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

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

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Edited by Sankar Das Sarma Aron Pinczuk

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ISBN-13: 978-0-471-11216-7 ISBN-10: 0-471-1 1216-X CONTENTS

Contributors xi Preface xiii

1 Localization, Metal-Insulator Transitions, and Quantum Hall Effect 1 S. Das Sarma 1.1. Introduction 1 1.1.1. Background 1 1.1.2. Overview 2 1.1.3. Prospectus 4 1.2. Two-Dimensional Localization: Concepts 5 1.2.1. Two-Dimensional Scaling Localization 5 1.2.2. Strong-Field Situation 7 1.2.3. Quantum Hall Effect and Extended States 9 1.2.4. Scaling Theory for the Plateau Transition 12 1.2.5. Disorder-Tuned Field-Induced Metal-Insulator Transition 16 1.3. Strong-Field Localization: Phenomenology 18 1.3.1. Plateau Transitions: Integer Effect 18 1.3.2. Plateau Transitions: Fractional Effect 21 1.3.3. Spin Effects 22 1.3.4. Frequency-Domain Experiments 23 1.3.5. Magnetic-Field-Induced Metal-Insulator Transitions 23 1.4. Related Topics 28 1.4.1. Universality 28 1.4.2. Random Flux Localization 30 References 31

2 Experimental Studies of Multicomponent Quantum Hall Systems 37 J. P. Eisenstein 2.1. Introduction 37 2.2. Spin and the FQHE 38 2.2.1. Tilted Field Technique 39 2.2.2. Phase Transition at v = 815 40 2.2.3. The v = 512 Enigma 45

V vi CONTENTS

2.3. FQHE in Double-Layer 2D Systems 49 2.3.1. Double-Layer Samples 50 2.3.2. The v= 1/2 FQHE 51 2.3.3. Collapse of the Odd Integers 56 2.3.4. Many-Body v = 1 State 58 2.4. Summary 66 References 67

3 Properties of the Electron Solid 71 H. A. Fertig 3.1. Introduction 71 3.1.1. Realizations of the Wigner Crystal 72 3.1.2. Wigner Crystal in a Magnetic Field 73 3.2. Some Intriguing Experiments 74 3.2.1. Early Experiments: Fractional Quantum 74 Hall Effects 3.2.2. Insulating State at Low Filling Factors: A Wigner Crystal? 75 3.2.3. Photoluminescence Experiments 79 3.3. Disorder Effects on the Electron Solid: Classical Studies 81 3.3.1. Defects and the State of the Solid 81 3.3.2. Molecular Dynamics Simulations 82 3.3.3. Continuum Elasticity Theory Analysis 86 3.3.4. Effect of Finite Temperatures 90 3.4. Quantum Effects on Interstitial Electrons 91 3.4.1. Correlation Effects on Interstitials: A Trial Wavefunction 92 3.4.2. Interstitials and the Hall Effect 95 3.5. Photoluminescence as a Probe of the Wigner Crystal 97 3.5.1. Formalism 97 3.5.2. Mean-Field Theory 99 3.5.3. Beyond Mean-Field Theory: Shakeup Effects 100 3.5.4. Hofstadter Spectrum: Can It Be Seen? 103 3.6. Conclusion: Some Open Questions 104 References 105

4 Edge-State Transport 109 C. L. Kane and Matthew P. A. Fisher 4.1 Introduction 109 4.2. Edge States 114 4.2.1. IQHE 114 4.2.2. FQHE 119 CONTENTS vu

4.3. Randomness and Hierarchical Edge States 126 4.3.1. The v = 2 Random Edge 127 4.3.2. Fractional Quantum Hall Random Edge 132 4.3.3. Finite-Temperature Effects 135 4.4. Tunneling as a Probe of Edge-State Structure 136 4.4.1. Tunneling at a Point Contact 138 4.4.2. Resonant Tunneling 145 4.4.3. Generalization to Hierarchical States 151 4.4.4. Shot Noise 152 4.5. Summary 154 Appendix: Renormalization Group Analysis 156 References 157

