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.