Dirac boundary condition at the reconstructed zigzag edge of graphene

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 $\ensuremath{\vartheta}$. Dispersive edge states exist generically, unless $|\ensuremath{\vartheta}|=\ensuremath{\pi}/2$. We compute $\ensuremath{\vartheta}$ 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.