5 Multicomponent Quantum Hall Systems: The Sum of Their Parts and More 161 S. M. Girvin and A. H. MacDonald 5.1. Introduction 161 5.2 Multicomponent Wavefunctions 165 5.3. Chern-Simons Effective Field Theory 169 5.4. Fractional Charges in Double-Layer Systems 169 5.5. Collective Modes in Double-Layer Quantum Hall Systems 172 5.6. Broken Symmetries 180 5.7. Field-Theoretic Approach 185 5.8. Interlayer Coherence in Double-Layer Systems 192 5.8.1. Experimental Indications of Interlayer Phase Coherence 193 5.8.2. Effective Action for Double-Layer Systems 196 5.8.3. Superffuid Dynamics 199 5.8.4. Merons: Charged Vortex Excitations 203 5.8.5. Kosterlitz-Thouless Phase Transition 206 5.9. Tunneling Between the Layers 209 5.10. Parallel Magnetic Field in Double-Layer Systems 213 5.11. Summary 216 References 218

6 Fermion Chern-Simons Theory and the Unquantized Quantum Hall Effect 225 B. I. Halperin 6.1. Introduction 225 6.2. Formulation of the Theory 227 6.3. Energy Scale and the Effective Mass 230 6.4. Response Functions 233 viii CONTENTS

6.5. Other Fractions with Even Denominators 238 6.6. Effects of Disorder 24 1 6.7. Surface Acoustic Wave Propagation 243 6.8. Other Theoretical Developments 247 6.8.1. Asymptotic Behavior of the Effective Mass and Response Functions 247 6.8.2. Tunneling Experiments and the One-Electron Green’s Function 249 6.8.3. One-Particle Green’s Function for Transformed Fermions 25 1 6.8.4. Physical Picture of the Composite Fermion 252 6.8.5. Edge States 253 6.8.6. Bilayers and Systems with Two Active Spin States 254 6.8.7. Miscellaneous Calculations 254 6.8.8. Finite-System Calculations 254 6.9. Other Experiments 255 6.9.1. Geometric Measurements of the Effective Cyclotron Radius R: 255 6.9.2. Measurements of the Effective Mass 256 6.9.3. Miscellaneous Other Experiments 257 6.10. Concluding Remarks 258 References 259

7 Composite Fermions 265 J. K. Jain 7.1. Introduction 265 7.2 Theoretical Background 267 7.2.1. Statement of the Problem 267 7.2.2. Landau Levels 268 7.2.3. Kinetic Energy Bands 269 7.2.4. Interactions: General Considerations 270 7.3 Composite Fermion Theory 270 7.3.1. Essentials 270 7.3.2. Heuristic Derivation 273 7.3.3. Comments 275 7.4 Numerical Tests 278 7.4.1. General Considerations 278 7.4.2. Spherical Geometry 279 7.4.3. Composite Fermions on a Sphere 280 7.4.4. Band Structure of FQHE 28 1 7.4.5. Lowest Band 282 7.4.6. Incompressible States 283 7.4.7. CF-Quasiparticles 284 CONTENTS ix

7.4.8. Excitons and Higher Bands 285 7.4.9. Low-Zeeman-Energy Limit 288 7.4.10. Composite Fermions in a Quantum Dot 290 7.4.11. Other Applications 292 7.5. Quantized Screening and Fractional Local Charge 293 7.6. Quantized Hall Resistance 294 7.7. Phenomenological Implications 295 7.7.1. FQHE 295 7.7.2. Transitions Between Plateaus 297 7.7.3. Widths of FQHE Plateaus 297 7.7.4. FQHE in Low-Zeeman-Energy Limit 297 7.7.5. Gaps 297 7.7.6. Shubnikov-de Haas Oscillations 298 7.7.7. Optical Experiments 298 7.7.8. Fermi Sea of Composite Fermions 298 7.7.9. Resonant Tunneling 299 7.8. Concluding Remarks 300 References 302

8 Resonant Inelastic Light Scattering from Quantum Hall Systems 307 A. Pinczuk 8.1. Introduction 307 8.2. Light-Scattering Mechanisms and Selection Rules 311 8.3. Experiments at Integer Filling Factors 317 8.3.1. Results for Filling Factors v = 2 and v = 1 319 8.3.2. Results from Modulated Systems 326 8.4. Experiments in the Fractional Quantum Hall Regime 33 1 8.5. Concluding Remarks 337 References 338

