Revised Calculation of $\bm \kappa_2(T)$ for s-wave Type-II Superconductors

This paper presents revised calculations for the Maki parameters $\kappa_{1}$ and $\kappa_{2}$ and the pair potential $\Delta({\bf r})$ of s-wave type-II superconductors near the upper critical field $H_{c2}$ with arbitrary impurity concentration. It is found that Eilenberger's well-known results on $\kappa_{2}$ [Phys. Rev. {\bf 153}, 584 (1967)] are not correct quantitatively, which are modified appropriately. Calculations are also performed for a two-dimensional system with an isotropic Fermi surface. The results in the clean limit differ substantially from those in three dimensions with an spherical Fermi surface. This fact indicates the necessity of considering detailed Fermi-surface structures for a quantitative understanding of the parameters. The coefficient of $\Delta({\bf r})\propto(H_{c2}-B)^{1/2}$, which is basic to any theoretical evaluation of the thermodynamic and transport properties near $H_{c2}$, is obtained accurately.