The Quantum Hall Effect

analyze the Hall effect in the plateau region relative to the fundamental value 2 h/e i expected in the simple one-electron picture for integer filling factors of Landau levels. Subsequent work in my laboratory in Wiirzburg using a super­ conducting solenoid confirmed the constancy of the Hall resistance both in Dorda's samples and in samples supplied by M. Pepper of the Cavendish Laboratory. With technical assistance from the Physikalisch-Technische Bundesanstalt in Braunschweig, an absolute measurement of the Hall resistance confirmed the 2 fundamental quantization relation RIJ = h/ei to an accuracy of about 1 part in ]05. Recalling the practical applications of the Josephson effect, my initial thinking was oriented toward the idea of a resistance standard, but various groups at national laboratories which are involved in high precision measurements of fun­ damental constants pointed out that, in addition, the quantized Hall resistance yields a new fundamental measure of the fine structure constant Ci. These then were the initial events which led to the remarkable surge of interest within both the metrology and condensed matter physics communities in quantum transport in inversion layer systems. Subsequent developments have been many and varied and are described in detail in this volume.

Contenu

1 Introduction.- 1.1. Introduction.- 1.2. Preview of Coming Attractions.- 1.3. The Ordinary Hall Effect.- 1.4. Measuring the Conductance.- 1.5. Introduction to the Quantum Case.- 1.6. Impurity Effects.- 1.7. Gauge Arguments.- 1.8. Inversion Layers.- 1.9. Acknowledgements.- 1.10. Notes.- 1.11. Problems.- A The Integer Effect.- 2 Experimental Aspects and Metrological Applications.- 2.1. Basic Principles.- 2.2. The Devices.- 2.3. The Basic Experiment.- 2.4. Initial Experiments.- 2.5. Precision Measurement Techniques.- 2.6. Quantum Hall Resistors.- 2.7. An Absolute Resistance Standard.- 2.8. The Fine-Structure Constant.- 2.9. Temperature Dependence of ?xx.- 2.10. Temperature Dependence of ?xy.- 2.11. Current Distribution and Edge Effects.- 2.12. Current Dependence and Breakdown.- 2.13. Hall Step Widths and Shapes.- 2.14. Thermomagnetic Transport.- 2.15. Magnetic Moment.- 2.16. Magnetocapacitance.- 2.17. Magneto-Photoresponse.- 2.18. Conclusions.- 2.19. Acknowledgements.- 2.20. Notes.- 3 Effects of Imperfections and Disorder.- 3.1. Introduction.- 3.2. The Impurity Potential.- 3.3. Weak Potential.- 3.4. Scattering Potential.- 3.5. Smooth Potential.- 3.6. Continuum Percolation Model.- 3.7. General Potential.- 3.8. Numerical Studies.- 3.9. Acknowledgements.- 3.10. Notes.- 3.11. Problems.- 4 Topological Considerations.- 4.1. Topological Quantum Numbers.- 4.2. Quantum Hall Effect in a Periodic Potential.- 4.3. Generalization of the Topological Interpretation.- 4.4. Notes.- 5 Field Theory, Scaling and the Localization Problem.- 5.1. Introduction.- 5.2. Basic Notions.- 5.3. The Search for the Principle.- 5.4. Structure of the Effective Field Theory.- 5.5. Instantons and Scaling.- 5.6. Notes.- B: The Fractional Effect.- 6 Experimental Aspects.- 6.1. Introduction.- 6.2. Conditions for the Observation of the FQHE and the Choice of Semiconductor Systems.- 6.3. The Fractional Quantum Hall Effect.- 6.4. Magneto-Transport in Two Dimensions.- 6.5. Future Experiments.- 6.6. Acknowledgements.- 6.7. Notes.- 7 Elementary Theory: The Incompressible Quantum Fluid.- 7.1. Introduction.- 7.2. Quantized Motion of Small Numbers of Electrons.- 7.3. Variational Ground State.- 7.4. Computational Methods.- 7.5. Fractionally Charged Quasiparticles.- 7.6. Exactness of the Fractional Quantum Hall Effect.- 7.7. More Than One Quasiparticle.- 7.8. Conclusions.- 7.9. Acknowledgements.- 8 The Hierarchy of Fractional States and Numerical Studies.- 8.1. Introduction.- 8.2. Pseudopotential Description of Interacting Particles with the Same Landau Index.- 8.3. The Principle Incompressible States: A Non-Variational Derivation.- 8.4. Quasiparticle and Quasihole Excitations.- 8.5. The Hierarchy of Quasiparticle-Quasihole Fluids.- 8.6. Translationally Invariant Geometries for Numerical Studies.- 8.7. Ground-State Energy Studies.- 8.8. Pair Correlations and the Structure Factors.- 8.9. Quasiparticle Excitations.- 8.10. Collective Excitations.- 8.11. Acknowledgements.- 9 Collective Excitations.- 9.1. Introduction.- 9.2. Density Waves as Elementary Excitations.- 9.3. Collective Modes in the FQHE.- 9.4. Magnetophonons and Magnetorotons.- 9.5. Further Superfluid Analogies: Vortices.- 9.6. Quasi-Excitons.- 9.7. Magnetoplasmons and Cyclotron Resonance.- 9.8. Other Collective Modes.- 9.9. Role of Disorder.- 9.10. Acknowledgements.- 9.11. Notes.- 9.12. Problems.- C: The Quantum Hall Effect.- 10 Summary, Omissions and Unanswered Questions.- 10.1. Integer Effect at Zero Temperature.- 10.2. IQHE at Finite Temperatures.- 10.3. Metrology.- 10.4. Basic Picture of the Fractional Effect.- 10.5. Remarks on the Neglect of Landau Level Mixing.- 10.6. Wanted: More Experiments.- 10.7. Towards a Landau-Ginsburg Theory of the FQHE.- 10.8. Acknowledgements.- 10.9. Notes.- 10.10. Problems.- References.