Evolution of two-gap superconductivity of MgB$_2$ by Al substitution

The temperature dependence of the upper critical fields $(H_{c2} (T )$) of the two-gap superconductors Mg$_{1-x}$Al$_x$B$_2$ not only for low doping level ($x$ $<$ 0.1) but also for high doping level (0.1 $\leq$ $x$ $\leq$ 0.3) was obtained by resistivity measurements. The $H_{c2}(T)$ values were analyzed, within the dirty-limit two-gap model, using the calculated electron-phonon coupling constants and Coulomb pseudo potentials. As the Al contents increased, both $T_c$ and $H_{c2}(0)$ decreased, which was the direct manifestation of the increasing number of electrons. At the same time, the three-dimensional $\pi$ bands became much dirtier when Al was doped. In contrast, the intraband scattering of the two-dimensional $\sigma$ bands was relatively unaffected by the Al doping. These behaviors could be understood when the characters of both bands were considered. In all the samples that we investigated, the $\sigma$ bands were dirtier than the $\pi$ bands.