Clar Sextet Analysis of Triangular, Rectangular, and Honeycomb Graphene Antidot Lattices

Pristine graphene is a semimetal and thus does not have a band gap. By making a nanometer. scale periodic array of holes In the graphene sheet a band gap may form; the size of the gap is controllable by adjusting the parameters Of the lattice. The,hole diameter, hole geometry, lattice geometry, and the separation of the holes are parameters that all play an important role in determining the size of the band gap, which, for technological applicatiens, should be at least of the order of tenths of an eV. We investigate four different hole configurations: the rectangular, the triangular, the rotated triangular, and the honeycomb lattice. It is found that the lattice geometry plays a crucial role for size of the band gap the triangular arrangement displays always a shable gap, while for the other types only particular hole separations lead to a large gap. This observation is explained using, Clear sextet theory, and we find that a sufficient condition for a large gap is that the number of sextets exceeds one-third of the total number of hexagons in the unit cell. Furthermore we investigate, nonisosceles triangular structures to probe the sensitivity of the gap in triangular lattices to small changes in geometry.