# Electron-Phonon Coupling Theory¶

Phonons can interact with the electrons in a variety of ways. For example, when an electron moves through the crystal, it can scatter off of a phonon, thereby transferring some of its energy to the lattice. Conversely, when a phonon vibrates, it can create an oscillating electric field that can interact with the electrons, inducing a change in their energies and momenta. The coupling between electrons and lattice vibrations is responsible for a range of interesting and important phenomena, from electrical and thermal conductivity to superconductivity.

The first order electron-phonon coupling matrix $$g_{mn}^\nu(\mathbf{k}, \mathbf{q})$$ couples the electronic states $$m(\mathbf{k}+ \mathbf{q}),n(\mathbf{k})$$ via phonons $$\nu$$ at wave vectors $$\mathbf{q}$$ and frequencies $$\omega_\nu$$:

$g_{mn}^\nu(\mathbf{k}, \mathbf{q}) = \sqrt{ \frac{\hbar}{2 m_0 \omega_\nu}} M_{mn}^\nu(\mathbf{k}, \mathbf{q}) .$

with

$M_{mn}^\nu(\mathbf{k}, \mathbf{q}) = \langle \psi_{m \mathbf{k}+ \mathbf{q}} \vert \nabla_u V^{KS} \cdot \mathbf{e}_\nu \vert \psi_{n\mathbf{k}} \rangle.$

Here $$m_0$$ is the sum of the masses of all the atoms in the unit cell and $$\nabla_u$$ denotes the gradient with respect to atomic displacements. For the three translational modes at $$\vert \mathbf{q} \vert = 0$$ the matrix elements $$g_{mn}^\nu = 0$$, as a consequence of the acoustic sum rule.

## Implementation¶

Within the PAW framework to Kohn-Sham potential can be split into a local part $$V(\mathbf{r})$$ represented on a regular grid and a nonlocal part $$\Delta H^a_{i_1 i_2}$$:

$V^{KS} = V + \Delta H^a_{i_1 i_2}.$

In GPAW $$\nabla_u V^{KS}(\mathbf{r})$$ is determined using the finite difference method in a supercell. The potential at the displaced coordinates is computed by the DisplacementRunner() class, which is based on ASEs ase.phonon.Displacement class. The central difference derivative, as evaluted in the Supercell() class, consists of four contributions:

$\langle \psi_{i} \vert \nabla_u V^{KS} \vert \psi_{j} \rangle = \langle \tilde \psi_{i} \vert \nabla_u V(\mathbf{r}) \vert \tilde \psi_{j} \rangle + \sum_{a, ij} \langle \tilde \psi_{i} \vert \tilde p^a_i \rangle (\nabla_u \Delta H^a_{i_1 i_2}) \langle \tilde p^a_j \vert \tilde \psi_{j} \rangle + \sum_{a, ij} \langle \tilde \psi_{i} \vert \nabla_u \tilde p^a_i \rangle \Delta H^a_{i_1 i_2} \langle \tilde p^a_j \vert \tilde \psi_{j} \rangle + \sum_{a, ij} \langle \tilde \psi_{i} \vert \tilde p^a_i \rangle \Delta H^a_{i_1 i_2} \langle \sum_{a, ij} \langle \tilde \psi_{i} \vert \nabla_u \tilde p^a_i \rangle \Delta H^a_{i_1 i_2} \langle \tilde p^a_j \vert \tilde \psi_{j} \rangle \tilde p^a_j \vert \tilde \psi_{j} \rangle$

Here we do not project the derivatives onto electronics states actually, not rather onto LCAO orbitals $$\Psi_{NM}$$, where $$N$$ denotes the cell index and $$M$$ the orbital index. We use a The Fourier transform from the $$\mathbf{k}$$-space Bloch to the real space representation so that we can can later to compute $$M_{mn}^\nu$$ for arbitrary $$\mathbf{q}$$:

$\begin{split}\mathbf{g}_{\substack{N M\\ N^\prime M^\prime}}^{sc} = FFT\left[ \langle \Psi_{NM}(\mathbf{k}) \vert \nabla_u V^{KS} \vert \Psi_{N^\prime M^\prime}(\mathbf{k}) \rangle \right].\end{split}$

Finally, the electron-phonon coupling matrix is obtained by projecting the supercell matrix into the primitive unit cell bands $$m, n$$ and phonon modes $$\nu$$ in ElectronPhononMatrix():

$\begin{split}M_{mn}^\nu(\mathbf{k}, \mathbf{q}) = \sum_{\substack{N M\\ N^\prime M^\prime}} C_{mM}^{\star} C_{nM^\prime} \mathbf{g}_{\mu}^{sc} \cdot \mathbf{u}_{q \nu} e^{2\pi i [(\mathbf{k}+\mathbf{q})\cdot \mathbf{R_N} - \mathbf{k}\cdot \mathbf{R_N^\prime}]},\end{split}$

where $$C_{nM}$$ are the LCAO coefficients and $$\mathbf{u}_{q \nu}$$ are the mass-scaled phonon displacement vectors.

Checkout Electron-phonon coupling for an example and exerice and Raman spectroscopy and Raman spectroscopy for extended systems for an application.