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.