## Electron-phonon Coupling

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

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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.

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