springer lecture Notes in Physics, Springer-Verlag, New York, 1995. J. Froehlich, T. Kerler, U. M. Studer, and E. Thiran, A classification of quantum Hall fluids, Nuc. Phys. B 453,670 (1995). D.-H. Lee and X.-G. Wen, Phys. Rev. B 49,11066 (1994). See, for example, A. H. MacDonald in Proc. Les Houches Summer School on Mesoscopic Physics, edited by E. Akkermans, G. Montambeaux, and J.-L. Pichard, Elsevier, New York, 1996. P. J. Forrester and B. Jancovici, J . Phys. (Paris) Lett. 45, L583 (1984). A. H. MacDonald, H. C. Oji and S. M. Girvin, Phys. Rev. Lett. 55,2208 (1985). The SMA is not reliable for compressible states or for hierarchical incompressible states. See A. Pinczuk, Chapter 8,this volume.

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79. W. Kohn, Phys. Rev. 123, 1242 (1961);C. Kallin and B. I. Halperin, Phys. Rev. B 31, 3635 (1985). 80. R. Renn and B. W. Roberts, Phys. Rev. B 48,10926 (1993). 81. H. MacDonald, unpublished notes. 82. An exception occurs for the Yl,l,l wavefunction, for which v = 1. In this case there are no states in the lowest Landau level, to which Yl*l,lis coupled by the total charge-density operator. The reasons for this will become clearer later in the chapter. 83. Y. W. Suen et al., Phys. Rev. Lett. 68, 1379 (1992); J. P. Eisenstein et al., Phys. Rev. Lett. 68, 1383 (1992). 84. E. H. Rezayi and F. D. M. Hddane, Bull. Am. Phys. SOC.32,892 (1987); S. He, S. Das Sarma, and X. C. Xie, Phys. Rev. B 47,4394 (1993). 85. T. Chakraborty and P. Pietilainen, Phys. Rev. Left. 59,2784 (1987). 86. The correlation functions were obtained by using a hypernetted chain approximation as described in Refs. [lOO]. 87. A brief discussion of these modes has been given: A. H. MacDonald, M. Rasolt, and F. Perrot, Bull. Am. Phys. SOC.33,748 (1988). 88. The absorption strength of this mode will be reduced by a factor of [d/cos(0)112, where 1 is the wavelength of the infrared light and O is the angle between the propagation direction and the normal to the 2D layers. 89. S.-C. Zhang, private communication, 1995. 90. On the other hand, we remind the reader that because of analyticity and Pauli principle constraints on the Landau level wavefunctions, Hund's rule does not always apply. We saw, for instance in Section 5.2, that in some cases, particularly those with even-denominator filling factors such as v = 5/2, that the interaction energy can be optimized by a spin liquid singlet(-like) state. 91. E. Fradkin, Field Theories of Condensed Matter Systems, Addison-Wesley, Reading, Mass., 1990. 92. R. Rajaraman, Solitons and Instantons, North-Holland, Amsterdam, 1982. 93. The precise definitions of the skyrmion and antiskyrmion energies in the presence of the long-range Coulomb interaction is subtle. For a more complete discussion, see Ref. [46]. 94. I. V. Kukushkin, N. J. Pulsford, K. von Klitzing, R.J. Haug, K. Ploog, H. Buhmann, M. Potemski, G. Martinez, and V. B. Timofeev, Europhys. Lett. 22,287 (1993). 95. A. Usher, R. J. Nicholas, J. J. Harris, and C. T. Foxon, Phys. Rev. B 41, 1129 (1990). 96. E. H. Rezayi, Phys. Rev. B 36,5454 (1987);Phys. Rev. B 43,5944 (1991). 97. X.-G. Wu and S. L. Sondhi, Phys. Rev. B 51, 14725 (1995). 98. M. V. Berry, Proc. Roy. SOC.London Ser. A 392,45 (1984). 99. C. Kallin and B. I. Halperin, Phys. Rev. B 31,3635 (1985). 100. 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);M. Rasolt, B. 1.Halperin, and D. Vanderbilt, Phys. Rev. Lett. 57, 126 (1986). 101. Kun Yang, L. K. Warman, and S . M. Girvin, Phys. Rev. Lett. 70, 2641 (1993).

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102. We require oxx= 0 so that the flux can be added adiabatically and reversibly. 103. T. Nakajima and H. Aoki, Phys. Rev. B 52, 13780 (1995); Phys. Rev. B 51, 7874 (1995). 104. A. Karlhede, unpublished. 105. A single-layer system at Landau level filling factor Y = 1/2 has no charge gap but does show interesting anomalies which may indicate that it forms a liquid of composite fermions. For a discussion of recent work, see Chapters 6 and 7 in this volume. 106. H. A. Fertig, Phys. Rev. B 40,1087 (1989). 107. G. S. Boebinger, H. W. Jiang, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 64, 1793 (1990); G. S. Boebinger, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 45, 11391 (1 992). 108. X. M. Chen and J. J. Quinn, Phys. Rev. B 45,11054 (1992);R. C k e , L. Brey, and A. H. MacDonald, Phys. Rev. B 46,10239 (1992). 109. This phrase was the original title of Ref. [24] (prior to the editorial process). 110. S. Datta, Phys. Lett. 103A, 381 (1984). 11 1. X. G. Wen and A. Zee, Sideways tunneling and fractional Josephson frequency in double layered quantum Hall systems, preprint (condmat/9402025), 1994. 112. M. B. Pogrebinskii, Fiz. Tekh. Poluprovodn. 11,637 (1977)[Sou. Phys. Semicond. 11, 372 (1977)l;P. M. Price, Physica (Amsterdam) 117B, 750(1983);P. M. Solomon, P. J. Price, D. J. Frank, and D. C. La Tulip, Phys. Rev. Lett. 63, 2508 (1989); T. J. Gramila, J. P. Eisenstein, A. H. MacDonald, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 66, 1216 (1991) and references therein. 113. B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B47,7312 (1993);V. Kalmeyer and S.-C. Zhang, Phys. Rev. B 46,9889 (1992). 114. R. Cate, L. Brey, H. Fertig, and A. H. MacDonald, Phys. Rev. B 51, 13475 (1995). 115. J. R. Schrieffer,in The k s s o n ofQuantum Theory,edited by J. De Boer, E. Dal, and 0. Ulfbeck, Elsevier, New York, 1986. 116. For simplicity we ignore the fact that different flavors may have different core energies. 117. I. Tupitsyn, M. Wallin, and A. Rosengren, Phys. Rev. B 53, R 7614 (1996). 118. H. Mooij, in Percolation, Localization, and Superconductivity, edited by A. M. Goldman and S. A. Wolf, Plenum Press, New York, 1984. 119. K. Yang and A. H. MacDonald, Phys. Rev. B 51,17297 (1995). 120. Note the order of limits here. The fact that the order matters really defines what we mean by spontaneously broken symmetry. 121. See, for example, George Griiner, Density Waves in Solids, Addison-Wesley, Reading, 1994, Chapter 7. 122. J. Yang and W.-P. Su, Phys. Rev. B 51, 4626 (1995); J. Y., Phys. Rev. B 51, 16954 (1995). 123. P. Bak, Rep. Prog. Phys. 45,587 (1982);M . den Nijs in Phase Transitions and Critical Phenomena, Vol. 12, edited by C. Domb and J. L. Lebowitz, Academic Press, New York, 1988, pp. 219-333. 124. Murphy et al. have recently revised slightly downward their estimates of Asas in their samples. Hence the numbers presented here are slightly different than those in

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previous publications. See Chapter 2 in this volume for further discussion of this point. 125. We are considering for the moment the case of zero temperature so that B,, is unrenormalized by thermal fluctuations.We also treat the problem classically. 126. N. Read, Phys. Rev. B, 52, 1926 (1995). 127. M. P. A. Fisher, private communication.

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

6

Fermion Chern-Simons Theory and the Unquantized Quantum Hall Effect B. I. HALPERIN Physics Department, Harvard University, Cambridge, Massachusetts

6.1. INTRODUCTION

During the 1980s, experimental and theoretical studies of two-dimensional electron systems in strong magnetic fields were focused on the integer and fractional quantized Hall effects. More recently, however, it has become clear that these systems are also extremely interesting in the vicinity of even-denominator Landau level filling fractions,such as v = 1/2, where the quantized Hall effect does not occur. The theoretical approach that has proved most useful for understanding the unquantized systems is the fermion Chern-Simons theory [l-81, itself an outgrowth of the composite fermion picture developed by J. K. Jain to describe the most prominent features of the fractional quantized Hall effect [9, lo]. However, many of the computational-techniques were originally developed by R. B. Laughlin and others to explore the properties of anyon systems advanced as a model for high-temperature superconductivity [ll-161. In the fermion Chern-Simons approach, the electron system is subjected to a mathematical transformation which converts it into a new system of fermions that has, in addition to the ordinary Coulomb interaction, an interaction via a fictitious vector potential, known as the Chern-Simons field. The transformation is sometimesdescribed by saying that an even number of fictitious magnetic flux quanta are attached to each electron. In many circumstances it appears that a mean-field approximation to the transformed Hamiltonian embodies much of the physics of the original electron problem and is an excellent starting point for further perturbative calculations. A remarkable consequence of the fermion Chern-Simons theory is that for v = 1/2, and for certain other even-denominator fractions, the ground state and Perspectives in Quantum Hall EfJects, Edited by Sankar Das Sarma and Aron Pinczuk. ~~

~

ISBN 0-471-11216-X

0 1997 John Wiley & Sons, Inc.

225

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FERMION CHERN-SIMONS THEORY

the low-energy excitations can be described by a modified Fermi-liquid theory, similar in many ways to the theory of electrons in zero magnetic field [S, 61. For a magnetic field B which deviates by a small amount AB from the field that corresponds to an even-denominator filling fraction such as v = 1/2, the elementary fermion excitations no longer move in straight lines but rather follow semiclassical circular orbits. The radius R,* of these orbits is just the cyclotron radius in an effective field B* proportional to AB. Specifically,near v = 1/2, we have

where the Fermi wavevector kF is related to the electron density by

kF = (4xne)'/'

(2)

and B* is just equal to AB in this case. This prediction of the theory has received dramatic support from several recent experimentsthat have essentially measured the effectivecyclotron diameter 2R:, as will be discussed in Sections 6.7 and 6.9.1. If the magnetic field B is chosen so that the mean-field ground state of the transformed fermions contains an integer number of filled Landau levels, the mean-field theory has an energy gap for excitations. If the energy gap is larger than the temperature and larger than any smearing effects due to scattering from impurities, the electron system should exhibit a quantized Hall plateau. As was noted originally by Jain [9], integer fillings for the transformed fermions correspond to electron filling fractions of the form v=-

P

2np

+1

where p and n are integers. These are precisely the filling fractions where the most prominent fractional quantized Hall plateaus have been seen experimentally. The mean-field theory, as noted above, is only a first approximation to a complete analysis. Interactions via the Chern-Simons field are very important at long wavelengths, and they must be taken into account, at least at the level of the random phase approximation (RPA) if one wishes to predict transport properties, dyanmic response functions, or collective excitation spectra at long wavelengths [1,5,7,12,17]. Fluctuations beyond the mean-field theory are also important, in a more subtle way, in determining the overall energy scale for excitations,including the values of the energy gaps at the principal fractional Hall states given by Eq. (3) [ 5 ] . The purpose of this article is to give an overview of results obtained using the fermion Chern-Simons theory of a partly filled Landau level, together with a very brief review of some of the experimentssupporting the theory. No attempt will be made to give details of the calculations leading to the theoretical results; the reader is referred to the original literature for these details.

FORMULATION OF THE THEORY

227

In Section 6.2 we begin with the mathematical formulation of the fermion Chern-Simons theory, and we obtain the mean-field results for the Fermi-liquid states and for the principal fractional quantized Hall states. In Section 6.3 we review the results of Halperin, Lee, and Read (HLR) [S] for the renormalization of the energy scale and the effective mass, due to fluctuations in the ChernSimons gauge field. Response functions obtained from the RPA and related approximations are reviewed in Section 6.4. Extensions of the theory to even-denominator filling fractions more complicated than v = 1/2 or v = 1/4 are discussed in Section 6.5. In Section 6.6 we discuss the effects of weak disorder, due to impurities, on the Fermi-liquid states of a partially filled Landau level, and we discuss the question of the magnitude of the longitudinal resistivity pxx.Applications to the theory of surface acoustic wave propagation are discussed in Section 6.7 and compared to the experiments of Willett and co-workers [18-201. In Section 6.8 we discuss other developments in the fermion Chern-Simons theory, including recent investigations of the validity of the HLR theory for the asymptotic low-energy behavior of the Fermi-liquid states, the one-electron Green’s function and applications to tunneling experiments, the one-particle Green’s function for the transformed fermions, bilayer systems,and inhomogenous geometries such as occur at the boundary of a fractional quantized Hall system. Some other experiments that bear on the validity of the fermion ChernSimons theory are reviewed briefly in Section 6.9, and concluding remarks are contained in Section 6.10.

6.2. FORMULATION OF THE THEORY We begin by considering a two-dimensional interacting electron system, in a uniform positive background, in a uniform external magnetic field B. We assume that all the electron spins are polarized parallel to B, so that we may ignore the spin degree of freedom. We then make a unitary transformation, originally employed by Leinaas and Myrheim in 1977, in the paper that developed the concept of fractional statistics in two-dimensional systems [21,22]. In first quantized notation, the transformation is written as

where Yeis the many-body wavefunction of the electrons, Ytris the transformed wavefunction, is an even integer, and zj is the position of the jth particle in complex notation:

6

Note that if the electron wavefunction obeys Fermi statistics so that Ye

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FERMION CHERN-SIMONS THEORY

6

changes sign under the interchange of two particles, and if is even, Ytrwill have the same property. If were chosen as an odd integer, the transformation (4) would convert fermions to bosons, or vice versa. A transformation from fermions to bosons was employed by Girvin and MacDonald and others to explore analogies between the quantized Hall effect and superfluidity [17]. In the present chapter, however, we restrict our attention to the case where the transformed particles are fermions. As a conceptual tool, we may extend our discussion to odd-integer values of 6, or even to fractional values of but this would then imply that the initial particles (“electronsy’)obey Bose statistics or fractional statistics (“anyons”). The unitary transformation (4)of the wavefunction,which may be described as a singular gauge transformation, requires a corresponding transformation of the Hamiltonian. Using second quantized notation, where ++(T)is the creation operator for a transformed fermion at point 7,we find

4

4,

where V is the potential energy operator and K is a kinetic energy of the form .

c

z(7)

Here mb is the electron band mass, is the electromagneticvector potential, satisfyingv x A’ = B, and Z(r) is a vector potential that depends on the positions of the particles and is given by

-a (-r ) = 4

1.

dzQ“7-F’)p(7)

and p ( 7 ) is the operator, which measures the density of particles at point 7. The vector potential Z(7) is referred to as a Chern-Simons field because in a Lagrangian formulation, it is obtained by including a term of the ChernSimons form in the field Lagrangian [22]. Throughout this chapter we use units where h = e/c = 1, and the electron charge is -e. The electron creation operator @L(T)is related to the transformed fermion operator ++(T)by

where arg(7 -7’) is the angle that vector (7-7’) forms with the x axis. The

FORMULATION OF THE THEORY

229

potential energy Y has the same form whether expressed in terms of $ or $e, as does the density p(F):

It follows from Eq. (8) that the Chern-Simons magnetic field, defined by

is related to the particle density by

In a translationally invariant system, the mean-field Hamiltonian H, may be obtained by replacing b ( 3 by its mean value,

where n, is the average electron density, and by ignoring the potential energy Y. Thus we have H , =2% where

V

x A*

- iV + A*(7)]2$(r)d2r

(15)

= B* = B - 2n&,

If we define an effective filling factor p by p=-

2zn, B*

the electron filling factor v = 2znJB is related to p by

If p is a positive or negative integer, the mean-field ground state has ( p i filled Landau levels, and the fraction v has the special form stated in Eq. ( 3 ) [1,9]. Within the mean-field approximation, we would identify the energy gap of the fractional quantized Hall state at filling fraction v with the cyclotron energy in the field B*:

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FERMION CHERN-SIMONS THEORY

In the case when v is an even denominator fraction of the form v = 1/2n, if we choose = 2n, we find, according to (16),B* = O-the external magnetic field B is precisely canceled by the average Chern-Simons field 6. Thus the mean-field ground state is a filled Fermi sea, with a Fermi momentum k, given by Eq. (2) [5,6]. The value of k, differs by a factor of from that of electrons in zero magnetic field, because we assume that only one spin state is occupied here. Of course, the mean-field approximation is at best a first estimate of the properties of the actual interacting Fermi system. What one hopes is that the mean-field state has the correct quantum numbers and symmetry properties, and that the exact ground state may be obtained (in principle) by a convergent perturbation expansion starting from the mean-field state. The perturbation Hamiltonian is the difference H - H,. In fact, there are two contributions: the Coulomb interaction V, which has been omitted from H,, and the difference between the actual Chern-Simons vector potential Z(i(r),which is an operator depending on the positions of all the particles, and the average value incorporated in Eq. (16). There is no obvious small parameter in this perturbation expansion; indeed, the coupling constant to the gauge field is the coefficient which is 2 2 in physical applications. However, there is at least some reason for optimism. In the case of fractional quantized Hall fractions, there is an energy gap separating the ground state from the lowest excited state; this is obviously favorable for the possible convergence of perturbation theory. For even-denominator fractions, such as v = 1/2, there is no energy gap in the mean-field theory; however, the existence of a Fermi surface is still very helpful. In general, perturbations about the mean-field theory produce intermediate states in which there are at least two holes below the Fermi energy and two particles above it. The density of states for such excitations tends to zero at low energies, which is the reason perturbation theory can be used to discuss the properties of an ordinary Fermi liquid, such as liquid 3He.Thus, if the matrix elements of the pertrbation are not too divergent at long wavelengths, some type of Fermi-liquid description may be valid for the two-dimensional electron system at even-denominator filling fractions.

4

fi

4,

6.3. ENERGY SCALE AND THE EFFECTIVE MASS As mentioned Section 6.1, one important effect of the gauge-field fluctuations is a large renormalization of the fermion effective mass, and consequently of the energy scale for excitations [S]. Indeed, the mean-field expression (19) for the energy gaps at the principal quantized Hall fractions is in clear violation of an important physical principle. In the limit where the electron-electron interaction is weak compared to the electron-cyclotron frequency, B/mb, the low-energy states have electrons entirely in the lowest Landau level. Then the energy gap must be independent of the electron band mass and must be linearly proportional to the Coulomb energy scale e2/& where E is the dielectric constant of the surrounding medium and lB = IBI- 1/2 is the magnetic length.

ENERGY SCALE AND THE EFFECTIVE MASS

231

If conventional Fermi-liquid theory applied to this system, we would expect that at least for large values of IpI, Eq. (19) should be replaced by [5,23]

where m* is the effective mass of the composite fermions. Thus, in the limit mbe2<< EB’”,we should find, for a fixed filling fraction v, that m* is proportional to (e21e/E)It should be noted that E,(v) is defined as the energy to add one quasiparticle and one quasi-hole, far away from each other, to the quantized Hall ground state at filling fraction v. If the creation operator I,V(T’) is applied to the quantized Hall ground state, it is equivalent, according to Eq. (lo), to simultaneously adding an electron and turning on a solenoid that pierces the layers at point 7 and contains quanta of magnetic flux. The impulse electromotive force from turning on the solenoid will result, after several cyclotron periods, in a hole at point 7,with v 4 electrons missing, while the system expands so that the missing charge appears at the boundary [I 3. The added electron then gives a net charge ( - e)/($p + 1) at point 7, consistent with the definition of the Laughlin quasiparticle [24]. Similarly,the annihilation operator I/@) creates a fractionally charged quasihole at point 7. Halperin, Lee, and Read (HLR) [S] considered explicitly the contribution to the fermion self-energy that arises from interactions with long-wavelengthfluctuations in the Chern-Simons gauge field. For the case where the electrowelectron interaction has the coulombic form, V(r)= e2/w for large separations I , they found that the contribution of the lowest-order Feynman diagram gave a logarithmic divergence to the effectivemass at low energies, which may be written as

’.

4

where o is the energy differencefrom the Fermi energy. They also gave a plausibility argument that this form might not be changed by higher-order Feynman diagrams. Thus perturbation theory gives a leading term in m* that is independent of mb, as required. However, nondiverging contributions to m* arise in principle from all orders in perturbation theory and from all ranges of intermediate wavevectors, and within the perturbation theory itself, there is no obvious way to see why these nondiverging terms should be independent of mb. In most applications of fermion Chern-Simons theory, therefore, one simply assumes that the nondivergent terms are independent of mb,in the limit where mb is small, and one fits the unknown constants to results of other calculations or experimental measurements. The mathematical reason for the appearance of (e2/c)- in the diverging part of m* is not difficult to understand. The large mass renormalization is produced by fluctuations in the Chern-Simons magnetic field, which are directly propor-

’

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tional to fluctuations in the electron density. The larger the value of the repulsive Coulomb interaction, the smaller are the quantum mechanical fluctuations of the density at long wavelengths and low frequencies.Moreover,it was found by HLR that if the electron-electron interaction falls off faster than l/r at large separations, low-frequency long-wavelength density fluctuations become more important, and m* is found to diverge even more strongly than logarithmically as w+O. Whether or not the behaviors obtained by HLR in the iong-wavelength limit are indeed correct has been the subject of some theoretical controversy, however, as will be discussed in Section 6.9. If the effectivemass indeed diverges logarithmicallyfor w -0 at v = l/$, then, of course, Landau Fermi-liquid theory does not strictly hold in its usual form. Nonetheless, it seems reasonable to use Eq. (20) to obtain the energy gap E, for large values of IpI if we evaluate m* self-consistentlyat an energy w that is equal to E,. (This procedure is supported by a recent analysis of the leading Feynmandiagram contributions [23].) Alternatively, we may consider Eq. (20) to provide a definition for the effective mass at energy w = E,. If we assume that m* has the divergence predicted by (21) and that there are no other divergingcorrections, we are led to the following asymptotic form for the energy gap E,, at v = p/(& + l), for large Ip I [S, 231: 1 2lI2n e2 Ep-$3/2 el, D(ln D + C)

(22)

where D = + 1I is the denominator of the filling fraction, and the constant C depends on the value of $ and on the short-distance behavior of the electronelectron interaction. (Unfortunately, there was a numerical error in the corresponding formula for the energy gap quoted in Ref. [5]. For = 2, the prefactor in Eq. (22) differs by a factor 1.23 from the value quoted in Ref.[5].) In deriving (22),we have made use of the relation(&%,) 1; 1(2/$)1'2, which is valid for large values of p. For the case of $ = 2, with pure Coulomb interactions and no Landau level mixing, if one compares Eq. (22) to existing numerical results from exact diagonalization of finite systems for the energy gap at v = 1/3,2/5, and 3/7, one finds the best fit for a choice C x 3. This value of C is larger than the value of In D for these fractions, and indeed the logarithmic term would not be expected to be much larger than C for any attainable filling fraction. In fact, the numerical data for the energy gaps could be fit perfectly well by ignoring the logarithmic dependence of m*,using instead Eq. (20) with the approximation (for $ = 2)

-

6

Recently, Morfand d'Ambrumenil[25] have extracted an estimate of m* from an analysis of the ground-state energies of finite systems on a sphere, close to v = 1/2, which do not correspond to quantized Hall fractions of the form

RESPONSE FUNCTIONS

233

+

v = p/(2p 1). They have obtained an estimate of m* which is larger than Eq. (23) by approximately 50%. In any case, one would expect that in real electron layers, finite thickness effects, which tend to soften the Colomb repulsion at short distances, and effects of Landau level mixing, will tend to increase the constant C in Eq. (22),thereby decreasing the energy gaps and increasing further the effective mass. In practice, for GaAs samples at typical densities, the effective masses obtained from the foregoing considerations should be of order of 5 to 20 times the band mass mb(or of the order of 0.3 to 1.2times the free electron mass), with a relatively weak dependence on the energy w or on the integer p. 6.4. RESPONSE FUNCTIONS

A very important task of any theory of the two-dimensional electron system is to calculate the linear response functions, which give the current and charge densities induced by an external electromagnetic field, at wavevector and frequency w. As mentioned in Section 6.1, the Hartree theory itself does not give good results for the response functions. However,a simple approximation, which gives surprisingly good results in the fermion Chern-Simons theory, is the random phase approximation (RPA),or time-dependentHartree approximation. The fact that a good description of the long-wavelengthresponsefunctions can be obtained using the RPA for systems with a Chern-Simons interaction was noted by Laughlin and co-workers in the context of anyon superconductivity [12-163 and by Girvin and MacDonald, Zhang et al., and Lee and Fisher [17] for the boson Chern-Simons theory of the fractional quantized Hall effect. The RPA was used for the fermion Chern-Simons theory of the fractional quantized Hall states by Lopez and Fradkin [1,3] and by HLR [S] for the Fermi-liquid states. In the RPA, the fermions are treated as free particles, having the band mass mb, responding to self-consistentelectric and magnetic fields which are the sum of the applied electric and magnetic fields, the Columb field produced by the induced charge density ( d p ) , and the Chern-Simons fields produced by the induced charge density and currents. Specifically,there will be an induced Chern-Simons electric field 6 2 and a magnetic field 6b, which are related to ( 6 p ) and the induced current (67)by 6b=24(6p)

(24)

The RPA response functions are related by.a pair of linear equations to the response functions of a noninteracting system in the uniform effective field B*. The effect of equation (24) and (25) is to couple together the longitudinal and transverse response functions, even at filling fractions such as v = 1/2, where B* = 0.

