Reverse process of usual optical analysis of boson-exchange superconductors: impurity effects on s- and d-wave superconductors

Abstract

We performed a reverse process of the usual optical data analysis of boson-exchange superconductors. We calculated the optical self-energy from two (MMP and MMP+peak) input model electron-boson spectral density functions using Allen's formula for one normal and two (s- and d-wave) superconducting cases. We obtained the optical constants including the optical conductivity and the dynamic dielectric function from the optical self-energy using an extended Drude model, and finally calculated the reflectance spectrum. Furthermore, to investigate impurity effects on optical quantities we added various levels of impurities (from the clean to the dirty limit) in the optical self-energy and performed the same reverse process to obtain the optical conductivity, the dielectric function, and reflectance. From these optical constants obtained from the reverse process we extracted the impurity-dependent superfluid densities for two superconducting cases using two independent methods (the Ferrel–Glover–Tinkham sum rule and the extrapolation to zero frequency of −ϵ1(ω)ω2); we found that a certain level of impurities is necessary to get a good agreement on results obtained by the two methods. We observed that impurities give similar effects on various optical constants of s- and d-wave superconductors; the greater the impurities the more distinct the gap feature and the lower the superfluid density. However, the s-wave superconductor gives the superconducting gap feature more clearly than the d-wave superconductor because in the d-wave superconductors the optical quantities are averaged over the anisotropic Fermi surface. Our results supply helpful information to see how characteristic features of the electron-boson spectral function and the s- and d-wave superconducting gaps appear in various optical constants including raw reflectance spectrum. Our study may help with a thorough understanding of the usual optical analysis process. Further systematic study of experimental data collected at various conditions using the optical analysis process will help to reveal the origin of the mediated boson in the boson-exchange superconductors.