Theory of the Thermal Conductivity of Superconducting Alloys with Paramagnetic Impurities

The electronic thermal conductivity ($K_s$) of a weakly coupled, isotropic superconductor doped with a small concentration of paramagnetic impurities is computed. The theory of such superconducting alloys has been given by Abrikosov and Gor'kov, and is based on the assumption that the static magnetic impurities are randomly distributed and that their spins are uncorrelated. Starting from a Kubo formula, $K_s$ is calculated by considering the electron-impurity interaction in the ladder approximation. A considerable simplification of the final expression for $K_s$ obtains if the exchange scattering time ${\tau }_S$ is much larger than the total scattering time. Numerical calculations have been made of the ratio of the thermal conductivity in the superconducting and normal states as a function of the reduced temperature ($\frac{T}{T_c}\equiv t$) for different impurity concentrations. Abrikosov and Gor'kov have shown that the energy gap function ${\omega }_0\mathrm{}\left(T\right)\mathrm{}$ is quite different from the Gor'kov order parameter $\Delta \mathrm{}\left(T\right)\mathrm{}$ in such alloys. It is found, however, that $\frac{K_s}{K_n}$ is less than unity even in the "gapless" region ($\Delta {\tau }_S<1\mathrm{}$). Moreover $\frac{K_s}{K_n}$ as a function of $t$ decreases with the paramagnetic-impurity concentration for $t\gtrsim 0.8$ and low concentrations. Some aspects of the Abrikosov-Gor'kov model are reviewed in an Appendix. The numerical values of $\Delta$, ${\omega }_0$, and the density of states that were used in the evaluation of $\frac{K_s}{K_n}$ are given separately.