Effect of Impurities upon Critical Temperature of Anisotropic Superconductors

The changes in critical temperature upon doping a superconductor are accounted for theoretically by solving a BCS-like model which includes anisotropy. This anisotropy is introduced via a factorable pairing potential $V_{p{p}^{\prime }}=[1+a(\hat{\Omega }\left)\right]V[1+a({\hat{\Omega }}^{\prime }\left)\right]$ which depends on the momenta of the paired particles relative to the crystal axes. The impurities have two effects upon $T_c$: They gradually change the gross properties of the system and, hence, produce a linear change in $T_c$ (valence effect), and in addition they abruptly reduce $T_c$ by reducing the anisotropy of the energy gap (mean-free-path effect). By adjusting parameters representing the strengths of these effects, we have fit theory to experiments on Sn, In, and Al alloy systems. We find $〈a^2〉\sim 0.01$, while "valence" parameters vary strongly with impurity and do not depend simply on valence. The major gap in the theory is in the unknown ratio of the mean collision times for smoothing out anisotropy and for transport.