Characterization of collective excitations in weakly coupled disordered superconductors

Isolated islands in two-dimensional strongly disordered and strongly coupled superconductors become optically active, inducing subgap collective excitations in the ac conductivity. Here, we investigate the fate of these excitations as a function of the disorder strength in the experimentally relevant case of weak electron-phonon coupling. An explicit calculation of the ac conductivity, that includes vertex corrections to restore gauge symmetry, reveals the existence of collective subgap excitations, related to phase fluctuations and therefore identified as the Goldstone modes, for intermediate to strong disorder. As disorder increases, the shape of the subgap excitation transits from peaked close to the spectral gap to a broader distribution reaching much smaller frequencies. Phase coherence still holds in part of this disorder regime. The requirement to observe subgap excitations is not the existence of isolated islands acting as nanoantennas but rather the combination of a sufficiently inhomogeneous order parameter with a phase fluctuation correlation length smaller than the system size. Our results indicate that, by tuning disorder, the Goldstone mode may be observed experimentally in metallic superconductors based, for instance, on Al, Sn, Pb, or Nb.