9 Case for the Magnetic-Field-Induced Two-Dimensional Wigner Crystal 343 M. Shayegan 9.1. Introduction 343 9.2. Ground States of the 2D System in a Strong Magnetic Field 347 9.2.1. Ground State in the v << 1 Limit and the Role of Disorder 347 9.2.2. Properties of a Magnetic-Field-Induced 2D WC 348 9.2.3. Fractional Quantum Hall Liquid Versus WC 352 9.2.4. Role of Landau Level Mixing and Finite Layer Thickness 352 x CONTENTS

9.3. Low-Disorder 2D Electron System in GaAs/AlGaAs Heterostructures 353 9.4. Magnetotransport Measurement Techniques 355 9.5. History of dc Magnetotransport in GaAs/AlGaAs 2D Electron Systems at Low v 355 9.6. Summary and Discussion of 2D Electron Data 357 9.6.1. Reentrant Insulating Phase 357 9.6.2. Nonlinear Current-Voltage and Noise Characteristics 358 9.6.3. Normal Hall Coefficient 361 9.6.4. Finite Frequency Data: Pinning and the Giant Dielectric Constant 362 9.6.5. Washboard Oscillations and Related Phenomena 365 9.6.6. Melting-Phase Diagram and the WC-FQH Transition 368 9.7. Summary and Discussion of the 2D Hole Data 370 9.7.1. Sample Structures and Quality 370 9.7.2. Insulating Phase Reentrant Around v = 1/3 370 9.7.3. Disappearance of the Reentrant Insulating Phase at High Density 372 9.7.4. Wigner Crystal Versus Hall Insulator 372 9.8. Bilayer Electron System in Wide Quantum Wells 375 9.8.1. Details of Sample Structure 376 9.8.2. Reentrant Insulating States Around v = 1/3 and 1/2 377 9.9. Concluding Remarks 380 References 380

10 Composite Fermions in the Fractional Quantum Hall Effect 385 H. L. Stormer and 0.C. Tsui 10.1. Introduction 385 10.2. Background 387 10.3. Composite Fermions 390 10.4. Activation Energies in the FQHE 392 10.5. Shubnikov-de Haas Effect of Composite Fermions 395 10.6. Thermoelectric Power Measurements 400 10.7. Optical Experiments Related to Composite Fermions 404 10.8. Spin of a Composite Fermion 405 10.9. How Real Are Composite Fermions? 410 10.10. Transport at Exactly Half Filling 415 10.11. Conclusions 418 References 419

Index 423 CONTRIBUTORS

SANKARDAS SARMA, Department of Physics, University of Maryland, College Park, MD 20742 J. P. EISENSTEIN,Division of Physics, Mathematics and Astronomy, Califor- nia Institute of Technology, Pasadena, CA 91 125 H. A. FERTIG, Department of Physics and Astronomy and Center for Com- putational Sciences, University of Kentucky, Lexington, KY 40506 MATTHEWP. A. FISHER, Institute for Theoretical Physics, University of Califor- nia, Santa Barbara, CA 93 106 S. M. GIRVIN, Department of Physics, Indiana University, Bloomington, IN 47405 B. I. HALPERIN,Department of Physics, Harvard University, Cambridge, MA 02138 J. K. JAIN, Department of Physics, State University of New York at Stony 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 This Page Intentionally Left Blank PREFACE

