Role of confined phonons in thin film superconductivity

We calculate the critical temperature $T_c$ and the superconducting energy gaps ${\mathrm{\Delta }}_n$ of a thin film superconductor system, where ${\mathrm{\Delta }}_n$ is the superconducting energy gap of the $n$-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 ${\mathrm{\Delta }}_n$ on the film thickness $d$ with this modified interaction.