We calculate the critical temperature $T_c$ and the superconducting energy gaps ${\Delta }_n$ of a thin-film superconductor system, where ${\Delta }_n$ is the superconducting energy gap of the $n\mathrm{th}$ subband. Since the quantization of both the electron energy and phonon spectrum arises due to dimensional confinement in one direction, the effective electron-electron interaction mediated by the quantized confined phonons is different from that mediated by the bulk phonon, leading to the modification of $T_c$ in the thin-film system. We investigate the dependence of $T_c$ and ${\Delta }_n$ on the film thickness d with this modified interaction.

• Received 27 August 1999

DOI:https://doi.org/10.1103/PhysRevB.61.8659