The Foldy-Wouthuysen and scattering matrix method for calculating the transmission of electrons through two dimensional graphene: electrons like light

Abstract

In this paper is provided a novel approach to solving the transmission of electrons through large graphene nano-structures, which is shown to be accurate both at high and low speeds. The model for graphene being solved is the continuum model governed by an analogue to the Dirac equation. For a solution, the Dirac equation is scalarised using the Foldy-Wouthuysen expansion approximation, to reduce the problem of calculating the electron wave propagation to a scalar differential equation. Also transformed is the exact solution of the Dirac equation in homogeneous space for the calculation of the propagation of electron waves. By analytically calculating the boundary conditions of the transformed wave functions, I have been able to generate transfer matrices for the scalar propagation equations. Furthermore, I have implemented the scattering matrix method upon these transfer matrices. Implementing the scattering matrix method makes a numerical stable propagation of the waves through the graphene. Finally I test the convergence and accuracy of the new method against analytic solutions.