PERSPECTIVES IN QUANTUM HALL EFFECTS Novel Quantum Liquids in Low-Dimensional Semiconductor Structures Edited by Sankar Das Sarma Aron Pinczuk WILEY- VCH Wiley-VCH Verlag GmbH & Co. KGaA This Page Intentionally Left Blank PERSPECTIVES IN QUANTUM HALL EFFECTS This Page Intentionally Left Blank PERSPECTIVES IN QUANTUM HALL EFFECTS Novel Quantum Liquids in Low-Dimensional Semiconductor Structures Edited by Sankar Das Sarma Aron Pinczuk WILEY- VCH Wiley-VCH Verlag GmbH & Co. KGaA 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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Printed in the Federal Republic of Germany Printed on acid-free paper Printing and Bookbinding buch biicher dd ag, Birkach ISBN-13: 978-0-471-11216-7 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.