Vacancy-induced low-energy states in undoped graphene

Abstract

We demonstrate that a nonzero concentration nv of static, randomly placed vacancies in graphene leads to a density w of zero-energy quasiparticle states at the band center ε=0 within a tight-binding description with nearest-neighbor hopping t on the honeycomb lattice. We show that wremains generically nonzero in the compensated case (exactly equal number of vacancies on the two sublattices) even in the presence of hopping disorder and depends sensitively on nv and correlations between vacancy positions. For low, but not-too-low, |ε|/t in this compensated case, we show that the density of states ρ(ε) exhibits a strong divergence of the form ρ_(Dyson)(ε)∼|ε|^(-1)/[log(t/|ε|)]^((y+1)), which crosses over to the universal low-energy asymptotic form (modified Gade-Wegner scaling) expected on symmetry grounds ρ_(GW)(ε)∼|ε|^(-1)e^(-b[log(t/|ε|)]2/3) below a crossover scale ε_c≪t. ε_c is found to decrease rapidly with decreasing nv, while y decreases much more slowly.