Discrete honeycombs, rational edges and edge states

TL;DR

This paper classifies edge types in graphene's tight binding model, proving flat band edge states occur only for zigzag edges and providing formulas for these states, while also suggesting dispersive edge states are common.

Contribution

It generalizes classical edge classifications in graphene, proving the existence of flat band edge states for zigzag edges and providing explicit formulas, along with evidence for dispersive edge states.

Findings

Flat band edge states occur only for zigzag edges.

Explicit formulas for flat band edge states are provided.

Dispersive edge states are likely present for most edge orientations.

Abstract

Consider the tight binding model of graphene, sharply terminated along an edge ${\bf l}$ parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges ${\bf l}$ into those of "zigzag type" and those of "armchair type", generalizing the classical zigzag and armchair edges. We prove that zero energy/flat band edge states arise for edges of zigzag type, but never for those of armchair type. We exhibit explicit formulas for flat band edge states when they exist. We produce strong evidence for the existence of dispersive (non flat) edge state curves of nonzero energy for most ${\bf l}$.