To date, several experimental groups reported measurements of the thickness dependence of $T_c$ of atomically uniform single-crystalline $\mathrm{Pb}$ nanofilms. The reported amplitude of the $T_c$ oscillations varies significantly from one experiment to another. Here we propose that the reason for this unresolved issue is an interplay of the quantum-size variations in the single-electron density of states with thickness-dependent oscillations in the phonon-mediated electron-electron coupling. Such oscillations in the coupling depend on the substrate material, the quality of the interface, the protection cover, and other details of the fabrication process, changing from one experiment to another. This explains why the available data do not exhibit one-voice consistency about the amplitude of the $T_c$ oscillations. Our analyses are based on a numerical solution of the Bogoliubov-de Gennes equations for a superconducting slab.

• Received 5 December 2011

DOI:https://doi.org/10.1103/PhysRevB.85.224517