  A variety of two-dimensional materials possess a band structure with an
energy extremal ridge along a ring in momentum space. Examples are biased
bilayer graphene, and surfaces and interfaces with a Rashba spin-orbit
interaction where at low doping the carriers fill an annulus. This topological
feature causes an anomalous screening behavior, which we study using the
Thomas-Fermi theory. Specifically, reducing the doping is predicted to enhance
the linear screening response, while at zero doping the size of the screening
cloud surrounding a Coulomb impurity is found to increase as the cube root of
the impurity charge.
