Proximity-induced superconducting transition temperature

We have calculated the proximity-induced superconducting transition temperature for a thin weak-coupling normal metal in contact with an infinitely thick strong-coupling superconductor. The model considered is that of aluminum in perfect contact with superconducting lead, where both metals are clean. The energy-dependent gap is calculated numerically as a function of temperature for the strong-coupling Pb superconductor with the \'Eliashberg equations, and this is used to calculate the induced energy-dependent gap in the weak-coupling Al metal through the modified \'Eliashberg equations for the normal-metal-superconductor configuration. In this latter calculation the presence of the Al metal is assumed to have a negligible effect on the energy gap of the Pb. We find the transition temperature of Al to be enhanced to 4.5 K, compared with its bulk transition temperature of 1.2 K. In addition, we find that the magnitude of the zero-temperature Al gap is enhanced from 0.17 to 0.68 meV.