Date of Award

Fall 2015

Project Type

Dissertation

Program or Major

Mathematics

Degree Name

Doctor of Philosophy

First Advisor

Jian-Ming Tang

Second Advisor

Kai Germaschewski

Third Advisor

John Gibson

Abstract

This dissertation reports on a novel theoretical and computational framework for calculating low-energy electron reflectivities from crystalline surfaces and its application to two layered systems of two-dimensional materials, graphene and molybdenum disulfide. The framework provides a simple and efficient approach through the matching of a small set of Fourier components of Bloch wave solutions to the Schrodinger Equation in a slab-in-supercell geometry to incoming and outgoing plane waves on both sides of the supercell. The implementation of this method is described in detail for the calculation of reflectivities in the lowest energy range, for which only specular reflection is allowed. This implementation includes the calculation of reflectivities from beams with normal or off-normal incidence. Two different algorithms are described in the case of off-normal incidence which differ in their dependence on the existence of a symmetry with a mirror plane parallel to the crystal surface. Applications to model potentials in one, two, and three dimensions display consistent results when using different supercell sizes and convergent results with the density of Fourier grids.

The design of the Bloch wave matching also allows for the accurate modeling of crystalline slabs through the use of realistic potentials determined via density functional theory. The application of the method to low-energy electron scattering from free-standing systems of a few layers of graphene, including the use of these realistic potentials, demonstrates this ability of the method to accurately model real systems. It reproduces the layer-dependent oscillations found in experimental, normal incidence reflectivity curves for a few layers of graphene grown on silicon carbide. The normal incidence reflectivity curves calculated for slabs consisting of few-layer graphene on 10 layers of nickel show some qualitative agreement with experiment. General incidence reflectivity spectra for free-standing few-layer graphene predict free-electron-like dispersion for the location of the lowest energy oscillations as a function of the in-plane Bloch wave vector, as well as other interesting features. The application of the method to scattering from free-standing systems of molybdenum disulfide predicts that low-energy electron reflectivities can provide a means for layer counting, especially in 2H- and 3R-molybdenum disulfide, and structural differentiation, especially between 1T- and 2H-molybdenum disulfide, in few-layer systems. An investigation of the issues and limitations of the method suggests that some modifications and improvements are likely necessary before widespread application is possible.

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