Statistics of random voltage fluctuations and the low-density residual conductivity of graphene

We consider a graphene sheet in the vicinity of a substrate, which contains charged impurities. A general analytic theory to describe the statistical properties of voltage fluctuations due to the long-range disorder is developed. In particular, we derive a general expression for the probability distribution function of voltage fluctuations, which is shown to be non-Gaussian. The voltage fluctuations lead to the appearance of randomly distributed density inhomogeneities in the graphene plane. We argue that these disorder-induced density fluctuations produce a finite conductivity even at a zero gate voltage in accordance with recent experimental observations. We determine the width of the minimal conductivity plateau and the typical size of the electron and hole puddles. We also propose a simple self-consistent approach to estimate the residual density and the nonuniversal minimal conductivity in the low-density regime. The existence of inhomogeneous random puddles of electrons and holes should be a generic feature of all graphene layers at low gate voltages due to the invariable presence of charged impurities in the substrate.

Figure 1

(Color online) This figure shows the representative probability distribution functions of random voltages arising from the screened potential [Eq. (13)] of randomly distributed charges in an electrically neutral substrate for three different densities (other parameters are fixed).

Figure 2

(Color online) The correlation function $C\left(r\right)$ [see Eq. (10) and text for the definition] for the charge impurity model with linear Thomas-Fermi screening. The solid line shows numerical integration for $q_{s}d=0.5$, while the dashed lines correspond to the asymptotic analytical expressions for small and large $q_{s}r$. The chosen value of $q_{s}d=0.5$ is determined by the density of carriers, where the experimentally observed behavior deviates from the predictions of the classical Drude-Boltzmann theory.

Figure 3

(a) The pictorial representation of the disorder-induced density fluctuations $⟨\delta n\left(\mathbf{r}\right)\delta n\left({\mathbf{r}}^{\prime }\right)⟩$, where the dashed line represents the Coulomb impurity potential and the shaded vertex is a ladder of these impurity lines. (b) Shown is a diagram, which contributes to the correlator $⟨\delta \Pi ({\mathbf{r}}_1,{\mathbf{r}}_2)\delta \Pi ({\mathbf{r}}_3,{\mathbf{r}}_4)⟩$ of the polarization operators, which determine the screening properties and the effective interaction (see text).

Figure 4

The first few diagrams in the perturbative expansion of the conductivity. The dashed lines correspond to the scattering off of Coulomb disorder potential.