Tunneling in an anisotropic cubic Dirac semi-metal

TL;DR

This paper investigates electron tunneling in an anisotropic cubic Dirac semi-metal, revealing conditions for Klein tunneling, effects of anisotropy and thickness, and how these influence electron propagation and tunneling probabilities.

Contribution

It provides a detailed analysis of tunneling phenomena in anisotropic cubic Dirac semi-metals, including effects of material thickness, anisotropy, and momentum, with new insights into Klein tunneling behavior.

Findings

Klein tunneling occurs in thin limits similar to graphene.

Increasing thickness suppresses Klein tunneling due to non-zero momentum.

Anisotropy parameter affects the number of propagating modes.

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 $k_z$ 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 $\lambda\neq 1$ leads to a breaking of rotational invariance. Furthermore, we observed that for different values of incident angle $\theta$ and anisotropy parameter $\lambda$ 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 $k_z$ but do not find any effect that is unique to the interplay between non-zero momentum $k_z$ and anisotropy parameter $\lambda$. Last, we studied the case of a barrier that was placed in the linearly dispersive direction and found Klein tunneling $T-1\propto \theta^6+\mathcal{O}(\theta^8)$ that is enhanced when compared to the Klein tunneling in linear Dirac semi-metals or graphene where $T-1\propto \theta^2+\mathcal{O}(\theta^4)$.