Electron-phonon coupling and phonon self-energy in MgB 2 : Interpretation of MgB 2 Raman spectra

TLDR: In this article, the authors consider a model Hamiltonian fitted on the ab initio band structure to describe the electron-phonon coupling between the electronic $\ensuremath{\sigma}$ bands and the phonon ${E}_{2g}$ mode in the Raman spectra.

Abstract: We consider a model Hamiltonian fitted on the ab initio band structure to describe the electron-phonon coupling between the electronic $\ensuremath{\sigma}$ bands and the phonon ${E}_{2g}$ mode in ${\mathrm{MgB}}_{2}$. 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 $\ensuremath{\sigma}$ 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 $\ensuremath{\Gamma}$. 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 ${\mathrm{MgB}}_{2}$ Raman spectra.