The unexpected observations of the quantized Hall phenomenon in 1980 and of the fractional quantum Hall effect in 1982 are among the most important physics discoveries in the second half of the twentieth century. The precise quantization of electrical resistance in the quantum Hall effect has led to the new definition of the resistance standard and had major impact in all of science and technology. From a fundamental viewpoint, studies of quantum Hall phenomena are among the most active research areas of physics, with vigorous contributions by researchers in condensed matter, low temperature, semiconductor materials science and devices, and quantum field theory. Striking new behaviors of two-dimensional electron systems in semiconductor quantum structures continue to be discovered, as researchers strive to achieve better and purer materials, higher magnetic fields, lower electron densities, lower temperatures, and new experimental methods. Examples of new discoveries in the quantum Hall regimes are the even denominator fractional quantum Hall effects, the anomalies at half-filled Landau level, and the long-elusive, spectroscopic finding of the fractional excitation gap. Theoretically, a number of new concepts and paradigms have emerged from studies of quantum Hall phenomena that are both elegantly beautiful and powerfully relevant to experiment. A basic picture that emerges from studies of quantum Hall effects over the past ten years is that a low-dimensional electron system, as occurring in semiconductor quantum structures in a high external magnetic field, is a fascinating quantum system for the exploration of fundamental electron interactions. Studies of such many-electron systems have already offered remarkable new insights, and the results of intense current research will continue to suprise us. The first phase in the study of the quantum Hall effect phenomena is the 1980-1986 period where the basic integer and the fractional quantization phenomena were observed and their fundamental understanding consolidated. This “classical” era is well covered by the book The Quantum Hall Effect,edited by R. E. Prange and S. M. Girvin (New York: Springer-Verlag, 1987). During the period 1987-1995 the subject entered its second phase when a large number of spectacular new developments have occurred that have considerably expanded and enhanced the scope of the field. In this second phase the fractional quantum Hall effect has emerged as an archetype of the novel electron quantum fluids that may exist in low-dimensional systems of man-made quantum structures. These studies have also cemented the conceptual links between quantum Hall phenomena and several areas at the current frontiers of physics-in particular, quantum field theory. xiii xiv PREFACE

The purpose of our book is to cover milestones of this second exciting phase by having a leading expert in each topic give a 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 experimental- ists and theorists alike. Each chapter is self-contained, with its own set of references guiding the reader to the original papers on the subject as well as to further reading on the topic. This book would have been impossible without prompt support from all the authors. We wish to thank them for their great care in preparing their chapters.

College Park, Maryland SANKARDAS SARMA Murray Hill. New Jersey ARONPINCZUK 1 Localization, Metal-Insulator Transitions, and Quantum Hall Effect

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

1.1. INTRODUCTION

1.1 .l.Background The close connection between the strong-field localization problem and the phenomenon of the quantum Hall effect was in some curious sense already appreciated before von Klitzing’s celebrated discovery [11 of the quantized Hall effect phenomenon in 1980. Klaus von Klitzing was, in fact, investigating the strong-field galvanomagnetic properties [2] of two-dimensional electrons con- fined in Si/SiO, MOSFETs with the specific goal of elucidating the nature of activated transport at the localized tails of Landau levels when he discovered the quantized Hall effect. (To be more precise, one of the issues being investigated [2] by von Klitzing was to ascertain the specific cause for the vanishing of the longitudinal conductivity, crxx, at the Landau level tails-in particular, whether oxx= 0 at T = 0 is caused by the number n of mobile carriers vanishing at the chemical potential or by a vanishing of the 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 rela- tionship between metal-insulator localization transitions and the phenomenon of the quantum Hall effect is an old subject of considerable fundamental significance.

Perspectives in Quantum Hall Effects, Edited by Sankar Das Sarma and Aron Pinczuk ISBN 0-471-1 1216-X 0 1997 John Wiley & Sons, Inc. 1 2 LOCALIZATION, METAL-INSULATOR TRANSITIONS

