Superconducting size effect in thin films under electric field: Mean-field self-consistent model

We consider the effects of an externally applied electrostatic field on superconductivity, self-consistently within a BCS mean-field model, for a clean three-dimensional (3D) metal thin film. The electrostatic change in superconducting condensation energy scales as Δ/μ close to subband edges as a function of the Fermi energy μ and follows 3D scaling (Δ/μ)² away from them. We discuss nonlinearities beyond the gate effect and contrast results with recent experiments on gating effects on Josephson junctions.