Upper critical field in dirty two-band superconductors: Breakdown of the anisotropic Ginzburg-Landau theory

We investigate the upper critical field in a dirty two-band superconductor within quasiclassical Usadel equations. The regime of very high anisotropy in the quasi-2D band, relevant for MgB₂, is considered. We show that strong disparities in pairing interactions and diffusion constant anisotropies for two bands influence the in-plane Hc2 in a different way at high and low temperatures. This causes temperature-dependent Hc2 anisotropy, in accordance with recent experimental data in MgB₂. The three-dimensional band most strongly influences the in-plane Hc2 near Tc, in the Ginzburg-Landau (GL) region. However, due to a very large difference between the c-axis coherence lengths in the two bands, the GL theory is applicable only in an extremely narrow temperature range near Tc. The angular dependence of Hc2 deviates from a simple effective-mass law even near Tc.