Boundary conditions for Dirac fermions on a terminated honeycomb lattice

TL;DR

This paper derives boundary conditions for Dirac fermions on honeycomb lattices, showing how edge types affect electronic states and gap opening, with implications for graphene nanoribbons.

Contribution

It provides a general derivation of boundary conditions for Dirac equations on honeycomb lattices with arbitrary edges, extending understanding of edge states and gap control.

Findings

Zigzag boundaries are generic for non-parallel edges.

Lattice termination does not always produce an insulating nanoribbon.

Edge states include propagating modes beyond localized states.

Abstract

We derive the boundary condition for the Dirac equation corresponding to a tight-binding model on a two-dimensional honeycomb lattice terminated along an arbitary direction. Zigzag boundary conditions result generically once the boundary is not parallel to the bonds. Since a honeycomb strip with zigzag edges is gapless, this implies that confinement by lattice termination does not in general produce an insulating nanoribbon. We consider the opening of a gap in a graphene nanoribbon by a staggered potential at the edge and derive the corresponding boundary condition for the Dirac equation. We analyze the edge states in a nanoribbon for arbitrary boundary conditions and identify a class of propagating edge states that complement the known localized edge states at a zigzag boundary.