Ginzburg-Landau Expansion and the Slope of the Upper Critical Field in Superconductors with Anisotropic Normal Impurity Scattering

Ginzburg-Landau expansion for superconductors with anisotropic $s-$ and $d-$wave pairing is derived in the presence of anisotropic normal impurity scattering which makes $d-$pairing state more stable under disordering. It is demonstrated that the slope of the upper critical field $|dH_{c2}/dT|_{T_{c}}$ in superconductors with $d-$wave pairing has nonlinear dependence on disorder, i.e. for the low anisotropic scattering rate it drops rather fast with concentration of normal impurities, but as anisotropy of scattering increases it features initial nonlinear growth, approaches a maximum and drops again, vanishing at the critical impurity concentration. In superconductors with anisotropic $s-$wave pairing $|dH_{c2}/dT|_{T_{c}}$ grows, approaching the known asymptotic behavior, characteristic of usual isotropic case irrespective to the scattering anisotropy.