Enhancement of Critical Temperature in Discrete Chaotic Superconducting Grains

Experimental evidence indicates that, for granular superconductors and superconducting films, the critical temperature, Tc, is often higher than in bulk samples of the same material. In the work https://arxiv.org/pdf/2408.03927, by Grankin, Hafezi, and Galitski, it is suggested that this enhancement is due to random matrix phonons mediating Cooper pairing better than bulk phonons. The earlier analysis treats the grain as a continuum, and it remains to be seen whether a realistic discrete grain has the same characteristics. To this end, we develop a realistic model of the phonon spectrum of a random grain by representing it as a system of springs and masses with nearest and next-nearest neighbor couplings. We then follow the earlier analysis, which develops the Eliashberg theory of superconductivity in chaotic grains, to calculate its critical temperature. We benchmark our results by scaling up the system size, and find our observations agree with the earlier analysis. We observe that the enhancement of Tc in discrete grains is markedly better than in the continuum case, and that the degree of enhancement scales linearly with the perimeter-to-area ratio of the shape. We conclude by discussing alterations to the model to improve accuracy or observe additional enhancement.