Foldy-Wouthuysen Green's function and WKB transfer matrix method for Dirac tunneling through monolayer graphene with a mass gap

TL;DR

This paper introduces a transfer matrix and WKB approximation for Dirac fermions in graphene with a mass gap, enabling accurate analysis of tunneling and band-gap engineering with verified methods.

Contribution

It develops a novel WKB approximation to all orders for Dirac fermions and extends it to 2D periodic structures, with detailed Green's function derivations.

Findings

WKB approximation matches analytic solutions across energies

Transfer matrix relates Dirac spinor coefficients to wavefunctions

Extension to 2D structures facilitates band-gap engineering

Abstract

I provide a transfer matrix method for the Foldy-Wouthuysen representation of the Dirac equation. I derive the relationship between the reflection and transmission coefficients of the Dirac spinors and the wavefunction in the transformed representation. I develop a WKB approximation for Dirac fermions that has the same elegant form as the WKB solution to Schr\"{o}dinger's equation. My WKB approximation is to all orders and includes the semi-classical turning point. I provide an extension to fully 2-dimensional periodic structures by Fourier methods for band-gap engineering. I verify my methods for all energies by comparison with analytic solutions developed in the Dirac spinor representation. Rich appendices detail my research into the Green's functions of Dirac fermions, where I rigorously derive the free space Green's functions for the Foldy-Wouthuysen representation of the Dirac equation.