Small Fermi energy and phonon anharmonicity in ${\mathrm{MgB}}_2$ and related compounds

The remarkable anharmonicity of the $E_{2g}$ phonon in ${\mathrm{MgB}}_2$ has been suggested in the literature to play a primary role in its superconducting pairing. We investigate, by means of local density approximation calculations, the microscopic origin of such an anharmonicity in ${\mathrm{MgB}}_2,$ ${\mathrm{AlB}}_2,$ and in heavily hole-doped graphite. We find that the anharmonic character of the $E_{2g}$ phonon is essentially driven by the small Fermi energy of the $\sigma$ holes. We present a simple analytic model which allows us to understand in microscopic terms the role of the small Fermi energy and of the electronic structure. The relation between anharmonicity and nonadiabaticity is pointed out and discussed in relation to various materials.