Spectral Approach to Transport Problems in Two-Dimensional Disordered Lattices (eBook)

Evdokiya Georgieva Kostadinova is a research assistant professor in the Center for Astrophysics, Space Physics & Engineering Research at Baylor University. She received her PhD from Baylor University in 2017. 

CHAPTER ONE ................................................................................................................. 1Introduction ..................................................................................................................... 11.1. Formulation of the Transport Problem ................................................................ 41.2. Nature of Disorder ............................................................................................... 71.3. Relevance to Physical Systems ........................................................................... 10CHAPTER TWO .............................................................................................................. 15Theoretical Background ................................................................................................ 152.1. Localization Criteria .......................................................................................... 152.2. Anderson Model ................................................................................................. 202.3. Edwards and Thouless Model ............................................................................ 252.4. Scaling Theory ................................................................................................... 29CHAPTER THREE .......................................................................................................... 32Spectral Approach ......................................................................................................... 323.1. Essence of the Spectral Method ......................................................................... 323.2. Simplified Numerical Model (“Toy Model”) ..................................................... 403.3. Physical Interpretation ...................................................................................... 453.4. Scope and Limitations of the Spectral Analysis ................................................. 49CHAPTER FOUR ............................................................................................................. 51Delocalization in 2D Lattices of Various Geometries .................................................. 514.1. Transport in the Honeycomb, Triangular, and Square Lattices ........................ 514.2. Orthogonality Check .......................................................................................... 564.3. Equation Fitting ................................................................................................. 574.4. Cluster Analysis ................................................................................................. 604.5. Comparison Between the Honeycomb and the Triangular Lattices .................. 63CHAPTER FIVE .............................................................................................................. 65Transport in the Two-Dimensional Honeycomb Lattice with Substitutional Disorder 655.1. Discrete Percolation .......................................................................................... 665.2. Formulation of the Transport Problem .............................................................. 695.3. Distribution of Variables ................................................................................... 785.4. 2D Honeycomb Lattice with Substitutional Disorder ........................................ 83CHAPTER SIX ................................................................................................................. 89Transport in 2D Complex Plasma Crystals ................................................................... 896.1. Complex Plasma Preliminaries ......................................................................... 906.2. Two-Dimensional Dust Crystal Analogue ......................................................... 946.3. Transport in the Classical Regime ..................................................................... 956.4. Numerical Simulations of Dust Particle Dynamics ........................................... 986.5. Spectral Analysis .............................................................................................. 105CHAPTER SEVEN ........................................................................................................ 107Conclusions ................................................................................................................. 107APPENDIX A ................................................................................................................. 113Basic Materials Science Terms ................................................................................... 113A.1. Comparison Between Fermi Energy and Fermi Level .................................... 114APPENDIX B ................................................................................................................. 118Mathematical Preliminaries ........................................................................................ 118B.1. Kindergarten Math .......................................................................................... 118B.2. Measure Theory ............................................................................................... 119B.3. Point-set Topology ........................................................................................... 121B.4. Group Theory .................................................................................................. 124B.5. Probability Theory ........................................................................................... 127BIBLIOGRAPHY ........................................................................................................... 129