# Diffraction catastrophes and semiclassical quantum mechanics for Veselago lensing in graphene

## TL;DR

This paper investigates how trigonal warping affects electron focusing in graphene n-p junctions, revealing valley-dependent caustics and polarization effects, supported by semiclassical theory and tight-binding simulations.

## Contribution

It introduces a semiclassical framework to analyze trigonal warping effects on Veselago lensing, predicting valley-specific focusing and transmission features in graphene.

## Key findings

- Trigonal warping causes cusp caustics and valley-dependent focusing.
- Transmission peaks split into two, indicating valley polarization.
- Semiclassical predictions match tight-binding simulation results.

## Abstract

We study the effect of trigonal warping on the focussing of electrons by n-p junctions in graphene. We find that perfect focussing, which was predicted for massless Dirac fermions, is only preserved for one specific sample orientation. In the general case, trigonal warping leads to the formation of cusp caustics, with a different position of the focus for graphene's two valleys. We develop a semiclassical theory to compute these positions and find very good agreement with tight-binding simulations. Considering the transmission as a function of potential strength, we find that trigonal warping splits the single Dirac peak into two distinct peaks, leading to valley polarization. We obtain the transmission curves from tight-binding simulations and find that they are in very good agreement with the results of a billiard model that incorporates trigonal warping. Furthermore, the positions of the transmission maxima and the scaling of the peak width are accurately predicted by our semiclassical theory. Our semiclassical analysis can easily be carried over to other Dirac materials, which generally have different Fermi surface distortions.