  We discuss the evolution from BCS to Bose superconductivity versus carrier
density in gated two-dimensional s-wave superconductors. We show that the
transition from the normal to the superconducting state is controlled by the
Berezinskii-Kosterlitz-Thouless vortex-antivortex pairing mechanism, and that
the evolution from high carrier density (BCS pairing) to low carrier density
(Bose pairing) is just a crossover in s-wave systems. We compare our results to
recent experiments on the superconductor LixZrNCl, a lithium intercalated
layered nitride, and obtain very good quantitative agreement at low and
intermediate densities.
