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

Abstract

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 ${H}_{c2}(\ensuremath{\theta})$ as a function of the angle $\ensuremath{\theta}$ between $\mathbf{H}$ 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 ${D}_{1}$ and ${D}_{2},$ and can be very different from the one-gap dirty-limit theory. In particular, value ${H}_{c2}(0)$ can considerably exceed ${0.7T}_{c}{\mathrm{dH}}_{c2}{/dT}_{c},$ which can have important consequences for applications of ${\mathrm{MgB}}_{2}.$ Based on a general equation for ${H}_{c2}(T,\ensuremath{\theta}),$ a scaling relation, which enables one to get the angular dependence of ${H}_{c2}(\ensuremath{\theta})$ from the equation for ${H}_{c2}$ at $\mathbf{H}\ensuremath{\Vert}c,$ is proposed. The behavior of ${H}_{c2}(\ensuremath{\theta})$ and ${H}_{c1}(\ensuremath{\theta})$ can deviate markedly from the results of the anisotropic Ginzburg-Landau theory, and the ratio of the upper critical field ${H}_{c2}^{\ensuremath{\Vert}}{/H}_{c2}^{\ensuremath{\perp}}$ for $\mathbf{H}\ensuremath{\Vert}\mathrm{ab}$ and $\mathbf{H}\ensuremath{\perp}\mathrm{ab}$ can both increase and decrease with temperature, depending on the relation between ${D}_{1}$ and ${D}_{2}.$ Implications of the obtained results for ${\mathrm{MgB}}_{2}$ are discussed.