1.1.2. Overview

It is now well accepted that the existence of localized states at Landau level tails (and of extended states at Landau level centers) is essential to the basic quantization phenomenon. Quantum Hall plateau transitions, where one goes from one quantized value of pxyto 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,), oxyis 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, CT~~(~T~,,)is zero (quantized) everywhere (i.e., for all values of the chemical potential) except at isolated energies E = E,", where E," is the critical energy at the Nth Landau level center, which is a set of measure zero. At nonzero temperatures (or, equivalently,at finite frequencies)the plateau transition is no longer infinitely sharp, and ox, (and consequently, pxx)is finite over a finite range of the chemical potential around E,". The foregoing scenario describing the quantum Hall plateau transition to be a quantum critical phenomenon is remarkably well verified experimentally. A fundamental detailed field-theoretical understanding of the quantum critical phenomenon, however, still eludes us even though a great deal of numerical work and experimental data provide a rather compelling phenomenology in its support. The quantized Hall effect requires the existence of a mobility gap and thereby directly implies the existence of localized and extended states with metal-insulator transitions between them at discrete values of the chemical potential, E = E:. Our current theoretical understanding of the two-dimensional localization problem, which is based on a perturbative renormalization group analysis of the nonlinear sigma model, rules out the existence of extended states in two dimensions. The quantized Hall effect is then a manifestly nonperturbative effect where extended states appear in a disordered two-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, concrete calculations (e.g., leading to critical exponents)including the topological term do not exist. The current understanding of the quantized Hall effect is thus based on the existence of localized states almost evergwhere except near the Landau level centers, with the plateau transition being the delocalization transition caused by the chemical potential passing through the extended state. Since there are no two-dimensional extended states in zero or weak magnetic fields (at least within the prevailing scaling theory or, more generally, within a weak-coupling pertur- bation treatment of the nonlinear sigma model), whereas the quantized Hall effect implies the essential existence of two-dimensional strong-field extended states, a question naturally arises about what happens to the extended states as the magnetic field is decreased with the eventual disappearance of the quantized Hall effect. There are two possible distinct scenarios consistent with the experimental situation:

1. The extended states float up in energy [10,16] as the magnetic field is decreased and eventually when the extended state corresponding to the lowest (N= 0) Landau level at E,N=O passes above the chemical potential, no quantized Hall effect can be observed and the system is localized because the chemical potential is necessarily in localized states. In this scenario the strong-field extended states float up to infinite energy as the magnetic field vanishes, making the zero-field (or, weak-field) two-dimensional system completely localized, consistent with the scaling theory of localization, 2. The second possibility is that the extended state at the middle of each Landau level eventually disappears (without floating up) at a characteristic magnetic field B:, and when the lowest extended state at vanishes, the quantized Hall effect disappears, with the system being completely localized without any delocalized states whatsoever.

Note that in the first scenario the extended states float up in energy (eventually to infinity), and in the second scenario they disappear (without floating) at (nonuniversal)critical magnetic fields. Currently, much of the community seems to be in favor of the first scenario, which, in fact, was proposed [lo, 16) right after the discovery of the quantized Hall effect (and has since been extended and resurrected under the rubric of a global phase diagram [17]) to reconcile the essential existence of extended states in strong magnetic fields as necessitated by the quantized Hall effect and the nonexistence of true two-dimensional extended states (based on weak-localization experiments and theories) in weak magnetic fields. In turns out, however, that despite a substantial fundamental theoretical difference between the two scenarios(i.e., the floating of extended states to infinity and the disappearance of extended states at finite magnetic fields without floating), it is not easy experimentally to distinguish directly between the two scenarios. It should be noted that from a practical viewpoint both scenarios lead to the existence of a (nonuniversal)critical magnetic field where a system that is 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-induced direct transition [181 between an insulator (at zero field) and a quantum Hall liquid (at finite field). The critical field for this transition is nonuniversal and depends on the system parameters, particularly the amount of disorder in the sample (which may not necessarily be characterized by a single experimental quantity). It has been claimed [ 171 that such field-induced metal-insulator transitions [181, where the system goes from an insulator at zero field to a “metal” at some finite fields, also belong to the same universality class (ie., the same critical exponents) as the quantum Hall plateau transitions. Current experimental evidence in support of this claim is not entirely conclusive. All of the foregoing localization issues, which so far have been discussed in the context of the integer quantized Hall effect, have their counterparts in the fractional quantized Hall situation, with the plateau transitions between fractional incompressible states behaving similar to the integer plateau transitions (albeit at lower temperatures and 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 electronic system).There is, however, a significant additional complication in the localization problem for the fractional state because of the theoretical possibility of the existence of a strong-field quantum Wigner solid phase at low filling factor (v 6 1/5). Indeed, the very high field reentrant metal-insulator transition [18] in the fractional situation (around v x 1/5 for electrons and v x 1/3 for holes) has been interpreted as a quantum phase transition between the Laughlin incompressible liquid and the quantum Wigner solid. This identification is based on theoretical calculations [19] which predict the Wigner solid phase as having lower energy than the Laughlin liquid phase for v 6 1/5, and on the expectation that a strongly pinned (due to disorder) Wigner solid should behave as an insulator with defect-mediated activated transport. (This issue is discussed in more detail in Chapters 3 and 9.)

