Clar sextets in square graphene antidot lattices

A periodic array of holes transforms graphene from a semimetal into a semiconductor with a band gap tuneable by varying the parameters of the lattice. In earlier work only hexagonal lattices have been treated. Using atomistic models we here investigate the size of the band gap of a square lattice, as the parameters of the lattice, that is, the width and height of the unit cell as well as the hole diameter, are varied. It is found that the size of the gap has a very intricate dependence on the width and height of the unit cell, and that even the smallest changes can cause large fluctuations in the gap. These findings are interpreted with the aid of Clar sextet theory and it is found that only when the number of Clar sextets exceeds one third of the total number of hexagons in the unit cell a large band gap is found.