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We shall concentrate here on the density response function K,,(q, a),which gives the response of the electron density to an external potential &7, t). For wavevectors q << k , the RPA response function has a sharp pole at the cyclotron frequency w, = B/m,, which exhausts the weight of the f-sum rule in the limit q 4 0 , as required by Kohn’s theorem. For B* = 0, one finds, in addition, that Im K J q , a)has a particle-hole continuum contribution that covers a range where

The weight in the particle-hole continuum is strongly concentrated at low frequencies, however, where there is an overdamped mode corresponding to a pole on the imaginary frequency axis at w = - iyg, with [S]

Here a,(q) is the longitudinal conductivity at wavevector ij’ (we take ij‘ (1 n), and &(q) is the Fourier transform of the electron-electron interaction, which in the Coulomb case is given by

The RPA result for a(q ,), is [ S ]

at v = 1/2, for a system without impurity scattering

Thus Eq. (26)leadsto a relaxation rate yq cc q2 for the Coulomb case. In the case of finite range interactions, where &(q) approaches a constant for q -0, Eq. (26) leads to a relaxation rate yq cc q3. The spectrum of density fluctuations represented by the RPA form of Im K p pis illustrated schematically in Fig. 6.1. In the presence of impurity scattering, the composite fermion motion will be cut off be a finite transport mean free path 1. Equation (28)then appliesfor q >> 2/1, while for q << 2/1 the RPA result is e2

a, = 4dkF1 At long wavelengths, then, Eq. (26) reduces to the classical formula for charge relaxation in a conducting sheet, yq = 2nqoX&.

RESPONSE FUNCTIONS

235

Figure 6.1. Frequency spectrum of density fluctuations at filling fraction v = 1/2, at long wavelengths, according to the RPA approximation. The heavy solid line shows the cyclotron mode, while the shaded area shows the particle-hole continuum, 0 <w <40, The dominant continuum contribution to Im K,,(4, a), however, comes from frequencies 0 <w 5 yp = q2e2/4~hk,,indicated by the heavily shaded region. (We have assumed here Coulomb interactions and no impurity scattering.)

What do we expect to happen to the density response function K,, when we go beyond the RPA, in particular if we take into account the mass renormalization discussed in the preceding section? The longitudinal conductivities (28) and (29) as well as the relaxation rate (26) are independent of the fermion mass, so it is reasonable to hope that these expressions will remain correct in an exact theory. Recent analyses based on perturbation theory, Fermi-liquid theory, bosonization methods, and a renormalization group approach support this expectation [23,26-321. If one were simply to substitute m* for mb in the RPA theory, one would find that the cyclotron frequency w, would be reduced by a factor mb/m*,in violation of Kohn’s theorem, and one would also find a violation of the f-sum rule: nn,q2 Im K,,(q, o ) w dw = llZb

These difficulties can be cured however, if we build on Landau Fermi-liquid theory and include the effects of the p-wave Landau interaction parameter f l . In appropriate units, for a Galilean invariant system in two dimensions, one finds that [23,33]

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FERMION CHERN-SIMONS THEORY

If the effects of f l are included in the analysis, the cyclotron pole occurs correctly at the bare cyclotron frequency B/mb,and the f-sum rule is satisfied [5,34]. A more subtle question concerns the edge of the particle-hole continuum, which occurs at w = qk,/mb in the RPA theory. For a conventional Fermi liquid, we would expect the edge to be shifted to w = uzq, with uz = k,/rn*. In addition, there could be one or more sharp contributions from zero sound modes, and also a weak continuum contribution arisingfrom multiple particle-hole excitationsat frequencies w > qu;, whose relative weight would be expected to vanish in the limit q +0. A peculiar feature of the current problem is that the interaction that leads to the diverging mass renormalization apparently also gives rise to an anomalous behavior of the Landau interaction parameters [23]. This apparently causes the interaction parameter to remain large at large angular momentum. In Landau Fermi-liquid theory, this would give rise to an infinite number of zero sound modes at frequencies above the continuum cutoff w = u;q. However, if the mass renormalization is divergent, as predicted by Eq. (21), we also expect that the quasiparticle scattering rates should be anomalously large compared to conventional Fermi-liquid theory [S]. It is then not clear whether there should actually be distinct zero sound modes or whether there would be a survivingsharp feature in the density response function at the renormalized continuum edge w = uzq. Recent calculations by Kim et al. [31] suggest that within the leading order of perturbation theory, the divergent contribution to the effective mass has no effect whatsoever on the density response: Im K,,(q, w ) has essentially the same form as the RPA in the limit q -0, o --t 0, for an arbitrary ratio of w/q. We interpret this as indicating that there is a cancellation between the effects of the mass renormalization and the Landau interaction parameters in this regime [23]. The density response function K,, can also be calculated for B* # 0. In this section we limit ourselves to the case where v is a principal quantized Hall fraction of the form (18). As at Y = 1/2, the RPA response function is related algebraically to the current and density response functions of noninteracting fermions in the magnetic field B*. At any wavevector ij', the noninteracting response functions have a series of poles at the frequencies o,= n Amc, where n is any integer other than 0, and Aw, is the cyclotron frequency in the effective field B* for the band mass mb; that is,

Of course, Amc is just equal to the energy gap (19),obtained in the mean-field theory. Only the residues of the poles depend on ij' for the noninteracting fermions. In the RPA, the response function KPP,at any given if, again contains a discrete set of poles, but the frequenciesare displaced from the noninteracting positions and generally depend on the wavevector ij' [5,7]. Of particular interest is the lowest excitation branch, which we shall identify with the quasi-exciton mode in

RESPONSE FUNCTIONS

237

the conventional theory of the fractional quantized Hall effect. At large values of q, in particular for q > 2k,, the lowest mode occurs essentially at the unshifted frequency A o c This is just the mean-field energy gap for producing a widely separated quasiparticle and quasihole with charges 2 e/J& + 11. It can be shown that a neutral excitation with wavevector Ti must have a mean electric dipole moment [35-381

Thus, for 1412 2k,, the quasiparticle and quasihole are separated on average by a distance that is 22R:. [See Eq. (l),and note that I, = kp at v = 1/2.] Thus, for large q, the orbits of the quasiparticle and quasihole have very little overlap, the residue of the corresponding pole in K,, should be very small, and the energy should be only slightly shifted from the energy at infinite separation, which is given by (32). For values of q smaller than 2k,, however, the quasi-exiton pole can be shifted considerablyfrom the energy Am,. In the limit q +0, one finds in the RPA that the lowest excitation branch comes in at an energy 2A0,. The weight of the lowest excitation branch vanishes in the limit q+O, however, as the entire weight appears in an excitation branch (or branches)whose frequency is equal to the true cyclotron frequency, o,= IJp 1 (do,[ 5 , 71. To take into account the expected renormalization of the effective mass rn* without violating the f-sum rule or Kohn's theorem, Simon and Halperin [34] have proposed a modification of the RPA which includes, in an approximation manner, the effects of the p-wave Landau interaction parameter f l , while setting all other Landau interactions equal to zero. Figure 6.2 shows the excitation spectra resulting from this modified RPA approximation for the cases v = 1/3 and 10/21 (correspondingto p = 1 and 10, with (p" = 2) with parameters appropriate to GaAs with n, x lo-' cm-' and the assumption that rn* x 4m,. The widths of the striped bands show the residue of the pole in K,,(q, a),multiplied by q-'. Excitations with residue smaller than a specified cutoff are not shown. The cyclotron

+

10.0

10.0

5.0 nn ".0.0

5.0

10.0

X5qRt * 2qk F

n o -.-

0.0

5.0 10.0 X=qR*=POq/k,

Figure 6.2. Poles in the density response function K J q , w), for two quantized Hall states of the form v = p / ( 2 p + l), calculated in the modified RPA approximation by Simon and Halperin [34]. Solid lines show pole positions,while the widths of striped bands show the residues divided by q2. Poles are not shown when strength falls below a cutoff value.

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FERMION CHERN-SIMONS THEORY

frequency is off scale in the second panel, and is therefore not shown in the figure. At v = 113, however, it is seen in the upper left corner as the mode that has the dominant weight. For large values of q, the lowest excitation frequency approaches Aw: = B*/m*, which is equal to the predicted energy gap at the fractional quantized Hall state as given by Eq. (20). Note the series of local minima in the energy of the lowest excitation branch, which become more pronounced with increasing IpI. (These minima are also present in the unmodified RPA, with a somewhat larger amplitude.)The minima occur when the effective cyclotron diameter 2R: is approximately equal to a multiple (n + 1/4) times the exciting wavelength 2n/q [34]. The reality of this geometrical effect is supported by comparisons between the modified RPA and exact numerical calculations of small systemson a sphere [39]. The minima in the quasi-exciton dispersion are also closely related to the geometric resonances observed in surface acoustic wave experiments, discussed in Section 6.7. Excitation modes above the lowest branch, which are sharp in the RPA or modified RPA, are presumably damped in the correct theory because they can decay to multiple excitations of the lower branches as a result of quasiparticle scattering. Spectral weights in the frequency regime w Eg z Jp1Am:, except for the cyclotron branch, are even less meaninful, as they are well outside the regime where Fermi-liquid theory should be valid. For v = 113, the lowest mode in the modified RPA is in qualitative agreement with quasi-exciton spectra obtained from the single-mode approximation and finite-size system calculations [40,41]. However, the modified RPA appears to underestimate the size of the dip in the dispersion curve, which is commonly described as a “magnetoroton minimum” in this case.

6.5. OTHER FRACTIONS WITH EVEN DENOMINATORS The Fermi-liquid-like theory described so far is directly applicable only to evendenominator filling fractions of the special form 1/2m, with m an integer. The fermion Chern-Simons theory can be generalized to describe a Fermi-liquid state, at least formally, for any filling fraction of even denominator [S]. In most cases, however, the resulting state will not be a good approximation to the actual ground state. Even for fractions of the form 1/2m the Fermi-liquid state may not be the correct ground state. For example, experimental and numerical evidence suggests that for 2m 3 6, the correct ground state is a Wigner crystal [42]. One obvious generalization of the Fermi-liquid theory is to states of the form v=n+-

1 2m

(34)

where n and m are integers. We assume here that we have n filled spin-split Landau levels, while the remaining electrons are all in the same spin state, in the next available level. Then we proceed by ignoring the electrons in the n filled

OTHER FRACTIONS WITH EVEN DENOMINATORS

239

4

levels, while making the gauge transformation (4) and attaching quanta of Chern-Simons flux only to the electrons in the partially occupied level. For the case n = 1, we have completely filled one spin state of the lowest Landau level and have placed the remaining electrons in the reversed spin state of the lowest Landau level. Assuming that the Zeeman energy is large enough to maintain the maximal spin alignment, and ignoring such effects as electrical polarizability of the filled Landau level, one finds that the Hamiltonian is essentially the same as if there were spinless electrons with v = 1/2m. Thus if a Fermi-liquid description is correct for, say v = 1/2 and v = 1/4, we also expect it to be valid for v = 3/2 and v = 5/4. Similarly, according to the principle of electron-hole symmetry for a fully spin-polarized Landau level, when one can neglect Landau level mixing, we expect to find Fermi-liquid states at v = 3/4 and v = 71’4. (We describe these states in the fermion Chern-Simons picture by attaching flux quanta to the holes in the Landau level.) In all of these cases, we must take into account one important modification of the discussions of Sections 6.2 to 6.4. The density a/. of composite fermions, interacting with the gauge field, is now different than the density of electrons n,, and Eq. (1) for the Fermi wavector must be replaced by k , =( 4 7 ~ ~ ) ” ~

(35)

In particular, for v = 3/4 or v = 3/2, we find that n/.= nJ3; for v = 5/4 we find that nf = nJ5; and so on. Moreover, if the applied magnetic field B is varied while n, is held fixed, nf will vary continuously as the number of electrons accommodated by the filled Landau level is changing simultaneously. Thus near an evendenominator fraction of form n f 1/2m, we have

The effectivefield B* seen by the fermions is given by

where B, is the magnetic field corresponding to v = n k 1/24 for the given value of n,, and = 2m. In a system with filling fraction 5/2, if the partially filled Landau level is fully spin polarized, there is an obvious formal similarity to the case v = 1/2, and one might therefore expect to have a Fermi liquid in this case as well. Morf and d’Ambrumenil have argued, however, based on numerical calculations of finite systemsofelectrons in the second Landau level, that the Fermi liquid is not stable near v=5/2, and that spin-polarized ground states in the entire range 7/3 < v < 8/3 have quantum numbers which are different from that predicted by the fermion Chern-Simons picture [24]. The nature of the correct ground state for an infinite system in this range of filling factors is not known. Morf and

4

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FERMION CHERN-SIMONS THEORY

d’ambrumenil characterize this state of affairs by saying that it is not energetically favorable for electrons to bind two flux quanta in the second Landau level, so that composite fermions are not formed. By contrast, they find that a composite fermion formed from an electron and four flux quanta is stable in the second Landau level, so that a Fermi liquid could exist at v = 9/4. As another example, Fermi-liquid states at fractions of the form v=-

2p+ 1 4p+4

may be formally constructed by attaching Chern-Simons flux to fractionally charged quasiparticles in the neighboring quantum Hall state, with filling v p = p/(2p + 1) [S]. It is plausible that a Fermi liquid constructed in this manner from a parent state at v1 = 1/3 may be the correct ground state at v = 3/8. However, it is unlikely that this type of Fermi-liquid state could be correct for large values of p. It seems more likely that the state would become inhomogeneous, as it would tend to break up into local regions with the stable filling fractions v p and V p i l C5lFractional quantized Hall states have been observed experimentally at evendenominator filling fractions under certain circumstances. For example, welldeveloped quantized Hall plateaus have been observed in single layer systems at v = 5/2 [43]. Haldane and Rezayi proposed that the ground state in this case is a spin singlet, and they suggested a specific trial wavefunction for the state [44]. Unfortunately,it has been found that the trial wavefunction does not have a good overlap with the spin-singlet ground states obtained from exact diagonalization studies of finite systems,which have been carried out in the limit where there is no mixing between Landau levels [45]. It is possible that mixing between levels will favor the Haldane-Rezayi state in actual samples [44], but the nature of the ground state at v = 5/2 remains an open question more generally. Within fermion Chern-Simons theory, it is natural to suggest that an energy gap in the spin-singlet ground state occurs because of a BCS pairing of composite fermions of opposite spin [4]. Such a state would be equivalent to the HaldaneRezayi state from the point of view of symmetry. However, there have been no calculations to investigate whether the interaction between composite fermions in the second Landau level would indeed be favorable to BCS pairing, let alone to estimate the size of the BCS gap. A fractional quantized Hall plateau has also been seen in bilayer systems at total filling fraction v = 1/2 [46], and in wide single-well systems, where it is believed that the electron system is effectively split into two layers by the selfconsistent Coulomb potential [47]. Numerical calculations [48] suggest that the ground state in these systems is well represented by the 33 1 trial wavefunction, originally discussed by Halperin in 1983 [49]. From the point of view of symmetry, this state is equivalent to the fermion Chern-Simon state, in which there are equal number of electrons in each layer, two flux quanta are attached to each electron regardless of the layer it is in, and the composite fermions form

EFFECTS OF DISORDER

241

a p-wave BCS state with pairing in a pseudospin triplet state with component S, = 0 [4, SO]. (The pseudospin operators s' act on the layer index, which corresponds to the eigenvalues S, = & 1/2 for a single particle. The actual spins are supposed to be completely polarized.) An alternative possibility for a single wide well, proposed by Greiter et al. [4], is that the electrons are fully condensed into the subband, which is an even combination of the two wells (i.e., the pseudospin state S, = 1/2). The energy gap in this case could arise from p-wave BCS pairing in pseudospin triplet state with S, = 1. This state has the same symmetry on a sphere or an infinite plane as the Pfaffian trial wavefunction originally proposed by Moore and Read [2]. However, the statistics of the quasiparticles and the ground-state degeneracy on a torus are different for the two states [2,51].

6.6. EFFECTS OF DISORDER In our discussion so far, we have generally assumed that the electrons move in a uniform positive background, and thus we have ignored scattering effects that arise from impurities or other sources of disorder, which are inevitably present in any actual sample. Samples that have the highest electron mobility in a zero magnetic field and which have been used most extensively in studies of the fractional quantized Hall effect are remotely doped GaAs/AlGaAs heterostructures, where the dopant impurities are separated from the two-dimensional electron gas by a spacer layer whose thickness d, is typically on the order of 700 to 2000 A. If the two-dimensionalFourier transform of the charge distribution in the the resulting electrostatic potential at the donor layer is denoted by electron layer, before electron screening, is given by

p,(a,

This potential is then reduced by the dielectric response function of the electron gas, which itself diverges as q-' for q +O. It can be seen from Eq. (39) that the random impurity potential is greatly attenuated for q > d,-' because of the setback d,. If d, is large compared to k; fluctuations in the impurity potential lead only to small-angle scattering of electrons at the Fermi surface, which is further reduced by electron-screening effects. This is the reason why the mobility of these samples is so high in the absence of a magnetic field. The situation for composite fermions at v = 1/2 (or at other even-denominator fractions) is rather different. A fluctuation in p,(agives rise to a screening fluctuation p ( 3 in the electron density, which is equal to - p , ( 3 in the longwavelength limit. The fluctuation in p ( q ) gives rise to static fluctuations in the Chern-Simons magnetic field b ( 3 , and thus in the vector potential Z(3.

',

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FERMION CHERN-SIMONS THEORY

Specifically, one finds that

The scattering effect of the vector potential is much larger than the scattering caused directly by the screened Coulomb potential, which can account for the experimental result that p, at v = 1/2 is very much larger than the resistivity of the same sample at B = 0. (The importance of this scattering mechanism was noted independently by Kalmeyer and Zhang [6] and by HLR [5].) To estimate the magnitude of p,.. HLR consider a simple model where the number of charged impurities is equal to the number of electrons in the ZDEG, and there is assumed to be no correlation in the positions of the charged impurities. If the scattering rate is calculated in the first Born approximation, the transport mean free path 1 for the compositefermions at v = 1/2 is found to be just equal to the setback distance d,. The conductivity c,, is then determined by Eq. (29). Experimental values of the resistivity at v = 1/2 in high-quality remotely doped samples have typically been smaller by a factor of 3 or more than the estimate obtained from this simple theoretical model [19,521. This implies that the actually mean free path may be significantlylarger than the estimate. Part of this discrepancy probably arises from a tendency of the Born approximation to overestimate scattering effects in the regime of interest. Also, there is probably a significant anticorrelation in the locations of the ionized donors in the actual samples, which should reduce the amount of scattering. (The total number of donors is significantlylarger than the number that are eventually ionized, and it is reasonable that the ionized sites are distributed in a way that tends to minimize charge fluctuations.) In some cases, however, values of pxx have been reported which are as much as 25 times smaller than that predicted by the simple model [53]. It would seem surprising that anticorrelations could be so effective as to account for a discrepancy of this magnitude. Analysis of resistivity measurements is further complicated by the fact that density fluctuations even at very long length scales can lead to an increase in the macroscopic resistivity of a system with (pxyl>>pxx.In fact, the measured resistivity might be largely determined by such inhomogeneities[54-581. Density fluctuations on multiple length scales have been suggested as a mechanism for explaining, at least in part [56], the remarkable linear magnetoresistancethat has been observed in two-dimensionalelectron systems over a wide range of magnetic fields at temperatures high enough so that the quantized Hall plateaus are suppressed [52]. A detailed understanding of the mechanism responsible for electrical resistivity in actual samples is still lacking, however. At any given even denominator v, for sufficiently high levels of disorder, the metallic Fermi-liquid phase should be destroyed because of localization effects. It is an open question whether in principle, even in samples with very low disorder, the metallic behavior should be destroyed in the limit of extremely large length

SURFACE ACOUSTIC WAVE PROPAGATION

243

scales and low temperatures because of logarithmic renormalization effects analogous to weak localization in conventional two-dimensionalsystems [59]. If one treats the composite fermions as a system of noninteracting particles subjected to a random Chern-Simons magnetic field and a random potential, one finds that the leading logarithmic correction, which is responsible for weak localization at B = 0, is suppressed at v = 1/2 due to the broken time reversal symmetry [S, 61. When interactions are taken into account, it appears that there is again alogarithmic correction to pxxsimilar to the case of B = 0 [S]. In practice, however, for high-mobility samples at v = 1/2, the logarithmic term will be very small for any realistic sample size or temperature, so that the metallic phase is not destroyed by weak disorder. If disorder is large enough so that the metallic phase at an even-denominator fraction vo is destroyed by localization, then as the magnetic field or electron density is varied so that v passes through yo, we expect to find a sharp direct transition between quantized Hall states corresponding to odd denominator fractionson either side of vo [ 5 , 6,10,60]. Considerable progress has been made in analyzing these disorder-dominated transitions [61], but this subject will not be discussed further here.

6.7. SURFACE ACOUSTIC WAVE PROPAGATION Experimental studies of surface acoustic wave (SAW) propagation in GaAs samples containing a 2DEG provided the first strong evidence that something quite unusual was happening near filling factor v = 1/2 [lS]. Subsequent refinements of these experiments have provided some of the strongest evidence supporting the fermion Chern-Simons theory [19,20]. We discuss here the theory of these effects. It is expected that interaction between the SAW and the 2DEG occurs by means of the longitudinal electric field due to the piezoelectriceffect in GaAs. The response of the 2DEG may be characterized by a longitudinal conductivity axk7j,a)at the frequency w and wavevector 7 j (parallel to the surface) of the acousticwave. The finite conductivityleads to an acoustic attenuation and a shift Av, in the velocity v, of the SAW, relative to the case of a 2DEG with infinite conductivity, which may be expressed as [18]

Here a is a coupling constant proportional to the piezoelectric coefficient,K is the amplitude attenuation coefficient, and 6, is given by

am=-Us& 27t

244

FERMION CHERN-SIMONS THEORY

'

If q - is smaller than the distance between the 2DEG and the free surface,then is essentially the dielectric constant of GaAs, and a, is found to be x7 x S2- '. At larger wavelengths, the electric field arising from the 2DEG penetrates partly into the vacuum, and a weighted average of the GaAs and vacuum dielectric constants must be used. In this case, a, can be as small as x 4 x 10-7S2-'. In general, a, is a function of? whose value can be calculated with reasonable accuracy if the geometry is known. The longitudinal conductivity ox&, w ) was evaluated by HLR using a semiclassical approximation, closely related to the RPA. The resistivity tensor pup(?,a), which is the matrix inverse of a,&, a),is related to a compositefermion resistivity P4&% w ) by

E

P = Pes + P

(43)

where pesis the Chern-Simons resistivity P c s 2= 7T 4( l 0

-

;)

[The origin of the Chern-Simons resistivity is the induced self-consistentChernSimons electric and magnetic fields, given by Eqs. (24) and (25).] The composite fermion conductivity, d,,(q,

0)

= CP(% 41;l

(45)

is approximated, as in the RPA, by the corresponding response function of a set of noninteracting charge fermionsin a uniform static field B*. Now, we use a further semiclassical approximation, which should be valid for q << k , and for RT sufficiently large (i.e., for B* sufficiently small) in which (7(q,o) is related to the current-current correlation functionfor a particle that moves in a circle of radius R: at the Fermi velocity o/*, and which undergoes velocity randomizing collisions at a rate determined by the mean free path 1 [62]. Moreover, because the sound velocity vs is small compared to of,the SAW frequency w = v,q is small, and it should be adequate to calculate d(q,w ) in the limit w = 0. The resulting expressions for the wave-vector-dependentlow-frequency conductivity ax&) were evaluated by HLR, and are indicated in Fig. 6.3, for magnetic fields near v = 1/2. Only positive values of B* (labeled AB in the figure) are shown, as the theory is symmetric about B* = 0. Several features of these results deserve attention. Precisely at v = 1/2 (B* = 0) the conductivity o,..(a varies smoothly as a function of 4 between the longwavelength limit given by Eq. (29) and the result of (28),valid for q >> 2/1. On the other hand, for fixed q 2 2/1, if B is varied sufficiently far away from the value at v = 1/2 so that the effective cyclotron diameter 2R: becomes smaller than the wavelength 2n/q, the enhanced nonlocal conductivity disappears and a,&) returns to the long-wavelength value axx(4= 0), which is independent of B* in the semiclassicalapproximation and is therefore given by (29).

SURFACE ACOUSTIC WAVE PROPAGATION

245

t

According to Eq. (41),an increase in a,,(q) should give rise to a decrease in Avr Thus one would predict that at a given SAW wavevector q, if the magnetic field B is varied, there should be a dip in the SAW velocity near v = 1/2 whose amplitude should become more pronounced with increasing q, and whose width in magnetic field AB should also be proportional to q. These features are in good agreement with the characteristics of the SAW anomaly originally observed by Willett et al. [lS] and confirmed by subsequent experiments [19,20]. Note, however, that the curves for a,&) shown in Fig. 6.3 are not monotonic functions 6f IB* I for ql > 2. In particular, we see a series of maxima in a&) which becomesmore pronounced for increasingql. These maxima can be understood as geometric resonances that occur when the effective cyclotron diameter 2R,* is related to the sound wavelength 1 = 2n/q by 2R,* x (n + 1/4)1

(46)

where n is a positive integer. (More precisely, the resonances occur at qR,* = X,,

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FERMION CHERN-SIMONS THEORY

where X, is the position of the nth zero of the Bessel function J1.)The most prominent peaks, at n = 1, correspond to an effective field B* =

hC

3.83 -qk,

(47)

e

It may be remarked that the semiclassical calculation for the composite fermion conductivity tensor 6(q,w)is very similar to calculations by Cohen et al. in 1960, of geometric resonance effects in ultrasonic attenuation in threedimensional metals [62]. However, the presence of the large off-diagonal ChernSimons contribution in Eq. (43) greatly changes the numerical values, and even the sign of the anomaly in ox, in the present case. Anomalies at the fields corresponding to (47) were not seen in the SAW data available prior to 1993. As SAW measurements have been extended to larger wavevectors q, however, the predicted structure has clearly emerged. In the 1993 data of Willett et al. [20], shown in Fig. 6.4, this is manifested by the appearance of double minima in the SAW velocity curves for frequencies 2 3.7GHz.

1

t " ' " " " ' J

3 .Ox10-4 2.5 2.0

/

2.4GHz 5.4 8.5

1

1.0 0.5

E

55

60 Magnetic field (kG)

65

Figure 6.4. Relative shift AuJus in the surface acoustic wave velocity, as a function of magnetic field, in a GaAs sample containing a two-dimensional electron layer for four values of the SAW frequency. Filling fraction v = 1/2, occurs at 57 kG.(Courtesy of R. L. Willett; data reported by Willett et al. in Ref. [20].)

