Anisotropy breaking and superconductivity in MgB$_{2}$

When magnesium is replaced by aluminium an extra 3p electron is added to the system, this 3p-electron perturbs the $\sigma $-band structure slightly, while the replacement of magnesium by scandium a 3d electron is added, this 3d electron has a large effect on the electronic structure. With aluminium replacement T$_{c}$ diminishes almost linearly and disappears at $\sim $0.53Al this corresponds to the point where the Fermi level fills up the $\sigma $-bands. The electrical conductivity in the direction of the plane due to the $\sigma $-bands ($\sigma _{a}^{\sigma })$ diminishes with a very similar trend. In this case these bands electrical anisotropy (a-direction vs. c-direction, $\sigma _{a}^{\sigma }$/$\sigma _{c}^{\sigma })$ also diminishes. On the other hand, in ScB$_{2}$ the $\sigma $-bands lose their electrical anisotropy ($\sigma _{a}^{\sigma }$/$\sigma _{c}^{\sigma }\approx $3.9), but the addition of an extra 3d electron does not raise the Fermi level above the $\sigma $-bands. Our results show that at first the anisotropy diminishes with scandium addition until $\sim $0.3Sc and then it remains almost constant, this shows a interesting parallelism with the T$_{c}$-experimental results of Agrestini et al. (2004 J. Phys. and Chem. Sol. 64, 1479), in which T$_{c}$ diminishes with scandium and disappears at 0.3Sc.