Enhanced Cooper Pairing via Random Matrix Phonons in Superconducting Grains

There is rich experimental evidence that granular superconductors and superconducting films often exhibit a higher transition temperature, $T_{c}$, than that in bulk samples of the same material. This paper suggests that this enhancement hinges on random matrix phonons mediating Cooper pairing more efficiently than bulk phonons. We develop the Eliashberg theory of superconductivity in chaotic grains, calculate the random phonon spectrum and solve the Eliashberg equations numerically. Self-averaging of the effective electron-phonon coupling constant is noted, which allows us to fit the numerical data with analytical results based on a generalization of the Berry conjecture. The key insight is that the phonon density of states, and hence $T_{c}$, shows an enhancement proportional to the ratio of the perimeter and area of the grain - the Weyl law. We benchmark our results for aluminum films, and find an enhancement of $T_{c}$ of about $10\%$ for a randomly-generated shape. A larger enhancement of $T_{c}$ is readily possible by optimizing grain geometries. We conclude by noticing that mesoscopic shape fluctuations in realistic granular structures should give rise to a further enhancement of global $T_{c}$ due to the formation of a percolating Josephson network.