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We discuss the evolution from BCS to Bose superconductivity versus carrierdensity in gated two-dimensional s-wave superconductors. We show that thetransition from the normal to the superconducting state is controlled by theBerezinskii-Kosterlitz-Thouless vortex-antivortex pairing mechanism, and thatthe 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 torecent experiments on the superconductor LixZrNCl, a lithium intercalatedlayered nitride, and obtain very good quantitative agreement at low andintermediate densities.

Density induced BCS-Bose evolution in gated two-dimensional  superconductors: The Berezinskii-Kosterlitz-Thouless transition as a function  of carrier density