Oscillations of the superconducting temperature induced by quantum well states in thin metallic films: Numerical solution of the Bogoliubov–de Gennes equations

Quantum confinement of electrons in a high-quality metallic nanofilm results in the formation of quantum-well states. The band of single-electron states is split into a series of subbands moving in energy with changing film thickness. When the bottom of such a subband passes through the Fermi surface, a shape resonance appears leading to oscillations of the critical temperature Tc as a function of film thickness. We present a quantitative description of recent experimental data on the film-thickness dependence of Tc in Pb nanofilms with atomic-scale uniformity in thickness [Science 306, 1915 (2004)] through a numerical solution of the Bogoliubov--de Gennes equations.