Foldy-Wouthuysen Green's function and WKB transfer matrix method for Dirac tunneling through monolayer graphene with a mass gap
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.
Keywords
Cite
@article{arxiv.1611.00247,
title = {Foldy-Wouthuysen Green's function and WKB transfer matrix method for Dirac tunneling through monolayer graphene with a mass gap},
author = {Mark Behzad Doost},
journal= {arXiv preprint arXiv:1611.00247},
year = {2024}
}
Comments
Revisions have been made throughout the text to enhance clarity and readability, with notable improvements to the derivations of the two- and three-dimensional free space Green's functions in Appendix F, introduced in the preceding version