Limits of the upper critical field in dirty two-gap superconductors

An overview of the theory of the upper critical field in dirty two-gap superconductors, with a particular emphasis on MgB$_2$ is given. We focus here on the maximum $H_{c2}$ which may be achieved by increasing intraband scattering, and on the limitations imposed by weak interband scattering and paramagnetic effects. In particular, we discuss recent experiments which have recently demonstrated ten-fold increase of $H_{c2}$ in dirty carbon-doped films as compared to single crystals, so that the $H_{c2}(0)$ parallel to the ab planes may approach the BCS paramagnetic limit, $H_p[T] = 1.84T_c[K] \simeq 60-70T$. New effects produced by weak interband scattering in the two-gap Ginzburg-Landau equations and $H_{c2}(T)$ in ultrathin MgB$_2$ films are addressed.Comment: Submitted to the special issue of Physica C on MgB$_2$ on Dec. 3, 200