Small Fermi energy and phonon anharmonicity in MgB_2

The remarkable anharmonicity of the E2g phonon in MgB2 has been suggested in literature to play a primary role in its superconducting pairing. We investigate, by means of LDA 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 E2g 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.