Category:Phonons

Phonons are the collective excitation of nuclei in an extended periodic system.

Here we will present a short summary with the complete derivation presented on the theory page. Let us start by making the Taylor expansion of the total energy ${\displaystyle E}$ in terms of the ionic displacement ${\displaystyle u_{I\alpha} = R_{I\alpha} - R^0_{I\alpha} }$ around the equilibrium positions of the nuclei ${\displaystyle R^0_{I\alpha}}$

${\displaystyle E(\{\mathbf{R}\})= E(\{\mathbf{R}^0\})+ \sum_{I\alpha} -F_{I\alpha} (\{\mathbf{R}^0\}) u_{I\alpha}+ \sum_{I\alpha J\beta} \Phi_{I\alpha J\beta} (\{\mathbf{R}^0\}) u_{I\alpha} u_{J\beta} + \mathcal{O}(\mathbf{R}^3) }$

with ${\displaystyle F_{I\alpha}}$ being the atomic forces and ${\displaystyle \Phi_{I\alpha J\beta}}$ the second-order force constants.

If the structure is in equilibrium (i.e. the forces are zero) then we can find the normal modes of vibration of the system by solving the eigenvalue problem

${\displaystyle \sum_{J\beta} \frac{1}{\sqrt{M_I M_J}} \Phi_{I\alpha J\beta} e^{i\mathbf{q} \cdot (\mathbf{R}_J-\mathbf{R}_I)} (\mathbf{q}) \varepsilon_{J\beta,\nu}(\mathbf{q}) = \omega_\nu(\mathbf{q})^2 \varepsilon_{I\alpha,\nu}(\mathbf{q}) }$

where the normal modes ${\displaystyle \varepsilon_{I\alpha,\nu}(\mathbf{q})}$ and corresponding frequencies ${\displaystyle \omega_\nu(\mathbf{q})^2}$ are the phonons in the adiabatic harmonic approximation.

The computation of the second-order force constants using the supercell approach can be done using finite-differences or density functional perturbation theory.

It is possible to obtain the phonon dispersion at different q points by computing the second-order force constants on a sufficiently large supercell and Fourier interpolating the dynamical matrices in the unit cell.

Electron-phonon interaction

The movement of the nuclei leads to changes in the electronic degrees of freedom with this coupling between the electronic and phononic systems commonly referred to as electron-phonon interactions. These interactions can be captured by perturbative methods or Monte-Carlo sampling to populate a supercell with phonons and monitor how the electronic band-structure changes.

How to

• Phonons from finite differences
• Phonons from density-functional-perturbation theory
• Computing the phonon dispersion
• Electron-phonon interactions from Monte-Carlo sampling