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

Abstract

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 $T_c$ are obtained. An equation for the upper critical field, which determines both the temperature dependence of $H_{c2}(T)$ and the orientational dependence of $H_{c2}(\theta )$ as a function of the angle $\theta$ between $\mathbf{H}$ and the c-axis is obtained. It is shown that the shape of the $H_{c2}(T)$ curve essentially depends on the ratio of the intraband electron diffusivities $D_1$ and $D_1$, and can be very different from the standard one-gap dirty limit theory. In particular, the value $H_{c2}(0)$ can considerably exceed $0.7T_{c}d{H}_{c2}/d{T}_c$, which can have important consequences for applications of $Mg{B}_2$. A scaling relation is proposed which enables one to obtain the angular dependence of $H_{c2}(\theta )$ from the equation for $H_{c2}$ at $\mathbf{H}\Vert c$. It is shown that, depending on the relation between $D_1$ and $D_2$, the ratio of the upper critical field $H_{c2}^{\Vert }/H_{c2}^{\perp }$ for $\mathbf{H}\Vert ab$ and $\mathbf{H}\perp ab$ can both increase and decrease as the temperature decreases. Implications of the obtained results for $Mg{B}_2$ are discussed.

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