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The splitting between the SAW velocity minima observed experimentallywas found to be in excellent agreement with the predictions of Eq. (47), near v = 1/2, for a range of SAW wavevectors q. More recent experiments, extended to 10.5GHz, have seen the predicted splitting of the velocity minima at filling fraction v=3/2, and have seen the secondary n = 2 minima at v = 1/2 [63]. A strong SAW anomaly has been observed near v = 1/4 and a weak anomaly near v = 3/4, but peak splitting has not yet been seen in these cases. Since the wavevector q of the sound wave is accurately known, the peak splitting of the velocity minima near v = 1/2 and v = 3/2 may be taken as a direct measure of the Fermi wavevector k , of the composite fermions. A noted in than k, in a zero Section 6.5, the value of k, at v = 1/2 is larger by a factor magnetic field, while at v = 3/2 it is reduced by (2/3)’/’ from the zero-field value. Since the effectivefield B* near v = 3/2 also differs by a factor of 3 from the actual field deviation AB, the field splitting near v = 3/2 is reduced by a factor of 33/2 relative to that at v = 1/2. Willett et al. have compared the experimental values of AuJu, near v = 1/2 with the quantitative predictions of the semiclassical calculation of the fermion Chern-Simons theory. They were able to obtain good agreement after making several adjustments. The theoretical predictions were convoluted with a Gaussian of width 1.5% FWHM in the magnetic field to account for presumed density inhomogeneities [20]. In addition, to obtain good agreement between the SAW measurements far from v = 1/2 (or at low frequencies at v = 1/2) and macroscopic dc resistivity measurement made on similar samples, it was necessary to use a value of the material parameter om that was several times larger than that expected from the simple theory discussed above [5, 20,633. It was then necessary to adjust the theoretical values of o,,(q) near v = 1/2 at large values of q by a factor of x 2 to get a good fit to the experimental data. The reasons for these discrepancies in the absolute magnitudes of o,(q) are not understood.

fi

6.8. OTHER THEORETICAL DEVELOPMENTS Many aspects of the fermion Chern-Simons theory are the subject of continuing theoretical investigations. One active area concerns the mathematical implications of the theory in the asymptotic “infrared”limit of long wavelengths and low energies, for an ideal system with no disorder, in the limit B* +O. Other areas concern applications of the theory to more complicated situations, such as bilayer systems or systems with two active spin states, and behavior near a sample boundary. We discuss a few of these issues below. 6.8.1. Asymptotic Behavior of the Effective Mass and Response Functions

There has been considerable interest in the question of whether the divergence of the effective mass m* obtained in the analysis of HLR is truly the correct asymptotic behavior, when all terms in the perturbation expansion are correctly

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summed. As discussed in Section 6.3, HLR found a logarithmic divergence in m* for the case of Coulomb interactions, and a power-law divergence for shorterrange interactions. Specifically,they found an energy-dependenteffective mass of the form m*(w)

-

(48)

w-”

where w = )E-E,I, and y = 1/3 for short-range interactions. For electronelectron interactions that fall off at large separations as

V(r)

-

l/rq

(49)

with 1< q <2, a similar analysis gives

The corresponding behavior of the energy gap at v = p/($p for large p, would be EfJ p -

+ l), according to (20), (51)

l A 1-”)

On the other hand, HLR predicted that at v = l/& linear response functions such as K,,(q, w ) should be substantially unaffected by the renormalization of m*,at least in the limit of small o/q. Interest in these problems has been increased by the fact that effective mass divergences analogous to (48) and/or other deviations from normal Fermi-liquid behavior have also been found in analyses of several other models in which a set of fermions interacts with a fluctuating gauge field [26-31,64-731. One example is the case where a set of fermions in two dimensions has a transverse-current interaction transmitted by a transverse gauge field with a field Lagrangian of the form

[a x Z(r)]u(7-7‘)[9

x Z(r‘)]d2rd2r‘

Related gauge models have also been advanced as models for high-temperature superconductivity [73,74,78,79]. To better understand the behavior of the diagrammatic expansion, it has been useful to generalize the models by considering a system with N distinguishable fermion species coupled to the gauge field. At least in the case of the currentinteraction models, there is a simplification in the analysis if one considers the limit N 00. Altshuler et al. [30] have studied the behavior of the theory at large N and have found, in general, results that are consistent with the conjectures of HLR. The leading diagrams in the limit N co give essentially the results of HLR for the singular effective mass in the one-fermion Green’s function and for the regular behavior of the density response function at long wavelengths and o << qu;. --f

--f

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249

(Altshuler et al. do not analyze the behavior of K,, for qu; 5 w 5 qu,.) They conclude that corrections of higher order in 1/N do not change the behavior of K,, at long wavelengths and do not change the nature of the singularity in fermion effective mass. They do not address, however, the question of how the singular effective mass might be directly or indirectly reflected in a physical quantity that may be measured physically. Altshuler et al. also study the singularity in K,, near q = 2 k , and w = 0, for T+O. Here they find a nonanalytic dependence whose exponents do depend on the parameter N. Their extrapolation to N = 1 does not give a clear indication of whether this susceptibility is actually divergent at q = 2kF or has a finite cusp singularity. It should be remarked that a weak divergence in the response function does not in itself imply an actual instability of the Fermi surface at T = 0. Several other authors have obtained results for the long-wavelength, lowfrequency linear response functions, which are consistent with the predictions of HLR. As discussed in Section 6.3, recent results by Kim et al. [31,75], and by Stern and Halperin [23], are also consistent with the HLR prediction for m* and for the energy gaps at the principal quantized Hall states, and with the feature that at v = l/& the singular renormalization of m* is canceled in the behavior of K,, by corresponding singularities in the Fermi-liquid parameters. On the other hand, there have also been calculations that suggest departures from the HLR predictions in some important respect. In particular, several groups have performed approximate calculations which gave forms of the one-fennion Green’s function quite different from the HLR predictions, different enough to raise questions about the fundamental stability or self-consistencyof the Fermi-liquid state [29,68,73]. There has also been a calculation of the energy gaps at the fractions v = p/($p + l), for large p , which obtained a form different from that predicted by HLR [76]. Several of the calculations leading to disagreements with HLR have made use of approximations that appear to be valid particularly in the limit where the number of fermion species N is taken to zero [29,68,73]. The recent analysis by Altshuler et al. mentioned above suggests, however, that the N = 0 case is special and that systems with N # 0 behave much more like N = 00 [30]. Nayak and Wilczek [28] have studied the v = 1/2 system using a renormalization group approach based on an expansion in powers of ( q - l), where q is defined by Eq. (49), without any assumption that N is large. Their results, obtained to lowest order in ( q - l), are consistent to this order with the conjectures of HLR and with the large N analyses. For Coulomb interactions (q = 0) they find a logarithmic divergence for the effective mass, consistent with Eq. (21). (An erroneous result, m* a Iln 011/2, contained in the short published version of Ref. [28] has been corrected in the longer version.) 6.8.2. Tunneling Experiments and the One-Electron Green’s Function

Although the two-dimensional electron system at v = 1/2 has many features in common with an ordinary Fermi liquid, according to the fermion Chern-Simons

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FERMION CHERN-SIMONS THEORY

theory, the one-electron Green’s functions should be quite different in the two cases. In fact, there are several one-particleGreen’s functions that may be defined in the fermion Chern-Simons theory which are quite different from each other. Two obvious ones are the Green’s functions for an electron and for the bare transformed fermion, defined, respectively, by

N t ( O , 0,)) GfF, t ) = - i ( @(7,

(54)

where T is the time-ordering operation, and $L and $t are the creation operators defined in Section 6.2. In this section we consider the electron Green’s function G,, leaving Gffor the following section. We shall focus on the low-energy behavior of the function G,@, E), which is the Fourier transform of (53) with respect to t. The imaginary part of G, determines the “density of states” measured in an ideal tunneling experiment, where an electron is transferred into the layer from a source sufficiently far away that one can neglect the Coulomb interaction between electrons in the source and in the layer. (For tunneling between two well-separated layers, the currentvoltage characteristic is determined by a convolution of G, for the two layers.) Since Fermi-liquid theory predicts a finite density of states for the composite fermions near the Fermi energy EF (neglecting the weak divergence in m* predicted by HLR),one might think, at first glance, that Im G, should be finite for E +E,. In fact, this is not the case; because of the phase factor in Eq. (lo),the state created by adding an electron to the composite-fermion ground state has vanishing overlap with a single composite-fermion excitation, and in fact has a rapidly vanishing overlap with any state of the system in the limit E EF. Using approximations based on the fermion Chern-Simons theory, He et al. [77] have obtained the following asymptotic form for an ideal system at v = 1/2, assuming Coulomb interactions between the electrons and negligible effects of disorder: Im Ge(O,E)

-

e-Odlwl

(55)

where o = E - E,, and oois given by

Using a somewhat different analysis, Kim and Wen obtained a result of the same form as (55) but with a value of oolarger than (56) by a factor of 2 [70]. Tunneling experiments by Ashoori et al. [78] and by Eisenstein et al. [79] show that the tunneling density of states decreasesas o tends to zero, and in fact is consistent with a “pseudogap” where the density of states is extremely small over an appreciable range of energy. In the case of tunneling between two identical layers, measured by Eisenstein et al. [79], an autoconvolution of (55) predicts

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251

a current-voltage curve with the asymptotic form [77]

The data of Eisenstein et al., on the low-voltage side of their current maxima, in the limit of low temperatures are well fit by an expression of the form of (57). However, the experimentalvalue of wo is six times smaller than predicted by (56), or 12 times smaller than predicted by Kim and Wen [70]. A possible explanation for this discrepancy is that for the voltages in question, the energies are still too high for the theoretical calculation to be applicable. In any case, the most that one can say at present is that the pseudogap observed in tunneling experiments is in qualitative agreement with the predictions of the fermion Chern-Simons theory. Moreover, the appearance of some kind of pseudogap is also predicted by various other theoretical models [80-841. In all cases the pseudogap is essentially a “Coulomb blockade” effect; the injected electron causes an increase in potential energy due to a pileup of charge near the origin, which relaxes only very slowly in the presence of a strong magnetic field. The specific form of ( 5 5 ) is a consequence of the predicted form (28) for the and the resulting charge relaxation rate (26). in-plane conductivity oxxm Eisenstein et al. [79] observe a relatively sharp maximum in the current when V is on the order of 0.4e2/&1,. At this high energy, Fermi-liquid theory is probably not very useful, although the position of the maximum is roughly where one would estimate using any reasonable model of the Coulomb blockade. A theoretical calculation of the shape and position of the current maximum by Johanson and Kinaret, based on a Wigner crystal model of the electrons, is in good agreement with the experimental results [83]. A model that assumes localization due to disorder has also been used to explain the data [82]. In our discussion thus far we have ignored the Coulomb interaction between layers. The layer separation in the experiments of Eisenstein et al. [79] are sufficiently small that the Coulomb interaction should lead to a significant downward shift in the position of the tunneling maximum. It is less clear what effect this might have on the region of the I-V curve, where the exponential behavior in 1/1 ll has been observed. For very low values of I V I, one predicts that the interlayer interaction will change the dependence to [70,77] I

-

eXP(

- const F )

6.8.3. One-ParticleGreen’s Function for Transformed Fermions

Let us now return to the Green’s function C, for the transformed fermion, defined by (54).As was noted by HLR, in an infinite system, at any value of v, the spectral density Im G,(r, E) should actually vanish identically for any finite E [S]. This is because the operator Il/f (0,O)has the physical effect of adding an electron at the origin and instantaneously turning on a solenoid containing two flux quanta. The

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FERMION CHERN-SIMONS THEORY

impulse electric field generated by the solenoid produces inter-Landau-level excitations at large distances from 7 (i.e., it produces long-wavelength magnetoplasma excitations at the cyclotron frequency 0,) whose mean number diverges logarithmically with the size of the system. The probability that the manyelectron state has remained entirely in the lowest Landau level (or in any other state with only a finite number of cyclotron excitations) is therefore found to vanish as a power of the size of the system. To observe low-energy structure in the one-fermion Green’s function that can reveal the dispersion relation of the composite fermions near the Fermi energy, it is necessary to study a renormalized version of the one-fermion Green’s function. One possibility would be to carry out calculations for a large but finite system, and to renormalizethe overall weight by a size-dependentfactor before taking the limit of infinite size. An alternative approach is to work in an infinitesystem in the limit where all operators are projected onto the lowest Landau level. This is equivalent to taking the limit where the band mass mb-+ 0 (hence w, + a), while the electron-electron interaction is held constant and the flux in the solenoid associated with J/+(O,O)is turned on at a rate that is slow compared to o,but fast compared to the scale of the electron-electron interaction. At v = 1/2, we expect that the renormalized Green’s function at a wavevector close to k, should have a singularity at (E - E F ) u;(k - k,) with weight that vanishes only slowly (ccl/lnlE - E J ) as one approaches the Fermi surface [5, 311. To the best of our knowledge, the hypothesized renormalization behavior of G, has never been checked explicitly in detail. The vanishing weight of G, due to cyclotron excitations is represented in the Chern-Simons theory by divergent contributions to the self-energy which arise from coupling to the longitudinal gauge field [ 5 ] . In most analyses to date, coupling to the longitudinal field has been ignored, so that this problem is not encountered.

-

6.8.4. Physical Picture of the Composite Fermion

A construction in which we take mb+O and turn on the solenoid flux slowly compared to o,also allows for an improved physical understanding of the nature of the composite fermion, as has been emphasized by Read [SSJ. If we consider a renormalized fermion creation oFerator in this limit, the transverse electric field during the addition of 4 flux quanta at point 7 leads to a buildup of a change deficit of - v 4 electrons in the vicinity of point 7. If v = l@, this “vortex hole,” together with the electron added by I&, (3, gives an object with zero net charge in the vicinity of 7,while one electron charge accumulates at the boundary. If the electron is placed off center in the hole, there will be a net electric dipole moment 2, and hence a nonzero wavevector T for the comglex, given by Eq. (33). Moreover, the complex should drift in the direction of k with a mean velocity determined by the mean value of the electric field at the electron position. According to the fermion Chern-Simons theory, if the complex has a wavevector slightly above k,, the complex will be stable and hence will travel in a straight line over a large distance scale. Of course, the renormalized creation operator J/fen(q,

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253

x,

which puts the electron in the center of the vortex holes, does not create a but rather, creates a quantumcomposite fermion with a defined value of mechanical linear superposition of states, with wavevectors in all directions and k > k,. In the case where v deviates slightly from l/& the complex created is not neutral and hence moves in a circle whose radius in just equal to the effective cyclotron radius R,*, given by Eq. (1). If the electron density varies in space on a long length scale, the charge associated with the electron is fixed, while the charge of the vortex hole varies with the local electron density. In this case the complex will be deflected from a straight-line path according to the deviation of the local density from the value at v = l/& just as one would expect for a charged particle moving in a local effective field B * O that fluctuates about zero. 6.8.5. Edge States

Several authors have used the Hartree approximation to the fermion ChernSimons theory to study the behavior of a principal fractional quantized Hall state near a sample boundary [86,87] and in small confined systems (quantum dots) [8]. In contrast to the case of a translationally invariant system, the Hartree approximation is nontrivial in this case, as there are spatially varying ChernSimons vector potentials and scalar potentials, corresponding to the fields (24) and (25), which must be calculated self-consistently according to profile of the charge density and the diamagnetic current near the edge [88,89]. Edge states [90,91] appear where the self-consistent dispersion relation for the composite fermions passes through the Fermi energy. Calculations have been carried out for several bulk filling factors, for edges with confining potentials of varying steepness. In general, extra edge states arise when the potential becomes less steep [92]. Predictions for the charge-density profile near the edge, and for the number of distinct edge states, seem to be very reasonable [86,87]. Oscillatory charge distributions are predicted in some cases, which are in qualitative agreement with the limited evidence available from exact diagonalizations or trial wavefunctions in finite systems [86,87]. To get reasonable estimates of the group velocities of the edge excitations, it appears necessary to go beyond the Hartree theory, at least to the level of RPA theory for the linear response functions. Since the edge states form a onedimensional metal where quantum fluctuations are known to be very important, the Hartree theory can be at best a starting point for more sophisticated calculations of the dynamic properties near an edge. Theories of edge states using the language of a chiral Luttinger liquid, and using methods of conformal field theory, have given us tools to study the behavior on long length scales,which one would want to join to the microscopic fermion Chern-Simons theory [93]. The Hartree approximation of fermion Chern-Simons theory has also been applied to discuss lateral transport across a potential border that separates two regions with v = 1/2 [94].

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FERMION CHERN-SIMONS THEORY

6.8.6. Bilayers and Systems with Two Active Spin States

As mentioned in Section 6.5, fermion Chern-Simons theory has been used to describe various quantized Hall states that can occur in electron systems with additional discrete degrees of freedom, such as systems with two active spin states, or electrons in bilayer systems. The theory may also be useful in discussing the possible existence and limits of stability of Fermi-liquid states in these systems. The fact that for two weakly coupled layers in a strong magnetic field, the amplitude for tunneling of electrons between layers should vanish at low energies (as discussed in Section 6.8.2) is an important ingredient of this analysis. For two coupled layers, each at v = 1/4, if tunneling between layers is negligible, there will be enhanced low-energy gauge fluctuations associated with density fluctuations that are out of phase in the two layers and have a relaxation rate y4 K q3 at long wavelengths. Bonesteel has discussed the possible effects of these fluctuations on the lifetimes of composite fermions on the layers and has speculated that these fluctuations may contribute to Cooper pairing between layers, in the more strongly _ _ coupled case where a Hall plateau is observed at vtotai = 1/2 P51. The presence of the spin degree of freedom has dynamic consequences even when the ground state is fully spin aligned, as there exist a variety of interesting excitations involving reversed spins [96]. Recently, Nakajima and Aoki have used the fermion Chern-Simons theory, at the level of RPA, to calculate the spin-wave dispersion relation at v = 1/3 and v = 1/5 [97]. 6.8.7. Miscellaneous Calculations Recently, Chen et al. [98], have used a fermion Chern-Simons approach to calculate the charge density near a quasiparticle or quasihole excitation at v = 1/3. This calculation was based on a Hartree approximation, applied to an inhomogeneous situation, as in the case of the edge-state calculations mentioned in Section 6.8.5. Again, results are in good qualitative agreement with exact diagonalizations and trial wavefunction calculations, including the prediction of a nonmonotonic variation near the origin in the case of the quasielectron. 6.8.8. Finitesystem Calculations There have been a large number of calculations comparing predictions of compositefermion theories with results of exact diagonalizations of finite systems (most commonly on a sphere).Indeed, calculations of this sort for quantized Hall ground states by Jain and co-workers provided the earliest support for the composite fermion approach [9]. More recent calculations ofground states of the sphere when the number of electrons and number of flux quanta do not correspond to one of the principal quantized Hall states, as well as studies of the low-lying excited states at various filling factors, have contributed to our understanding of the validity and of the limitations of the fermion Chern-Simons

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255

description. Some of this work was mentioned earlier in the chapter [24,39], but the reader is referred to Chapter 7 for a further discussion [99]. 6.9. OTHER EXPERIMENTS In addition to the surface acoustic wave measurements discussed in Section 6.7 and tunneling experiments mentioned in Section 6.8.2, a number of other experiments have provided varying degrees of support for various aspects of the fermion Chern-Simons theory. We present here only a very brief summary of the experimental situation. Many additional details will be found elsewhere in this volume, particularly in Chapter 10. 6.9.1. Geometric Measurements of the Effective Cyclotron Radius RT The existence of the predicted effective cyclotron radius RT, and consequently of a well-defined Fermi wavevector k,, has been confirmed by several experiments in addition to the SAW measurements. An experiment contemporaneous with the SAW observations of Ref. [20] was the geometric resonance experiments of Kang et al. [lOO]. In this experiment, a set of very high mobility GaAs samples were lithographicallypatterned with a square array of antidots-smaller circular regions where the two-dimensional electron gas is absent. The macroscopic resistivity of each sample was monitored as the applied magnetic field was varied near v = 1/2, and resistance maxima were observed, on either side of v = 1/2, at a field such that the effective cyclotron diameter 2R:, given by Eq. (l), was approximatelyequal to the lattice constants of the dots. Similar maxima had been observed earlier by Weiss et al. [loll near B = 0, when the true cyclotron diameter 2R, coincided with the spacing of the antidot array, as well as when the cyclotron orbit fit nicely around a neighboring pair of antidots, around four antidots, and around nine antidots. A semiclassicaltheoretical calculation, based on numerical simulations of projectile trajectories in the potential of the antidot array, had been found to give reasonable agreement with experimental observations in the weak-field case [102]. There is no quantitative theory availablefor the resistivity of the antidot array near v = 1/2. Nevertheless, observation of the resistivity maxima by Kang et al. is a striking conformation of the composite fermion picture. In particular, the field splitting between the resistivity maxima near v = 1/2 was found to vary linearly with the lattice constant of the array and was found to be larger by a factor of $than the field splitting in the same sample near B = 0. This is in accord with Eq. (1) and the fact that the Fermi wavevector k, is larger by the factor near v = 1/2 than near B = 0, where both spin states are occupied. Other differencesbetween v x 1/2 and B x 0 result from the fact that the mean free path of the composite fermions near v = 1/2 is much smaller than the mean free path of electrons in small B. this makes the experiment much more difficult near v = 1/2 and imposes very stringent conditions on the sample quality. The

3

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FERMION CHERN-SIMONS THEORY

fact that the effect was seen at all suggests that at least some significant fraction of the composite fermions could travel, without large scattering, over a distance equal to the circumference 2nR: of the effective cyclotron orbit, which was of order 3 pm in these experiments. A more recent experiment to study the trajectories of quasiparticles near v = 1/2 was the “magnetic focusing” experiment of Goldman et al. [103]. Here, a nonlocal conductance measurement was made on a set of specially constructed samples where one of the voltage contacts is separated by a small distance (w4pm) from a current contact on the same edge of the sample. Under suitable conditions, the experimenterswere able to see a series of maxima in the measured signal when the effective cyclotron diameter was equal to an integer submultiple of the spacing between contacts. As for the experiment of Kang et al., the magnetic focusing experiment has its counterpart in experiments previously performed in much weaker magnetic fields, where maxima occur where the true cyclotron diameter 2R, is a submultiple of the contact separation [1041.Again, a quantitative theory of the magnetic focusing near v = 1/2, which takes into account effects of disorder and the actual potential profile near the sample boundary, has not been carried out. A semiclassical interpretation of the observations implies that a significant fraction of the composite fermions closely follows the trajectory of an ideal cycloidal skipping orbit for a net distance equal to the contact separation of 4 pm. It should be emphasized that the distance scale of 4pm in the magnetic focusing experiments, and the comparable distances over which the composite fermions appear to travel in the geometric resonance experiments of Kang et al. [1001, are several hundred times larger than the electron-electron separation or the cyclotron radius for the electrons in these samples, which is essentially the magnetic length l p Thus the success of the experiments,together with the surface acoustic wave experiments, give a remarkable confirmation of the existence of composite fermions and of at least one aspect of the fermion Chern-Simons theory at large length scales. 6.9.2. Measurements of the Effective Mass

Neither the surface acoustic wave experiment nor the geometric resonance experimentsjust discussed given direct information about the size of the effective mass m* of the composite fermions As discussed in Section 6.3, the effective mass at v = 1/2 may be determined in principle by measuring the energy gaps E,(v) at the quantized Hall states whose filling fraction has the form v = p/(2p + l), for large values of p. In turn, the energy gap may be identified,in principle, with twice the thermal activation energy E , for the resistivity px,, at the center of the Hall plateau, in the limit of vanishing disorder. In practice, however, it has been necessary to make a large correction for the presence of disorder, which makes interpretation of the experiments particularly difficult for larger values of p. Measurements of the thermal activation energy by Du et al. [1051 were found to be in reasonable agreement with the predictions of the fermion Chern-Simons

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theory, for a finite value of the effective mass m*, under the assumption that the effect of disorder is to reduce the measured value of 2E, below the ideal energy gap E, by a constant energy r, which is independent of p. (The data could be equally well fit if the effective mass m* had a contribution proportional to In p, in addition to the constant term, as predicted by the HLR theory.) Unfortunately, however, current theories are not developed sufficientlyto either effectively support or contradict the assumption of Du et al. with regard to the effects of disorder. More recently, experimental values of the effective mass have been extracted by several groups from an analysis of the Shubnikov-de Haas oscillationsin pxx, at finite temperatures near v = 1/2, in both electron-doped [1061 and hole-doped [lo71 samples. If the standard theory of Shubnikov-de Haas oscillations is applied to the compositefermions near v = 1/2, one expects to find for the leading Fourier component of the resistivity oscillations [106]:

X

Ap,,a-expsinh X

(

)::;,T ‘OS

(h)

(59)

where zq is a composite fermion scattering time, assumed to be independent of temperature, Am: = eB*/m*c as in Section 6.3, and

XZ-

2n2k,T h Am:

The overall magnitude of m* obtained from the Shubnikov-de Haas measurements using Eq. (59) has generally been consistent with the values extracted from the thermal activation energies, as interpreted in the composite fermion picture. However, various of the experimentsappear to show a sharp increase in m* close to v = 1/2, very much larger than would occur if there was a logarithmic behavior as predicted by HLR. Further theoretical work is necessary to see if an interpretation based on Eq. (59) is indeed justified for the actual experiments near v = 1/2. Again, a proper understanding of the effects of disorder is crucial to this enterprise.

