Modeling Klein tunneling and caustics of electron waves in graphene

TL;DR

This paper uses a numerical method to study electron wave propagation, Klein tunneling, and caustics in graphene, revealing how potential barrier sharpness and energy regimes affect tunneling and wave interference.

Contribution

It introduces an efficient tight-binding propagation method to model Klein tunneling and interference in large graphene systems, including effects beyond the Dirac regime.

Findings

Klein tunneling occurs mainly with smooth potentials larger than atomic scale.

Sharp n-p junctions lose Klein tunneling due to intervalley scattering outside the Dirac regime.

Caustic patterns match classical trajectories for broad Gaussian potentials.

Abstract

We employ the tight-binding propagation method to study Klein tunneling and quantum interference in large graphene systems. With this efficient numerical scheme, we model the propagation of a wave packet through a potential barrier and determine the tunneling probability for different incidence angles. We consider both sharp and smooth potential barriers in n-p-n and n-n' junctions and find good agreement with analytical and semiclassical predictions. When we go outside the Dirac regime, we observe that sharp n-p junctions no longer show Klein tunneling because of intervalley scattering. However, this effect can be suppressed by considering a smooth potential. Klein tunneling holds for potentials changing on the scale much larger than the interatomic distance. When the energies of both the electrons and holes are above the Van Hove singularity, we observe total reflection for both sharp and smooth potential barriers. Furthermore, we consider caustic formation by a two-dimensional Gaussian potential. For sufficiently broad potentials we find a good agreement between the simulated wave density and the classical electron trajectories.