  We have used density functional theory to study the nonlinear screening
properties of a two-dimensional (2D) electron gas. In particular, we consider
the screening of an external static point charge of magnitude Z as a function
of the distance of the charge from the plane of the gas. The self-consistent
screening potentials are then used to determine the 2D stopping power in the
low velocity limit based on the momentum transfer cross-section. Calculations
as a function of Z establish the limits of validity of linear and quadratic
response theory calculations, and show that nonlinear screening theory already
provides significant corrections in the case of protons. In contrast to the 3D
situation, we find that the nonlinearly screened potential supports a bound
state even in the high density limit. This behaviour is elucidated with the
derivation of a high density screening theorem which proves that the screening
charge can be calculated perturbatively in the high density limit for arbitrary
dimensions. However, the theorem has particularly interesting implications in
2D where, contrary to expectations, we find that perturbation theory remains
valid even when the perturbing potential supports bound states.
