Comparison of $H_{c2}(T,\theta )$ in Mg${}_{1-x}$Al${}_x$B${}_2$ single crystals with the dirty-limit two-gap theory

We studied the temperature and the angular dependences of the upper critical field ($H_{c2}(T,\theta$)) of Mg${}_{1-x}$Al${}_x$B${}_2$ single crystals ($x=0.12$ and 0.21) and compared with the dirty-limit two-gap theory. We found that $H_{c2}(T,\theta )$'s were well described in a unified way by this theory. The obtained values of the parameters indicated that as the Al concentration was increased, anisotropic impurity scattering increased, making the $\sigma$ bands less anisotropic. Accordingly, the temperature dependence of the anisotropy ratio of $H_{c2}$ (${\gamma }_H$) systematically decreased, and for $x=0.21$, ${\gamma }_H$ was nearly constant. Our results imply that Mg${}_{1-x}$Al${}_x$B${}_2$ single crystals are in dirty-limit and that two-gap nature survives until $x = 0.21$.