arXiv 2103.018072021-03cond-mat.mes-hall

各向异性立方Dirac半金属中的隧穿

Tunneling in an anisotropic cubic Dirac semi-metal

Ahmed Bouhlal, Ahmed Jellal, Hocine Bahlouli, Michael Vogl

DOI：: 10.1016/j.aop.2021.168563

摘要：

基于第一性原理对真实材料Tl（TeMo）$_3$中各向异性立方狄拉克半金属的预测，我们研究了电子隧穿势垒的行为.为了清楚地研究不同贡献的哈密顿量的影响，我们研究了各种限制的模型。首先，在非常薄的材料的极限中，其中线性色散的$z$方向在零动量下被冻结，并且在$x$ - $y$平面中的色散是旋转对称的。在这个极限下，我们发现了一个克莱因隧道让人想起在单层石墨烯和线性色散狄拉克半金属中观察到的。其次，材料厚度的增加导致非零动量本征值$k_z$的可能性，其充当哈密顿量中的有效质量项。我们发现，这些导致抑制克莱因隧道。第三，包含各向异性参数$λ\neq 1$导致旋转不变性的破坏。此外，我们观察到，对于不同的值的入射角$θ$和各向异性参数$λ$的哈密顿量支持不同数量的模式传播到无穷大。我们以类似于远距离探测器的相位图的形式显示这种效应。第四，我们考虑各向异性和非零$k_z$的共存，但没有发现任何效果，是唯一的非零动量$k_z$和各向异性参数$λ$之间的相互作用。最后，我们研究了放置在线性色散方向上的势垒的情况，并发现当与线性狄拉克半金属或石墨烯中的Klein隧穿相比时，Klein隧穿$T-1\propto θ^6+\mathcal{O}(θ^8)$被增强，其中$T-1\propto θ^2+\mathcal{O}(θ^4)$。

英文摘要：

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

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