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

The Helfand-Werthamer (HW) scheme (Helfand and Werthamer 1966 Phys. Rev. 147 288; another part of this work published as a separate paper by Werthamer et al 1966 Phys. Rev. 147 295) 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) (Miranović et al 2003 J. Phys. Soc. Japan 72 221, Dahm and Schopohl 2003 Phys. Rev. Lett. 91 017001) along with the existing data on H(c2)(T) and its anisotropy γ(T) = H(c2,ab)(T)/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 γ(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 γ(T) decreases on warming; however, relatively weakly. For order parameters of the form Δ(k(z)) = Δ(0)(1 + η cos k(z)a) (Xu et al 2011 Nature Phys. 7 198), according to our simulations γ(T) may either increase or decrease on warming even for a single band depending on the sign of η. Hence, the common belief that the multi-band Fermi surface is responsible for the temperature variation of γ is proven incorrect. For two s-wave gaps, γ decreases on warming for all Fermi shapes examined. For two order parameters of the form Δ(k(z)) = Δ(0)(1 + η cos k(z)a), presumably relevant for pnictides, we obtain γ(T) increasing on warming provided both η(1) and η(2) are negative, whereas for η > 0, γ(T) decreases. We study the ratio of the two order parameters at H(c2)(T) and find that the ratio of the small gap to the large one does not vanish at any temperature, even at H(c2)(T), an indication that this does not happen at lower fields.