Edge states and flat bands in graphene nanoribbons with arbitrary geometries

TL;DR

This paper introduces a universal, calculation-free method to predict edge states and flat bands in graphene nanoribbons of any shape, based on geometric rules and hybridization principles.

Contribution

It provides a novel, general prescription for predicting edge states in arbitrary graphene edges without complex calculations.

Findings

Accurately predicts localization and degeneracy of zero-energy bands.

Validates predictions with tight-binding and first-principle calculations.

Qualitatively predicts non-zero energy bands in certain edges.

Abstract

We prescribe general rules to predict the existence of edge states and zero-energy flat bands in graphene nanoribbons and graphene edges of arbitrary shape. No calculations are needed. For the so-called {\it{minimal}} edges, the projection of the edge translation vector into the zigzag direction of graphene uniquely determines the edge bands. By adding extra nodes to minimal edges, arbitrary modified edges can be obtained. The edge bands of modified graphene edges can be found by applying hybridization rules of the extra atoms with the ones belonging to the original edge. Our prescription correctly predicts the localization and degeneracy of the zero-energy bands at one of the graphene sublattices, confirmed by tight-binding and first-principle calculations. It also allows us to qualitatively predict the existence of $E\ne 0$ bands appearing in the energy gap of certain edges and nanoribbons.