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:

\begin{equation} \Pi^{\prime\prime}_{{\bf q}\nu} = {\rm Im} \sum_{mn,{\bf k}} w_{\bf k} |g_{mn,\nu}^{SE}({\bf k,q})|^2 \frac{ f(\epsilon_{n{\bf k}}) - f(\epsilon_{m{\bf k+q}}) }{ \epsilon_{m{\bf k+q}} - \epsilon_{n{\bf k}} - \omega_{{\bf q}\nu} - i\eta}. \end{equation}

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

\begin{equation} g_{mn,\nu}^{SE}({\bf k,q}) = \bigg( \frac{\hbar}{2m_0 \omega_{{\bf q}\nu} } \bigg)^{1/2} g_{mn}^{\nu}({\bf k},{\bf q}) \end{equation}


\begin{equation} g_{mn}^{\nu}({\bf k},{\bf q}) = \langle \psi_{m{\bf k+q}} | \partial_{{\bf q}\nu}V | \psi_{n{\bf k}}\rangle, \end{equation}

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

\begin{equation} \lambda_{{\bf q}\nu} = \frac{1}{N_{\rm F}\omega_{{\bf q}\nu}}\sum_{mn,{\bf k}} w_{{\bf k}} |g_{mn,\nu}^{SE}({\bf k,q})|^2 \delta(\epsilon_{n{\bf k}})\delta(\epsilon_{m{\bf k}+{\bf q}}), \end{equation}

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:

\begin{equation} \lambda_{{\bf q}\nu} = \frac{1}{\pi N_{\rm F}} \frac{\Pi^{\prime\prime}_{{\bf q}\nu}}{\omega^2_{{\bf q}\nu}} \end{equation}

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

\begin{equation} \lambda = \sum_{{\bf q}\nu} w_{{\bf q}} \lambda_{{\bf q}\nu}. \end{equation}

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:

\begin{equation} \alpha^2F(\omega) = \frac{1}{2}\sum_{{\bf q}\nu} w_{{\bf q}} \omega_{{\bf q}\nu} \lambda_{{\bf q}\nu} \, \delta( \omega - \omega_{{\bf q}\nu}). \end{equation}

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 \(\lambda_{{\rm T},{\bf q}\nu}\):

\begin{equation} \alpha^2F_{\rm T}(\omega) = \frac{1}{2}\sum_{{\bf q}\nu} w_{{\bf q}} \omega_{{\bf q}\nu} \lambda_{{\rm T},{\bf q}\nu} \delta(\omega - \omega_{{\bf q}\nu}), \end{equation} \begin{equation} \lambda_{{\rm T},{\bf q}\nu} = \frac{1}{N_{\rm F}\omega_{{\bf q}\nu}}\sum_{mn,{\bf k}} w_{{\bf k}} |g_{mn,\nu}^{SE}({\bf k,q})|^2 \delta(\epsilon_{n{\bf k}})\delta(\epsilon_{m{\bf k}+{\bf q}}) \left (1 - \frac{{\bf v}_{n{\bf k}} \cdot {\bf v}_{m{\bf k+q}}}{ |{\bf v}_{n{\bf k}}|^2}\right), \end{equation}

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

\begin{equation} \Sigma^{}_{n{\bf k}} = \sum_{{\bf q}\nu,m} w_{{\bf q}} |g_{mn,\nu}^{SE}({\bf k,q})|^2 \left[ \frac{n(\omega_{{\bf q}\nu})+ f(\epsilon_{m{\bf k+q}})}{\epsilon_{n{\bf k}} - \epsilon_{m{\bf k+q}} + \omega_{{\bf q}\nu} - i\eta} + \frac{n(\omega_{{\bf q}\nu})+ 1 -f(\epsilon_{m{\bf k+q}})}{\epsilon_{n{\bf k}} - \epsilon_{m{\bf k+q}} - \omega_{{\bf q}\nu} -i\eta} \right], \end{equation}

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