Category:Phonons: Difference between revisions

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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 [[Phonons: Theory|theory page]].
Let us start by making the Taylor expansion of the total energy <math>E </math> in terms of the ionic displacement
<math>
u_{I\alpha} = R_{I\alpha} - R^0_{I\alpha}
</math>
around the equilibrium positions  of the nuclei <math>R^0_{I\alpha}</math>
:<math>
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)
</math>
with <math>F_{I\alpha}</math> being the atomic forces and
<math>\Phi_{I\alpha J\beta}</math> 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
:<math>
\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})
</math>
where the normal modes <math>\varepsilon_{I\alpha,\nu}(\mathbf{q})</math>
and corresponding frequencies <math>\omega_\nu(\mathbf{q})^2</math> are the phonons in the adiabatic harmonic approximation.
The computation of the second-order force constants using the supercell approach can be done using [[Phonons from finite differences|finite-differences]] or [[Phonons from density-functional-perturbation theory | density functional perturbation theory]].
It is possible to [[Computing_the_phonon_dispersion_and_DOS |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_theory | electron-phonon interactions]].
These interactions can be captured by perturbative methods or [[Electron-phonon_interactions_from_Monte-Carlo_sampling | Monte-Carlo sampling]] to populate a supercell with phonons and monitor how the electronic band-structure changes.
== How to ==
== How to ==
*<math>\Gamma</math> phonons from finite differences: {{TAG|Phonons from finite differences}}.
* [[Phonons from finite differences]]
* [[Phonons from density-functional-perturbation theory]]
* [[Computing the phonon dispersion and DOS]]
* [[Electron-phonon interactions from Monte-Carlo sampling]]


----
----
[[Category:Lattice Vibrations]]
[[Category:VASP|Phonons]][[Category:Linear response]]

Latest revision as of 07:59, 8 October 2024

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 in terms of the ionic displacement around the equilibrium positions of the nuclei

with being the atomic forces and 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

where the normal modes and corresponding frequencies 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