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

We provide a transfer matrix method for the Foldy-Wouthuysen representation of the Dirac equation. We develop the relationship between the reflection and transmission coefficients of the Dirac spinors and the wavefunction in the transformed representation. We develop a WKB approximation for Dirac fermions which has the same elegant form as the WKB solution to Schrodinger’s equation. Our WKB approximation is to all orders and includes the semi-classical turning point. We provide an extension to fully 2-dimensional periodic structures by Fourier methods for band engineering. We verify our methods for all energies by comparison with analytic solutions developed in the Dirac spinor representation. Also included is a rich appendix detailing our research into the Green’s functions of Dirac fermions.