Topological property of graphene with triangular array of nanoholes

The nontrivial band topology for graphene with regular arrays of nanoholes with C_6v symmetry is investigated theoretically. For the case of the $3\sqrt{3}\times 3\sqrt{3}$ triangular array of nanoholes, we find an energy gap at the Γ point around the Fermi level associated with a band inversion which induces a change in parity indices, whereas deep below the Fermi level there is a bunch of valence bands (VBs) characterized as an obstructed atomic limit (OAL) which also accommodates an imbalance in parity indices. This band structure renders the gap at the Fermi level topologically trivial and carrying no edge states, while the nontrivial band topology of the OAL manifests in two flat bands in the ribbon structure associated with localized electronic states at the ribbon edges. The present results exhibit rich topological behaviors in graphene derivatives waiting for exploration.