# Symmetry breaking and (pseudo)spin polarization in Veselago lenses for massless Dirac fermions

## TL;DR

This paper investigates how polarization in Veselago lenses for massless Dirac fermions causes symmetry breaking and spin polarization effects, with analytical and semiclassical methods revealing displacement and current differences.

## Contribution

It introduces a semiclassical approach to analyze symmetry breaking and spin polarization effects in Veselago lensing for Dirac fermions, providing analytical formulas and identifying scattering regimes.

## Key findings

- Polarization causes spatial symmetry breaking in Veselago lensing.
- Analytical formula for vertical displacement of the main focus.
- Identification of two scattering regimes in current injection.

## Abstract

We study Veselago lensing of massless Dirac fermions by n-p junctions for electron sources with a certain polarization. This polarization corresponds to pseudospin for graphene and to real spin for topological insulators. Both for a point source and for injection into a sample through a narrow lead, we find that polarization leads to spatial symmetry breaking. For the Green's function, this results in a vertical displacement, or even complete vanishing of the main focus, depending on the exact polarization. For injection through a lead, it leads to a difference between the amounts of current emitted with positive and negative transversal momenta. We study both systems in detail using the semiclassical approximation. By comparing the results to the exact solutions, we establish that semiclassical methods provide a very effective way to study these systems. For the Green's function, we derive an easy-to-use analytical formula for the vertical displacement of the main focus. For current injection through a lead, we use semiclassical methods to identify two different scattering regimes.