Non-empirical Calculations of the upper-critical field $H_{c2}$ for Nb, NbSe$_{2}$, and MgB$_{2}$

Detailed Fermi-surface structures are essential to describe the upper critical field $H_{c2}$ in type-II superconductors, as first noticed by Hohenberg and Werthamer [Phys.\ Rev.\ {\bf 153},\ 493 (1967)] and shown explicitly by Butler for high- purity cubic Niobium [Phys.\ Rev.\ Lett.\ {\bf 44},\ 1516 (1980)]. However, most of $H_{c2}$ calculations performed so far have used simplified model Fermi surfaces and/or phenomenological fitting parameters. Due to this lack of {\em ab}-{\em initio}-type calculations, our understanding on $H_{c2}$ remains at a rather unsatisfactory level. With these observations, we have derived an $H_{c2}$ equation for classic type-II superconductors which is applicable to systems with anisotropic Fermi surfaces and/or energy gaps under arbitrary field directions. Based on the formalism, we have calculated $H_{c2}$ curves for clean type-II superconductors Nb, NbSe$_{2}$, and MgB$_{2}$ using Fermi surfaces from {\em ab initio} electronic structure calculations. The results for Nb and NbSe$_2$ excellently reproduce both temperature and directional dependences of measured $H_{c2}$ curves, including marked upward curvature of NbSe$_{2}$ near $T_{c}$. As for MgB$_2$, a good fit is obtained for a $\pi$/$\sigma$ gap ratio of $\sim\! 0.3$. Our results indicate essential importance of Fermi surface anisotropy for describing $H_{c2}$.