# Minimal Geometry for Valley Filtering in Graphene

## TL;DR

This paper demonstrates that breaking mirror symmetry in minimal impurity configurations in graphene can induce valley splitting and the valley Hall effect, providing a simple geometric approach to valley filtering.

## Contribution

It introduces a minimal model showing how mirror symmetry breaking in impurity arrangements causes valley splitting and the valley Hall effect in graphene.

## Key findings

- Mirror symmetry breaking leads to valley splitting.
- Differential cross sections show valley-dependent scattering.
- The skew parameter indicates non-zero valley Hall effect.

## Abstract

The possibility to effect valley splitting of an electronic current in graphene represents the essential component in the new field of valleytronics in such two-dimensional materials. Based on a symmetry analysis of the scattering matrix, we show that if the spatial distribution of multiple potential scatterers breaks mirror symmetry about the axis of incoming electrons, then a splitting of the current between two valleys is observed. This leads to the appearance of the valley Hall effect. We illustrate the effect of mirror symmetry breaking in a minimal system of two symmetric impurities, demonstrating the splitting between valleys via the differential cross sections and non-vanishing skew parameter. We further discuss the role that these effects may play in transport experiments.