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Vertical absorption edge and universal onset conductance in semi-hydrogenated graphene

We show that for graphene with any finite difference in the on-site energy between the two sub-lattices ($Δ$), The optical absorption edge is determined by the $Δ$. The universal conductance will be broken and the conductance near the band edge varies with frequency as $1/ω^2$. Moreover, we have identified another universal conductance for such systems without inversion symmetry, i.e., the onset conductance at the band edge is $% σ_c=2σ_0=πe^2/2h$, independent of the size of the band gap. The total integrated optical response is nearly conserved despite of the opening of the band gap.