6.9.3. Miscellaneous Other Experiments At the International Conference on High Magnetic Fields in Semiconductor Physics held in Cambridge, Massachusetts,in August 1994, a number of experiments were reported, complementaryto those discussed above, whose interpretation also involved comparison with predictions based on the composite fermion picture [1081. Since the connection to composite fermions is less direct in most of these experiments or the theory is less well developed than in the cases discussed above, we list them only briefly here. Several groups have measured the thermopower of the partially filled Landau level in various regimes of temperature [l09]. Results seem to be qualitatively consistent with a picture based on composite fermions whose effective mass is

258

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consistent with the values obtained from other experiments. Both theoretical work and further experiments are required here. The group of Digby et al. [1 lo] reported measurementsof phonon absorption by a two-dimensional electron system, using thermal phonons from a heater in the temperature range 1 to 4 K. Results were found to be consistent with absorption by creation of magnetoexciton,at v = 213,315, and 417, with a composite fermion effective mass m* sz 0.6~1,. Luminescence experiments by Kukushkin et al. [l 1 13 and Turberfield et al. [1121 showed discontinuities in the chemical potential p at filling factors of the form v = p/(2p + l), which are much smaller than predicted by the compositefermion theory, with a reasonable value of m*, for an ideal system with no disorder. However, effects of disorder might reduce the experimentaldiscontinuities by a large amount. Previous measurements of dp/dn, by Eisenstein et al. [113] using a capacitive technique also found that variations near the principal fractional quantized Hall states were generally much smaller than might be expected based on theoretical energy gaps. These authors also attributed the discrepancy to effects of disorder or inhomogeneity in the sample, but a quantitative understanding of these effects is still lacking. 6.10. CONCLUDING REMARKS As we have seen, the fermion Chern-Simons theory has proved to be a very powerful tool for predicting the behavior of a two-dimensionalelectron systems with a partially filled Landau level. Various aspects of the theoretical predictions are strongly confirmed by experiments and by comparison with exact calculations of finite-size systems. Other aspects, such as the asymptotic low-energy behavior of the effective mass in Fermi-liquid systems, are on a weaker footing, and are currently subjects of active research. The effects of disorder in actual samples are only partially understood, as remarked in Section 6.6, and more work in this area is clearly needed. Very little work has been done so far on extensions of the fermion Chern-Simons theory to discuss effects of nonzero temperature [114], and further effort is needed in this area as well. Undoubtedly, the next few years will see numerous additional applications of the fermion Chern-Simons theory to properties of electrons in a partially filled Landau level. It remains to be seen to what extent fermion Chern-Simions theories will turn out to be useful in the theory of high-temperature superconductors or in other areas of physics, and to what extent knowledge gained from the study of quantum Hall systems will ultimately be applicable in these areas.

ACKNOWLEDGMENTS The author has benefited greatly from discussionswith many colleagues.Particular thanks are due to P. A. Lee, N. Read, R. Morf, R. L. Willett, H. L. Stormer, J.

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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

/ Composite Fermions J. K. JAIN Department of Physics, State University of New York at Stony Brook, Stony Brook, New York

7.1. INTRODUCTION Two-dimensional electron systems (2DESs) exhibit spectacular phenomena when subjected to an intense transverse magnetic field. Most notable is the quantum Hall effect (QHE), in which the Hall resistance [l] exhibits plateaus at the quantized values

h R" =se" where f is either an integer [2] or a simple rational fraction [3]. The former is called the integer QHE (IQHE) and the latter the fractional QHE (FQHE). The prominent fractions appear in certain sequences, some of which are:

f = n = 1,2,3, ...

-=

All observed fractions with f 1 have odd denominators. The longitudinal resistance in the plateau region vanishes exponentially with temperature. ~

Perspectives in Quantum Hall Eflects, Edited by Sankar Das Sama and Aron Pinczuk. ISBN 0-471-11216-X 0 1997 John Wiley &Sons, Inc.

265

266

COMPOSITE FERMIONS

These astonishingly clean experimental results in a dirty solid-state system stimulated an intense theoretical investigation into the properties of 2DES in a magnetic field, and substantial progress has been made in understanding the underlying physical principles. While the IQHE can be explained in terms of noninteracting electrons [4], the FQHE state is a strongly correlated state of interacting electrons [S]. It has become clear in recent years that its essential features can be understood straightforwardly in terms of a new kind of particle, called a compositefermion (CF) [ 6 ] ,which is a bound state of an electron and an even number of vortices of the many-body quantum-mechanical wavefunction. Composite fermions are formed because, in a range of filling factors, electrons avoid each other most efficiently by capturing an even number of vortices of the wavefunction. The residual interaction between composite fermions is weak and can be neglected altogether to a good first approximation. The strongly correlated liquid of interacting electrons thus resembles a gas of noninteracting composite fermions. A crucial property of composite fermions is that they experience an effectivemagnetic field that is different from the external magnetic field. This is a direct consequenceof the binding of vortices to the electrons: As the composite fermions wind around one another, the phases generated by the vortices partly cancel the Aharonov-Bohm phases originating from the external magnetic field, thereby effectively reducing the field. This is the fundamental cause of the unusual experimental properties of the FQHE system. It is often intuitively useful, though not literally correct, to view the composite fermion as an electron carrying an even number of flux quanta; this model produces the correct phases as the composite fermions move around. In this picture, electrons absorb part of the external magnetic flux to become composite fermions, which then move in the weaker residual magnetic field. The existence of composite fermions was originally postulated to provide a unified description of the QHE, in which the FQHE of electrons is equivalent to the IQHE of composite fermions [S]. Subsequently, certain anomalies were observed near the half-filled Landau level (LL) [7], where no FQHE is observed. In an insightful work, Halperin et al. [8] interpreted these as the signature of a Fermi sea of composite fermions. There is now good experimental evidence for the existence of such a Fermi sea [9-113. The composite fermions thus provide a simple unifying description of the various liquid states ofelectrons in the lowest LL, which could collectivelybe termed the compositefermion state. A detailed and accurate microscopic theory of the CF state also exists. The wavefunctions based on the CF theory have been found to have a high degree of overlap with the exact eigenstates in small system studies [12-18], and several quantitative predictions of Ref. [8] have been verified experimentally [9]. That the composite fermion is the appropriate collective variable for the FQHE problem follows from the feature that, as discussed later, the complex system of interacting electrons looks like a system of noninteracting composite fermions. This provides an effectively single-particle description of the physics and brings the qualitative understanding of the FQHE to the same level as that of a normal metal, which is viewed as a system of weakly interacting Landau

THEORETICAL BACKGROUND

267

quasiparticles. Such understanding is often a crucial step and forms the basis of a more quantitative theory. Unlike in the Landau Fermi liquid, the relevant particles in the FQHE are intrinsically different from electrons and are not accessible from the noninteracting electron system by a naive perturbation theory. A number of earlier ideas contributed to the development of the CF theory. Many of the key concepts were introduced by Laughlin in a seminal paper [S], where he identified incompressibility [19] as the physical origin of the FQHE, recognized the importance of Jastrow-type correlations, proposed elegant trial wavefunctions for the FQHE states at f = 1/(2m + l), carried out numerical studies to test their microscopic validity, and showed that the quasiparticle excitations have fractional charge. Other important contributions were the quasiparticle-hierarchy schemes of Haldane [20] and Halperin [21]; the cooperative-ring-exchangeideas of Kivelson et al. [22]; mapping of the Laughlin wavefunction to a bosonic ground state by Girvin and MacDonald [23]; Laughlin’s mean-field approximation for a many-anyon system [24]; and the bosonic Chern-Simons formulations of the FQHE by Zhang et al. [25] and Read [26]. All these works share with the CF scheme the basic philosophy that an important part of the physics of the FQHE lies in the winding phases of particles.

7.2. THEORETICALBACKGROUND 7.2.1. Statement of the Problem Theoretically, the aim is to solve the many-body quantum mechanical problem defined by the Hamiltonian

j+kErjk

j

The first term on the right-hand side is the kinetic energy in the presence of a constant external magnetic field (where 2 is the vector potential), the second term is the Coulomb interaction energy, and the third term is a one-body potential due to a uniform positive background. The electrons are constrained to move in the x-y plane. In GaAs, where most experiments are carried out, the band mass of the electron (me) is 0.07 times the free electron mass and the dielectric constant E = 13. We will assume no disorder to begin with, which can be incorporated through UF).It should not come to anyone as a surprise that this problem is not exactly solvable. However, as will be seen below, considerable insight into its eigenspectrum and the structure of the low-energy eigenstates can be gained based on the single assumption that composite fermions are formed.

268

COMPOSITE FERMIONS

7.2.2. Landau Levels

The problem of a single electron in a magnetic field was solved by Landau a long time ago, and the solution may be found in standard textbooks [27]. We enumerate here some principal results.

1. The kinetic energy scale in the problem is the cyclotron energy ho,=

mec

x 20B[T]

K

(3)

the length scale is the magnetic length ro=(z)

l/'

z

250

m

A

(4)

and the interaction energy scale is (5)

The last term in each equation gives the values in a convenient form for GaAs heterostructures. 2. In the circular gauge 2 = (B/2)(- y , x, 0), the eigenfunctions are given by [5]

where the electron position is expressed in terms of z = x - iy, all lengths are expressed in units of ro, and i3 E i3/az. The Landau level (LL) index s = 0,1,2, .. ., the angular momentum index 1 = - s, - s + 1,. . .,and the eigenenergies are

All 1 states in a given LL are degenerate.The eigenstates have an especially simple form in the lowest LL (LLL)(s= 0);

3. The eigenstate Po., has its weight located at the circle of radius R, x ro,/5

(9)

This allows a calculation of the degeneracy of the LLL as follows. Consider a disk

THEORETICAL BACKGROUND

269

of radius R centered at the origin. The largest value off for which the eigenstate falls within the disk is given by I,,, = R2/2r& which is also the total number of eigenstates in the LLL that fall within the disk (neglecting corrections of order unity). The degeneracy per unit area is therefore

where 4o= hc/e is the magnetic flux quantum. Thus there is one state per flux quantum in the LLL. Apart from corrections of order unity, the degeneracyis the same for all LLs. 4. The filling factor is defined to be

v =P 4o B where p is the (area) density of electrons. The quantity v is the nominal number of filled LLs, also equal to the ratio of the number of electrons to the number of flux quanta penetrating the sample. As the magnetic field is increased, there are more and more states available in each LL, and the electrons occupy fewer and fewer LLs, thereby decreasing the filling factor. 5. In the discussion above, we have taken the magnetic field to be pointing in the + z direction. For a magnetic field in the - z direction, everything remains the same except that the eigenstates are This is seen by noting that a complex conjugation of the Schrodinger equation is equivalent to reversing the direction of the magnetic field.

7.23. Kinetic Energy Bands Let us now consider the system of many noninteracting electrons. For later purposes, we need to get used to thinking in terms of many-particle states rather than single-particlestates. Let us specialize to the case of spinless electrons, for reasons that will become clear later. The ground state is obtained when electrons occupy the lowest-energy states, without occupying any state twice, as required by the Pauli principle. At filling factor n < v < n + 1,the lowest energy is obtained when the lowest n LLs are fully occupied, the (n + 1)st LL is partially occupied, and the higher LLs are unoccupied. The ground state is highly degenerate since all different configurations of electrons in the partially filled (n + 1)st LL produce the same energy. The excitations are obtained when one or more electrons are promoted across one or more LLs. The energy spectrum of the many-electron system thus consists of bands of degenerate states separated by ha,, which will be called kinetic energy bands (KEBs). Note that the KEBs refer to many-body eigenstates, as opposed to LLs, which are the energy levels of a single electron. For noninteracting electrons, all states in each KEB can be enumerated straight-

270

COMPOSITE FERMIONS

forwardly for any arbitrary filling factor. At the special filling factors given by v = n, the ground state contains n fully occupied LLs and is nondegenerate.In this case, the lowest KEB (LKEB) contains only one (many-body) state.

7.2.4. Interactions: General Considerations Now switch on the Coulomb interaction. The FQHE is observed at relatively large B and believed to occur in the limit B + 00, where the Hilbert space can be restricted to the LKEB. (This limit will be assumed throughout this article. For a fixed v, this also requires that p = Bv/c#J, -,co.)This follows since the Coulomb interaction energy ( $) is small compared to the LL spacing (- B) in this limit, hence does not cause any LL mixing. For v < 1, restriction to the LKEB is equivalent to confining all electrons to the LLL. The advantage of working in the limit of B-, 00 is that the size of the LKEB Hilbert space is finite for a finite number of electrons,and the Coulomb Hamiltonian can in principle be diagonalized exactly (numerically) to obtain all eigenstates and eigenenergies of the interacting system. We will further assume, except when explicitly mentioned otherwise, that the Zeeman energy is also large compared to the interaction energy.The spin degree of freedom is then completely frozen,so that electronscan be taken to be spinless. Within the LKEB, the kinetic energy is an irrelevant constant that can be dropped. The Hamiltonian in the LKEB thus consists only of the Coulomb interaction term. All LKEB states are degeneratewithout interaction, and there is no small parameter in the problem, since the interaction strength is the only energy scale. This is the reasons why the interaction cannot be treated perturbatively in the FQHE problem. N

7.3. COMPOSITE FERMION THEORY 7.3.1. Essentials The composite fermion theory is based on the single hypothesis that in a certain range of filling factors, electrons capture an even number of vortices of the wavefunction to become composite fermions, which can be treated as weakly interacting. The basic ideas of the composite fermion theory are rather straightforward, and I start by stating them. Later, I discussthe initial motivation behind these ideas. Wawefunctions. The wavefunctions of noninteracting composite fermions are obtained from those of noninteracting electrons by attaching 2m vortices to each electron, which amounts to multiplying of the noninteracting electron wavefunction by the Jastrow factor

D"

n

j
( z j - zk)*"

COMPOSITE FERMION THEORY

271

To see how this factor binds 2m vortices to each electron, fix all z i s except z l . As z1 taken in a closed loop around any other electron, D" contributes a phase of 4mn, i.e., each electron sees 2m vortices on every other electron. (By definition, a closed loop around a unit vortex produces a phase of f 2 n . ) Denote the many-particle Slater-determinant states of noninteracting electrons at filling factor v* by @v.,b,ar where b is the KEB index and a labels the eigenstates within a KEB. (Some or all of the subscripts v*, b, and a will often be suppressed in the following.)This produces

This wavefunction is not restricted to the LLL of electrons,however. Since we will be interested in the B + 00 limit, we project r,j on to the LLL to obtain the wavefunctions of composite fermions at v*, @ ,' which are identified with the LLL eigenstates of interacting electrons, x. Thus

or, in shorthand, x = OCF= BD"@.Here, S is the LLL projection operator. The right-hand side of this equation is completely known; it involves the eigenstatesof the solvable problem of noninteracting electrons at v*. Relationship Between u and u*. The careful reader may have noticed that different filling factors have been associated with x and 0 in Eq. (14). This is because the system of composite fermions at v* describes an electron system at a different filling factor, v, given by V=

V*

2mv* + 1

It is instructive to derive this equation. From the definition of the filling factor, the angular momentum of the outermost occupied state in @., is ,.:I = N/v*. The angular momentum of the outermost single particle state in x is then I, = I:, + 2m(N - l), since the largest power of, say zl,in the Jastrow factor is 2m(N - 1).Therefore, the filling factor of x, v = N/I,,,, is given by Eq. (15) in the limit N -P co. The Jastrow factor thus pushes electrons farther apart, thereby decreasing the density and reducing the filling factor. (In actual experiments, the magnetic field changes rather than the system size.) Equation (15) can also be written as a relationship between the effective magnetic field experienced by the composite fermions, B* = p&,/v*, and the external field B = p&,/v as

272

COMPOSITE FERMIONS

Equation (14), along with the relationship between B and B* (or, alternatively, between v and v*) implied by it, is the defining equation of the CF theory. B* and v* are allowed to be either positive or negative (the negative values correspond to the case where B* and B are antiparallel).Therefore, Eq. (15) can be also written as V=

lv*l

2mlv*I f 1

We note that &,, = [@,.I*. An illuminatingway of understanding why the compositefermions experience a different magnetic field than electrons is as follows. Let a composite fermion move around a closed loop enclosing an area A. The net phase associated with this path is given by

where the first term is the usual Aharonov-Bohm phase due to the external magnetic field By and the second term is generated by the vortices of N,,, composite fermions inside the loop. In a mean-field approximation, N,,, is replaced by its average value, which is pA for a uniform density system. Equating the resulting phase to an effective Aharonov Bohm phase 27rB*A/4, gives the effective field B* = B - 2mp4,. It should be emphasized that while the CF wave functions are necessarily only approximate, the physics represented by them, in particular, Eqs. (16) and (17),are expected to be exact in the regime of validity of the C F theory. Low-Energy Spectrum. The energy levels of composite fermionsare analogous to the LLs of electrons at B* and are called CF-LLs or quasi-LLs. Hence, a fundamental consequence of the formation of composite fermions is that the LKEB of interacting electrons splits into the KEBs of composite fermions. The interaction energy of electrons manifests itself effectivelyas the cyclotron energy of composite fermions. The formation of composite fermions thus causes a drastic reduction of low-energy Hilbert space, as only the lowest CF-KEB is relevant at low energies, whose dimension is much smaller than that of the LKEB of electrons that we began with. Equation (14) provides correlated bases for the various CF-KEBs. The Hilbert space reduction is most dramatic at the special electron filling factors: v=-

n 2mn k 1

These map into noninteracting composite fermions at v* = n, where the lowest CF-KEB contains only one state, with n filled CF-LLs. Here, the Coulomb interaction removes completely the enormous degeneracy of the noninteracting electron system to produce a nondegenerate ground state, separated from other

COMPOSITE FERMION THEORY

273

states by a gap. The C F ansatz for the ground-state wavefunction is

where is the Slater-determinant state for n filled LLs. It contains no adjustable parameters.

7.3.2. Heuristic Derivation In this section we repeat the argument of Ref. [6], which motivated the C F theory via a flux-attachment transformation followed by a mean-field approximation. It proceeds along the following steps:

1. Start with the many-body Slater-determinant states in the LKEB of noninteracting electrons at B*. 2. Now imagine attaching to each electron an infinitesimally thin massless solenoid carrying an even number of flux quanta. It will become clear below that the bound state of an electron and an even number of flux quanta is a (mean-field) representation of the composite fermion, since a flux quantum is also a vortex. So, to avoid proliferation of names, we call it composite fermion (which was incidentally how the composite fermion was first defined). This produces a C F state at B*. The crucial point here is that this state is identical to the electron state at B*: Adding the flux in this manner does not alter either the Aharonov-Bohm phases [22] or the statistics of the particles [28]. (This is why we need to add an even number of flux quanta-the composite of an electron and odd number of flux quanta is a boson.) In particular, composite fermions exhibit the IQHE. 3. The reason for redefining the problem in this way is that it suggests a mean-field approximation (MFA) that was not manifest in the original problem; this type of MFA was first alluded to by Arovas et al. [29] and employed by Laughlin in the anyon model of high-T, superconductors [24]. In this approximation, the flux, which is a part of the composite fermions, is smeared so that it becomes a part of the external magnetic field. We thus again recover electrons, but at a new magnetic field, B. The validity of the MFA requires that as the flux is slowly spread, the gap between the lowest two KEBs does not close, even though it may evolve in some unspecified, complicated manner. Then, it is a reasonable assumption that the states of the LKEB do not mix appreciably with the states of the higher bands during the flux-smearing process (although they may mix among each other). 4. It is straightforward to see that the new magnetic field is given by B = B* + 2m+,p, Eq. (16),since 2m flux quanta have been added per electron. Thus the composite fermions at B* are related to electrons at B* on the one hand and to electrons at B on the other. Figure 7.1 illustrates these ideas pictorially. 5. There are two calculational schemes based on the intuitive physics above. One constructs explicit wavefunctions. The first guess for the C F wavefunction

274

COMPOSITE FERMIONS

0

0

I I; 0

0

Figure 7.1. (Top)Electrons (dots) in a uniform magnetic field, B*, with arrows representing magneticflux quanta. Each electronis converted into a composite fennion (middle)by attachment of two flux quanta. A mean-field approximationsmears this flux to produce a system of electrons at a larger magnetic field B = B* 2p4, (bottom).

+

would be [6]

where the Jastrow factor is a pure phase factor, as would be produced by flux tubes attached to electrons. A moment’s thought tells us that this is not the most appropriate wavefunction for the actual electron state: it has a high occupation of higher LLs and no special short-distance correlations. Motivated by the Laughlin wavefunction and by a number of other considerations,it was conjectured in Ref. [6] that a more reasonable wavefunction for the FQHE state is

COMPOSITE FERMION THEORY

275

@zrproj.

obtained by throwing away the denominator of Eq. (21), which gives The Jastrow factor Dmin generates the same phases as the Jastrow factor in and thus also describes composite fermions, but at the same time also servesto keep electrons apart. The second scheme employs a Chern-Simons (CS) field theory [8,33] to investigate the CF state. The wavefunction 0:; is obtained at the mean-field level of the CS field theory, starting from which one seeks to approach the true state perturbatively, using techniques developed earlier in the contexts of Bose-condensate-like description of the Laughlin wavefunction [23,25] and anyon superconductivity [34]. Although the two schemes are based on the same physics, a precise quantitative relationship between them is not clear at the present.

@:rproj

7.3.3. Comments 1. The formation of composite fermions is a nonperturbativeeffect. All LKEB states are degenerate in the absence of interactions, and the composite fermions and their KEBs are formed in the presence of arbitrarily weak repulsive interaction. The prefix CF or quasi differentiates the energy levels of composite fermions from the real LLs of electrons. Composite fermions can occupy several CF-LLs even when the electrons are entirely within their lowest LL. 2. The Laughlin wavefunction [5] at v = 1/(2m + l),

is a special case of the states in Eq. (20), since

It represents one filled LL of composite fermions. 3. Noninteracting composite fermions describe strongly correlated electrons. The Jastrow To see this, consider first the unprojected CF wavefunction @::proj. factor ensures that the probability of electrons coming close to one another is small, since the wavefunction vanishes with a high power of the distance between two particles, which provides an intuitive argument for why the formation of composite fermions might lead to a low interaction energy. The FQHE is observed at relatively large B, where the low-energy states are expected to be predominantly in the lowest LL. Since involves higher LLs in general, one might expect that @t:proj also has a large component in higher LLs, which was one of the main initial objections against the CF theory. However, the Jastrow factor, D, in addition to keeping the electrons apart, also pulls them down to the LLL to a surprisingly large extent [30], as shown in Table 7.1. For the B -+ 00 limit, which is very convenient for numerical calculations,it is desirable to

276

COMPOSITE FERMIONS

TABLE 7.1. Kinetic Energy of Certain Unprojected CF Wavefunctions, D"8," V

1/3 215 317 419 If5 219 3/13

Wavefunction

DQ, DO2 D@3

Da.4 DZ#, D2#,

D2#,

E, 0

0.041(5) 0.0635) O.OSO(6)

0 0.019(3) 0.025(6)

Source: Results from Ref. [30]. "The kinetic energy per electron is defined to be ( E , + 1/2)hwc; a small value for E, implies that the wavefunction is predominantly in the LLL. All results are estimates for the thermodynamic limit obtained in variational Monte Carlo calculations; the statistical uncertainty in the least significant digit is given in parentheses. Note that the kinetic energy of the IQHE state is given by E, = (n - 1)/2.

write wavefunctions strictly confined to the LLL. We would like to find an adiabatic continuation of the unprojected CF wavefunction to this limit. We assume that this can be done simply by throwing away the part that involves higher LLs (i.e., by projecting the wavefunction on to the LLL). Since the wavefunction was mostly in the LLL to begin with, it is plausible that such projection will preserve its essential correlations. 4. A crucial new feature of the CF theory is its use of the higher LLs. Earlier theories of the FQHE restricted the Hilbert space strictly to the LLL. There is no fundamental reason for such a restriction, since the FQHE is indeed possible in the presence of some LL mixing; it was motivated principally by the hope that the physics would simplify in the LLL. However, the CF theory shows that the relevant correlations of the FQHE state can be understood in a transparent manner if a small amount of admixture with higher LLs is allowed. It also tells us how to obtain, eventually, states confined strictly to the LLL. The use of higher LLs is not merely a mathematical trick but has a real physical significance, as it brings out the quasi-LL structure of compositefermions. There exists compelling theoretical evidence (e.g., see Sections 7.4.5 and 7.4.10) that the low-energy spectrum of interacting electrons at v, conjned to the LLL, bears a striking resemblance to that of noninteracting fermions at v*, occupying several LLs in general. Most experimentsare also understood straightforwardly in terms of LLs of composite fermions. The use of higher LLs for the FQHE also has the aesthetically pleasing quality that it unifies the IQHE and the FQHE.

COMPOSITE FERMION THEORY

277

5. The CF theory gives wavefunctions more naturally than energies. (Of course, given wavefunctions,energies can, in principle, also be computed.) The reason is that the eigenfunctionsare in general rather insensitive to the form or the strength of the interaction, provided that it is repulsive, whereas the eigenenergiesare not. As noted earlier, the C F wavefunctions for the incompressible ground states contain no fitting parameters. It is an extraordinary feature of the FQHE state it lends itself to such a parameter-free description. The energy gaps, on the other hand, depend on the interaction. A simple example is the state at v = n,which is well approximated by then filled-LL wavefunction,independent of the cyclotron energy gap. In this sense, the eigenfunctions of the FQHE problem are “universal,” whereas the eigenenergies are a matter of detail. Some attributes of the energy spectrum are also universal, namely the existence of bands, and the presence of gaps at certain filling factors. The C F theory makes statements about these features. The analogy between the FQHE and the IQHE has sometimes been overinterpreted. It applies only to the universal fcatures, enumerated above. The fact that the IQHE gap is determined primarily by the cyclotron energy, whereas the FQHE gap is determined by the Coulomb interaction, is not evidence against the CF theory. The gap of the FQHE state ought to be computed using the explicit CF wavefunctions for the ground state and the excited state. 6. Since the physics of the FQHE is to a certain extent not sensitive to the form of the repulsive interaction, it would be useful to have analytic solutions for any reasonable model interaction. No such exactly solvable model is known. The Laughlin wavefunction is the exact ground state for a short-range interaction potential [20,3 I], and certain artificial models involving two- and three-body interactions have been constructed for which @:fproj are also the exact ground states [32]. However, each of these models can be solved only at one specialfilling factor, and even then only for the ground state. Rather than attempting to construct approximate models with exact solutions, the philosophy of the CF scheme is to write approximate solutions for the realistic Coulomb problem in a manner that brings out the essential physics. 7. Consider the polynomial part of the LLL wavefunctions of the incompresible states. Fix all z j except zl, and ask where the zeros of z1 are. The fundamental theorem of algebra tells us that the total number ofzeros is equal to the degree of the polynomial, N / v (neglecting corrections of order unity). The answer is simple for the Laughlin wavefunctions: All the zeros are bound to electrons,with (2m + 1) zeros on each electron. This makes one wonder if there is something special about the distribution of zeros of the other FQHE states. It is elear that all zeros can be placed on electrons only at v = 1/(2m + 1). In general, there is only one zero on each electron, as required by the Pauli principle, but the arrangement of the other zeros is rather complicated and depends on the positions of the fixed electrons. According to the C F theory, the v = n/(2mn & 1) incompressible state in the LLL is special since it is adiabatically connected to Dm@*,,, whose zeros have the same structure as those of the IQHE state @*,,

278

COMPOSITE FERMIONS

(aside from the 2m additional zeros on each electron due to the Jastrow factor). Unfortunately, little is known at the moment about the nodal structure of the IQHE states. What is a composite fermion, then? It is a collective quantum-mechanical object that behaves like a particle, with the defining properties that (a) its charge is - e (and spin 1/2)(b)it obeys fermionic statistics,and (c)it experiencesan effective magnetic field B* = B - 2mp4,. It is the last property that distinguishes it from an electron. As shown below, the most accurate approximation to the truth is provided by the projected CF wavefunctin,which, however, does not lend itself to useful intuition. The simpler way of thinking about the composite fermion is as an electron carrying an even number of vortices, which clarifies the origin of the effective field. There are two approaches for modeling the vortices, given by the wavefunctions in Eqs. (13) and (21). Both produce identical phase factors as the composite fermions wind around one another, and it is likely that they both are perturbatively connected to the projected CF wavefunction.

