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Dimensional Quantum H all Systems DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Graduate School of the Ohio State University By Jonghyun Kahng, B.S., M.S. )fc $ jfc jfc $ The Ohio State University 1992 Dissertation Committee: Approved by Prof. Charles A. Ebner Prof. Ciriyam Jayaprakash f l X a o U s Advisor Prof. Charles H. Pennington Department of Physics To my mom, Chung Wha, and Hisoo ii ACKNOWLEDGEMENTS I thank Prof. Ho for his guidance throughout the research and for the preparation of this thesis. In particular, his concern about the thesis even in his very critical moment is greatly appreciated. It is unfortunate that he could not be in my dissertation committee in the last moment. However, with no doubt, he has everything to do with this thesis. I also thank Prof. Ebner for his generosity to lead the committee as a chairman in Prof. Ho’s absence. In addition, his advises on computational techniques during my early work in this graduate program have been indispensable for the rest of my graduate study as well as for my future career. This thesis would not have been complete without Prof. Jayaprakash’s critical reading of and comments on the manuscript. His kindness to do so is much appreciated. I am indebted to Prof. Patton for his continuous and warm encouragement for the last four years. I thank Prof. Pennington for serving in the committee in spite of his compact schedule. My special thanks goes to my mom and my wife Chung Wha for their patient support and encouragement throughout the period of my graduate work. I like to attribute much credit to my daughter Hisoo for the tremendous amount of joy that she has brought me during the time of pressure. Without her, the last two years would have been even more painful. VITA February 2, 1959 .......................... Born - Seoul, Korea February, 1982 .............................. B.S., Physics, Seoul National University, Seoul, Korea 1982-1985 ....................................... System Programmer, Korean Air Lines, Seoul, Korea 1985-1988 ....................................... University/Graduate Fellow, The Ohio State University, Columbus, Ohio 1986-1992 ....................................... Graduate Teaching/Research Associates, Department of Physics, The Ohio State University, Columbus, Ohio August, 1991 ................................. M.S., Physics, The Ohio State University, Columbus, Ohio PUBLICATIONS J. Kahng, C. Ebner, Melting of multilayer films: Further studies of a Potts lattice- gas model, Phys. Rev. B 40, 11269 (1989) T.-L. Ho, J. Kahng, Macroscopic angular momentum of the crystal states in two- dimensional quantum Hall systems, Phys. Rev. B 45, 9481 (1992) FIELD OF STUDY Major Field: Physics Theoretical Condensed Matter Physics TABLE OF CONTENTS ACKNOWLEDGEMENTS ........................................................................... iii VITA ..................................................................................................................... iv LIST OF TABLES ........................................................................................... vii LIST OF FIGURES ........................................................................................ viii CHAPTER PAGE I. Introduction 1 1.1 Overview of Crystal States in Two Dimensional Electron Systems 1 1.2 Fundamentals of Two Dimensional Electron System s ................ 4 II. Review of Experiments and Theories 11 2.1 Experiments on Quantum Hall Systems ....................................... 11 2.1.1 Two Dimensional Electron System s ..................................... 11 2.1.2 Quantum Hall E ffe c ts ............................................................ 15 2.1.3 Crystal States ......................................................................... 17 2.2 Theories of Quantum Hall S y stem s ................................................. 22 2.2.1 Integer Quantum Hall E ffe c ts ............................................... 22 2.2.2 Incompressible Quantum Fluids at v — 1 / m ...................... 25 2.2.3 Hierarchical Schemes ............................................................... 33 2.2.4 Crystal States ........................................................................ 40 III. Crystal States 47 3.1 Crystals of Q u asip articles ................................................................ 48 3.2 Energy of Crystals ............................................................................. 52 3.3 Angular Momentum and Induced Magnetic Field in Crystals . 60 3.4 Monte Carlo M e th o d s ....................................................................... 69 3.5 Particle-Hole Symmetry ................................................................... 74 3.5.1 Characteristics of Particle-Hole Conjugate States .... 76 3.5.2 Quasihole Crystal vs. Wigner Crystal ....................... 80 3.6 Quasiparticle Crystals in Hierarchical S ta te s ................................ 81 IV. Conclusions 91 APPENDICES A. Lists of Computer Programs 95 A.l Program for the Density of Wigner Crystals ................................ 95 A.2 Program for the Pair Correlation Function of Quasihole Crystals 99 B IB L IO G R A PH Y ............................................................................................ 110 vi LIST OF TABLES Monte Carlo data for quasihole crystals LIST OF FIGURES 1.1 The density of states and the eigenstates of a two-dimensional electron in a magnetic field ............................................................................................ 7 1.2 A schematic diagram of the Hall effect ........................................................ 9 2.1 Electron energy level diagrams for a Si MOSFET and a GaAs-AlGaAs heteroj unction ................................................................................................... 12 2.2 A typical geometry of samples in magnetotransport experiments.. 14 2.3 Magnetotransport measurements in a GaAs-AlGaAs heterojunction. 18 2.4 A schematic phase diagram of two-dimensional electron systems. 21 2.5 The density of states for two-dimensional electrons and the geometry of a metal strip in Laughlin’s thought experiment .......................... 23 2.6 The electron density profile of quasiparticle excitations at v — 1/3. 32 2.7 The estimated energy of hierarchical quantum Hall states ............ 36 2.8 The relative stability of fractional quantum Hall states.................. 41 2.9 The comparison of energies of fractional quantum Hall states and Wigner crystals ................................................................................................. 45 3.1 Pair correlation functions of quasihole crystals .......................................... 54 3.2 The energy of quasiparticle crystals near v = 1/5..................................... 55 3.3 The energy of quasiparticle crystals near v = 1/3...................................... 56 3.4 The energy of quasiparticle crystals near v — 1/2...................................... 57 viii 3.5 The angular momentum in Wigner crystals and quasihole crystals 67 3.6 The angular momentum in quasihole crystals .................................. 68 3.7 The induced magnetic field in quasihole crystals and Wigner crystals. 70 3.8 The density contour plot of the quasihole crystal at v = 1/2. 73 3.9 The extrapolation of the energy of a quasihole crystal ............................ 75 ix CH A PTER I Introduction 1.1 Overview of Crystal States in Two Dimensional Elec­ tron Systems In 1934, Wigner [1] predicted that the electron gas in a uniform neutralizing positive background crystallizes at sufficiently low densities. His argument is based on the comparison of the kinetic and potential energies of the electrons. At low temperatures, the kinetic