# ELECTRON-PHONON COUPLING¶

In this section we describe some basic quantities relating to the electron-phonon interaction which can be calculated using EPW.

The imaginary part of the phonon self-energy within the Migdal approximation is calculated as:

In this equation the electron-phonon matrix elements are given by

and

with \(\psi_{n{\bf k}}\) the electronic wavefunction for band \(m\), wavevector \(\bf k\), and eigenvalue \(\epsilon_{n{\bf k}}\), \(\partial_{{\bf q}\nu}V\) the derivative of the self-consistent potential associated with a phonon of wavevector \(\bf q\), branch index \(\nu\), and frequency \(\omega_{{\bf q}\nu}\). The factors \(f(\epsilon_{n{\bf k}}), f(\epsilon_{m{\bf k+q}})\) are the Fermi occupations, and \(w_{\bf k}\) are the weights of the \({\bf k}\)-points normalized to 2 in order to account for the spin degeneracy in spin-unpolarized calculations. A very common approximation to the phonon self-energy consists of neglecting the phonon frequencies \(\omega_{{\bf q}\nu}\) in the denominator and taking the limit of small broadening \(\eta\). The final expression is positive definite and is often referred to as the “double-delta function” approximation. This approximation is no longer necessary when using EPW. The imaginary part of the phonon self-energy corresponds to the phonon half-width at half-maximum \(\gamma_{{\bf q}\nu}\).

The electron-phonon coupling strength associated with a specific phonon mode and wavevector \(\lambda_{{\bf q}\nu}\) is given by

with \(\delta\) being the Dirac delta function. In the double-delta function approximation the coupling strength \(\lambda_{{\bf q}\nu}\) can be related to the imaginary part of the phonon self-energy \(\Pi^{\prime\prime}_{{\bf q}\nu}\) as follows:

The total electron-phonon coupling \(\lambda\) is calculated as the Brillouin-zone average of the mode-resolved coupling strengths \(\lambda_{{\bf q}\nu}\):

Here the \(w_{\bf q}\) are the Brillouin zone weights associated with the phonon wavevectors \({\bf q}\), normalized to 1 in the Brillouin zone. The Eliashberg spectral function \(\alpha^2 F\) can be calculated in terms of the mode-resolved coupling strengths \(\lambda_{{\bf q}\nu}\) and the phonon frequencies using:

The transport spectral function \(\alpha^2 F_{\rm T}\) is obtained from the Eliashberg spectral function \(\alpha^2F\) by replacing \(\lambda_{{\bf q}\nu}\) with \(\lambda_{{\rm T},{\bf q}\nu}\):

with \({\bf v}_{n{\bf k}} = \nabla_{\bf k}\epsilon_{n{\bf k}}\) the electron velocity.

The real and imaginary parts of the electron self-energy \(\Sigma_{n{\bf k}} = \Sigma_{n{\bf k}}^{\prime} + i\Sigma_{n{\bf k}}^{\prime \prime}\) can be calculated as

with \(n(\omega_{{\bf q}\nu})\) the Bose occupation factors.