Graphene with vacancies: Supernumerary zero modes

The density of states $\varrho \left(E\right)$ of graphene is investigated within the tight-binding (Hückel) approximation in the presence of vacancies. They introduce a nonvanishing density of zero modes $n_{\text{zm}}$ that act as midgap states, $\varrho \left(E\right)=n_{\text{zm}}\delta \left(E\right)+\text{smooth}$. As is well known, the actual number of zero modes per sample can, in principle, exceed the sublattice imbalance, $N_{\text{zm}}\ge |N_{\text{A}}-N_{\text{B}}|$, 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, $n_{\text{zm}}>|c_{\text{A}}-c_{\text{B}}|$, 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.

Figure 1

The simplest example of a balanced lattice animal $N_{\text{A}}=N_{\text{B}}=3$ that carries SZMs. There are no predictable zero modes because $N_{\text{A}}=N_{\text{B}}$. However, two double bonds can be placed and therefore we have two SZMs: $\zeta =6-4=2$.

Figure 2

Edge motifs: (a) Dangling site. (b) Double-dangling site that is associated with a zero-energy state, as can be seen from the fact that one site without double bonds is identified. (c) Double-dangling chain. (d) “$U$ structure.”

Figure 3

Supernumerary zero modes (SZMs) in balanced graphene sheets with a single dangling site and two isolated vacancies. The plot shows that the number of possibilities $N_{\text{pos.}}$, for all placements of the pair of isolated vacancies that yield two SZMs, grows proportional to the sample area (number of lattice sites $N_{\text{sites}}$). Inset: Example with size $10\times 20$ that exhibits two SZMs with double-periodic boundary conditions, which is seen by either exact diagonalization or by bond placing.

Figure 4

Average number of zero modes associated with the percolation cluster obtained in systems of size $L=20,22,...,128$ over its (average) mass $s$. The number of zero modes ${\zeta }_{\text{p}}^{\text{pre}}$ based on sublattice imbalance and based on the combined effects of sublattice imbalance and edge structures are compared ($n_{\text{vac}}=20\%$; average of 1000 realizations for systems up to 10 000 sites and over 100 realizations for systems exceeding that size). In order to highlight the effect of edge motifs (Fig. 3) at large concentrations, we also show a curve ($▵$) where they have been added to the predictable modes.

Figure 5

Average number of supernumerary zero modes associated with the percolation cluster obtained in systems of size $L=100,110,...,140$ over the vacancy concentration $n_{\text{vac}}$ in a double logarithmic scale. The dashed-dotted-dotted line is a guide indicating a conservative upper bound ($\alpha =4$) for the power after finite size extrapolation. A value $\alpha =2$ corresponds to the horizontal line as shown in the figure (dashed-dotted-dashed).

Figure 6

Zero modes on a percolation cluster. The total number of modes ($▿$) and the fraction assigned to only sublattice imbalance ($\circ$) is shown. Data are normalized to the total number of zero modes in the graphene sample (size: $L=100$; average over 500 disorder realizations).

Figure 7

Cluster labeling. Left: An exemplary graphene sample with vacancies (black). Right: Schematics of the Hoshen-Kopelman algorithm. The sites are labeled linewise: A given site is attributed the label of the sites to the left or the line above if one of them is nonvacant or they both have the same label. The marked site points out a conflict where the label to the left and to the top do not match and the clusters with labels 2 and 3 get merged.

Figure 8

The final labeling after the Hoshen-Kopelman algorithm has been performed.

Figure 9

Illustration of the algorithm for the identification of edge motifs using a coloring scheme. First step (red): Identification of corner sites. Second step (blue): Coloring of sites adjacent to corner sites. In case an already blue site is to be colored, this site is colored green. Third step (yellow): Coloring of sites which are adjacent to one vacancy and two blue (or green) sites.