Resonance Scattering in Impure Superconductors

A model of Anderson's is used to obtain the effect of transition-metal impurities on simple superconductors due to resonance scattering. If it is assumed that the impurities do not sustain a magnetic moment, then a sharp decrease in the critical temperature is found when $k{T}_c\gg \Gamma \gg E_d$. Here $\Gamma$ is the broadening of the bound $d$ state on each impurity due to Anderson's $s-d$ interaction and $E_d$ is the energy displacement of the $d$ state from the Fermi level. This region is, in fact, unphysical since the $s-d$ interaction is too small to swamp out the magnetic moment of each impurity. In consequence, the theoretical analysis of Merriam et al. for the In-Mn alloys is incorrect. The decrease in $T_c$ is much weaker when $\Gamma \gg k{T}_c$. This is the physical region in which the magnetic moments are swamped out by the $s-d$ interaction. It is also shown that the superconducting state in such alloys displays a nonzero energy gap at all impurity concentration. When the $s-d$ interaction is weak the impurity has a nonzero magnetic moment and the problem is shown to reduce to one already considered by Gor'kov and Rusinov. The results are used to comment on the experimental results of Merriam et al. and of Boato et al. Also, reasonable agreement with Friedel's predictions is found.