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.
|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|
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