Influence of scattering on the optical response of superconductors
Fan Yang, M. W. Wu
Abstract
By using the gauge-invariant kinetic equation [Yang and Wu, Phys. Rev. B 98, 094507 (2018); Yang and Wu, Phys. Rev. B 100, 104513 (2019)], we analytically investigate the influence of the scattering on the optical properties of superconductors in the normal-skin-effect region. Both linear and second-order responses are studied under a multicycle terahertz pulse. In the linear regime, we reveal that the optical absorption ${\ensuremath{\sigma}}_{1s}(\ensuremath{\omega})$, induced by the scattering, exhibits a crossover point at $\ensuremath{\omega}=2|\mathrm{\ensuremath{\Delta}}|$. Particularly, it is further shown that when $\ensuremath{\omega}<2|\mathrm{\ensuremath{\Delta}}|, {\ensuremath{\sigma}}_{1s}(\ensuremath{\omega})$ from the scattering always exhibits a finite value even at low temperature, in contrast to the vanishing ${\ensuremath{\sigma}}_{1s}(\ensuremath{\omega})$ in the anomalous-skin-effect region as the Mattis-Bardeen theory [Mattis and Bardeen, Phys. Rev. 111, 412 (1958)] revealed. In the second-order regime, responses of the Higgs mode during and after the optical pulse are studied. During the pulse, we show that the scattering causes a phase shift in the second-order response of the Higgs mode. Particularly, this phase shift exhibits a significant $\ensuremath{\pi}$ jump at $\ensuremath{\omega}=|\mathrm{\ensuremath{\Delta}}|$, which provides a very clear feature for the experimental detection. After the pulse, by studying the damping of the Higgs-mode excitation, we reveal a relaxation mechanism from the elastic scattering, which shows a monotonic enhancement with the increase of the impurity density.