Nonlinear screening and charge redistribution in periodically doped graphene

The screening problem for the Coulomb potential of a charge located in a two-dimensional (2D) system has an intriguing solution with a power law distance screening factor due to out-of-plane electrical fields. This is crucially different from a three-dimensional case with exponential screening. The long-range action of electric fields results in the effective inflow of electrons from high-doped regions to low-doped regions of a 2D heterostructure. In graphene and other materials with a linear energy spectrum for electrons, such an inflow in low-doped regions also occurs, but its effectiveness is dependent on the doping level. It can be used to fabricate high-conducting channels. We propose a theory for determining electron potential and concentration in a graphene sheet periodically doped along one dimension, taking into account all effects of long-range 2D screening. This results in a substantially nonlinear integro-differential problem, which is solved numerically via a computationally cheap algorithm. Similar nonlinear problems arise in a wide range of doped 2D heterostructures made of linear spectrum materials.