7.4. NUMERICAL TESTS 7.4.1. General Considerations As remarked earlier, it has not been possible to derive in a rigorous manner the composite fermions, their wavefunctions, or their effective field, starting from a (model)Hamiltonian. However, strong support for the validity of the CF ansatz comes from numerical calculations and experiments, to which the rest of the chapter is devoted. In this section we describe the results of a comparison of the CF theory with numerical diagonalization studies. While such studies have the disadvantage that they deal with finite systems of usually fewer than 10electrons, they have the virtue of being exact. Ofcourse, exact does not always mean useful, but it is a remarkable feature of the FQHE problem that systems with as few as eight electrons already exhibit several features of the real expeiments; for example, they produce incompressible states at precisely those filling factors where the FQHE is observed. The reason for this good fortune is that the relevant correlation length of the FQHE system is finite nue to the presence of a gap, and in a range of parameters, the numerical systems are already large compared to this length, i.e., effectively thermodynamic.It thereforemakes sense to take the results of the numerical calculations seriously. Of course, a satisfactory confirmation of the theory will also require agreement with experiment, which is the subject of a later section. We treat the numerical calculations as a little laboratory in the computer where a theorist can measure every imaginable detail, and test various aspects of a given theory. The CF theory will be compared with numerical calculations at two levels. First is the verification of the qualitative predictions of the theory, regarding the existence of bands, the number of states in each band, and their quantum numbers. The second concerns microscopicCF wavefunctions. Instead

NUMERICAL TESTS

279

of considering several correlation functions, we simply compute the overlap between the CF wavefunction and the corresponding Coulomb eigenstate obtained numerically. A near-unity overlap will guarantee that all correlation functions as well as the energies of the Coulomb system are well approximated by those of the CF system. All results given below are exact, in the sense that both the Coulomb eigenstates and the CF wavefunctions are evaluated exactly. 7.4.2. Spherical Geometry A convenient geometry is the spherical geometry, studied in detail by Wu and Yang [35] and first used by Haldane in the studies of the FQHE [20], in which N electrons move on the surface of a sphere under the influence of a radial magnetic field produced by a magnetic monopole of strength q at the center. The flux through the surface of the sphere is equal to 2q flux quanta, where 2q is an integer, and the filling factor is given by v = N/2q in the limit N + 00. We summarize here some basic results, and refer the interested reader to two very useful papers by Wu and Yang [35] for further information. The eigenstates in the spherical geometry are called monopole harmonics Yq,l,,r. For q = 0 these are the usual spherical harmonics. The quantity I = 141, I q ) + 1,. .. denotes the orbital angular momentum, and 1, = k I , k (I - l),... is its component along the z direction. Note the special features that the minimum value of 1 is (41, and it can take either integer or half-integer values. The angular momentum shells of the spherical geometry are the analogs of the LLs of the planar geometry, with the LLL given by the 1 = ( qIshell.The degeneracy of the Ith shell is 21 + 1;it is 21ql+ 1 in the LLL and increases by two in each successive higher LL. The Coulomb interaction (see Ref. [14] for explicit matrix elements) commutes with the total orbital angular momentum L of the many-particle state, which will be used to label the eignestates. All eigenstates in a given L multiplet (i.e., states with different L,) have the same energy. Therefore, it is sufficient to work in the subspace of smallest L,, provided it is remembered that each state in this subspace denotes a multiplet of 2L + 1 degenerate states of the full Hilbert space. (Recall that electrons are assumed to the spinless unless specified otherwise.) The L = 0 state has uniform density. There is a subtlety in assigning a filling factor to a finite system. For a given system (N, q), many equations

N = 2qvi + Ci

(24)

can be written with different choices of vi and C i ,apparently leading to different filling factors vi = N/2q in the limit N + 00. Which one should be used? The answer is that only that choice of vi and Ci is meaningful for which all systems (N, q) satisfying Eq. (24) exhibit qualitative similarity, allowing a smooth extrapolation to the N-rco limit. Equation (24) must be tested for several (N,q) for a given vi and Ci before it is used to assign a filling factor to a finite system.

280

COMPOSITE FERMIONS

7.4.3. Composite Fermions on a Sphere The Jastrow factor D in the spherical geometry is the square of the wavefunction of the lowest filled LL of spinless electrons, that is,

(This is also true in the planer geometry, aside from the exponential factors.) Here it is more convenient to label the wavefunctions by q rather than the filling factor v, and Eq. (14) is written as

The effective flux experienced by composite fermions is 2q* (in units of &), with q* = q - m ( N - 1)

(27)

The last term follows since for any given composite fermion, each of the remaining N - 1 composite fermions reduces the flux by 2m flux quanta. In the thermodynamic limit N -+ co,this equation coincides with Eq. (15) with v = N/2q and v* = N/2q*. Note that for finite systems, the C F mapping can be applied independent of the question of what filling factor it represents. The C F transformation conserves L. It is easy to see that multiplication by CP,, which represents a filled shell with zero angular momentum, does not alter L. The projection operator 8also commutes with L, which becomes apparent by writing it as [17] N

8 =fl gi i= 1

gihas the property that it produces a zero when applied to any single-particlestate in a higher LL, and unity when yplied to any state in the lowest LL. Since the total angular momentum operator Lz commutes with the angular momentum operator of an individual electron, 72,it also commutes with the projection operator. The C F theory asserts that the LKEB of interacting electrons at q splits into bands, which can be understood as the KEBs of composite fermions. The prescription for generating the correlated basis states for the CF-KEBs is: (1)start with the orthogonal states of noninteracting electrons at q*;(2) multiply them by the appropriate Jastrow factor; (3)project the product states onto the LLL; and (4) orthonormalize. We will need to compare the resulting C F basis with the exact Coulomb eigenstates. Denote the exact Coulomb eigenstates by xi and the C F withj = 1,2,. .. ,J . Diagonalize the Coulomb Hamiltonian in basis states by

my,

NUMERICAL TESTS

281

the C F basis, and call Oj the overlap of thejth CF eigenstate with xi. Then (njOj)'lJprovides a measure of how close the two bases are and will be called the overlap between the two bases. For J = 1, it is the usual overlap. An advantage of this definition is that its calculation does not require any diagonalization in the CF basis, since it is equal to [Det < 0TIxk>]'IJ. 7.4.4. Band Structure of FQHE

Now we are ready to compute the exact Coulomb eigenstates and compare them to the CF theory. We restrict the Hilbert space to the LKEB at q and switch on the Coulomb interaction, which will remove the degeneracy of states in the LKEB. One might naively expect a broad featureless band. However, an exact diagonalization of the Coulomb Hamiltonian for few electron systems shows that the structure of the low-energy Hilbert space is much more interesting [12,36]. Several bands are formed, as shown in Fig. 7.2. In particular, a few states split off from the rest to form a low-energy band, well separated from the other higher-

-1.0-

-1

3

7 L

11

-0.11

.

.

I

-1

3

7

11

-0.1

L

-1

3

7 L

11

Figure 7.2. Low-energy spectra of eight interacting electrons on the surface of a sphere at various values of q. The states below the dash-dotted line form the lowest band. The energies are in units of e2/wo,where E is the dielectric constant of the background material and ro is the magnetic length. Panel ( a ) corresponds to v = 1/3 and (f)to v = 2/5. (From Ref. [36], with permission.)

282

COMPOSITE FERMIONS

L (a) 2q*= 7

(e) 2q*= 3

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

L L Figure 7.3. Lowest kinetic energy band of eight noninteractingcomposite fermions at the values of q* correspondingto the q of Fig. 7.2. There is a complete one-to-one correspondence between the lowest bands of Figs. 7.2 and 7.3. The overlaps between the CF states and the correspondingCoulomb states of Fig. 7.2 are shown on the figure. (Overlapsfrom Ref. LIZ].)

energy states. At some special filling factors, the lowest band contains only one state (see, e.g., Fig. 7 . 2 ~and f). Some higher bands may also be identified, although they are usually less well defined. 7.4.5. Lowest Band

In Fig. 7.3, we plot the LKEB of noninteracting fermions at q* = q - N + 1. Remarkably, there exists a complete one-to-one correspondence between the lowest bands of Figs. 7.2 and 7.3: There is an equal number of states in both bands with identical quantum numbers. The low-energy dynamics of interacting electrons at q thus resembles that of non-interacting fermions at q*. This has been confirmed for a large number of other systems (with no known exceptions in the relevant filling factor range) and establishes, in a transparent, model-independent way, the formation of composite fermions. We next test the microscopic CF wavefunctions. The overlaps of the Coulomb eigenstates with the corresponding CF wavefunctions [121 are shown on Fig. 7.3. Similar tests have been carried out for the lowest-band states for numerous other cases (including situations when there are several multiplets at a given L), with similar results [12-14,183. If the composite fermions were truly noninteracting, all lowest-band states in Fig. 7.2 would be degenerate. The amount by which the degeneracy is lifted is a measure of the residual interaction between the composite fermions. The

NUMERICAL TESTS

283

relatively small energy splitting between the lowest-band states, compared to the gap separating them from the higher band states, demonstrates that the composite fermions are only weakly interacting. 7.4.6. IncompressibleStates

Special attention is due to the states containing filled LLs of composite fermions. The state with n filled CF-LLs is obtained for q*=

*-N -2nn 2

It is predicted that the system of interacting electrons (N,q),with

has an incompressible ground state. This has been fully confirmed in numerical studies. All systems (N, q) studied to date, which satisfy Eq. (31),have been found to have an L=O ground state, separated from the excited states by a gap, generallyevident by inspection. In some cases (e.g., for 1/3,2/3,2/5), more detailed studies have shown that the gap varies smoothly with N and has a nonzero extrapolation in the thermodynamic limit [37]. Thus Eq. (31) satisfies the criterion stated earlier, and the corresponding finite systems (N, q) can be assigned the filling factor v = n/(2mn & 1) [38]. The CF wavefunction for the ground state is given by Eq. (20). Table 7.2 gives the overlaps between the CF wavefunctions with the corresponding exact Coulomb ground states for several values of n and N. The near-unity overlaps substantiate the description of the incompressible states as filled LLs of composite fermions.

with the Corresponding Coulomb Eigenfunction for Several Incompressible States”

TABLE 7.2. Overlap of

-

v

n

3

1

N

1

1

Overlap

1:ll:: 9 ~~

0.9941

-

v

n

4

2

f

3

N

Overlap

1:Il I 9

0.9994

1 I

v

4

n

N Overlap

2

8 0.9982 6 0.9965

10 0.9940

~

Source: Results from G. Fano, F. Ortolani, and E. Colombo, Phys. Rev. B 34,2670 (1986) for v = 1/3; Ref. [12] for v = 2/5; and Ref. [13] for v = 3/7 and 2/3. *N is the number of electrons.

284

COMPOSITE FERMIONS

7.4.7. CF-Quasiparticles

A composite fermion in the topmost partially Jilled CF-LL may be viewed as a (CF)-quasielectronof the background FQHE state. Similarly,a hole in @ maps into a quasihole of the FQHE state x. The results of Section 7.4.5 show that the CF theory provides a good description of not only a single quasiparticle (quasihole or quasielectron) but also of several (interacting) quasiparticles. In addition, the CF picture also leads to an intuitive understanding of several qualitative properties of the quasiparticles. For example, it has been known for some time that there is a fundamental asymmetry between the quasihole and quasielectron of a FQHE state. Such an asymmetry is evident in the CF theory, since vortices are attached only to the electrons of @ but not to the holes. To see what further light the CF theory sheds on this issue, let us consider the example of the 1/3 state. Its quasihole and quasielectron states are related, respectively, to states with the LLL completely occupied except for a hole, and the LLL completely occupied plus an additional electron in the second LL. Figure 7.4 shows the charge densities associated with these (IQHE) states. The density profiles are different simply because the hole is in the LLL, while the additional electron resides in the second LL. The charge densities of the quasihole and the quasielec-

1.5

0

.5

1

1.5

2

2.5

3

3.5

4

r Figure 7.4. Charge density of the lowest filled LL with a hole at the origin (solid line) and the lowest filled LL plus an electron in the second LL at the origin (dashed line). The distances are measured in unit of the magnetic length.

NUMERICAL TESTS

285

tron of the Laughlin 1/3 state [39] are similar to those in Fig. 7.4. In particular, the quasielectron of the 1/3 state also has the smoke-ring shape with a dip at the origin [37], which nicely illustrates the appearance of a second-LL-likestructure within the LLL. Interestingly, the maxima in the density of the quasielectrons of the v = 1 and v = 1/3 states occur at very nearly the same distance from the origin. The analogy to IQHE also provides an intuitive explanation of why the quasielectron energy is greater than the quasihole energy.

7.4.8. Excitons and Higher Bands For the basic phenomenon of the FQHE, a satisfactory description of the lowest band, discussed above, is sufficient. We now proceed to the excited bands. The one-to-one correspondence between the IQHE and the FQHE states cannot be expected to continue to arbitrarily high energies even as a matter of principle. We have three systems here: (1) interacting electrons at q, (2)noninteracting electrons at q*, and (3) noninteracting composite fermions at q*. These will be called the x, 0,and OCF,respectively. For a finite number of electrons, the LKEB Hilbert space of x is finite, whereas the Hilbert space of 0 is infinite, since all LLs are allowed. Therefore, there cannot exist a complete one-to-one correspondence between Q, and x. (This underscores the nontriviality of the fact that the lowest bands of the two systems are analogous.)On the other hand, by construction, OCF and x have identical Hilbert spaces, and a correspondence between them can possibly be more robust. Let us first consider the second band at the special filling factors corresponding to incompressible states. It contains precisely one multiplet at each L in a certain range of L (see, e.g., Fig. 7 . 2 and ~ f), called the (intra-LL) collective mode or the exciton branch. The angular momentum L is roughly proportional to the linear momentum p of the planar geometry [MI. Soon after Laughlin's theory of v = 1/(2m + l), Girvin, MacDonald, and Platzman (GMP) [41] developed a single-mode approximation for the intra-LLL collective mode of the Laughlin state. With a straightforward generalization, the GMP wavefunction for the collective mode at v = n/(2mn + 1) is constructed by applying the projected electron density operator to the ground state as Sp'Q,FF = SpL[SD"Q,n]

where

= SD"Q,,, is the

ground state at v = n/(2mn + l), and N

is the density operator at L. It has been shown that this wavefunction provides a good description of the intermediate-L states of the collective-mode branch at v = 1/(2m + 1)[40]. It, however, does not work as satisfactorily at other fractions [42,43].

286

COMPOSITE FERMIONS

In the CF description, it is more natural to view this branch in terms of excitons of compositeferrnions [14]. In contrast to Eq. (32), which gives the collectivemode of electrons,the CF-exciton of the v = n/(2n + 1) state is constructed by analogy to the exciton of the IQHE state @, [45]. Its wavefunction is given by

where pL(n--t n + 1) creates an exciton at v* = n, which involves excitation of one electron from the nth LL to the (n + 1)st LL. The angular momentum of the topmost filled shell of is 1q* I + n - 1, and the lowest empty shell is lq*( n, where q* is given in Eq. (30). Since this excitation creates one electron in the lowest empty shell and one hole in the topmost filled shell, the allowed L values are L = 1,2,..,, 2( I q* I + n) - 1. The wavefunction of the exciton of the IQHE state is determined completely by symmetry for any given L; therefore, the wavefunction of the CF exciton also does not contain any fitting parameter. The angular momentum range predicted by the CF theory agrees with that found numerically, with the exception that there is no exciton state at L = 1 in the numerical spectrum. Remarkably, OCFconstructed from the L = 1 exciton wavefunction of the v* = n IQHE state vanishes identically upon projection for all cases tested so far [12], bringing the CF range in complete agreement with the numerical range. The overlaps of the Coulomb states with the CF states are shown in Fig. 7.5 for the exciton branch at v = 1/3, as also are the Coulomb energies of the CF wavefunctions. The CF theory works similarly well for 2/5 and 3/7 [14]. The minimum energy to create a C F exciton at v = 1/3 has been estimated to be 0.063(3) e2/ero [14], which is lower than the SMA result of 0.078 e2/&ro[41]. Better estimates are obtained at 1/5 and 1/7 as well. Furthermore, an excitonic instability is found to occur at v = 1/9; here the energy of the CF exciton at a finite wavevector is actually less than the energy of the Laughlin state. The value of this wave vector suggests that the FQHE state is unstable to the formation of a Wigner Crystal, which is indeed believed to be the ground state for v < 1/7. This is an example of how the C F theory can even shed light on the luck of FQHE at certain filling factors. The fermion-Chern-Simons theory also allows a computation of the collective-mode dispersion for all FQHE states [MI. This is discussed in Chapter 6. Higher bands have been studied away from the special filling factors as well [141. Let us consider the example of (N, q) = (8,8.5),for which a second band can be identified by inspection (see Fig. 7.2e). Figure 7 . 6 ~shows the spectrum of noninteracting electrons at (N, q*) = (8,1.5). The integer near each line shows the number of independent degenerate multiplets. The second band consists of states in which an electron is promoted to the next-higher LL, either from the lowest to the second, or from the second to the third. Clearly, this does not provide a good description of the second band in Fig. 7.2e. We nevertheless proceed to construct the OCFbasis for the second band, as outlined earlier. Surprisingly,the states obtained after the projection are not linearly independent,

+

NUMERICAL TESTS

n

287

-30

-.4 U

-.42 -.44

-.46 (b)

@

E

u1

3 -1 1

non-interactingelectrons

2 s s s s L s a l l

-1 I L L 1 1 1

1

1

1

15

2

.95

9)

-1 .99

1

1

1

1

1

1

1

1

non-interactlng composite fermions

(c)

6

1

N = 6, q' = 2.5

1 4 2 6 2 . 6 3 2 1 1

.w

36 . I .98 . I .96

1 l l 1 1

.95 .99

1

1

.w 1

.w .w .w

.99 . I

1

1

1

1

. I

N = 6. q8 = 2.5

1

1

1

1

1

and an orthonormalization produces a smaller basis than the one we started with. Some of the states in this basis are the same as those in the lower CF bands. We remove these states and identify the remaining as the CF basis for the second band (see Ref. [141 for further details). The number of independent multiplets in this basis is shown near each Lin Fig. 7.6b.The second band of OCFmatches with

288

COMPOSITE FERMIONS

non-Interactingelectrons

(a)

3 4 4 4 3 2 1 1

@ 2!

w

1

-1

-1

I

I

I

I

I

I

I

I

I

I

0

1

2

3

4

5

6

7

8

9 1011 121314

~

2!

1 3 2 3 1 2 1 1 .99 .96 .97 .9!3 37 .96 .9a .99

1

-1

.99

I

I

I

I

non-Interactingcomposite fennions

-

@

I

~~

(b)

W

N = 8, q' = 1.5

-1

.98

I

I

N

39

I

I

I

I

I

I

I

-

8, q' = 1.5

I

I

I

the second band of x. The overlaps between the CF basis and the actual Coulomb eigenstates,shown in Fig. 7.6b, are reasonable, indicating the microscopic validity of the CF description for higher bands. We have repeated this exercise for several systems for two and three bands (see Fig. 7.5 for the third band of the 1/3 state) and found in all cases that the correspondence between @ and x breaks down beyond the lowest band, but that between QCF and x continues to higher energies. It is nontrivial that the projected CF states obtained from orthogonal CP states are not linearly independent, since the number of states in thelowest two bands of OCFis only a small fraction of the total number of available states. This indicates certain hidden mathematical properties of the CF wavefunctions. It is also noteworthy that the elimination of states in going from @ to CPcF happens precisely so as to bring it in agreement with the actual system x. The study of higher bands has two important messages. First, the mean-field description fails beyond the lowest band. Second, while the procedure of obtaining CF-KEB states is admittedly complicated, the microscopic CF theory is, at least in principle, capable of describing the higher bands. 7.4.9. Low-Zeeman-EnergyLimit We have been working with spinless electrons, which is relevant to completely polarized FQHE state. However, the Zeeman splitting is quite small in GaAs-

NUMERICAL TESTS

289

approximately 1/60 of the cyclotron energy [46]. Therefore, at relatively low magnetic fields, which are still large to suppress LL mixing, a finite fraction of electrons may find it energetically favorable to reverse their spins. This regime is theoretically investigated by setting the Zeeman energy to zero, while restricting all electrons to the LLL. With zero Zeeman energy, the Hamiltonian commutes with the spin operator, and it is sufficient to work in the subspace of the smallest L, and the smallest S,. One state in this subspace represents a multiplet of (2L + 1)(2S-t- 1) degenerate states. The spin quantum number S is also preserved under the C F transformation. Generalization of the CF scheme to the low-Zeeman energy limit is straightforward [6,47]: the liquid of interacting electrons at q is still represented by composite fermionsat q*, except that they are now spinful. As before, electrons at v = n/(2rnn & 1) are equivalent to composite fermions at v* = & n. When n is an even integer, the CF state has n/2 CF-LLs fully occupied; it is a spin-singlet incompressible state. When n is an odd integer, the topmost CF-LL is only half full. In the absence of interaction, there are many degenerate ground states, but with a residual repulsive interaction, the multiplet with fully polarized topmost CF-LL becomes the ground state, in analogy to the Hund's rule of atomic physics. For the FQHE of electrons, the discussion above implies that at zero Zeeman energy, this incompressible states at even-numerator fractions are spin-singlet, while those at odd-numerator fractions are partially polarized [except the state at 1/(2m + l), which is fully polarized]. The ground-state quantum numbers predicted above have been fully confirmed in numerical studies so far. The C F wavefunctions for the incompressible states are constructed in the usual manner, again with no adjustable parameters. Table 7.3 compares them with the TABLE 7.3. Comparison of the CF Wavefunctions with the Incompressible Coulomb States at US,213, and 3/5 in the Limit of Small Zeeman Splitting" n

V 2

2

3

-2

3

3

-

3

N

S

Overlap

6

0

0.9998

4

0

0.9998

6

0

0.9990

8

0

0.9980

5 8

4 2

1

0.9918

Source: Overlaps from Ref. [13].

"S is the spin quantum number of the ground state.

The states at 2/3 and 2/5 are spin-singlet, while the one at 3/5 is partially polarized.

290

COMPOSITE FERMIONS

corresponding Coulomb ground state for several filling factors. In the special case of n = 2, the CF state is identical to the 2/5 spin-singlet state considered by Halperin [46] and Haldane [40]. The structure of states at general filling factors has also been investigated. The approximation of noninteracting composite fermions has been found to be not valid at low Zeeman energies [47]. This is easy to understand since the interaction of two composite fermions is rather strong when they coincide (which is not possible for spinless compositefermions)and cannot be neglected.We have found that imposing a hard-core constraint on composite fermionsis sufficientto identify the low-energy states. In fact, a considerableamount of useful information can be obtained by applying Hund's maximum spin rule to compositefermions(which is a consequence of the short-range part of the residual repulsive interaction), an example of which we have seen above. In particular, the CF theory provides [47] a natural description of the skyrmion excitation near v = 1/(2m + 1) [48]. Nakajima and Aoki [49] have given a quantitative theory of the spin-wave dispersion of the Laughlin 1/(2m + 1) states.

