Dirac electrons in the presence of a matrix potential barrier: application to graphene and topological insulators

Scattering of 2D Dirac electrons on a rectangular matrix potential barrier is considered using the formalism of spinor transfer matrices. It is shown, in particular, that in the absence of the mass term, the Klein tunneling is not necessarily suppressed but occurs at oblique incidence. The formalism is applied to studying waveguiding modes of the barrier, which are supported by the edge and bulk states. The condition of the existence of the uni-directionality property is found. We show that the band of edge states is always finite with massless excitations, while the spectrum of the bulk states, depending on the parameters of the barrier, may consist of the infinite or finite band with both, massive and massless, low-energy excitations. The effect of the Zeeman term is considered and the condition of the appearance of two distinct energy-dependent directions corresponding to the Klein tunneling is found.