Dirac boundary condition at the reconstructed zigzag edge of graphene

TL;DR

This paper develops a low-energy theory for reconstructed zigzag edges in graphene, deriving boundary conditions for the Dirac equation, and predicts measurable effects on edge states and Landau levels.

Contribution

It introduces a boundary condition parameterized by an angular variable to describe reconstructed zigzag edges in graphene, linking microscopic models to observable electronic properties.

Findings

Dispersive edge states depend on the boundary parameter $ heta$.

The boundary parameter $ heta$ can be extracted from local density of states measurements.

In a magnetic field, three distinct edge modes appear in the lowest Landau level.

Abstract

Edge reconstruction modifies the electronic properties of finite graphene samples. We formulate a low-energy theory of the reconstructed zigzag edge by deriving the modified boundary condition to the Dirac equation. If the unit cell size of the reconstructed edge is not a multiple of three with respect to the zigzag unit cell, valleys remain uncoupled and the edge reconstruction is accounted for by a single angular parameter $\vartheta$. Dispersive edge states exist generically, unless $|\vartheta| = \pi/2$. We compute $\vartheta$ from a microscopic model for the "reczag" reconstruction (conversion of two hexagons into a pentagon-heptagon pair) and show that it can be measured via the local density of states. In a magnetic field there appear three distinct edge modes in the lowest Landau level, two of which are counterpropagating.