Vacancy-induced low-energy states in undoped graphene

Abstract

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 $\epsilon =0$ 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} $|\epsilon |/t$ in this compensated case, we show that the density of states (DOS) $\rho (\epsilon )$ exhibits a strong divergence of the form ${\rho }_{1D}(\epsilon )\sim |\epsilon {|}^{-1}/[\log (t/|\epsilon |){]}^{(y+1)}$, which crosses over to the universal low-energy asymptotic form expected on symmetry grounds ${\rho }_{\mathrm{GW}}(\epsilon )\sim |\epsilon {|}^{-1}{e}^{-b[\log (t/|\epsilon |){]}^{2/3}}$ below a crossover scale ${\epsilon }_c\ll t$. ${\epsilon }_c$ is found to decrease rapidly with decreasing $n_v$, while $y$ decreases much more slowly.

revised in response to referee reports

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