Calculating Tc for noble-metal thin films away from the critical thickness Lc

Determine the superconducting critical temperature Tc predicted by the generalized Eliashberg equations for gold (Au), silver (Ag), and copper (Cu) thin films at thicknesses deviating from the critical thickness Lc, where Tc becomes very small and the authors’ current numerical solver fails to converge within reasonable computational time.

Background

The paper introduces a generalized Eliashberg framework that incorporates quantum confinement effects through a thickness-dependent density of states and electron-phonon coupling to predict superconductivity in ultra-thin films of Au, Ag, and Cu. Numerical solutions of the modified Eliashberg equations indicate that Tc exhibits a sharp peak near a critical thickness Lc (~0.5 nm), consistent with a confinement-driven topological transition of the Fermi surface.

While Tc values were successfully computed at thicknesses close to Lc (yielding finite Tc for Au, Ag, and Cu), the authors report that moving away from Lc leads to very small Tc values whose computation becomes infeasible due to numerical convergence limitations. This leaves unresolved the quantitative characterization of Tc in the thin-film regime away from the critical thickness.