Graphene with vacancies: Supernumerary zero modes

Abstract

The density of states $\ensuremath{\varrho}(E)$ of graphene is investigated within the tight-binding (H\"uckel) approximation in the presence of vacancies. They introduce a nonvanishing density of zero modes ${n}_{\text{zm}}$ that act as midgap states, $\ensuremath{\varrho}(E)={n}_{\text{zm}}\ensuremath{\delta}(E)+\text{smooth}$. As is well known, the actual number of zero modes per sample can, in principle, exceed the sublattice imbalance, ${N}_{\text{zm}}\ensuremath{\ge}|{N}_{\text{A}}\ensuremath{-}{N}_{\text{B}}|$, where ${N}_{\text{A}},{N}_{\text{B}}$ denote the number of carbon atoms in each sublattice. In this paper, we establish a stronger relation that is valid in the thermodynamic limit and that involves the concentration of zero modes, ${n}_{\text{zm}}>|{c}_{\text{A}}\ensuremath{-}{c}_{\text{B}}|$, where ${c}_{\text{A}}$ and ${c}_{\text{B}}$ denote the concentration of vacancies per sublattice; in particular, ${n}_{\text{zm}}$ is nonvanishing even in the case of balanced disorder, ${N}_{\text{A}}/{N}_{\text{B}}=1$. Adopting terminology from benzoid graph theory, the excess modes associated with the current carrying backbone (percolation cluster) are called supernumerary. In the simplest cases, such modes can be associated with structural elements such as carbon atoms connected with a single bond, only. Our result suggests that the continuum limit of bipartite hopping models supports nontrivial ``supernumerary'' terms that escape the present continuum descriptions.