Superconductive critical temperature of Pb/Ag heterostructures

Recent experimental data (H. Nam et al., Phys. Rev. B 100, 094512 (2019)) of critical temperature and gaps measured on superconductor/normal metal heterostructure Pb/Ag, epitaxially grown, shown a interesting not usual behaviour. The critical temperature decreases strongly but, despite the large differences in the lattice constants and electronic densities of states in the separate components, this heterostructure shows a spatially constant superconducting gap. In the paper it is demonstrated that the proximity Eliashberg equations, whit no free parameters, cannot explain the dependence of critical temperature from the rate $d_{Pb}/d_{Ag}$. However it is sufficient to assume that the density of states at the Fermi level of silver is equal to that of lead in a layer adjacent to the separation interface, presumably of a thickness less than the coherence length of the lead, to perfectly explain the decrease in the critical temperature and the gap value in function of the rate $d_{Pb}/d_{Ag}$ always without free parameters.