Tunneling in an anisotropic cubic Dirac semi-metal

Abstract

Motivated by a recent first principles prediction of an anisotropic cubic Dirac semi-metal in a real material Tl(TeMo)3, we study the behavior of electrons tunneling through a potential barrier in such systems. To clearly investigate effects from different contributions to the Hamiltonian we study the model in various limits. First, in the limit of a very thin material where the linearly dispersive z-direction is frozen out at zero momentum and the dispersion in the x-y plane is rotationally symmetric. In this limit we find a Klein tunneling reminiscent of what is observed in single layer graphene and linearly dispersive Dirac semi-metals. Second, an increase in thickness of the material leads to the possibility of a non-zero momentum eigenvalue kz that acts as an effective mass term in the Hamiltonian. We find that these lead to a suppression of Klein tunneling. Third, the inclusion of an anisotropy parameter λ = 1 leads to a breaking of rotational invariance. Furthermore, we observed that for different values of incident angle θ and anisotropy parameter λ the Hamiltonian supports different numbers of modes propagating to infinity. We display this effect in form of a diagram that is similar to a phase diagram of a distant detector. Fourth, we consider coexistence of both anisotropy and non-zero kz but do not find any effect that is unique to the interplay between non-zero momentum kz and anisotropy parameter λ. Last, we studied the case of a barrier that was placed in the linearly dispersive direction and found Klein tunneling T − 1 ∝ θ 6 + O(θ 8 ) that is enhanced when compared to the Klein tunneling in linear Dirac semi-metals or graphene where T − 1 ∝ θ 2 + O(θ 4 ).