Role of confined phonons in thin-film superconductivity

We calculate the critical temperature T-c and the superconducting energy gaps Delta(n) of a thin-film superconductor system, where a,, is the superconducting energy gap of the nth 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-him system. We investigate the dependence of T-c and Delta(n) on the film thickness d with this modified interaction.