1.1.3. Prospectus The goal of this chapter is to provide a broad and critical perspective on the subject of two-dimensional strong-field localization and metal-insulator transitions as they relate to the phenomena of quantum Hall effects. While the experimental and theoretical status of the subject is reviewed, no attempt is made to discuss the various topics in detail; that is left to the original articles (and several detailed reviews [20-231) cited throughout this chapter. The emphasis here is on providing a critical (and necessarily somewhat subjective) perspective TWO-DIMENSIONAL LOCALIZATION: CONCEPTS 5 on what is understood and what is not and the extent to which theory and experiment combine to form a unified description of localization phenomena in the quantum Hall effect. The introduction essentially sets the tone and provides an overview of the topics discussed in subsequent sections. In Section 1.2 the basic ideas of two- dimensional localization are discussed, emphasizing the underlying theoretical concepts. In Section 1.3 the phenomenology is discussed by describing the current status of experimental studies on localization aspects of the quantum Hall effect phenomena, with a critical assessment of the agreement between experiment and theoretical concepts. The issues of universality in two-dimensional Landau level localization and random flux localization are discussed in Section 1.4, based on a critical evaluation of the experimental results and various numerical studies.

1.2. TWO-DIMENSIONAL LOCALIZATION: CONCEPTS

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

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

where E = d - 2 with d as the spatial dimension, and the subscripts u, 0, and s refer to unitary, orthogonal, and symplectic ensembles, respectively. The symplectic ensemble applies to the situation where the disorder is associated with spin-flip and spin-orbit scattering, and is, therefore, not of much relevance to us. In the absence of an external magnetic field, there is time-reversal symmetry and the orthogonal ensemble applies, whereas the application of an external magnetic field destroys time-reversal symmetry. It follows from Eq. (3) that the zero-field two-dimensional situation is logarithmically localized with /3(g) - g - ',implying that the resistance R = g- ' has logarithmiccorrections, R(L)= R, + Cln(L/L,). The zero-field orthogonal ensemble can be characterized by a localization length, <,in the weak disorder limit, which is exponential in the mean-field conductance go: < - ego (5) where go may be the dimensionless mean-field Drude conductance calculated, for example, in the self-consistent Born approximation. Note that for weak disorder go can be very large, making the localization length exponentially large, implying that the weak-localization correction to conductance is logarithmically small. For strong disorder, go is extremely small, also making <( - 1) small, which is the usual strong localization situation. [Note, however, that Eq. (5) does not apply when go < 1 in dimensionless units because only the leading order dependence in g-' has been kept.] It is important to realize that within the nonlinear sigma model scenario (or, within the scaling theory of localization in a narrower sense) the transition from a weakly localized two-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.] TWO-DIMENSIONALLOCALIZATION CONCEPTS 7

The consensus that all two-dimensional electron states are at least logarithmically localized (orthogonal ensemble), no matter how weak the disorder is based on weak-coupling perturbative renormalization group arguments [25-271. There is no exact or rigorous argument (even in a narrowly defined context) to rule out potential strong-coupling problems in the renormalization group pertur- bation expansion (which, despite being carried out [27] to an amazing five-loop order exploiting heterotic superstring theoretic techniques, is somewhat ill behaved). Although there is definite experimental evidence [7) in favor of weak logarithmic increase in the sheet resistance of thin-metal films and Si MOSFETs at very low temperatures, it is probably correct to state that experimental evidence supporting weak localization in good-quality GaAs heterostructures is not abundant. On the other hand, there is a great deal of experimental data [28] showing strong localization behavior in GaAs heterostructures at low electron densities (equivalently, at high disorder). Thus some questions remain about the universal applicability of Eq. (3) to all two-dimensional situations in the orthogonal ensemble. It may also be relevant in this context to mention that the localization exponents calculated by direct numerical simulation [29] do not agree with those obtained from the nonlinear sigma model theory for the three- dimensional orthogonal ensemble and the two-dimensional symplecticensemble. 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 time- reversal symmetry in a two-dimensional electron system so that, at least in the weak-field situation (assuming the only effect of the magnetic field to be the breaking of the time-reversal symmetry), the unitary ensemble nonlinear sigma model result, Eq. (2), applies. Thus all states are still localized except that the Cooperon channel associated with the maximally crossed backscattering diagram is suppressed, and the leading order term (the diffusion channel) in the /? function instead of being -g-' is -g-', implying the unitary ensemble localization to be even weaker than that in the zero-field orthogonal situation. The effective weak disorder localization length in the unitary ensemble behaves as

