MgB$_{2}$; Al and C doping, $\sigma $-band filling and anisotropy reduction

Al and C-MgB$_{2}$ doping adds an electron to the system for each atom. This extra electron fills up the $\sigma $-bands thus diminishing the number of $\sigma $-carriers; this has been the usual explanation for the $T_{c}$ reduction. Nevertheless in this work we show that there is also a large reduction of anisotropy in the electrical conductivity due to the $\sigma $-carriers which should also have an effect on the $T_{c}$ reduction. Al and C doping produce a different $T_{c}$ pattern; this difference can be largely explained by the relative shift between the $\sigma $-bands and $\pi $-bands. After adjusting to this shift there is a small but visible difference, at low doping $T_{c}$ in the Al compounds drops faster than in the C compounds, this can be directly related to the faster loss of conductivity anisotropy in the Al compounds.