Effect of Localized and Resonant Impurity Modes on Energy Gap and Transition Temperature of Isotropic Superconductors

Abstract

The valence effect caused by a small concentration of light or heavy impurity atoms, giving rise to high-frequency localized modes or to low-frequency resonant modes, respectively, is studied in the framework of the \'Eliashberg electron-phonon model for an isotropic superconductor. The impurity atoms modify the phonon-induced electron-electron interaction and also the pseudo-Coulomb potential. The corresponding change ${K}_{1}(\ensuremath{\omega},{\ensuremath{\omega}}^{\ensuremath{'}})$ of the interaction kernel in the integral equation for the gap function of the impure metal $\ensuremath{\Delta}(\ensuremath{\omega})$ is assumed to be small. With a perturbation calculation, a linear integral equation is derived at zero temperature for the impurity-gap function ${\ensuremath{\Delta}}_{1}(\ensuremath{\omega})=\ensuremath{\Delta}(\ensuremath{\omega})\ensuremath{-}{\ensuremath{\Delta}}_{0}(\ensuremath{\omega})$. Assuming a single Lorentzian (or a super-position of two) for the phonon distribution of the host lattice, and an Einstein distribution for that of the impurity atoms, the integral equation is solved by Neumann's iteration procedure. The change of the gap parameter, ${\ensuremath{\Delta}}_{10}={\ensuremath{\Delta}}_{1}(\ensuremath{\omega}={\ensuremath{\Delta}}_{00})$, is calculated as a function of the impurity-mode frequency ${\ensuremath{\omega}}_{11}$, the electron-impurity-mode coupling parameter ${\ensuremath{\alpha}}^{2}({\ensuremath{\omega}}_{11})$, and the change ${U}_{1}$ of the pseudo-Coulomb potential. For a special case, dilute lead-indium alloys, ${\ensuremath{\Delta}}_{1}(\ensuremath{\omega})$ is evaluated using the phonon spectra found from tunneling experiments. To determine the effect of impurity atoms on the transition temperature ${T}_{c}$, one starts from \'Eliashberg's gap equation for finite temperatures, which has solutions for $T\ensuremath{\le}{T}_{c}$ and which becomes linear near ${T}_{c}$. The transition temperature is considered as an eigenvalue parameter. An exact formula for the change of this parameter $\ensuremath{\delta}{T}_{c}$, caused by ${K}_{1}(\ensuremath{\omega},\ensuremath{\omega})$, is derived by applying a theorem of Fredholm to the inhomogeneous integral equation for the impurity-gap function at the transition temperature. The final result for $\ensuremath{\delta}{T}_{c}$ contains, besides the interaction kernels of host and impurity lattice, the solutions ${\ensuremath{\Delta}}_{0}(\ensuremath{\omega},{T}_{c})$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}}_{0}(\ensuremath{\omega},{T}_{c})$ of the gap equation and of the transposed gap equation of the pure metal, respectively. The theory is applied to dilute alloys of lead with In, Sn, Sb, Hg, Tl, and Bi. The results for $\ensuremath{\delta}{T}_{c}$ are discussed using the available experimental data for these alloys.