Counterpropagating Topological Interface States in Nano Punctured Graphene

It has been known that superstructures such as a regular array of holes (antidot lattice) on a graphene sheet can induce a gapped state. We analyze the band structure of graphene with a hole array in more detail, and reveal that different arrangements of holes, triangular or honeycomb, can lead to topologically distinct states. The topological nontriviality is shown by explicitly calculating interface states between two regions with triangular and honeycomb hole arrangements, where the interface state appears as cross-shape bands mimicking helical edge states in a quantum spin Hall state. The topological character is also discussed in terms of the parity of the wave function at the high symmetry point in the Brillouin zone, in relation to the state-of-art argument on topological crystalline insulators.