Gapless insulator and a band gap scaling law in semihydrogenated graphene

We demonstrate two unusual electronic properties of semihydrogenated graphene with variable sized A- or B-hydrogenated domains within the tight-binding formalism as follows: (i) a universal band gap scaling law which states that the band gap depends linearly upon the ratio of the number of A- to B-hydrogenated atoms, NA/NB, reaching zero gap at NA = NB, but independent of the domain size, and (ii) an insulating state with zero band gap at NA = NB, a rare phenomenon in nature. We confirm this gapless insulator state by the zero optical conductance at low frequencies.