Superconducting Geometric Potential and Curvature-Enhanced Superconductivity in Curved Thin Films

We derive the linearized Ginzburg-Landau (GL) equation for a curved ultra-thin superconducting film in the presence of a magnetic field. By introducing a novel transverse order parameter that varies slowly along the film with the superconducting/vacuum boundary condition, we decouple the linearized GL equation into a transverse component and a surface component in the thin-layer quantization scheme. A superconducting geometric potential (GP) is present in the surface equation, which can substantially affect the nucleation of the superconducting state in the curved ultra-thin superconducting film. From the perspective of the GL free energy, the superconducting GP reduces the coefficient of the quadratic term of the order parameter, which enables the curved film to stay in the superconducting state even when the superconducting parameter $\alpha$ becomes positive. Based on our equivalent equation, for a superconducting thin film with uniform curvature, the relative increase of the critical temperature is proportional to the magnitude of the superconducting GP. As an example, we numerically investigate the phase transition of a rectangular superconducting film bent around a cylindrical surface, with the numerical results in good agreement with the theoretical expectations. We further propose a strain-free experimental validation using ultracold atomic condensates, where nested superfluid shells enforce Neumann boundary conditions to isolate the superconducting GP.