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

Quasiclassic Usadel equations for two-band superconductors in the dirty limit are derived with the account of both intraband and interband scattering by nonmagnetic impurities. From these equations, the Ginzburg-Landau equations, and the equations for the critical temperature T_c and the upper critical field H_c2 are obtained. The equation for H_c2 determines both the temperature dependence of H_c2(T) and the orientational dependence of H_c2(θ) as a function of the angle θ between H and the c axis. It is shown that the shape of the H_c2(T) curve essentially depends on the ratio of the intraband electron diffusivities D₁ and D₂, and can be very different from the one-gap dirty-limit theory. In particular, value H_c2(0) can considerably exceed 0.7T_cdH_c2/dT_c, which can have important consequences for applications of MgB₂. Based on a general equation for H_c2(T,θ), a scaling relation, which enables one to get the angular dependence of H_c2(θ) from the equation for H_c2 at H‖c, is proposed. The behavior of H_c2(θ) and H_c1(θ) can deviate markedly from the results of the anisotropic Ginzburg-Landau theory, and the ratio of the upper critical field H_c2^‖/H_c2^⊥ for H‖ab and H⊥ab can both increase and decrease with temperature, depending on the relation between D₁ and D₂. Implications of the obtained results for MgB₂ are discussed.