Disorder-dependent slopes of the upper critical field in nodal and nodeless superconductors

We study the slopes of the upper critical field S = delta H-c2 /partial derivative T at the superconducting transition temperature T-c in anisotropic superconductors with transport (nonmagnetic) scattering employing the Ginzburg-Landau theory, developed for this case by Pokrovsky and Pokrovsky [Phys. Rev. B 54, 13275 (1996)]. We find unexpected behavior of the slopes for a d-wave superconductor and, in a more general case, of materials with line nodes in the order parameter. Specifically, the presence of line nodes causes S to decrease with increasing nonmagnetic scattering parameter P = (h) over bar /2 pi T-c0 tau (T-c0 is for the clean limit, tau is the scattering time), unlike the nodeless case where the slope increases. In a pure d-wave case, the slope changes from decreasing to increasing when the scattering parameter approaches P approximate to 0.91 P-crit, where P-crit approximate to 0.28, at which T-c -> 0, which implies the existence of a "gapless" state in d-wave superconductors with transport scattering in the interval, 0.91 P-crit < P < P-crit. Furthermore, we consider the mixed (s + d)-wave order parameter with four nodes on a cylindrical Fermi surface when the d part is dominant, or no nodes at all when the s-wave phase dominates. We find that the presence of nodes causes the slope S(P) to decrease initially with increasing P, whereas in the nodeless state, S(P) monotonically increases. Therefore relatively straightforward measurements of the disorder dependence of the slope of H-c2 at T-c can help distinguish between nodal and nodeless order parameters, which is particularly useful for quickly assessing newly discovered superconductors.