# 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:

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

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

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

and

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

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

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

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:

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

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

$$\lambda = \sum_{{\bf q}\nu} w_{{\bf q}} \lambda_{{\bf q}\nu}.$$

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:

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

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 $$\lambda_{{\rm T},{\bf q}\nu}$$:

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

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

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

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