Goos-H{\"a}nchen Shift for Relativistic Particles Based on Dirac's Equation

The Goos-H{\"a}nchen (GH) shift is a specifical optical phenomenon that describes a shift parallel to the reflected light inside the plane of incidence, when a finite-width light undergoes total internal reflection at the interface of medium. Although the GH shift in optics has been widely observed experimentally, its generalization remains uncovered completely in relativistic quantum mechanics for the existence of Klein's paradox. Recently, Wang has solved Klein's paradox based on the different solutions adpoted for Dirac's equation with step potential in corresponding energy regions \href{https://dx.doi.org/10.1088/2399-6528/abd340}{[J. Phys. Commun. {\bf 4}, 125010 (2020)]}. In the light of Wang's method, we calculate the GH shift for Dirac fermions under relativistic conditions when they are incident obliquely on a three-dimensional infinite potential barrier. Furthermore, we find that the relativistic quantum GH shift can be negative, which is different from the non-relativistic case.