Resistance networks as a model for conduction on the nano-scale
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In this thesis, we calculate transport properties of amorphous materials in one, two, and three dimensions. We take into account site disorder, manifest as a random variation of the locations of atomic species. We employ a resistor network model as a theoretical framework for calculating transport characteristics. The numerical calculations we employ are based on an iterative algorithm used as an improvement over the direct solution of the relevant linear systems. The Monte Carlo calculations are used to validate analytical perturbative treatment valid in the bulk limit. In approaching random resistor networks, we discuss and apply a paradigm based on the connectivity of nodes instead of mesh currents where the applicability is limited to a specific set of geometries. We argue that this perspective is very useful in strongly disordered systems, especially for three-dimensional cases.
Table of Contents
Introduction -- An application of charge conservation and node connectivity -- Periodic resistor networks with current injected at specific nodes -- Iterative algorithm for the numerical calculations of transport characteristics -- A resistor model for transport characteristics in regular lattices -- Introduction of disorder: an analytical perturbative calculation in one dimension -- Random resistor networks in two dimensions: analytical and numerical results -- Three-dimensional geometries: analytical perturbative calculations and numerical results -- Conclusions and suggestions for future research