Dissertation/Thesis Abstract

Coulomb Potential on a Lattice Using Graph Theory
by Burgess, Christopher Michael, M.S., California State University, Long Beach, 2020, 73; 28086484
Abstract (Summary)

To describe the electronic properties of a crystal one must understand the Coulomb interactions within the lattice. Often, two simplifying limits are considered. At long wavelengths the details of the lattice are irrelevant and the Coulomb potential is taken in the continuum limit. The other limit considers purely onsite interactions with no spatial dependence of the potential. The present work aims at describing the Coulomb potential in the regime between these two limits, where the lattice structure is relevant. We solve the Poisson equation for the Coulomb potential on a discrete lattice. By expanding upon tools from graph theory used in network analysis, we develop a methodology for finding the specific form of the discrete Laplace operator on a lattice. The Coulomb potential is then calculated for the two-dimensional graphene structure, and for two representative three-dimensional models, the AAA and the ABA Bernal stacked graphene multilayers. The accuracy of the discretized Laplace operator is analyzed for various graph theory stencils by comparing equipotential surface plots for the three-dimensional crystals.

Indexing (document details)
Advisor: Bill, Andreas
Commitee: Peterson, Michael, Ojeda-Aristizabal, Claudia
School: California State University, Long Beach
Department: Physics and Astronomy
School Location: United States -- California
Source: MAI 82/6(E), Masters Abstracts International
Subjects: Theoretical physics, Computational physics, Condensed matter physics
Keywords: Coulomb, Graph theory, Graphene, Graphite, Laplacian, Poisson, 3D modeling
Publication Number: 28086484
ISBN: 9798698595403
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