Enhancement of the upper critical field by nonmagnetic impurities in dirty two-gap superconductors

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Quasiclassic Uzadel equations for two-band superconductors in the dirty limit with the account of both intraband and interband scattering by nonmagnetic impurities are derived for any anisotropic Fermi surface. From these equations the Ginzburg-Landau equations, and the critical temperature Tc are obtained. An equation for the upper critical field, which determines both the temperature dependence of Hc₂(T) and the orientational dependence of Hc₂(θ) as a function of the angle θ between H and the c-axis is obtained. It is shown that the shape of the Hc₂(T) curve essentially depends on the ratio of the intraband electron diffusivities D₁ and D₁, and can be very different from the standard one-gap dirty limit theory. In particular, the value Hc₂(0) can considerably exceed 0.7TcdHc₂/dTc, which can have important consequences for applications of MgB₂. A scaling relation is proposed which enables one to obtain the angular dependence of Hc₂(θ) from the equation for Hc₂ at H c. It is shown that, depending on the relation between D₁ and D₂, the ratio of the upper critical field Hc₂/Hc₂⊥ for H ab and H⊥ ab can both increase and decrease as the temperature decreases. Implications of the obtained results for MgB₂ are discussed.