Energy bandstructures of MgB2 and the related compounds

Energy band structures are calculated for the new superconductor MgB${}_2$ and the related compounds by using the LDA and an FLAPW method. It is found that the strong three dimensional network in low-lying $\pi$ bands brings about two dimensional $\sigma$ holes in MgB${}_2$, which should be responsible for the superconductivity. The de Haas-van Alphen frequencies and the cyclotron masses are obtained for the Fermi surfaces. The $\sigma$ hole is not found in LiBC due to the less three dimensional network. MgB${}_2$C${}_2$ is found a semiconductor, but the top of the valence bands are similar to the $\sigma$ bands in MgB${}_2$. The total energy calculations are also performed for Mg${}_{1-x}$Al${}_x$B${}_2$ to investigate the structural phase instability, experimentally observed around $x=$10%.