Transport entropy of vortices in superconductors with paramagnetic impurities

The entropy SD transported by a unit segment of a moving vortex line, in a type-II superconductor in the flux-flow regime, is calculated microscopically for gapless superconductors containing arbitrary amounts of paramagnetic and nonmagnetic impurities, assuming low average magnetic induction B0, and large Ginzburg-Landau parameter κ1. The calculation is based on a new prescription for calculating certain heat-current-related transport properties in magnetic conductors, recently derived by the author to resolve a contradiction with either the third law of thermodynamics or Onsager's principle. For high concentrations of magnetic impurities, when the gaplessness condition is satisfied at absolute zero temperature, we successfully show for the first time in the low-field limit that a requirement from the third law, viz., SD→0 as T→0, is exactly satisfied, thereby giving strong support to the underlying new method for calculating SD and other related transport properties. Combining the present results with our earlier results on SD in the high-field limit BHc2, we predict SD to first rise linearly as B is lowered from Hc2 and then to bend upward to approach a finite limit as B is further lowered to approach zero, for practically all concentrations of magnetic and nonmagnetic impurities. The exact amount of upward bending depends on the concentrations of the two types of impurities, but is generally larger for dirtier systems. As a side product of this work, we also give a plausible identification of the physical meaning of an anomalous quantity u₁ in the set of dynamic equations first derived by Eliashberg as being proportional to the local temperature deviation from equilibrium.