Super-Klein tunneling in 2D Lorentzian-type barriers in graphene

Abstract

We introduce a two-dimensional model of spin-1/2 Dirac fermions in graphene subjected to a highly tunable electric field, which exhibits super-Klein tunneling. The electric field can be continuously interpolated between two limiting configurations: a uniform electrostatic Lorentzian barrier with translational invariance and a chain of well-separated electrostatic scatterers. We demonstrate that super-Klein tunneling arises naturally as a direct consequence of the intrinsic connection of the model to free-particle dynamics, a relation that is established through methods of supersymmetric quantum mechanics, which provide an elegant and analytically tractable framework. Besides the mentioned super-Klein tunneling, scale invariance of the model and invisibility of the potential for particles of specific energy are revealed, and possible routes toward experimental realization are discussed.

Figures (2)

• Figure 1: (left column) Potential $\tilde{V}(x,y)$. (middle column) Probability density of $\tilde{\psi}=\mathcal{L}\!\left(\psi_{0}+\psi_{\pi/4}\right)$ together with its current $\vec{j}=(\widetilde{\psi})^{\dag} \vec{\sigma}\widetilde{\psi}$. The red lines in c) are borders of the areas where the probability current changes direction. (Right column) Probability density of localized state $\tilde{\psi}_\beta=\mathcal{L}\psi_\beta$ together with its current $\vec{j}=(\widetilde{\psi}_{\beta})^{\dag} \vec{\sigma}\widetilde{\psi}_{\beta}$ (right column). We fixed a)-c) $\alpha=0$, d)-f) $\alpha=0.25$, g)-i) $\alpha=0.4$ and j)-l) $\alpha=1.64$. In all figures, $m=1$ and $\beta=1$.
• Figure 2: a) Line-charged tip parallel to the graphene sheet and grounded plate. b) Potential $V_{el}(x,y)$ in (\ref{['potentialel']}) with charge density (\ref{['tau']}) (blue dashed) and $\tilde{V}$ for $\alpha=0$ (red dotted). c) The difference $\frac{|V_{el}-\tilde{V}|}{\hbox{max} \tilde{V}}$. We fixed $z_0=1$, $z_1=3$, $m=0.173$.