Origin of anomalous mass renormalizations inferred from proximity tunneling experiments

Experiments with thin Ag layers ($d_N\sim 100-700$ Å) backed by thicker Pb ($d_S\sim 3500$ Å) have led Khim and Tomasch to infer large effective renormalizations for Ag that increase with decreasing $d_N$, even though the electron-phonon interaction presumably retains its bulk character. Arnold has recently shown that dirty proximity superconductors violate Anderson's theorem and that, in this regime, the usual treatment of elastic scattering is inappropriate for Andreev interference phenomena. He finds that both the pairing potential, ${\Delta }_N\mathrm{}\left(E\right)\mathrm{}$, and the effective layer renormalization, $Z_N\mathrm{}\left(E\right)\mathrm{}$, are influenced strongly by scattering. Our reanalysis, based on Arnold's theory and on Gallagher's compound-resonance theory, assumes bulk behavior for the Ag electron-phonon interaction. We obtain successful quantitative agreement with the Ag-Pb data when the Ag mean free path nearly equals the Ag film thickness. Our best estimates of the dressed Fermi velocities are $v_{\mathrm{FS}}^{*}\left(\mathrm{Pb}\right)=(0.72\mathrm{}\pm 0.06)\times {10}^6$ m/s and $v_{\mathrm{FN}}^{*}\left(\mathrm{Ag}\right)=(1.07\mathrm{}\pm 0.06)\times {10}^6$ m/s.