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Vacancy-induced low-energy states in undoped graphene

We demonstrate that a nonzero concentration $n_v$ 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-neighbour hopping $t$ on the honeycomb lattice. We show that $w$ remains 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 $n_v$ and correlations between vacancy positions. For low, {\em but not-too-low} $|ε|/t$ in this compensated case, we show that the density of states (DOS) $ρ(ε)$ exhibits a strong divergence of the form $ρ_{\rm 1D}(ε) \sim |ε|^{-1}/ [\log(t/|ε|)]^{(y+1)} $, which crosses over to the universal low-energy asymptotic form expected on symmetry grounds $ρ_{\rm GW}(ε) \sim |ε|^{-1}e^{-b[\log(t/|ε|)]^{2/3} }$ below a crossover scale $ε_c \ll t$. $ε_c$ is found to decrease rapidly with decreasing $n_v$, while $y$ decreases much more slowly.