and can be macroscopically large for large values of the mean-field conductivity go. But the states are, in principle, localized for any disorder, albeit with extremely large localization lengths for weak values of disorder (large values of go). The unitary ensemble, therefore, does not allow quantization of Hall conductance, which requires the existence of true extended states (along with bands of localized states). It is not surprising that a perturbative renormalization group theory misses the basic essence of the quantized Hall effect phenomenon, which is manifestly a nonperturbative (topological) macroscopic quantum effect. To 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 topological term 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(essentially the same as the dimensionless conductance g introduced in Section 1.2.1) and the Hall conductivity oxy(which determines the topological term). Thus the strong-field localization theory, in contrast to the usual weak-field (orthogonal or unitary ensemble) nonlinear sigma model situation [which is a one-parameter scaling theory with the fl function, fl(g), being dependent only on a single parameter g], is claimed [lS, 161 to be a two-parameter scaling theory which depends on the renormalization flow for ox, and oxy The renormalization group flow is postulated to be characterized by two different fixed points. One is a stable fixed point connected with the localized states where oxx= 0 and 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 framework may be considered somewhat of a tautology where the introduction of the topological term (and the consequent saddle-point fixed point) into the Lagrangian is equivalent to postulating the existence of strong-field extended states, which experiments unambiguously demonstrate to be there by virtue of the nontrivial quantization of 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 critical exponent and the critical value of ox, at the saddle point based on the field theory. Unfortunately, such concrete calculations have been singularly lacking [30) within this topological field theory 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 oCz,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 intro- ducing Landau quantization into the standard weak-localization self-consistent diagram technique, which, however, by definition cannot access the strong- coupling situation.

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

oxx= 0 (7)

ve oxy = 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 = Er, where it jumps from one quantized plateau to the next, and the longitudinal conductance u,, is zero everywhere except at E,“. The quantized values of Hall conductance are given by

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

e2 axx= - 2h

a,,, = (v + i)f

Note that Eqs. (9), (lo), and (1 1) characterize, respectively, the stable and saddle fixed points. It must be emphasized that while Eq. (9) about the quantization phenomenon itself is absolutely beyond any doubt (and in fact now serves as the definition of the unit of resistance), the same cannot be said about Eq. (1 1). The experimental support for it is at best weak (many experimental results flatly contradict the universality of a,, at the peak, but the zero-temperature limit is always a problematic issue). There is some numerical support [34] for a universal value a,, = e2/2hat 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 effective length scale, L,into the problem, which now competes with the diverging localization length (5) at the critical point. When 5 2 L, the system behaves as a metal because the localization length has exceeded the effective system size. (For the true phase transition in the thermodynamic limit, 5 -, 00 exactly at the critical point where the extended states of measure zero reside.) Thus at finite temperatures (and/or finite system sizes, frequencies, etc.) the 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 identically zero 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. 12 LOCALIZATION, METAL-INSULATOR TRANSITIONS

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

where x (often referred to as v, which is used as the symbol for the Landau level filling factor in this chapter) is the localization critical exponent. Most of the experimental work on the plateau transition in quantum Hall effect can be quantitatively understood on the basis of the scaling law defined in Eq. (13). Somewhat confusingly [35], the scaling theory defined by Eq. (13) has sometimes been referred to as the one-parameter scaling theory [20,35], presumably to emphasize (somewhat redundantly) the fact that there is only one diveriging length in the problem, <(E).It is to be noted that the terminology single-parameter scaling theory in the context of Eq. (13) is totally disconnected from the terminology two-parameter scaling [ 15,161used 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 terminologies is totally different, and they do not contradict each other. As emphasized above, the absolute validity of Eq. (13) for the quantum Hall effect localization transition has not yet been established from an underlying field theory. It is, in fact, by no means obvious that the quantum Hall effect localization critical behavior must necessarily be characterized by a power law divergence in the localization length-in classical two-dimensional critical phenomena the well-known Kosterlitz-Thouless transition, for example, has a different behavior. Since no rigorous or exact (or even an analytic) derivation of Eq. (13) currently exists, it may be helpful to summarize the great deal of phenomenological support [20] in its favor:

