Electron-phonon coupling and phonon self-energy in MgB2: interpretation of MgB2 Raman spectra

We consider a model Hamiltonian fitted on the ab initio band structure to describe the electron-phonon coupling between the electronic sigma bands and the phonon E-2g mode in MgB2. The model allows for analytical calculations and numerical treatments using very large k-point grids. We calculate the phonon self-energy of the E-2g mode along two high symmetry directions in the Brillouin zone. We demonstrate that the contribution of the sigma bands to the Raman linewidth (Landau damping) of the E-2g mode via the electron-phonon coupling is zero. As a consequence the large resonance seen in Raman experiments cannot be interpreted as originated from the E-2g mode at Gamma. We examine in details the effects of Fermi surface singularities in the phonon spectrum and linewidth and we determine the magnitude of finite temperature effects in the phonon self-energy. From our findings we suggest several possible effects which might be responsible for the MgB2 Raman spectra.