  The screening problem for Coulomb potential of a charge located in a
two-dimensional system has an intriguing solution with power-law screening
factor decreasing with distance, which is crucially different from a 3D case
with common exponential screening law. This stems from the outplane electrical
fields leading to the long-range action on charge distribution. An effective
screening length controlling the power law of potential drop far from the
charge depends on material parameters of a 2D semiconductor, although it does
not prevent from electron flooding of low-doped regions of a heterostructure.
In Graphene and other materials with linear energy spectrum for electrons this
flooding effects also occur, but the effective screening length is
doping-level-dependent now. We built the theory for electronic potential and
electrons concentration in a 1D-periodically doped Graphene sheet taking into
account all effects of a long-range 2D screening and linear spectrum influence.
We show that due to linear spectrum such problem results in a substantially
nonlinear integro-differential equation, which can be solved numerically.
Similar nonlinear problems arise in a wide range of doped structures using 2D
linear spectrum materials.
