Effective Theory of Superconductivity in Strongly-Coupled Amorphous Materials

A theory of phonon-mediated superconductivity in strong-coupling amorphous materials is developed based on an effective description of structural disorder and its effect on the vibrational spectrum. The theory accounts for the diffusivelike transport of vibrational excitations due to disorder-induced scattering within the Eliashberg theory of strong-coupling superconductivity. The theory provides a good analytical description of the Eliashberg function α2F(ω) in comparison with experiments and allows one to disentangle the effects of transverse and longitudinal excitations on the Eliashberg function. In particular, it shows that the transverse excitations play a crucial role in driving an increase or excess in the Eliashberg function at low energy, which is related to the boson peak phenomenon in vibrational spectra of glasses. This low-energy excess, on the one hand, drives an enhancement of the electron-phonon coupling but at the same time reduces the characteristic energy scale ω log in the Allen-Dynes formula. As a consequence, the nonmonotonicity of Tc as a function of alloying (disorder) in Pb-based systems can be rationalized. The case of Al-based systems, where disorder increases Tc from the start, is also analyzed. General material-design principles for enhancing Tc in amorphous superconductors are presented.