Disorder Induced Localized States in Graphene

We consider the electronic structure near vacancies in the half-filled honeycomb lattice. It is shown that vacancies induce the formation of localized states. When particle-hole symmetry is broken, localized states become resonances close to the Fermi level. We also study the problem of a finite density of vacancies, obtaining the electronic density of states, and discussing the issue of electronic localization in these systems. Our results also have relevance for the problem of disorder in $d$-wave superconductors.

Figure 1

Sketch of the geometry considered in the text for the study of a single $B$-site vacancy ($\otimes$). The system has periodic boundary conditions along the vertical axis, and is infinite along the horizontal axis. Only some $A$-site amplitudes are shown.

Figure 2

(a) Comparison between the local DOS in the vicinity of a vacancy (blue/solid) with the bulk DOS (red/dashed) in clean systems. (b) Total DOS in the vicinity of the Dirac points for clusters with $4\times {10}^6$ sites, at selected vacancy concentrations. Numerical results in (a), (b) obtained for $t^{\prime }=0$ (top panels), $t^{\prime }=0.1t$ (center) and $t^{\prime }=0.2t$ (bottom). [Notice the scale truncation in the upper part of the first panel in (a)].

Figure 3

Inverse participation ratio for a concentration of vacancies of 0.5% in systems with ${10}^4$ sites. The dashed curve is the total DOS for the same dilution. Top: $t^{\prime }=0$; center: $t^{\prime }=0.1t$; bottom: $t^{\prime }=0.2t$. Only the low-energy region is shown.