Electronic optics in graphene in the semiclassical approximation

TL;DR

This paper develops a semiclassical approximation for Dirac electrons in graphene under electrostatic potential and mass variations, analyzing focusing effects, caustics, and phase influences with good agreement to numerical tight-binding results.

Contribution

It introduces a comprehensive semiclassical framework for Dirac electrons in graphene, including phase effects and caustic analysis, extending prior scalar wave approaches.

Findings

Semiclassical approximation accurately predicts focusing and caustics.

The semiclassical phase significantly affects focus position and intensity.

Results align well with tight-binding numerical simulations.

Abstract

We study above-barrier scattering of Dirac electrons by a smooth electrostatic potential combined with a coordinate-dependent mass in graphene. We assume that the potential and mass are sufficiently smooth, so that we can define a small dimensionless semiclassical parameter $h \ll 1$. This electronic optics setup naturally leads to focusing and the formation of caustics, which are singularities in the density of trajectories. We construct a semiclassical approximation for the wavefunction in all points, placing particular emphasis on the region near the caustic, where the maximum of the intensity lies. Because of the matrix character of the Dirac equation, this wavefunction contains a nontrivial semiclassical phase, which is absent for a scalar wave equation and which influences the focusing. We carefully discuss the three steps in our semiclassical approach: the adiabatic reduction of the matrix equation to an effective scalar equation, the construction of the wavefunction using the Maslov canonical operator and the application of the uniform approximation to the integral expression for the wavefunction in the vicinity of a caustic. We consider several numerical examples and show that our semiclassical results are in very good agreement with the results of tight-binding calculations. In particular, we show that the semiclassical phase can have a pronounced effect on the position of the focus and its intensity.