Neutrality Point of Graphene with Coplanar Charged Impurities

The ground state and the transport properties of graphene subject to the potential of in-plane charged impurities are studied. The screening of the impurity potential is shown to be nonlinear, producing a fractal structure of electron and hole puddles. Statistical properties of this density distribution as well as the charge compressibility of the system are calculated in the leading-log approximation. The conductivity depends logarithmically on alpha, the dimensionless strength of the Coulomb interaction. The theory is asymptotically exact when alpha is small, which is the case for graphene on a substrate with a high dielectric constant.