7.4.10. Composite Fermions in a Quantum Dot The initial numerical calculations for the FQHE were performed for an electron droplet on a disk [5, 501. Several downward cusps were observed in the plot of energy as a function of the total angular momentum. However, except for the Laughlin states, it was not clear how to interpret the other states. In particular, it was not known how to assign filling factors to various cusps. This was the main reason why compact geometries (e.g., spherical) became more popular. In this section we show that the CF theory explains the positions of the cusps and clarifies the microscopicstructure of the correspondingstates. These results are also relevant to the high4 properties of parabolic quantum dots containing a few electrons. In the disk geometry, it is convenient to label states by their total angular momentum. Then XL

=0 :: = 9 J-J(Zj - Zk)2m@L. I
L = L* + m N ( N - 1)

(35) (36)

where QL. is the wavefunction of noninteracting electrons with total angular momentum L*. Any arbitrary L can be related to L* in the range - (1/2)N(N- 1) < L* < (1/2)N(N- 1) for a suitable choice of m. First consider the limit of large Zeeman energy, when all electrons are fully polarized [Sly 521. Figure 7.7 plots the kinetic energy of the ground state of noninteracting spinless composite fermionsat L* (which is the same as the kinetic energy of spinless electrons at L* but with the cyclotron energy replaced by an effective cyclotron energy) and the interaction energy of interacting electrons at L (computed with the LLL restriction). A striking correspondence between the two curves is evident. In particular, all cusps are predicted correctly. The CF

NUMERICAL TESTS

291

L'

7 6.5

'

N

e

v

J

E

3

e

6 5.5

Non-interacting Composite Fermions

5

3

E 4.5 Interacting electrons 4

3.5 20

25

30

35

40

L

45

50

55

60

65

Figure 7.7. Coulomb energy of seven interacting electrons at L, along with the kinetic energy of noninteracting composite fermions at L*. The kinetic energy of noninteracting composite fermions at L* is the same as that of noninteracting electrons at L*,but with the cyclotron energy of electrons replaced by a suitable effective cyclotron energy of composite fermions. The quantity a is the effective magnetic length depending on the external magnetic field and the confining potential of the quantum dot (it is equal to the magnetic length r, in the absence of confinement). The CF curve has been vertically shifted for clarity. (From Ref. [52].)

wavefunctions associated with several cusps have been compared with the exact eigenstates, and found to be quite accurate [15,52]. These facts establish the relevance of composite fermions to few electrons systems on a disk. In the other interesting limit of small Zeeman energy [53], the cusps are much weaker. Figure 7.8 plots the cusp sizes, defined as A(L) = E(L 1) E(L - 1) - 2E(L), for interacting electrons at L and noninteracting composite fermions at L*. Again, the CF theory obtains the prominent cusps. The weaker structure is attributable to the residual interaction between the composite fermions [53]. The results above can be translated to parabolic quantum dots at high magnetic fields by replacing the magnetic length by an effective magnetic length and adding a confinement energy [54]. The confinement energy is a linear function of L and does not alter the positions of the cusps. The ground state is in general one of the states where there is a downward cusp, and shifts progressively to lower L with increasing confinement. The CF theory thus predicts the

+ +

292

COMPOSITE FERMIONS

.03 n

0

.02

5

.01

.# p

0

N '

v)

a

-.01

-.02 15

20

25

30

35

40

45

Total Angular Momentum (L)

Figure 7.8. The cusp size at angular momentum Lis defined as A(L)= E(L + 1) + E(L - 1) - 2E(L),where E is the energy of the lowest state in the Lsubspace. The lower panel shows the cusp sizes for a system of six interacting electrons.The upper panel shows the cusp sizes for a system of six noninteracting composite fermions in units of the effective cyclotron energy. (From Ref. [53).)

quantum numbers of the possible ground states in a quantum dot, which are generally found to be in good agreement with exact results. See, for example, Fig. 7.7, where the predicted and the actual possible ground states are indicated by bigger symbols. To assign filling factors to these states, we consider the corresponding CF states. Except for the lowest filled CF-LL state, which has v* = 1 (v* = 2) in the limit of infinite (zero) Zeeman energy, other states at cusps do not necessarily have an integer number of filled CF-LLs; they occur as composite fermions are promoted one by one to higher CF-LLs. Consequently, no thermodynamic filling factor may be associated to the cusps in general. 7.4.11. Other Applications

I have discussed in some detail numerical studies of the CF theory. The CF ideas have been applied to several other situations. It has been argued [55] that the CF mapping remains valid in the presence of disorder. Kivelson et al. [56] and

QUANTIZED SCREENING AND FRACTIONAL LOCAL CHARGE

293

Halperin et al. [8] have constructed a global phase diagram in the FQHE regime by analogy with that in the IQHE regime. Halperin et al. [8] and Kalmeyer and Zhang [57] have investigated the state at v = 1/2 in terms of a Fermi sea of composite fermions. Chklovskii and Lee [SS] have developed a theory of transport in the FQHE regime. Wen [59], Brey [60], Chklovskii [61], and Kirczenow and Johnson [62] have studied edge states in the FQHE regime in terms of the CF theory. Many of these theories extend the CF scheme to situations with density variations, which leads to a position-dependent effective magnetic field for composite fermions, as can be seen from the mean-field approximation of Eq. (18). It is also worth pointing out that Halperin’s state for the FQHE at v = 1/2 in a two-layer system [46], which has been observed experimentally 163,641, also belongs to the class of CF wavefunctions, as it is of pure Jastrow-Laughlin form. It is straightforward to construct a larger class of FQHE states for the two-layer system in the spirit of the CF appproach. In the following, we will accept the validity of the CF description and explore its consequences.

7.5. QUANTIZED SCREENING AND FRACTIONAL LOCAL CHARGE Laughlin [5] explained the f = 1/(2m + 1) plateau as a consequence of the fractional quasiparticle charge. Since then, fractional charge has appeared naturally in the theories of the FQHE [21,65]. We identified earlier a composite fermion in the topmost partially filled CF-LL as a (CF)-quasielectron.Since it is simply an electron carrying an even number of vortices, its intrinsic charge is - e, as implicit in its coupling to the magnetic field in Eq. (18). It is, however, screened by the CF medium, which produces a “correlation hole” around each quasielectron. We define as “local charge” the sum of the intrinsic charge of the CFquasielectron and the charge of the correlation hole, measured relative to the charge density of the background FQHE state, Equivalently, the local charge of a quasielectronis the charge in a sufficientlylarge area containing a quasielectron minus the charge in the same area if it contained no quasielectron. The local charge of a quasielectronis only a fraction of - e, with its precise value depending on the screening properties of the background CF state. There are several ways of obtaining the local charge [S, 211; we briefly repeat here a counting argument [66]. Start with the n/(2mn + 1) FQHE state, x = P D V , , confined to a disk of a given radius, which fixes the largest allowed power of z,. Now add an electron to this system while insisting that the size of the system not change. The product in the Jastrow factor D now goes from 1 to N + 1, which increases the largest power of zjin the Jastrow factor by 2m. Therefore, to stay within the original area, the largest power of z j in CD must be reduced by 2m, which requires taking 2m electrons from the boundary of each LL and putting them in the interior of the (n + 1)st LL. Including the new electron, the (n + 1)st LL of CD now has 2mn 1 electrons, i.e., the (n + 1)st CF-LL of x now has 2mn + 1 composite fermions (or

+

294

COMPOSITE FERMIONS

quasielectrons). Since a charge - e was added, each quasielectron has a local charge of -e

- e* = -

2mn + 1

(37)

Thus, as the added electron gets screened by the medium into a CF-quasielectron of local charge - e*, 2mn additional CF-quasielectrons are excited out of the “vacuum”. The quantization of the local charge is indicative of a quantization of screening by the CF state. It owes its origin to the fact that the power of the Jastrow factor (2m)cannot be changed continuously (i.e., is quantized) due to the special nature of the lowest LL physics. The reader is urged to reformulate the above argument in the spherical geometry, and also to verify that: 1. Absence of a composite fermion in any one of the lower filled CF-LLs (j.e.y a quasihole) has a local charge + e/(2mn + 1) associated with it. This follows since a composite fermion can be excited from a filled CF-LL to the (n 1)st CF-LL without changing the size of the system. 2. The argument above extends to the presence of disorder and away from the special filling factors, when there are several (possibly localized) quasiparticles. The local charge of the quasielectron is still give by - e* = - e/ (2mn + l), where n is the number of CF-LLs whose (extended) edges are occupied. 3. For the n/(2mn - 1)states, a composite fermion in the ( n + 1)st CF-LL has a local charge e/(2mn + l)(note the sign).

+

+

These examples show that the system rearranges itself in a fantastically complicated manner under the addition (or removal) of electrons,which is, however, straightforwardly understandable in terms of composite fermions. 7.6. QUANTIZED HALL RESISTANCE The existence of quantized Hall plateaus was explained by Laughlin by a simple and elegant argument [4]. The principal assumption in this argument is the existence in the energy spectrum of bands of extended states separated by bands of localized states. The only way we know of obtaining such a spectrum is to introduce weak disorder in a state that has bands separated by gaps in the absence of disorder. It can be argued that disorder creates localized states in the gap, turning it into a mobility gap. Laughlin showed that the Hall resistance is quantized when the Fermi level lies in the mobility gap. More specifically, consider a Corbino sample [67], in which an azimuthal current I flows when a voltage V is applied across the sample. Laughlin calculated the current I as

PHENOMENOLOGICAL IMPLICATIONS

295

follows. Insert a thin solenoid at the center and change the (test) flux through it adiabatically. The extended and localized eigenstates behave rather differently. The former, which encircle the test flux, evolve adiabatically, since the BohrSommerfeld phase matching conditions change as the test flux is varied. The localized states, on the other hand, remain unaffected. When the test flux is changed by one flux quantum, each extended state evolves into the next one. Now consider a filling factor for which all extended states (in the interior of the sample) of nth and lower LLs are occupied (i.e.¶the Fermi level lies in the localized states). It can be shown that the current is given by the relation I = c d U / d 4 , where U is the energy of the system [4]. Laughlin approximated the derivative by AU/4,. When the flux is changed by one flux quantum, each extended state goes into the next one, carrying its electron with it, which amounts to a transport of a total of n electrons across the sample. This implies that AU = neV, which yields RH = V/I = h/ne2.Thus the Hall resistance is quantized, and its value depends on the number of extended bands below the Fermi level. In the absence of interaction it is sufficient to know the behavior of each single-particle state as the flux is varied. In the FQHE regime, the problem is more complicated, since now one must determine the evolution of the ground state of interacting electrons,in the presence of disorder, as the test flux is varied. In the C F theory, Laughlin’s argument can be generalized provided it is assumed that the CF description remains valid in the presence of disorder [SS]. Then the ground state of interacting electrons at v is equivalent to that of noninteracting composite fermions at v*. Consider v* such that the CF-Fermi level is in the mobility gap of composite fermions, with n extended CF bands below. Now, as the test flux is changed by 4,¶ n composite fermions go across the sample. Since each composite fermion carries a local charge of e* = e/(2mn l), the change in energy is AU = ne*I/, leading to the quantized Hall resistance of RH = V / I = h/ nee* = h/ f ez, with f = n/(2mn + 1).

+

7.7. PHENOMENOLOGICAL IMPLICATIONS

The CF theory provides simple, back-of-the-envelopeexplanations for the essential experimental facts. These are discussed next. 7.7.1. FQHE

As seen above, a gap at filling factor v =f in a disorderless system results in a plateau of quantized Hall resistance at h/ fez in the presence of sufficientlyweak disorder. In the CF scheme, a gap appears whenever composite fermions fill an integer number of LLs. Filled CF-LLs correspond to electron filling factors of f = n/(2rnn_+ 1). These, along with the states f = 1 - n/(2mn f l), related by particle-hole symmetry in the LLL [68], are called the principal fractions and produce the prominent sequences of observed fractions [69]. The FQHE of electrons is thus simply the IQHE of composite fermions. Note that only odd-

296

COMPOSITE FERMIONS

u

-

u

-

u

-

u

-

u

-

U

Figure 7.9. Hierarchy of states in the CF theory. New QHE states are obtained from old ones using the particle-hole symmetry (f + 1 -f, denoted by C) and vortex attachment l), denoted by 03. (From Ref. [69].) transformation [f +f/(2f

+

denominator fractions are obtained, in agreement with experiments. The origin of the odd denominator can be traced back to the feature of the CF theory that an even number of vortices are attached to each electron, i.e., the power of the Jastrow factor must be an even integer to satisfy the Pauli principle. Figure 7.9 shows the CF hierarchy of the FQHE states using the operators D, which attaches two vortices to each electron, and C, which is the particle-hole conjugation operator, relating v to l - v [68]. Each application of D produces a state at a higher level of the hierarchy, while C connects states at the same level. The IQHE states are at the lowest level of the hierarchy. The QHE states are expected to become weaker as one goes either toward right within a given level or to the next-higher level. This is in agreement with experiment. Fractions other than the principal ones are also allowed in the CF theory. They would be related to the FQHE of composite fermions in higher LLs. For example, the FQHE of composite fermions at f * = 1 + 1/3 = 4/3 and f* = 1 + 2/3 = 513 corresponds to FQHE of electrons at f = 4/11 and 5/13. Since 1/3 and 2/3 are the strongest FQHE states in the second LL (recall that we are working with spinless particles here; in actual experiments, these correspond to 2 + 1/3 and 2 + 2/3), it is anticipated that 4/11 and 5/13 should be the first nonprincipal fractions to be observed. Since FQHE is rare in the higher LLs even for electrons, observation of very few fractions other than the principal ones is to be expected in the lowest LL. It is suspected that in the second LL there is no FQHE of electrons at the principal fractions 1 + n/(2rnn + 1) and 2 - n/(2rnn+ 1) for n > 1 and m > 2 [70]. This suggests that in the range 2/5 > v > 1/3, fractions other than 4/11, 5/13, 6/17, and 9/23 will not occur (for spin-polarized FQHE). No FQHE has been observed in the third or higher LL, indicating that fractions other than the principal ones are not possible in the range 315 > v > 215.

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297

7.7.2. TransitionsBetween Plateaus It is believed that at zero temperature, the transition from one FQHE plateau to the next occurs sharply at some filling factor. At the transition, the Hall resistance jumps from one plateau to another and the longitudinal resistance exhibits a peak. In the CF theory, this transition is simply the transition between f* = n IQHE of composite fermions to f* = n 1 IQHE [SS]. It is expected to occur when the CF-Fermi level passes through the C F extended states, which happens in low disorder samples at v* = n 1/2. This corresponds to the electron filling factor v = (2n 1)/[2rn(2n 1) 21. This prediction has been found to be in excellent agreement with the experimentalpositions of the longitudinal resistance peaks [71]. Engel et al. [72] have found that the temperature dependence of the width of the transition region is characterizedby the same exponent in the FQHE regime as in the IQHE regime, which is also naturally understandable within the CF framework [SS]. Recently, Rokhinson et al. [73] have investigated the conductivity peak heights in the FQHE regime and confirmed their relation with the peak heights in the IQHE regime, derived in Refs. [56, 581.

+

+

+ +

+

7.7.3. Widths of FQHE Plateaus The knowledge of transition positions also tells us the widths of various plateaus in low-disorder samples.

7.7.4. FQHE in Low-Zeeman-Energy Limit It is well established experimentally that the ground states at 2/3,3/5,4/3, and 8/5 are not fully spin polarized in low-density samples, where these states are observed at relatively low magnetic fields [74]. Also, no mixed-spin state has been observed at v = 1/3. These facts are consistent with the CF theory. While the degree of polarization, predicted by the theory, cannot be measured experimentally at present [75], the observed evolution of the FQHE states in the range 1 < v < 2 as a function of the zeeman energy is explained effectively in terms of spin split CF Landau levels [76].

7.7.5. Gaps An important experimentally measurable quantity is the gap of a FQHE state. Since the CF theory gives accurate wavefunctions for the ground and excited states, it can, in principle, provide reasonable estimates for the gaps in the small disorder limit. For the 1/3 state the gap computed using the CF wavefunctions [16,77] agrees well with that obtained by other means [37]. However, a computation of the gaps of the other FQHE states has not yet been possible due to technical difficulties. Halperin et al. [8] have suggesteda simple picture in which the gap of a FQHE state is viewed as the quasi-cyclotron energy of the composite fermions. The gap

298

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of the n/(2mn f 1) state is

where m* is the effective mass of the composite fermion, and hoFF is the CF cyclotron energy. Du et al. [78] determined the gaps of five members of the principal sequences for two different samples and found that they seem to be well approximated by hoFF - r,where r is a constant. This fits nicely with the above prediction, provided r is interpreted [78] as the disorder-induced broadening of the CF-LLs. The effective mass estimated for these experimental parameters is roughly an order of magnitude larger than the band mass of the electron in GaAs. 7.7.6. Shuboikov-de Haas Oscillations

Leadley et al. and Du et al. [79] have successfully analyzed the minima and maxima around v = 1/2 in terms of the Shubnikov-de Haas oscillations of composite fermions, in analogy to the Shubnikov-de Haas oscillations of electrons near B = 0. This also allows a determination of the effective mass of composite fermions, which is in general agreement with the mass obtained from the gap measurements. 7.7.7. Optical Experiments

Raman experiments probe the collective mode at small wavevectors, which may be interpreted as either the GMP or the CF collective mode. Pinczuk et al. [SO] have reported observation of the collective mode of the 1/3 state in Raman experiments. No collective modes have yet been detected at other fractions. Recently [Sl], a broad secondary peak has also been observed at finite wavevectors at v = 1/3;this may be an indication of the third band of composite fermions. Chen and Quinn [82] have suggested that the appearance of multiple-peak structure in photoluminescenceexperiments [83] is also due to the formation of bands, arising naturally within the CF framework. All these experiments provide support to the picture in which the FQHE of electrons is viewed as the IQHE of composite fermions. The FQHE results from the existence of composite fermions in the same way as the IQHE from the existence of electrons and can therefore be considered an observation of composite fermions. 7.7.8. Fermi Sea of Composite Fermions

Even though there is no kinetic energy in the problem of interacting electrons in the LLL, the interaction energy between electrons effectively acts as the kinetic energy of composite fermions,as evidenced by the formation of LLs of composite fermions. In a limiting case, which corresponds to v = 1/2m, composite fermions

PHENOMENOLOGICALIMPLICATIONS

299

fill an infinite number of CF-LLs, or experience a vanishing effective magnetic field. Infinite filled LLs is another name for the Fermi sea. This limit was not taken seriously for some time, since, unlike in the case of filled LLs, there is no gap in the excitation spectrum here to stabilize the mean-field state. However, motivated by certain anomalies observed near v = 1/2, Halperin et al. [8] proposed that the composite ferrnions do form a Ferrni sea at v = 1/2 with a sharp Fermi surface. It is analogous to the Fermi sea of electrons at B = 0, with the trivial difference that the composite fermions at B* = 0 are fully spin polarized, while the electrons at B = 0 are spin unpolarized. As a result, the Fermi wavevector of composite fermions is kg = ,,hk,, where k, is the Fermi wavevector of electrons at B = 0. As the magnetic field is moved slightly away from B* = 0 (i.e., B = 2p4,), composite fermions are expected to execute a cyclotron orbit with radius R* = hk$/eB*. Three beautiful experiments [9-113 have observed such cyclotron motion of composite fermions. Since kg = f i k , , the cyclotron radius of composite fermions at B* is equal to that of electrons at B = B*/,,h, and, if the state at 1/2 is indeed a Fermi sea of composite fermions, the structures near B = 0 and B* = 0 should look similar provided they are plotted on scales differing by a factor of The experiments of Refs. [lo, 111 find a close correspondence between the structures near B = O and B* = O . This demonstrates that the cyclotron dynamics of the charge carriers is described by the effective field B* rather than the external field B. Ref. [9] has verified certain qualitative and quantitative predictions made by Halperin et al. [8]. These experimental results in fact provide a direct evidence for the intrinsic charge (- e ) and the statistics (fermionic) of composite fermions. (Note that the local charge of a composite fermion is not well defined for the 1/2 state, due to the absence of a gap.) Other features at v = 1/2 have also been analyzed in terms of composite fermions. Ying et al. [84(a)] have found that the thermopower can be understood nicely using the independent CF picture. Ballistic transport of composite fermions through narrow channels has also been studied [84(b)].

8.

7.7.9. Resonant Tunneling Finally, consider a resonant tunneling experiment in which a composite fermion tunnels from one edge of the sample to the other through a path that goes around a potential hill (an antidot) containing no electrons. According to Eq. (18),with N,,,= 0, the phase associated with this path is the usual Aharonov-Bohm phase, BA 2n -

40

(39)

which implies that successive resonant tunneling peaks are expected when the flux through the electron-freeregion changes by &,independent of the surrounding QHE state, consistent with the intrinsiccharge - e of the composite fermions. This period was anticipated theoretically [85] and has also been observed experimentally [86,87]. The charge deficiency of the electron-free region, how-

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ever, is an integer multiple of the local charge e*, as measured by Goldman and Su [86] in an elegant experiment. 7.8. CONCLUDING REMARKS

An earlier theory of the FQHE, the quasiparticle-hierarchy (QPH) scheme [20,21], had postulated that given a “parent”state, its fractionallycharged quasiparticles (which were assigned either fractional [21] or bosonic [20,25] statistics)

form a Laughlin-like state, as a result of the repulsive Coulomb interaction between them, to produce new “daughter” FQHE states. An iteration of this procedure leads to a possibility of FQHE at all odddenominator rational fractions starting from the Laughlin states. While the QPH scheme originated many novel ideas, one of its unsatisfactory features was the fact that it required a large density of quasiparticles to create daughter states. At these densities, the quasiparticles are strongly overlapping,since thedistance between them is of the same order as their size, and therefore the quasiparticle description itself becomes questionable, as also does their treatment as point particleswith well-definedcharge and statistics. With the composite fermions in the topmost partially filled CF-LL identified as the (CF)-quasielectrons,the quasiparticles in the CF scheme also have the same fractional local charge. Further, in a mean-field approximation, the winding phases in the various approaches are also in agreement [88]. To see this, consider the CF state with n filled CF-LLs and two CF-quasielectrons in the (n + 1)st CF-LL. Let one of the CF-quasielectrons go around a closed loop. The phase associated with the loop is given by Eq. (18).In the mean field approximation, N,,,is replaced by the average number of electrons inside the loop. When this loop does not enclose the other CF-quasielectron, the average phase (for a uniform density) is 2aBAlq5, -4mapA, and when it does, it is 2nBAIq5, 4mn(pA + e*/e).The difference between the two is written as 2 d * , where

The total winding parameter, 8 = 1 + 8*, where the term of unity is due to the fermionic exchange statistics of the composite fermion, is identical to that obtained in the QPH theory. It may seem surprising that these different approaches provide the same answers for the local charge e* and the winding parameter 8. However, the underlying reason is that the very assumption of incompressibility puts strong constraints on these quantities. In fact, Su has shown that they are uniquely determined by the filling factor of an incompressible state, with the weak assumptions that there is only one type of quasiparticle, which has the largest allowed charge [89].Therefore,all not-too-convoluted theories should produce the same values of e* and Bas obtained by Su from his kinematical arguments, independent of what scenario is ascribed to the origin of incompressibility in the LLL.

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The composite fermion approach is both conceptually simpler and more general. As indicated earlier, the (anyonic or bosonic) quasiparticles in the QPH schemes must be treated as interacting in order to produce incompressibility.On the other hand, most of the qualitative physics can immediatelybe understood in terms of noninteracting composite fermions: this model produces the numerous FQHE states in a completely equivalent fashion, clarifies their relationship with the IQHE, and allows an understanding of numerous other features, notably the nontrivial nature of the compressible states. Another indication of the simplicity of the CF scheme is that it requires only two types of composite fermions for the explanation of the present experiments, electrons carrying two and four vortices, in contrast to the QPH theory, which associates a different quasiparticle with each FQHE state. Conceptually, the C F scheme is quite similar to the conventional Landau Fermi liquid theory of interacting electrons. Like the Landau quasiparticle, the composite fermion is also simply a dressed electron, and consequently has the same intrinsic charge and statistics as an electron; these quantum numbers have effectively been measured in several experiments. The dramatic properties (e.g., the effectivemagnetic field)arise in the FQHE problem due to the fact that the dressing (or screening) occurs in a very special manner, through binding of an even number of vortices to electrons, which produce additional phases, partly canceling the Aharonov Bohm phases. The quantization of various quantities (e.g., the Hall resistance or the local charge) is ultimately a consequence of the fact that the vorticity bound to electrons is quantized to be an even integer, as required by the single-valuedness and the antisymmetry of the many electron wavefunction. In addition to offering a simple insight into the structure of the correlated electron states in the lowest LL, the CF theory also gives rather accurate wavefunctions for the low-energy states. While these have been used to obtain thermodynamic estimates in certain cases, their full quantitative potential awaits new techniques allowing treatment of large systems at general fractions. In summary, the following succinct picture has emerged. First, electrons form Landau levels due to a quantization of the kinetic energy. Within the lowest Landau level, in a broad range of filling factors, electrons find it energetically favorable to transform into composite fermions by capturing an even number of vortices of the wavefunction. Despite their quantum mechanical,topological and collective character, the composite fermions behave, to a great extent, as ordinary, noninteracting particles moving in an effectivemagnetic field. They provide a simple explanation for the FQHE and related phenomena. ACKNOWLEDGMENTS This work was supported in part by the National Science Foundation under Grant No. DMR93-18739. I am indebted to L. Belkhir, G. Dev, A. S. Goldhaber, V.J. Goldman, R. K. Kamilla, T. Kawamura, S. A. Kivelson, D. J. Thouless, N. Trivedi, and X.G. Wu for valuable discussions and collaboration.

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+

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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

8

Resonant Inelastic Light Scattering from Quantum Hall Systems A. PINCZUK Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey

8.1. INTRODUCTION

More than 10 years after the discoveries of the integer [l] and fractional [2] quantum Hall effects, studies of two-dimensional electron systems in semiconductor quantum structures continue to uncover intriguing behaviors that arise from combined effects of strong electron-electron interactions and the reduction in the physical dimensions. Experimentsin modulation-doped semiconductorsin conjunction with theoretical studies of electron correlation explore the beautiful physics that resides in two-dimensional electron systems of artificial quantum structures. The fractional quantum Hall effect (FQHE) is an archetype of the class of quantum liquids that occur in electron systems of reduced dimensions [2-41. The transport anomalies of the FQHE, seen at “magic” values of the Landau level filling factor, can be regarded as a sequence of metal-insulator transitions that emerge when free electrons of two-dimensional systems are in partially occupied Landau levels. The observed insulating behaviors in the FQHE reveal that the electron quantum fluid states are incompressible and that an energy gap separates the ground state from its excited states [3-51. Given the absence of single-particlegaps at fractional values of Landau level occupation, the FQHE manifests the startling behaviors associated with electron-electron interactions in two dimensions. At integer filling factors, energy gaps arise from quantization of kinetic energy into Landau levels and the Zeeman splitting of the spin states [3,6]. In contrast, gaps of the FQHE are novel phenomena that are primary manifestations of the strong interactions that lead to condensation of the electron system into incompressible quantum fluids [S, 7,8]. Gap excitations of the quantum Hall states are represented by neutral density excitations of the two-dimensional electron gas in the large perpendicular 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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(4

~

..

