Klein paradox between transmitted and reflected Dirac waves on Bour surfaces

TL;DR

This paper investigates how the geometry of Bour surfaces influences Dirac fermion behavior in curved graphene, revealing geometry-induced potentials, transmission phenomena, and Klein paradox effects through scattering analysis.

Contribution

It introduces a model of Dirac fermions on Bour surfaces, showing how surface geometry induces potential barriers and affects electronic transmission and Klein paradox phenomena.

Findings

Total transmission at high energies for B0 and B1 surfaces.

Existence of a specific energy point E_K where Klein's paradox occurs.

Conductance suppression at energies much higher than E_K.

Abstract

It is supposed the existence of a curved graphene sheet with the geometry of a Bour surface $B_{n}$, such as the catenoid (or helicoid), $B_{0}$, and the classical Enneper surface, $B_{2}$, among others. In particular, in this work, the propagation of the electronic degrees of freedom on these surfaces is studied based on the Dirac equation. As a consequence of the polar geometry of $B_{n}$, it is found that the geometry of the surface causes the Dirac fermions to move as if they would be subjected to an external potential coupled to a spin-orbit term. The geometry-induced potential is interpreted as a barrier potential, which is asymptotically zero. Furthermore, the behaviour of asymptotic Dirac states and scattering states are studied through the Lippmann-Schwinger formalism. It is found that for surfaces $B_{0}$ and $B_{1}$, the total transmission phenomenon is found for sufficiently large values of energy, while for surfaces $B_{n}$, with $n\geq 2$, it is shown that there is an energy point $E_{K}$ where Klein's paradox is realized, while for energy values $E\gg E_{K}$ it is found that the conductance of the hypothetical material is completely suppressed, $\mathcal{G}(E)\to 0$.