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: .. math:: :nowrap: \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 .. _epmatel: .. math:: :nowrap: \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 .. math:: :nowrap: \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 :math:`\psi_{n{\bf k}}` the electronic wavefunction for band :math:`m`, wavevector :math:`\bf k`, and eigenvalue :math:`\epsilon_{n{\bf k}}`, :math:`\partial_{{\bf q}\nu}V` the derivative of the self-consistent potential associated with a phonon of wavevector :math:`\bf q`, branch index :math:`\nu`, and frequency :math:`\omega_{{\bf q}\nu}`. The factors :math:`f(\epsilon_{n{\bf k}}), f(\epsilon_{m{\bf k+q}})` are the Fermi occupations, and :math:`w_{\bf k}` are the weights of the :math:`{\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 :math:`\omega_{{\bf q}\nu}` in the denominator and taking the limit of small broadening :math:`\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 :math:`\gamma_{{\bf q}\nu}`. The electron-phonon coupling strength associated with a specific phonon mode and wavevector :math:`\lambda_{{\bf q}\nu}` is given by .. math:: :nowrap: \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 :math:`\delta` being the Dirac delta function. In the double-delta function approximation the coupling strength :math:`\lambda_{{\bf q}\nu}` can be related to the imaginary part of the phonon self-energy :math:`\Pi^{\prime\prime}_{{\bf q}\nu}` as follows: .. math:: :nowrap: \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 :math:`\lambda` is calculated as the Brillouin-zone average of the mode-resolved coupling strengths :math:`\lambda_{{\bf q}\nu}`: .. math:: :nowrap: \begin{equation} \lambda = \sum_{{\bf q}\nu} w_{{\bf q}} \lambda_{{\bf q}\nu}. \end{equation} Here the :math:`w_{\bf q}` are the Brillouin zone weights associated with the phonon wavevectors :math:`{\bf q}`, normalized to 1 in the Brillouin zone. The Eliashberg spectral function :math:`\alpha^2 F` can be calculated in terms of the mode-resolved coupling strengths :math:`\lambda_{{\bf q}\nu}` and the phonon frequencies using: .. math:: :nowrap: \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 :math:`\alpha^2 F_{\rm T}` is obtained from the Eliashberg spectral function :math:`\alpha^2F` by replacing :math:`\lambda_{{\bf q}\nu}` with :math:`\lambda_{{\rm T},{\bf q}\nu}`: .. math:: :nowrap: \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 :math:`{\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 :math:`\Sigma_{n{\bf k}} = \Sigma_{n{\bf k}}^{\prime} + i\Sigma_{n{\bf k}}^{\prime \prime}` can be calculated as .. math:: :nowrap: \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 :math:`n(\omega_{{\bf q}\nu})` the Bose occupation factors.