Position

(b)

%

CJ)

tions with wavevector q. (b) Wavevector dispersion of inter-Landau-level excitations at v = 2. The singlet is the magnetoplasmon

magnetic field. In these excitations a particle (or quasiparticle)is promoted to an empty state leaving behind an oppositely charged hole (or quasihole), as shown . Landau gauge is used when one of the quantum schematically in Fig. 8 . 1 ~The numbers is the excitation wavevector. For a wavevector there is a displacement between the centers of the two cyclotron orbits that represent the single-particle states. The displacement is given by xo = d

o2

where lo = (hc/eB)'I2is the magnetic length and the direction of the displacement

INTRODUCTION

309

xo is orthogonal to that of T. Collective modes are constructed with such pair states as a basis, and the excitationscan be regarded as magnetic excitons with an average dipole length xo [9-131. The magnetic exciton concept is consistent with the classical description of the motion of a neutral particle-hole pair in the perpendicular magnetic field. In the classical framework, with the Coulomb attraction balanced by Lorentz forces, there is linear motion of the center of mass of the pair with a constant velocity proportional to the separation between the particles [12]. Measurements of thermal activation of magnetotransport yield gap energies in the quantum Hall regimes. The method is well established and has given major insights into the physics of the FQHE [14]. The gaps obtained from the thermal activation experiments are interpreted as the energy required to create a widely separated quasiparticle-quasihole pair. It follows from Eq. (1) that such a pair can be regarded as a gap excitation having q-+ 00. In the domain of optical experiments, measurements of the magnetic field dependence of acceptor luminescencehave been used to extract the energies of widely separated quasiparticle-quasihole pairs [151. The study of gap excitations is the principal task of spectroscopy in the quantum Hall regimes. The quest for the direct observation of dispersive gap excitations is particularly significant in the extreme magnetic quantum limit, where spectroscopy experiments could test the existence of the collective excitations predicted by theories of the electron quantum fluid of the FQHE. Collectiveexcitations of the FQHE state at v = 1/3 were observed by resonant inelastic light scattering [161. These experiments rely on the enhanced sensitivity that is associated with resonant excitation of the spectra. The success in the fractional quantum Hall regime suggests that resonant inelastic light-scattering methods, proposed in 1978 for studies of two-dimensional electron systems at semiconductorinterfaces [171, will continue to have major impact in research of strongly correlated electron quantum fluids of artificial semiconductor structures. We are interested in two-dimensional electron layers supported by GaAs quantum structures. In these systems the conduction band states of electrons have spin 1/2 and orbital angular momentum is zero. As an example,consider the case of Landau level fillingfactor v = 2, when the two spin states of lowest Landau quantization (1 = 0) are fully occupied. The neutral particle-hole pairs in the collective excitations can be classified according to the total angular momentum of the states. The pair states are either singlets (J = 0) or triplets (J = 1). The singlets are charge-density excitations in which there is no change in the spin degree of freedom. The triplets are spin-density excitations without a fluctuating charge density. Figure 8.lb shows schematically a theoretical prediction for the dispersive inter-Landau-levelexcitations with Landau level index charge A1 = 1. These modes are gap excitations for the quantum Hall state at v = 2. At q = 0 the energy of charge-density excitations (magnetoplasmons) equals the cyclotron energy mb as required by Kohn’s theorem. The dispersive modes in Fig. 8.lb display characteristic roton (or magnetoroton) minima at wavevectors 4 2 l/l,,.

310

RESONANT INELASTIC LIGHT SCATTERING

The rotons, caused by the dependences of particle-hole interactions on q (and on xo according to Eq. (1)) are signatures of electron-electron interactions of twodimensional electron systems in a large magnetic field [lo, 121. In gap excitations of the FQHE, electrons do not change Landau level quantum number. These collective intra-Landau-levelmodes are associated with fractionally charged quasiparticles that obey fractional statistics [S, 20,211. In seminal work, wavevector dispersions of gap excitations were calculated by means of numerical evaluations of small finite systems and within a single-mode approximation (SMA) [7,8,22,23]. The SMA calculation incorporates the oscillator strengths and structure factors obtained with the Laughlin wavefunction [S]. The calculated wavevector dispersions also exhibit distinctive magnetoroton minima at 4 2 &'lo, as shown schematically in Fig. 8.1~.Numerical evaluations of finite systems and the single-mode approximation continue to be employed in calculations of dispersive collective excitations of the FQHE (see Refs. [24-291 and references therein). The long-wavelength limit of the collectivegap excitation has also been considered within the framework of Landau-Guinzburg theories [30-321. In current interpretations the FQHE is regarded as the quantum Hall effect of composite fermions [14,33,34]. These remarkable composite objects are quasiparticles in which an even number of magnetic flux quanta are attached to each electron. Composite fermions experience effective magnetic fields that are described by fermion Chern-Simons theory [35-381. These advances in conceptual frameworks led to a renewed interest in the theory of dispersive collective excitations of FQHE states [39-471. The first light-scattering observations of q = 0 gap excitations at v = 1/3 are displayed in Fig. 8.2. The mode is the sharpest peak in the spectra (full width at half maximum FWHM = 0.04 meV). The broader (but still quite sharp) bands

Figure 8.2. Temperature dependence of lightscattering spectra of a low-energy excitation of the FQHE at v = 3. The sharp peak at 1.18 meV is interpreted as the q = 0 collective gap excitation of the incompressible electron liquid. The sample is a 250-1(GaAs/Alo., Ga,., As single quantum well with n =8.5 x 10" an-'. The lowtemperature mobility is p = 3 x 106cmZ/V.s. The inset shows the magnetic field dependence of the 0.5-Kspectra. The temperature dependence of the Lo and intensities is a characteristic optical anomaly at v =$(see Ref. [93]). (After Ref. [16].)

LIGHT-SCATTERING MECHANISMS AND SELECTION RULES

311

labeled Lo and Loare due to luminescencefrom recombination of the electron gas with weakly photoexcited holes in the highest-valence Landau level of the GaAs quantum well. Strong luminescence occurs in the spectra of Fig. 8.2 because in these resonant inelastic light-scattering experiments the photon energies overlap with the fundamental optical transitions of the quantum structure that hosts the twodimensional system. In measurements carried out in the fractional quantum Hall regime the luminescencebackground creates serious difficulties,clearly displayed in Fig. 8.2, in the unambiguous identification of inelastic scattering spectral lineshapes. The light-scattering observations displayed in Fig. 8.2 are surprising because the dynamical structure factor, the function that enters in conventional inelastic scattering cross sections, vanishes as q4 for the small wavevectors in optical experiments [S, 8,231. The results are also intriguing because the character of the lowest q = 0 gap excitation is still a subject of debate [8, 32,40,41]. Resonant inelastic light-scattering experiments in the regime of the integer quantum Hall effect have also demonstrated observations of inter-Landau-level excitations that have vanishing dynamical structure factors for q +O [18]. Furthermore, resonant light-scattering measurements in doped three-dimensional semiconductors [47,48] and extensive light-scattering studies of the two-dimensional electron systems of semiconductor quantum structures in the absence of external magnetic fields [50-531 have clearly shown access to excitations that have the character of single-particle transitions. Light scattering by these excitations is not predicted by conventional response functions of the electron gas. These experimental results suggest that resonant inelastic lightscattering methods enable studies of electron gas excitations that are not within reach of other experimental tools. This chapter presents a review of resonant light-scattering studies in the quantum Hall regimes [16,18,19]. In Section 8.2 we incorporate a short description of mechanisms and selection rules that apply in resonant inelastic light scattering by electron systems in semiconductor quantum structures. In Sections 8.3 and 8.4 we consider results in the quantum Hall regimes. In Section 8.5 we present concluding remarks.

8.2. LIGHT-SCATTERING MECHANISMS AND SELECTION RULES This section highlights the conceptual framework that applies to inelastic light scattering by the two-dimensional electron fluids of semiconductor quantum structures. We also present a short review of resonant inelastic light-scattering processes that may occur in these electron systems. We focus on the resonant enhancements of the scattering cross sections, which play the key roles of creating the extremely high sensitivity required to measure the excitations of dilute twodimensional electron systems [17,48].

312

RESONANT INELASTIC LIGHT SCATTERING

3He or 3He/4He

SUPERCON

1

SPECTROMETER

Figure 8.3. Schematic representation of the low-temperature inelastic light-scattering experiment.The magnetic field is perpendicular to the two-dimensionalelectron layer. In this configuration the in-plane component of the light-scattering wavevector is k < 104cm-'.

The experiments considered in this chapter employ the backscattering geometry shown in Fig. 8.3, where the magnetic field is along the direction perpendicular to the two-dimensional electron layer. In this configuration the and are the wavevectors of scattering wave vector is 52 = k, - k,, where the incident and scattered light. In the geometry of Fig. 3, 52 is nearly perpendicular to the plane of the quantum structure and the two-dimensional electron layer. The in-plane component of 52, called here is quite small. Typical values are lxl= k < 104cm-'. The kinematics of inelastic light-scattering processes follow from conservation of momentum. Translational invariance, expected in quantum structures with weak residual disorder, implies that conservation of momentum is equivalent to the conservation of wavevector. Single-layer two-dimensional systems considered here have only in-plane components ?, and conservation of wavevector is simply written as

- -

x,

x,

q=k

4

-

xs

LIGHT-SCATTERING MECHANISMS AND SELECTION RULES

313

The range of wavevectors relevant to dispersive collective excitations is determined by l/lo.For magnetic fields in the quantum Hall regimes 1,s 100 A. Thus experiments that use the configuration of Fig. 8.3, where k < 104cm-', have kl, << 1. Wavevector conservation, given by Eq. (2), dictates that only longwavelength dispersive excitations should be active in these experiments. In doped bulk semiconductors the resonant inelastic light-scattering spectra display coupled plasmon-phonon collective models at large wavevectors [54]. These observations were explained by the breakdown of wavevector Conservation caused by loss of translational invariance due to the random potential of ionized dopants [54,55]. In modulation-doped quantum structures [56], in which extremely high electron mobilities are achieved, the impact of breakdown of wavevector conservation due to disorder is greatly reduced [57]. In fact, at B = 0 breakdown of wavevector conservation is not significant in spectra of singleparticle and collective excitations of two-dimensional systems when electron mobilities exceed lo5 cm2/v.s [SO-53,583. Breakdown of wavevector conservation in resonant inelastic light scattering has been invoked to explain the observation of large wavevector collective excitations in the integer quantum Hall regime [l8, 59-61]. This interpretation relies on the loss of translational symmetry due to the well-known effects of localization and residual-disorder that prevail in the insulating states of the two-dimensional electron system at integer filling factors. Following Burstein et al. [17,623, resonant inelastic light-scattering processes are considered within the effective-mass approximation. In this framework the resonant light-scattering mechanisms are similar to those active in bulk semiconductors [63-681. Such resonant processes invoke two intermediate virtual optical transitions across the forbidden energy gap of the semiconductor that hosts the free-electronsystem. Calculations of the light-scattering matrix element yif should incorporate final-state interactions (exciton states) in the virtual optical transitions. In lowest order these effects are excluded and the expression for the matrix element is written as [48]

where 1 i) and If ) are the initial and final states of the transitions of the electron gas in the conduction band. I/?) are intermediate states in the valence band and jj is the electron momentum operator. The intermediate virtual optical transitions Ip > + I f ) have energy E,. The incident photon has frequency aLand unit polarization vector kL. The scattered photon has a, and ks. The matrix element in Eq. (3) gives access to single-particle transitions t i ) + If) that satisfy wavevector conservation, as given by Eq. (2), and the Pauli principle. The transitions 1 i ) + If ) are the neutral particle-hole pairs that enter in the construction of collective excitations of the electron system. The neutral pairs can be classified according to changes in the charge (J = 0) or spin ( J = 1) degrees offreedom [55, 62,64,68,69]. Inelastic light scattering by spin excitations

314

RESONANT INELASTIC LIGHT SCATTERING

is made possible by the spin-orbit coupling in the valence-band states of the semiconductor that supports the electron system [68,69]. When photon energies are smaller than the energies of optical transitions of the host semiconductor (ho,< E,), the denominator in Eq. (3)can be taken out of the summation, and Eq. (3) is written as

where E(n)is the optical energy gap at the free-electron density n. The lightscattering cross sections are calculated from Eq. (4)by well-established methods that incorporate electron-electron interactions in the responses of the electron system [55, 64-71]. For charge-densityexcitations the result is

d2a

( w L ) 2

dQdw OC [ h a , - E(n)12

In Eq. (5) 0 = w,

s(q,a)

- as

is the scattering frequency. Conservation of energy requires that w be equal to the energy of the elementary excitation active in the light-scattering process. The dynamical structure factor is [55, 69,701

where

In the evaluations of Eqs. (4)to (8) the modes active in inelastic light-scattering experiments are the collective excitations predicted by conventional chargedensity response functions of the electron gas [%,69,703. In the presence of changes in the spin degrees of freedom, a similar calculation yields cross sections proportional to spin response functions that predict collective spin-densityexcitations [64,69,70]. Light-scattering cross sections proportional to conventional response functions of the electron gas occur because of the assumption of weakly resonant conditions implicit in the use of Eq. (4).The matrix element in Eq. (4)states that all accessibleparticle-hole pairs have the same resonant enhancement factor. Under such weakly resonant conditions the light-scatteringresponses are related only to dynamical behavior of the electron gas. When E , N haL, simple expressions like those in Eqs. (4) and (5) are only firstorder approximations. Under strong-resonant conditions a group of Ii) + If)

LIGHT-SCATTERING MECHANISMS AND SELECTION RULES

315

transitions, the ones with the largest resonant enhancement factor ( E , - hwJ ', can make the dominant contribution to the matrix element yip This selective resonant enhancement creates strong-coupling channels for a subset of accessible Ii ) + If ) transitions. Excitations activated by strong-coupling channels are expected to emerge in resonant light-scattering spectra with unique energies and wavevector dispersions. New spectral features due to strong resonant enhancement could be vastly different from those of collectiveexcitations predicted by the conventional response functions of the electron gas. The emergence of features characteristic of strongly resonant light-scattering channels has been invoked to explain the observations of spectral bands of singleparticle excitations in doped bulk semiconductors [48,49]. A similar mechanism could explain measurements of features that disperse like single-particle excitations in the resonant inelastic light-scattering spectra of modulation-doped quantum wells [48-50). While qualitative arguments like the ones presented above offer plausible interpretations, at this time we do not have detailed evaluations of strongly resonant light-scattering cross sections. In the case of electron gases of bulk semiconductors, light-scattering cross sections that could interpret experiments carried out under extremely resonant conditions have been written in terms of response functions that incorporate the resonant denominators [49,71]. However, these functions are difficult to evaluate. The capabilities to access excitations that are not predicted by conventional response functions of the electron gas are particularly significant in the spectroscopy of excitations of the electron gas in the quantum Hall regimes. This impact of the light-scattering method in studies of incompressiblequantum Hall states is related to the parity selection rules, which require that collective gap excitations have S(4,o)+0 for the small wavevectors in optical experiments [ 5 , 8,23,81]. The projiles of resonant enhancement, the dependencies of light-scattering intensities on a,., are particularly striking in measurements with photon energies that overlap the fundamental optical gap of the quantum well. Sharp (about 1 meV) excitonic resonant enhancement profiles were discovered in measurements of Raman scattering by optical phonons of undoped GaAs quantum wells [72]. Similar resonances were later found in the inelastic light-scattering intensities of intersubband and inter-Landau-level excitations in the high-mobility two-dimensional electron systems of modulation-doped GaAs quantum structures [59,73]. Much sharper excitonic resonant enhancement profiles were discovered in the fractional quantum Hall regime [16,74]. An example is displayed in Fig. 8.4, which shows measurements of resonant inelastic light scattering by inter-Landaulevel excitations at the cyclotron energy o, [74]. The spectra in this figure demonstrate an extremely sharp resonant enhancement maximum of width close to 0.2 meV. Inelastic light-scatteringmechanisms that invoke intermediate excitonicstates are third-order (or three-step)processes [62,72,73,75]. A diagrammatic representation of a strongly resonant third-order process is shown in the upper inset to

316

RESONANT INELASTIC LIGHT SCATTERING

ENERGY (mew

Figure 8.4. Resonant inelastic light scattering by the q = 0 charge-density inter-Landaulevel excitation at v = i.In addition to light scattering by the ma netoplasmon at wc,the spectra show the Lo and Columinescence.The sample is the 250- GaAs quantum well of Fig. 8.2. Spectra measured at five incident photon energies hw, are shown. The results reveal a sharp profile of resonant enhancement. (After Ref. [74].)

1

Fig. 8.7. In the first step the incident photon is annihilated with creation of a virtual exciton. In the second step this exciton creates the collective excitation of the two-dimensional system and is scattered to another virtual exciton state. In the third step the exciton is annihilated with creation of the scattered photon. Strongly resonant three-step light-scattering processes are well known in Raman scattering by optical phonons in semiconductors [17,62,67,72,75]. Following Danan et al. [73], virtual exciton states in three-step light-scattering processes are regarded as superpositions of electron-hole pair states. The electron is in one of the confined conduction states of the quantum well and the hole is in one of the valence states. The term in the light-scattering matrix element that contributes with the strongest resonant enhancement is written as [73]

where M, and M, are the optical matrix elements for the creation of optical exciton I v ) and I p ) . The intermediate exciton states have energies E, and E,. (He),, is the matrix element of the interaction between virtual excitons and collective excitations of the two-dimensional electron system.

EXPERIMENTS AT INTEGER FILLING FACTORS

317

(He),,,,arises here from electron-electron interactions and has two different types of terms. These terms are classified according to which component of the exciton,the conduction electron or the valence hole, changes state in the step that creates the elementary excitation. Further, to each of these terms there are two contributions arising from the direct and exchange parts of electron-electron interactions. In this formulation, resonant inelastic light-scatteringchannels exist for excitations that contribute terms to (He),,p[73]. The matrix element in Eq. (9) has resonances when incident and scattered photon energies overlap the energies of exciton states. The one at oLis the incoming resonance and that at wsis the outgoing resonance. Such resonant enhancement doublets are characteristic signatures of three-step light-scattering mechanisms [18,49, 59, 64-67, 71-73, 75, 761. Three-step processes were considered in the discussion of experimental results at integer filling factors [18,59]. Three-step and higher-order processes also enter in a theory of resonant inelastic light scattering in the two-dimensional Wigner crystal regime 1773. Extensive evidence of outgoing resonances is found in the resonant inelastic light-scattering experiments reported in the quantum Hall regimes [18,19,59,74]. One striking example is seen in the spectra of Fig. 8.4, in light scattering by the inter-Landau-levelexcitation at w,. The outgoing resonance manifests here as the enhancement maximum in the spectra in which the light-scatteringpeak overlaps the luminescence due to optical recombination at the fundamental gap of the GaAs quantum structure. The cross sections of inelastic light scattering by quantum Hall fluids have been examined in recent calculations [78,79]. Platzman and He make use of an approximation similar to the one employed in the derivation of Eq. (4)and obtain cross sections of gap excitations of the FQHE [78]. Govorov [79] incorporates a series expanaion of the resonant denominators to derive cross sections in the regime of the integer quantum Hall effect. The evaluations in Refs. [78] and [79] yield cross sections that are proportional to response functions of the quantum Hall states. These calculations, however, do not predict the observed outgoing resonances. 8.3. EXPERIMENTS AT INTEGER FILLING FACTORS In this section we review resonant inelastic light-scattering results reported for Landau level filling factors v = 2 and v = 1. We consider measurements of interLandau-level excitations with change in Landau level quantum number A1 = 1 [18,19,59]. These experimentsenable observations of charge and spin excitations at q 0 and yield spectroscopic determinations of exchange Coulomb interactions [191. Through breakdown of wavevector conservation, due to residual disorder at integer filling factors, the spectra display direct evidence of roton densities of states in dispersivecollective modes [18,59]. We also consider recent resonant light-scattering experiments in GaAs quantum structures with patterns fabricated by means of current nanotechnologies [SO]. In these samples a weak

-

318

RESONANT INELASTIC LIGHT SCATTERING

periodic in-plane modulation of the electron density activates light scattering by collective modes at large wavevectors in the interval 0 5 4 5 l/lo. These measurements give direct insights into the wavevector dispersions of the excitations [80]. To highlight some of the issues involved in the experiments reviewed in this section, we consider first the long-wavelength limit of the matrix element (iI f ) that enters in the expression of S(q,w ) given in Eq. (7).In lowest order we write (i l e i g + l f )

- iT*(

il71f)

(10)

for the dipole-active charge-density inter-Landau-level excitations [Sl]. Since Eqs. (7) and (10) predict a q2-dependenceof S(q, a)at long wavelengths, observations of q = 0 cyclotron excitations, like the visually striking spectra shown in Fig. 8.4, imply the existence of strong resonant contributions to the lightscattering matrix elements that are not taken into account in the processes described by Eqs. ( 5 ) to (7). Mixing between valence Landau levels has been suggested as an explanation for the light-scatteringobservations of q = 0 inter-Landau-levelexcitations [19]. In GaAs quantum structures nonparabolicity of the valence bands and spin-orbit coupling result in extensive mixing of orbital and spin angular momentum eigenstates [82]. It is well known that mixing between valence Landau levels has a substantial impact on optical properties of semiconductor quantum structures immersed in magnetic fields [83,84]. Such effects should also allow, in the dipole approximation, the two optical transitions in y i f , y;, and vif required for light scattering by excitations with A1 = 1. We consider next the energies of dispersive collective modes. At integer filling factors collective excitations of two-dimensional electron systems are described within the time-dependent Hartree-Fock approximation [12,133. When hw,>> E,, where E, = e2/Eol0 and E~ is the dielectric constant of the semiconductor, only the lowest-order terms in Ec/hwcare kept. In this limit the energies of dispersive collective excitations are written as [12,13,19,29]

where Ed(qy B) and E,(q, B) represent the coupling between the particle and hole in the neutral pairs of the collectiveexcitations.Ed, which arises from direct terms of electron-electron interactions, occurs only in charge-density excitations. It represents the energy in the macroscopic electric field associated with the modes. Ex is due to the exchange terms. Ex exists in charge as well as spin excitations and represents an excitonic binding energy of the electron-hole pair. Ed and E , have pronounced q-dependenciesand both vanish for 4 + co.In this limit collectiveexcitations represent uncoupled, infinitelyseparated [see Eq. (2)], particle-hole pairs [12]. Thus, in Eq. (11) the q-independent term wsp is the energy of the widely separated electron-hole pair. In this limit particle-hole

319

EXPERIMENTS AT INTEGER FILLING FACTORS

interactions vanish, and asp is regarded as a single-particle energy given by

where o,,is the transition energy in the Hartree approximation [ 6 ] , E, the Zeeman energy and C(B) the self-energy that arises from exchange Coulomb interactions when an electron changes state leaving behind a hole.

8.3.1. Results for Filling Factors v = 2 and v = 1 We consider first inter-Landau-level excitations at v = 2. The results shown in Fig. 8.5 were obtained with incident photon energies in resonance with optical transitions of the GaAs quantum well. The participating Landau levels are l

~

l

~

l

~

1500

-

GaAs (Alo.3Gao.7)As Single Quantum Well n=2.49x101' cm-2 m=3.5x106cm2/Vsec

l

-

2

i

~

. .\..

6=5.5T T=1.7K

-

~

E. fw

I I.' 1555 1557 fw[mevl I

1553

-

..

1553.4

a .

1554.9

1555.4 1556.3 1557.3 1

7

1

1

8

1

1

9

l

1

1

1

10 11 Energy [mev]

1

1

12

1

1

13

1

1

14

Figure 8.5. Inelastic light scattering spectra by inter-Landau-level excitations at four incident photon energies. The filling factor here is v = 1.88. The inset shows the profile of resonant enhancement at energy hw = 8.8 meV. (After Ref. [59].)

l

320

RESONANT INELASTIC LIGHT SCATTERING

associated with the ground-valence subband h, and the first excited conduction subband c1 of the 250 8, well (see the lower inset to Fig. 8.7) [59]. The profile of resonant enhancement has a peak of FWHM = 1.1meV, as shown in the inset to Fig. 8.5. The spectra in Fig. 8.5 consist of continua with three well-defined peaks. The relative intensities of the maxima in the spectra have a striking dependence on incident photon energy. The distinctive evolution of the spectral lineshapes reveal a dependence of resonant light-scattering intensities on scattered photon energy ho,.This is a characteristic signature of outgoing resonances such as those in the three-step light-scattering processes considered in Eq. (9).Three-step processes of the class shown in the upper inset to Fig. 8.7 have been proposed to explain the measurements shown in Fig. 8.5 [18,59]. In Fig. 8.6 we compare the measurements with the predictions of the timedependent Hartree-Fock approximation for v = 2. Figure 8 . 6 shows ~ a spectrum that displays clearly the three major structures observed at v = 2 [60]. The

-2 \

Y

150

i

>-

c

p

w

100

c

z

0

Y

50

3

Y F

0:

0

c

8 > w

2 1

7

8

9

I0 11 12 ENERGY [meV]

13

Figure 8.6. (a) Inelastic light-scattering spectra of inter-Landau-level excitations at v = 2; (b)calculated collective mode dispersions.(After Ref. [a].)