1. A great deal of experimental evidence (discussed in Section 1.3) exists supporting the scaling law. 2. A great deal of numerical simulation [20], where the noninteracting strong-field two-dimensional Schrodinger equation is solved directly in the presence of random quenched disorder (e.g., the two-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-size numerical simulation [20,21] has emerged as a powerful tool to study the interplay between localization and quantum Hall effect phenomena and has provided the most significant evidence supporting Eq. (13). TWO-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 two- dimensional percolation transition [36-381, where the disorder potential is extremely smooth on magnetic length scales. This then leads to a percolation length or a percolation cluster size (which can also be thought as the size of a typical quantum Hall droplet), which diverges at the percolation transition, and an identification of the localization length ( with the percolation cluster size leads to behavior similar to that in Eq. (13). The most compelling evidence in favor of the critical scaling behavior defined by Eq. (13) for the delocalization-localization transition associated with the quantum Hall plateau transition comes from a great deal of direct numerical simulation of the strong-field two-dimensional Anderson model using various types of disorder potential. The most direct numerical technique has been the calculation [20,21,35,39-421of 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 f(x >> 1) - l/x and f (x << 1) - constant. In calculating A,(E) numerically, one considers a disordered two-dimensional strip of length L and width M, with L typically so large that the system is self-averaging. In practice, La 10’ (in units of magnetic length) usually suffices. Periodic boundary conditions are used in the width direction, and the largest value of the width that is accessible with the currently available massively parallel processing machines is M = 5 12, even though M = 32 - 128 is more typical. A recursive Green’s function technique or a transfer matrix technique is used to solve the Schrodinger equation, leading to the Lyapunov exponent or the inverse localization length. For a given finite system of width M, the calculation of &(E) is exact. Details of the calculational method are given in the literature [20,21]. Obtaining A,(E) numerically as a function of M and E, one can then calculate e(E)by plotting A,(E)/M against M/C;(E),with ((E)being adjusted to achieve the best scaling behavior in accordance with Eq. (14).The values of ((E)that give the best finite-size scaling fit [i.e., Eq. (14)] for the numerical data are also found to obey the scaling law defined by Eq. (13),providing the most direct numerical evidence for the quantum critical behavior of the plateau transition. 14 LOCALIZATION. METAL-INSULATOR TRANSITIONS

Several different groups [20,21,35,39-421 have carried out finite-size scaling calculations using somewhat different models of the random disorder potential. The agreement among the numerical results of different groups is excellent, and the localization critical exponent for short-range white-noise disorder is found to be (neglecting Landau level coupling) ~=2.3fO.1 for N=O w 5.5 k 0.5 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-level calculations. Thus the calculated critical exponent xo=2.3 in the lowest Landau level is more reliable than the calculated exponent = 5.5 in the first excited level. Systematic studies [20,21] of the dependence ofx1 the critical exponent on the strength and type (attractive/repulsive, etc.) of disorder have been carried out. These studies are somewhat incomplete, and their results are consistent with the conclusion that the localization exponent x is universal and independent of the strength or type of disorder potential. In practice, however, a weak dependence of the calculated exponent on the details of disorder potential is found in numerical simulations [21] and attributed (rather uncritically) to crossover effects. The localization critical exponent for the plateau transition has also been calculated using a quantum percolation network model [43,44], where the random potential is assumed to vary slowly on the scale of the magnetic length and the electron guiding centers follow the semiclassical equipotential contours of the smooth disorder potential. If quantum tunneling effects are neglected,the problem reduces to a two-dimensional classical percolation problem, where in the symmetric situation, there is exactly one energy at the percolation threshold, where the critical percolation cluster encompasses the entire system. If one identifies [36,43,44] this percolation transition as the delocalization transition at E,, onegets x = 4/3. It turns out that quantum tunnelingeffectscan be included in this percolation picture through a quantum network model where the nodes of