Characterization of collective excitations in weakly-coupled disordered superconductors

TL;DR

This paper investigates how weak disorder affects collective excitations in two-dimensional superconductors, revealing the persistence and evolution of Goldstone modes related to phase fluctuations as disorder varies.

Contribution

It provides explicit calculations of ac conductivity including vertex corrections, demonstrating the existence and behavior of sub-gap collective excitations in weakly-coupled disordered superconductors.

Findings

Goldstone modes exist at intermediate to strong disorder levels.

Sub-gap excitation shape broadens and shifts to lower frequencies with increasing disorder.

Observation of these modes depends on inhomogeneous order parameter and phase fluctuation length scales.

Abstract

Isolated islands in two-dimensional strongly-disordered and strongly-coupled superconductors become optically active inducing sub-gap 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 sub-gap excitations, related to phase fluctuations and therefore identified as the Goldstone modes, for intermediate to strong disorder. As disorder increases, the shape of the sub-gap 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 sub-gap excitations is not the existence of isolated islands acting as nano-antennas 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.