Foldy-Wouthuysen WKB transfer matrix method for Dirac tunneling through monolayer graphene with a mass gap

I provide a transfer matrix method for the Foldy-Wouthuysen representation of the Dirac equation. I develop 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 which has the same elegant form as the WKB solution to Schrödinger’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 engineering. I verify my methods for all energies by comparison with analytic solutions developed in the Dirac spinor representation. I also include a rich appendix detailing my research into the Green’s functions of Dirac fermions