Nature of single-particle states in disordered graphene

We analyze the nature of the single particle states, away from the Dirac point, in the presence of long-range charge impurities in a tight-binding model for electrons on a two-dimensional honeycomb lattice which is of direct relevance for graphene. For a disorder potential $V(\vec{r})=V_0\exp(-|\vec{r}-\vec{r}_{imp}|^2/\xi^2)$, we demonstrate that not only the Dirac state but all the single particle states remain extended for weak enough disorder. Based on our numerical calculations of inverse participation ratio, dc conductivity, diffusion coefficient and the localization length from time evolution dynamics of the wave packet, we show that the threshold $V_{th}$ required to localize a single particle state of energy $E(\vec{k})$ is minimum for the states near the band edge and is maximum for states near the band center, implying a mobility edge starting from the band edge for weak disorder and moving towards the band center as the disorder strength increases. This can be explained in terms of the low energy Hamiltonian at any point $\vec{k}$ which has the same nature as that at the Dirac point. From the nature of the eigenfunctions it follows that a weak long range impurity will cause weak anti localization effects, which can be suppressed, giving localization if the strength of impurities is sufficiently large to cause inter-valley scattering. The inter valley spacing $2|\vec{k}|$ increases as one moves in from the band edge towards the band center, which is reflected in the behavior of $V_{th}$ and the mobility edge.