Breakdown of Superconductivity in Small Metallic Grains

Superconductivity in small metallic grains is carefully checked as their size is decreased to a few nm when the average level spacing d could be compared with the bulk gap Δ. Using random matrix theory to the mean field, we find that the average theoretical values of the critical level spacing for both odd and even numbers of electrons and the transition temperature T_c in three Gauss ensembles are quite different for those from the model of uniformly spaced levels. For S_z = 1/2, as grain size is reduced, the transition temperature or the granular gap decreases monotonously, and the relation 2Δ ( 0 ) /кB T_c≤ 3.53 always exists.