Small Fermi energy and phonon anharmonicity in MgB₂ and related compounds

The remarkable anharmonicity of the E_2g phonon in MgB₂ 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 MgB₂, AlB₂, 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 σ 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.