Two-particle spectral function for disordered <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="italic">s</mml:mi></mml:math>-wave superconductors: Local maps and collective modes

Abstract

We make testable predictions for the local two-particle spectral function of a disordered $s$-wave superconductor, probed by scanning Josephson spectroscopy (SJS), providing complementary information to scanning tunneling spectroscopy (STS). We show that SJS provides a direct map of the local superconducting order parameter that is found to be anticorrelated with the gap map obtained by STS. Furthermore, this anticorrelation increases with disorder. For the momentum-resolved spectral function, we find the Higgs mode separates from the continuum at arbitrarily small disorder and appears as a nondispersive subgap feature at low momenta, spectrally separated from phase modes for all disorder strengths. The amplitude-phase mixing remains small at low momenta even when disorder is large. Remarkably, even for large disorder and high momenta, the amplitude-phase mixing oscillates rapidly in frequency and hence does not significantly affect the purity of the Higgs and phase-dominated response functions.