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I provide a transfer matrix method for the Foldy-Wouthuysen representation ofthe Dirac equation. I develop the relationship between the reflection andtransmission coefficients of the Dirac spinors and the wavefunction in thetransformed representation. I develop a WKB approximation for Dirac fermionswhich has the same elegant form as the WKB solution to Schr\"{o}dinger'sequation. My WKB approximation is to all orders and includes the semi-classicalturning point. I provide an extension to fully 2-dimensional periodicstructures by Fourier methods for band engineering. I verify my methods for allenergies by comparison with analytic solutions developed in the Dirac spinorrepresentation. I also include a rich appendix detailing my research into theGreen's functions of Dirac fermions.

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