Quantitative Eliashberg theory of the superconductivity of thin films.

A quantitative theory of the superconductivity of materials confined at the nanoscale in parameter-free agreement with experimental data has been missing so far. We present a generalization, in the Eliashberg framework, of a BCS theory of superconductivity in good metals which are confined along one of the three spatial directions, such as thin films. In this formulation of the Eliashberg equations the approximation of taking the normal density of states (DOS) as its value at the Fermi level has been removed. By numerically solving these new Eliashberg-type equations, we find the dependence of the superconducting critical temperature $T_{c}$ on the confinement size $L$, in quantitative agreement with experimental data of Pb and Al thin films with no adjustable parameters. This quantitative agreement provides an indirect confirmation that, upon increasing the confinement, a crossover from a spherical-like Fermi surface, which contains two growing hole pockets caused by the confinement, to a strongly deformed Fermi surface, occurs. This topology of the Fermi sea is implemented in the new Eliashberg-type equations to reproduce the experimentally observed maximum in the critical superconducting temperature vs film thickness of ultra-thin Pb films.