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

We study the slopes of the upper critical field $\partial_{T}H_{c2}|_{T_{c}}\equiv\partial H_{_{c2}}/\partial T$ at $T_{c}$ in anisotropic superconductors with transport (non-magnetic) scattering employing the Ginzburg-Landau theory, developed for this situation by S. Pokrovsky and V. Pokrovsky, Phys. Rev. B 54, 13275 (1996). We found 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 $\partial_{T}H_{c2}|_{T_{c}}$ to decrease with increasing non-magnetic scattering parameter $P$, unlike the nodeless case where the slope increases. In a pure $d-$wave case, the slope $\partial H_{c2}|_{T_{c}}$ changes from decreasing to increasing when scattering parameter approaches $P\approx0.91\,P_{{\rm crit}}$, where $P_{{\rm crit}}\approx0.2807$ at which $T_{c}\to0$ that implies the the existence of a gapless state in $d-$wave superconductors with transport scattering in the interval, $0.91\,P_{{\rm {crit}}}<P<P_{{\rm crit}}$. Furthermore, we have considered the mixed $s+d$ order parameter that has 4 nodes on a cylindrical Fermi surface when a $d-$part is dominant, or no nodes at all when an $s-$phase is the major one. We find that presence of nodes causes the slope $\partial_{T}H_{c2}|_{T_{c}},$ to decrease initially with increasing $P$, whereas in the nodeless state, $\partial_{T}H_{c2}|_{T_{c}}$ monotonically increases. Therefore, fairly straightforward experiments make it possible to decide whether or not the order parameter of a superconductor has nodes by measuring the disorder-dependence of the slope of $H_{c2}$ at $T_{c}$.