Orbital upper critical field and its anisotropy of clean one- and two-band superconductors

The Helfand-Werthamer (HW) scheme\cite{HW} of evaluating the orbital upper critical field is generalized to anisotropic superconductors in general, and to two-band clean materials, in particular. Our formal procedure differs from those in the literature; it reproduces not only the isotropic HW limit, but also the results of calculations for the two-band superconducting MgB${}_2$\cite{MMK,DS} along with the existing data on $H_{c2}(T)$ and its anisotropy $\gamma (T)=H_{c2,ab}(T)\mathrm{/}{H}_{c2,c}(T)$ ($a,c$ are the principal directions of a uniaxial crystal). Using rotational ellipsoids as model Fermi surfaces we apply the formalism developed to study $\gamma (T)$ for a few different anisotropies of the Fermi surface and of the order parameters. We find that even for a single band d-wave order parameter $\gamma (T)$ decreases on warming, however, relatively weakly. For order parameters of the form $\mathrm{\Delta }(k_z)={\mathrm{\Delta }}_0(1+\eta \cos k_{z}a)$,\cite{Xu} according to our simulations $\gamma (T)$ may either increase or decrease on warming even for a single band depending on the sign of $\eta$. Hence, the common belief that the multi-band Fermi surface is responsible for the temperature variation of $\gamma$ is proven incorrect.