321

EXPERIMENTS AT INTEGER FILLING FACTORS

calculation in Fig. 8.66 incorporates the effect of the finite width of the twodimensional layer. This is done by writing the Fourier components of the electron-electron interaction as V ( q )= 2ne2F(q/b)/c,q, where F(q/b) is a form factor that takes into account the spread of the electron wavefunction in the perpendicular direction [6] (b is the inverse length parameter that enters in a variational envelope function introduced to describe the states of the electrons confined to the two-dimensional layer). For the 250 8, GaAs quantum structures of the experiments reviewed in this chapter, typical values of b are within the interval 1.5 5 bl, 5 3 [19]. The lower-energy branch in Fig. 8.6b is for spindensity (or triplet) excitations, and the other branch corresponds to chargedensity (or singlet) excitations. Equations (11) and (12) offer several insights on the physics underlying the characteristic dispersions shown in Fig. 8.6b. For inter-Landau-level transitions with A1 = 1, we set w, = 0,. Further, in the absence of spin polarization, as it occurs for v = 2, the exchange self-energy term C(B) is identical for spin and charge excitations [12]. At long wavelengths Ed q [9] and E,(O, B) = - C(B) [12,13]. Thus the q = 0 charge-density excitation (or magnetoplasmon) is at w,. This is the single-particleenergy in the absence of electron-electron interactions, as required by translational invariance (Kohn's theorem). The roton in the dispersion of spin excitations is due to a maximum in the excitonic binding E x that occurs at q 1/1, [lo, 12). The maximum in the dispersion of charge excitations is due to the q-dependence of Ed, which has a maximum at q 1/1, [9,12]. The roton minimum follows from the reduction in excitonic binding at wavevectors q > 1/1,. Figure 8.6 shows that the three structures resolved in the spectra have energies close to the positions of the critical points in the calculated mode dispersions (where the densities of states peak because dw/dq = 0). Particularly striking in the data is the doublet in the energy range of the charge-density excitations, which agrees with the prediction of a characteristic pair of critical points. Since the critical points occur at wavevectors q 2 1/l, lo6 cm- ',much larger than the inplane component of the scattering wavevector k, the results in Fig. 8.6 imply a massive breakdown of wavevector conservation. Breakdown of wavevector conservation in these experiments has been attributed to the loss of translational symmetry due to unscreened residual disorder in the insulating states of the integer quantum Hall effect [181. In the high-quality quantum structures employed in these studies, residual disorder, even if unscreened, is relatively weak. Thus we anticipate that its impact on resonant inelastic light-scattering processes can be considered in the context of perturbation theory [54,55,57]. Within this conceptual framework, resonant lightscattering spectra could be interpreted as a superposition of features due to processes that conserve wavevector (q = k) and structures that arise from processes with breakdown of wavevector conservation (q # k). Figures 8.5 and 8.6 illustrate resonant inelastic light-scattering studies of gap excitations in the quantum Hall regimes. These results offer direct evidence of magnetorotons in the collective-mode dispersions. The rotons, as we have seen,

-

-

-

-

322

RESONANT INELASTIC LIGHT SCATTERING

are major predictions of theories of electron interactions in two-dimensional systems. There are relatively small (about 0.5 meV) differences between measured and calculated critical points in the mode dispersions. The discrepancy could be explained by the approximation ho,>> E, that is used in the calculation. This approximation ignores the coupling to higher inter-Landau-level transitions (AZ = 2) that at these relatively small magnetic fields could reduce the energies of q # 0 magnetoplasmons [13]. Figure 8.7 shows the positions of the peaks of profiles of resonant enhancement of light scattering at o,as a function of magnetic field. The four points correspond to the incoming - resonance for filling factors v = 1,2,3, and 4. The energies in Fig. 8.7 are written as E,(B) = ELO) + (r + 1/2)hwi

c Filling Factor

1570

-

!s

/'c2-6mev'T -

E

Y

' eE01= 1542.4meV

EXPERIMENTS AT INTEGER FILLING FACTORS

323

where 0I

=-eB

p*c

is the effective cyclotron energy for the virtual optical transitions. The slope of the line, 2.6 meV/T, yields I' = 1 for the Landau level number and a value of p* = 0.064m0 for the reduced effective mass in the optical excitations. An interpretation of the results in Fig. 8.7 is offered by three-step processes shown in the upper inset to the figure. In this diagram we require that EAO) = Eel. The intermediate transitions take place in the sequence indicated by the numbers. Steps 1 and 3 are the virtual optical transitions. The inter-Landau-level excitations are created in step 2. The requirement of wavevector conservation is relaxed for optical transitions that are broadened by residual disorder. The impact of breakdown of wavevector conservation in light-scattering spectra was evaluated in the integer quantum Hall regime [61,79]. For q 2 l/lo, when S(4,o)represents the scatteringcross section for charge-density excitations, Marmorkos and Das Sarma write [61]

In Eq. (1 5) a Lorentzian function a

f (k, 47 a) = ( k - q)2 + a2 replaces the conventional wavevectorconservation condition f ( k , q, 0) = 6(k -4). The adjustable parameter a is a measure of the extent of breakdown of wavevector conservation. The effect of disorder on the collective-mode energies is incorporated by a phenomenological broadening parameter r and the replacement o2-,w(o + ir)in S(q, a). Figure 8.8 displays evaluations of Marmorkos and Das Sarma for filling factor v = 2 [61]. In agreement with experiment, the calculation predicts the major spectral features near the positions of the critical points. The relative intensities of the two maxima in the calculated spectra depend on values of a and r. While a detailed comparison with experiment requires consideration of the effects of the outgoing resonance, it emerges from Fig. 8.8 that values (r/oc) =5 x and al, N 0.1 would explain measured spectra. Of particular interest is the adjusted value of (a)-' lWA, which is close to the width of the spacer layer that separates the doping layer of ionized impurities in the AlGaAs barrier from the electron layer in the GaAs quantum well. This result is consistent with the interpretation of residual disorder as the cause of breakdown of wavevector Conservation. N

324

RESONANT INELASTIC LIGHT SCATTERING

16

I

1

I

1

I

1

I

14

w

- wc[e2/q,hl

Figure8.8. I ( o , k , a ) calculated with Eqs. (15) and (16).The results are for the chargedensity inter-Landau-level excitations (magnetoplasmons) at v = 2. The lines shown are four sets of values of the broadening parameters tl and r.The heavy solid line is for a\, = 1 and T/o, = lo-’, the dotted line has a\, = 0.1 and T/w, = 5 x the dashed line is for or\, = 2 and T/w, = 5 x and the light solid line is for tll, = 2 and T/o, = The inset shows the calculated magnetoplasmon dispersion. (After Ref. [61].)

We consider next the results for the spin-polarized case at v = 1. Figure 8.9 compares measured spectra with the theoretical prediction for the dispersive collective modes [60]. In the interpretation of these results we should highlight the fact that in the presence of spin polarization the exchange self-energiesin Eq. (12)are different for spin-flip and non-spin-flip (or magnetoplasmon)excitations [12,19,29]. The difference in exchange self-energies,AX(@ >> E,, results in a splitting ASF between the two 4 = 0 excitations that can be written as

Figure 8.9a displays spectra of inter-Landau-level excitations with A1 = 1. The incident photon energies are in resonance with excitonic transitions of higher Landau levels (I’ = 2) of the lowest confinement states c, and h, of the 250A quantum well [E,(O)= E,,]. Figure 8.9b is a rendition of the dispersive collective modes calculated within the time-dependent Hartree-Fock approximation. The

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325

Single Quantum Well ( 250A )

A

n -2.49 x lof1 0 = 10.67T

* = 3.5 x l@cm2Nsec T = 0.5K v = 0.97

Figure 8.9. Resonant inelastic light scattering of inter-Landau-level excitations at v = 1:

(a) spectra measured with four incident photon energies ha,; (b)calculated mode disper-

sion (private communication of C. Kallin). (Ref. [59].)

strongest peaks in the spectra are the q-0 modes. They are the magnetoplasmon = 22.3 meV. at 0,-17.7 meV and the spin-flip inter-Landau-level mode at asp The weaker structures centered at o = 19.8 meV are interpreted as critical points of magnetoplasmons, observed here, as in the case of v = 2, because of breakdown of wavevector conservation due to residual disorder. The q = 0 results in Fig. 8.9 are a direct spectroscopicmeasurement of ASP The observations reveal the unique exchange interactions in the spin-polarized electron gas. These results also display the excellent agreement of the prediction of Hartree-Fock theory with the value of ASFdetermined from the experiment. In the measurements at v = 1 the light-scattering intensities have a marked dependenceon temperature. At T = 5 K the intensity has decreased by a factor of 10 from its value at 1.8 K and the spectral features are no longer observable above 10 K. It is interesting that such temperature dependence is similar to that of N

326

RESONANT INELASTIC LIGHT SCATTERING

the occupation of the two lowest spin-split Landau levels as determined by interband optical absorption [85]. The similaritiescan be considered as evidence that the temperature dependence of the light-scatteringintensity is related to the loss of spin polarization in the electron gas.

8.3.2. Results from Modulated Systems The wavevector dispersions of gap excitations and other collective modes are among the major predictions of current theories of electron fluids in systems of reduced dimensions. The novel insight that could be reached from the comparison of theoretical predictions with experimental results stimulates great interest in direct measurements of mode dispersions. However, in typical light-scattering experiments the in-plane scattering wavevectors are too small (kli << 1) to reach a significant segment of the mode dispersions. Large in-plane wavevectors were accessed in resonant light-scattering experiments at B = 0 by means of couplers in which a short-period metallic grating was evaporated on the sample surface [86]. These experiments enabled the observation of two-dimensional plasmon excitations at wavevectors q,=k+k, where k, is the grating wavevector m2n k m =a

m = 1,2,3,.

and a is the period of the grating. The use of such grating couplers was proposed for studies of dispersive gap excitations in the FQHE regime [78]. Grating couplers work by modulating the incident and scattered electric fields, thus introducing wave vector components k,. However, the strengths of the induced electric fields decay exponentially as exp( - k,L), where L is the distance between the grating coupler and the twodimensional system. This characteristic decay restricts applications of grating couplers to experiments in which L s WOA, a requirement that has negative impact on studies of the quantum Hall regimes where high electron mobilities are achieved with L 2 1500 A. To circumvent the limitations intrinsic to grating couplers, Sohn et al. have carried out resonant inelastic light scattering experiments in systems with weak in-plane modulation of the free electron density [80]. In this work a periodic variation of the density is introduced in high-mobility two-dimensional systems of GaAs quantum structures by means of state-of-the-art nanofabrication technology. In these samples, the weak superimposed in-plane periodicity creates a modulation in the wavefunctions of the conduction and valence states [87]. The modulated Landau levels enter in the intermediate transitions of light-scattering

EXPERIMENTS AT INTEGER FILLING FACTORS

327

processes and are expected to create the access to wave vectors q, in resonant light-scattering experiments. Prior to processing, the quantum structures employed by Sohn et al. are similar to those in previous light-scattering work [16,19,80]. The modulationdoped GaAs quantum wells are 250 8, wide and the Si doping layer is 11008, below the surface of the sample. In the nanofabrication procedure a periodic array of lines is initially written onto the sample surface by means of electronbeam lithography. The width of the lines is d = 38508, and the period is a = 45008,. To create the weak density modulation the much narrower partial periods of widths a - d = 6508, are processed using a mild reactive-ion-etching procedure to remove the top 300 8, of material. Two-dimensional electron gas photoluminescence measurements at B = 0 confirm the creation of a density modulation. Sohn et al. found that these spectra have two components. The one identical to that measured before processing is assigned to the wider regions, of width d, that were not exposed to etching. The other component is assigned to the narrower regions of width a - d. The two densities differ by 7%, with the etched semiperiods having the higher one. Inelastic light-scattering spectra measured at B = 0 display sharp peaks due to dispersive two-dimensional plasmons. The plasmon energies show the wellknown q,!," dispersion that confirms the activation of modulation wavevectors 4,.

The results in Fig. 8.10 display the access to large wavevector collective excitations at v = 1 [80]. The spectra show charge-density (non-spin-flip)intersubband excitations with A1 = 1. These modes are associated with electron

800

600

(a)

.

1555.0 mV 1555.2

400 ?

?

Q

W

200

h

.CI Y

9 E

E Y 200 7-

0

1556.3

1556.5

22

23 24 Bncrgy Shift (mcV)

25

Figure 8.10. Resonant inelastic light-scattering spectra of charge-density intersubband excitations at v = 1. Spectra are labelled according to incident photon energy (in meV). (a) Results for the weakly modulated two-dimensional electron system at B = 5.9 T and T = 0.2 K. Numbers above peaks indicate the assignments to wavevector components m. (b) Results for the unmodulatedtwo-dimensionalsystem. The q = O mode is dominant. (After Ref. L-801.)

328

RESONANT INELASTIC LIGHT SCATTERING

transitions c, +cl between the confined conduction states shown in the lower inset to Fig. 8.7. The magnetic field is B = 5.9 T, which corresponds to v = 1 for the higher-electron-density regions below the etched lines. The filling factor in the wider lower-density regions is v = 0.95. The spectra from the modulated area of the sample are shown in Fig. 8 . 1 0 ~ As . seen in Fig. 8.10b, the sharper peak at 23.75 meV is also present in the spectra from the unmodulated area of the sample. This peak corresponds to the q = 0 charge-density intersubband excitation of the 250 A GaAs quantum well. The additional peaks seen in the spectra of Fig. 8 . 1 0 ~observed only in the spectra from the modulated area of the sample, correspond to the multiple wave-vector components induced by the modulation. The induced harmonic peaks are relatively sharp, with a FWHM N 0.2 meV comparable to their spacing. The harmonics are assigned sequentially to m = 1,2,3,4,5, and 6. The dependencies of the relative intensities and spectral lineshapes on photon energies are further manifestations of outgoing resonances in the scattering cross sections. Figure 8.11 shows results for charge-density (non-spin-flip) inter-landaulevel excitations at v = 1. In a field of B = 5.7 T the lowest Landau level filling factors are0.97 for the low-density regions and 1.03 for the high-density regions of the modulated sample. In the spectra we find a weak peak due to the q = 0 mode at w, = 9.65 meV. The low intensity of this peak is consistent with the parity selection rule of Eq. (10). More intense higher-order dispersion modes are seen in the energy range 9.9 to 11.5 meV and are numbered sequentially. The greater intensity of the harmonic modes is consistent with the prediction that such large wavevector excitations should have S(q, o)# 0. The experimental results displayed in Fig. 8.10 and 8.11 demonstrate the existence of a well-defined dispersive excitations at v = 1, despite the localization and residual disorder that prevail in this insulating state of the integer quantum Hall effect. Sohn et al. have interpreted these measurements within the framework of the predictions for a homogeneous two-dimensional systems. To examine this assumption one needs to consider the range of the dispersive terms of electron-electron interactions Ed and Ex. This range should be related to x, [see Eq. (l)], the dipole length of the magnetic exciton that represents the collective mode. For the wave vectors of interest we have x,, lo N 100 A. Since the dipole lengths of interest are much smaller than the widths d and a - d of the two semiperiods of the modulation, the use of calculations for homogeneous twodimensional systems was considered a reasonable approach to the interpretation of these experiments. The light-scattering measurements in the modulated sample have been used to construct dispersions of the collective modes [80]. Figure 8.12 presents the results for magnetoplasmons and compares the experimental dispersions with the prediction of the time-dependent Hartree-Fock approximation for a homogeneous two-dimensional system. The calculation is for B = 5.7 T, the field of v = 1 for the wider semiperiods of the modulation. The calculation also incorporates a finite-width parameter b = 3 [88].

-=

EXPERIMENTS AT INTEGER FILLING FACTORS

"

8

9

10 Energy shift (meV)

11

329

12

Figure 8.11. Resonant inelastic light-scattering spectra of charge-density inter-landaulevel excitations at v = 1. The results are for the weakly modulated two-dimensional electron system of Fig. 8.10. B = 5.7 T and T = 0.2 K. The numbers above the peaks are the wavevector component assignments m. L indicates weak luminescence. The spectra are labeled according to incident photon energy (in MeV). (After Ref. [80].)

In Fig. 8.12 we find that the calculated dispersion is in excellent agreement with the measured energies of the critical points (activated by residual disorder). In the cases of the strongly dispersive modes at smaller wavevectors, however, there are substantial discrepancies with the dispersion constructed by assuming the modulation wavevectors q m defined by Eqs. (18) and (19). Sohn et al. highlighted the unexpected finding, clearly displayed in Fig. 8.12, that excellent agreement between experiment and theory is obtained with modulation wave-

330

RESONANT INELASTIC LIGHT SCATTERING

0

disorder-activated

Wavevector (qlo)

.5

Figure 8.12. Dispersions of charge-density inter-Landau-level excitations at v = 1. The experimentally obtained dispersions are shown as triangles for q, assignments and as open circles for q$. The disorder-activated critical points are shown as open squares. The numbers below the dispersionpoints indicatevalues of m.The solid line is the prediction of the Hartree-Fock approximation with b = 3 (after C. Kallin, private communication). (After Ref. [80].)

vectors where k::-

mn d

m = 1,2,3, ...

is the “confinement” wavevector of the wider semiperiods. A transition in modulation wavevectors from q,,, at B = 0 to q: at v = 1 is intriguing. Sohn et al. proposed that the transition results from having a modulation with two well-differentiated semiperiods [SO]. Such systems were studied theoretically to provide a picture of edge states in the quantum Hall regime [89]. It was pointed out that because the filling factor v = 1 occurs at different fields in the two regions, the density-modulated system divides into alternating strips of different quantum Hall fluids with slightly different filling factors v N- 1. Sohn et al. suggested that in such systems electron states may become confined within individual strips of different electron quantum fluids. Thus the electron states acquire the modulation wavevectors associated with confinement within semiperiods, and the intermediate optical transitions that enter in the light-scattering matrix elements will allow spectra of collective excitations at confinement wavevectors. In this scenario the emergence of collective modes at the wavevectors kz follow from the fact that the wider semiperiods obviously make the dominant contribution to the light-scattering signal.

EXPERIMENTS IN THE FRACTIONAL QUANTUM HALL REGIME

331

8.4. EXPERIMENTS IN THE FRACTIONAL QUANTUM HALL REGIME The light-scattering observations displayed in Fig. 8.2 offered the first direct evidence that a state of the FQHE supports the collective gap excitations predicted by theories of the incompressible quantum fluid [16]. More recently, measurements of ballistic phonon propagation have uncovered evidence of absorption assigned to magnetoron minima of collective gap modes of FQHE states [go]. In this section we consider light-scattering results in the regime of the FQHE. Gap excitations of the FQHE are constructed from intra-Landau-level transitions. In spin-polarized electron fluids, as in the v = 1/3 state, the gap modes are charge-density excitations without changes in the spin degree of freedom. Figure 8.lc is a schematic representation of the lowest dispersive gap excitation predicted by the SMA [8]. The q --+ co limit, at energy AA(see Fig. 8 . 1 ~represents )~ the energy of the infinitely separated quasiparticle-quasihole pairs. These are the energies determined from activated magnetotransport measurements [141. Direct measurement of dispersive gap excitations at finite wavevector requires spectroscopy methods. Optical experiments such as absorption and inelastic light scattering under conservation of wavevector, access only the long-wavelength mode near A. (see Fig. 8 . 1 ~ )These . methods, however, were judged to be ineffective in the FQHE regime. The negative predictions were based on the parity selection rule requiring that intra-Landau-level excitations have vanishing oscillator strength and dynamical structure factors for q +0 [S, 8,23,8l]. The resonant light-scattering experiments were stimulated by the ideas introduced for understanding of the character of the long-wavelength gap excitation of the FQHE state with v = 1/3. In their seminal work Girvin et al. speculated that two gap modes, each near the magnetoroton minimum at wavevectors close to l/lo, could pair to produce a two-roton state with q = 0 [8]. The long-wavelength excitation was also considered within Landau-Guinzburg theory [30-321. Working in the Landau-Guinzburg framework, Lee and Zhang proposed that the q = 0 excitation consists of two dipoles, each a gap excitation at wavevector near the magnetoroton minimum, arranged in a configuration that has a quadrupole moment but no net dipole moment [32]. These conjectures about the possible structures of long-wavelength gap modes suggested that inelastic light scattering, which as a two-photon process is sensitive to excitations that lack an electric dipole moment, might be the optical method to observe the collectivegap excitations of the FQHE [16]. To be specific, we look into the scenario in which the 4 = 0 gap mode is the two-roton state proposed by Girvin et a]. [8]. Resonant inelastic light scattering by such states can be considered within the frameworks discussed in Section 8.2. We focus on the one in which resonant light scattering takes place with intermediate virtual exciton transitions. In this framework inelastic light scattering by two-roton states can be thought of as four-step processes in which the interaction

332

RESONANT INELASTIC LIGHT SCATTERING

(He),,,,acts twice. In such light-scattering processes the interaction between the intermediate virtual optical excitons and the collective excitation, (He),,,,,could arise from the electric dipole moment associated with each roton state. The matrix elements of these four-step processes have features in common with those in second-order Raman scattering. The difference is that (H& assumes the role played by the electron-phonon interaction in second-order Raman scattering by optical phonons [76]. This class of four-step processes have resonant enhancement denominators at the incoming and outgoing scattering channels. Such incoming and outgoing resonances are responsible for the strong second-order Raman scattering by optical phonons in semiconductor quantum structures [9l]. The scattering geometry described in Fig. 8.3 was employed in experiments in the FQHE regime. In this setup, 3Heor dilution 3He/4Hecryostats are inserted in the cold bore of a superconductive magnet. The system has silica windows for optical access. The conventional backscattering geometry depicted in Fig. 8.3 allows only small values k 6 lo4 cm- of the light-scattering wavevector. For measurements in the fractional quantum Hall regime the power densities of incident tunable dye laser light are kept low, at power levels of to lo-’ W cm-2. This experimental conjiguration, in conjunction with almost perfect sample surfaces, is associated with the low stray light that enables light-scattering measurements at energy shijtsfrom the exciting laser line that can be smaller than 0.2 me V. The light-scattering measurements of collective excitations at v = 1/3 take advantage of sharp and strong resonant enhancements of the cross sections [16,741.An example of spectra measured with sharp enhancement profiles can be seen in Fig. 8.4. In these experimentsthe outgoing photon energies are close to the fundamental gap of the 250 8, GaAs quantum well. The intermediate optical excitations are associated here with excitonic transitions between valence Landau levels and the states of the electron quantum liquid [92]. In Figs. 8.2 and 8.4 the bands labeled Lo and Lo with FWHM N 0.2meV are the characteristic doublets of intrinsic photoluminescencedue to recombination of the electron gas with weakly photoexcited holes in the highest valence Landau level [93]. The remarkably sharp light-scattering peak at w,is the q = 0 magnetoplasmon of the two-dimensional system [28,29]. The data in Fig. 8.4 were the first to uncover astonishing evidence that intensitiesof light scattering by excitations of the dilute two-dimensionalelectron gas could be comparable to peak intensities of photoluminescenceof the GaAs quantum well. Results such as these offered valuable experimental insights into mechanisms of resonant enhancement of inelastic light scattering in the fractional quantum Hall regime. The sharp peak at energy 1.18 meV in the spectra shown in Fig. 8.2 has been assigned to a low-energy excitation of the incompressible state at v = 1/3. This conclusion is supported by the striking temperature and magnetic field dependences shown in the figure. In these results the light-scattering peak is observed for temperatures T < 1.3 K and in the small magnetic field interval AB 2: 0.5 T near

EXPERIMENTS IN THE FRACTIONAL QUANTUM HALL REGIME

333

v = 1/3. Such temperature and magnetic field dependences are characteristic of magnetotransport signatures of the FQHE states. The narrow widths of the light-scattering peaks in Fig. 8.2, of FWHM = 0.04meV, suggests that the wavevector is conserved (k = q 0) for resonant inelasticlight scattering by these excitations.If this were not the case, much larger widths would be expected [57]. Since the measured energy is in the range of gap excitations of theincompressiblestate, the peaks were interpreted as the k = q 0 gap excitation of the FQHE state at v = 1/3 [16]. Comparison of the energy measured in Fig. 8.2 with theoretical predictions for A, requires consideration of the effect of the finite width of the two-dimensional layer along the perpendicular direction. The impact of the finite width effect has been assessed within the single-mode approximation [8] and by means of numerical evaluations of finite systems with small numbers of electrons "941. The two calculations are in good agreement with the value of A, obtained in the light-scattering measurement. Resonant inelastic light scattering by the long-wavelength gap excitation of the FQHE state at v = 1/3 was measured in modulation-doped quantum structures with free electron densities in the range 0.7 < n < 1.3 x 10" cm-2. Figure 8.13 shows results obtained at B = 15.5 T. In these spectra we find a sharp peak (FWHM N 0.07 meV) that overlaps a broader structure. This sharp peak is observed only for fields near v = 1/3. The broader structure (FWHM N 0.25 mew is largely from luminescencedue to recombination between Landau levels above the fundamental gap of the 250-w GaAs quantum well. This nonequilibrium luminescence,here labeled L,, is at energy 0.7 meV higher than the fundamental optical emission Lb [95]. The energy of the sharp light-scattering peak in Fig. 8.13 is also in excellent agreement with a finite system calculationof the q = 0 gap excitation of the incompressible quantum liquid state [94]. The numerical evaluation considers a nine-electron system and incorporates the effect of the finite width of the electron wavefunction along the perpendicular direction. The value of the finite-width parameter, determined from electron density and magnetic field [6], is here b = 1.75. The intensities of the L , luminescenceat B = 15.5 T are about 40 times weaker than the fundamental Lb recombination. Thus the resonant light-scattering intensities in Fig. 8.13 are comparable to those of the nonequilibrium luminescence. These spectra display enhanced contrast of the light-scattering intensities against the luminescence and offer new insights into spectral lineshapes. In fact, examination of the spectra in Fig. 8.13 suggests that the broad structure that overlaps the sharp peak interpreted as the 4 = 0 gap excitation may be due in part to inelastic scattering. This is particularly clear in the spectrum measured with incident photon energy h o , = 1530.39 meV, where the L , luminescence only partially overlaps the light-scattering signal. This spectrum hints that light scattering by the long-wavelength gap excitation of the v = 1/3 state may consist of a sharp peak followed by a continuum at higher energy. The existence of such a continuum at higher energies is indeed predicted by numerical evaluations of systems with small numbers of electrons [44,46,47,78,94].

-

-

334

RESONANT INELASTIC LIGHT SCATTERING

T = 0.2K

---....-

hwL = 1530.49

1530.58 meV

1

1.o

*

*

1

1

1

(

,

1

1

1.5

1

,

,

,

,

2.0

Energy (mew Figure 8.13. Resonant inelastic light-scattering spectra of long-wavelength gap excitations of the FQHE at v = 3. The sharp peak at 1.38meV is interpreted as the q = 0 mode. The sample is a 250 A GaAs quantum well. The dotted line is the estimated background, the dashed line is the estimated nonequilibrium luminescenceL , and the vertical bars give the position of the peak of L,. (After Ref. [SS].)

Resonant inelastic light scattering by inter-Landau-levelexcitations and spin waves have also been reported for the FQHE state at v = 1/3 [16]. The results in the energy range of long-wavelength spin waves are presented in Fig. 8.14. Figure 8 . 1 4 ~shows a sharp inelastic scattering peak at an energy shift o = 0.26meV. This energy can be understood as a Zeeman splitting E, = gp& where g = 0.43 is close to the g-factor value reported for the two-dimensional electron gas in GaAs quantum structures [96]. The mode at the Zeeman energy can be regarded as the q --+ 0 limit of the dispersive collective spin-wave excitation [lZ, 29,451. This mode is required to have energy E, by the rotational invariance in the twodimensional system (Larmor’s theorem) [l 1,121.

EXPERIMENTS IN THE FRACTIONAL QUANTUM HALL REGI