Effect of Magnetic Impurities on Suppression of the Transition Temperature in Disordered Superconductors

We calculate the first-order perturbative correction to the transition temperature $T_c$ in a superconductor with both non-magnetic and magnetic impurities. We do this by first evaluating the correction to the effective potential, $\Omega(\Delta)$, and then obtain the first-order correction to the order parameter, $\Delta$, by finding the minimum of $\Omega(\Delta)$. Setting $\Delta=0$ finally allows $T_c$ to be evaluated. $T_c$ is now a function of both the resistance per square, $R_\square$, a measure of the non-magnetic disorder, and the spin-flip scattering rate, $1/\tau_s$, a measure of the magnetic disorder. We find that the effective pair-breaking rate per magnetic impurity is virtually independent of the resistance per square of the film, in agreement with an experiment of Chervenak and Valles. This conclusion is supported by both the perturbative calculation, and by a non-perturbative re-summation technique.