Scaling of the energy gap in pattern-hydrogenated graphene

Recent experiments show that a substantial energy gap in graphene can be induced via patterned hydrogenation on an iridium substrate. Here, we show that the energy gap is roughly proportional to N_H^1/2/N _C when disorder is accounted for, where N_H and N _C denote concentrations of hydrogen and carbon atoms, respectively. The dispersion relation, obtained through calculation of the momentum-energy resolved density of states, is shown to agree with previous angle-resolved photoemission spectroscopy results. Simulations of electronic transport in finite size samples also reveal a similar transport gap, up to 1 eV within experimentally achievable N_H^1/2/N_C values.