Electron-phonon coupling and phonon self-energy in MgB$_2$: do we really understand MgB$_2$ Raman spectra ?

TL;DR

This study models the electron-phonon interactions in MgB$_2$ to clarify the origins of its Raman spectra, revealing that the $ ext{E}_{2g}$ phonon mode's contribution to Raman linewidth is negligible, challenging previous interpretations.

Contribution

The paper introduces an analytical and numerical model of electron-phonon coupling in MgB$_2$, providing new insights into phonon self-energy and Raman spectral features.

Findings

Sigma bands do not contribute to Raman linewidth via electron-phonon coupling.

Large Raman resonance cannot be attributed to the $ ext{E}_{2g}$ mode at $ ext{Γ}$.

Finite temperature effects influence phonon self-energy.

Abstract

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 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 $\sigma$ bands to the Raman linewidth 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 the phonon self-energy. From our findings we suggest several possible effects which might be responsible for the MgB$_2$ Raman spectra.