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:

\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}

and

\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.

Installing Math Fonts
In order to properly display math fonts it might be necessary to install some Tex fonts. This can done by copying/pasting the following line into your terminal:
mkdir -p ~/.fonts; cd ~/.fonts; wget www.math.union.edu/~dpvc/jsMath/download/TeX-fonts-linux.tgz;
tar xfz TeX-fonts-linux.tgz; mv TeX-fonts-linux/* ./; sudo fc-cache -fv; rm -rf TeX-fonts-linux*