Shiba-Rusinov theory of magnetic impurities in superconductors beyonds-wave scattering: Eliashberg formalism

Abstract

Shiba-Rusinov theory of magnetic impurities in isotropic superconductors beyond the $s$-wave scattering is generalized by using the Eliashberg formalism. The analytical expressions for the transition temperature ${T}_{c}$ and the specific-heat jump $\ensuremath{\Delta}C$ are given using the square-well model for the electron-phonon interaction. Taken as a function of the impurity concentration, the quantities $\frac{{T}_{c}}{{T}_{c0}}$ and $\frac{\ensuremath{\Delta}C}{\ensuremath{\Delta}{C}_{0}}$ depend on the microscopic parameters $\ensuremath{\lambda}$, ${\ensuremath{\mu}}^{*}$, and ${\ensuremath{\omega}}_{D}$ of the host material. However, this dependence is absent if one plots the above properties versus the normalized impurity concentration $\frac{\ensuremath{\alpha}}{{\ensuremath{\alpha}}_{\mathrm{cr}}}$ or if $\frac{\ensuremath{\Delta}C}{\ensuremath{\Delta}{C}_{0}}$ vs $\frac{{T}_{c}}{{T}_{c0}}$ is studied. (${T}_{c0}$ and $\ensuremath{\Delta}{C}_{0}$ are values of ${T}_{c}$ and $\ensuremath{\Delta}C$, respectively, in the absence of impurities; $\ensuremath{\lambda}$ is the electron-phonon interaction parameter, ${\ensuremath{\mu}}^{*}$ the Coulomb pseudopotential, and ${\ensuremath{\omega}}_{D}$ the Debye cutoff frequency; $\ensuremath{\alpha}$ is the spin-flip scattering rate; ${\ensuremath{\alpha}}_{\mathrm{cr}}$ is the value of $\ensuremath{\alpha}$ for which $T$ becomes zero.)