Relationship between superconductivity and calculated \textit{$\sigma $}--band conductivity in MgB$_{2}$

Four different relaxation times were needed to adjust the electrical conductivity and Hall coefficient to experimental values in MgB$_{2}$. Al-doping was analyzed in the \textit{rigid band approximation}; this permitted a detailed study of the partial substitution of magnesium for aluminium (Mg$_{1{\-}x}$Al$_{x}$B$_{2})$. Other substitutions such as AB$_{2}$ (A=Be, Sc, Zr, Nb and Ta) were also discussed. In (Mg$_{1{\-}x}$Al$_{x}$B$_{2})$ $T_{c}$ diminishes with Al content, the other compounds are not superconductors or have a low $T_{c}$. In this work it was found that with electron doping, such as Al substitution, the \textit{$\sigma $}{\-}band conductivity decreases in a very similar way as T$_{c}$ and the corresponding bands become less anisotropic. \textit{$\sigma $}{\-}band contribution for BeB$_{2}$ and ScB$_{2}$ at $E_{F}$ is very small and the anisotropy is much lower. For Zr, Nb and Ta there are no \textit{$\sigma $}{\-}bands at $E_{F}$. These results give a clear connection between superconductivity and the character of the \textit{$\sigma $}-band, band conductivity, and band anisotropy. This gives a plausible explanation for the diminution of $T_{c}$ with different doping of MgB$_{2}$