Diffusion and Criticality in Undoped Graphene with Resonant Scatterers

A general theory is developed to describe graphene with an arbitrary number of isolated impurities. The theory provides a basis for an efficient numerical analysis of the charge transport and is applied to calculate the Dirac-point conductivity $\sigma$ of graphene with resonant scatterers. In the case of smooth resonant impurities the symmetry class is identified as DIII and $\sigma$ grows logarithmically with increasing impurity concentration. For vacancies (or strong on-site potential impurities, class BDI) $\sigma$ saturates at a constant value that depends on the vacancy distribution among two sublattices.

Figure 1

Mean conductivity calculated numerically from Eqs. (7, 9), as a function of concentration of resonance scalar impurities (data points). The solid line shows the result of the sigma model in class DIII.

Figure 2

(a) Mean conductivity vs vacancy concentration, $n=n_A+n_B$, found numerically from Eqs. (8, 9), for different values of $\delta =(n_A-n_B)/n$ and a single color of vacancies; solid diamonds: $\delta =0$ and uniform coloring. (b) Conductivity as a universal scaling function of $(L/\xi {)}^2$. Inset: power-law scaling of the length $\xi \sim n^{-1/2}{\delta }^{-0.36}$.