Phonons: Theory

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

To compute them we start by Taylor expanding the total energy ($E$) around the set of equilibrium positions of the nuclei ($\{\mathbf{R}^0\}$)

$E(\{\mathbf{R}\})= E(\{\mathbf{R}^0\})+ \sum_{I\alpha} \frac{\partial E(\{\mathbf{R^0}\})}{\partial R_{I\alpha}} (R_{I\alpha}-R^0_{I\alpha})+ \sum_{I\alpha J\beta} \frac{\partial E(\{\mathbf{R}^0\})}{\partial R_{I\alpha} \partial R_{J\beta}} (R_{I\alpha}-R^0_{I\alpha}) (R_{J\beta}-R^0_{J\beta})+ \mathcal{O}(\mathbf{R}^3),$

where $\{\mathbf{R}\}$ the positions of the nuclei. The first derivative of the total energy with respect to the nuclei corresponds to the forces

$F_{I\alpha} (\{\mathbf{R}^0\}) = - \left. \frac{\partial E(\{\mathbf{R}\})}{\partial R_{I\alpha}} \right|_{\mathbf{R} =\mathbf{R^0}}$,

and the second derivative to the second-order force-constants

$\Phi_{I\alpha J\beta} (\{\mathbf{R}^0\}) = \left. \frac{\partial E(\{\mathbf{R}\})}{\partial R_{I\alpha} \partial R_{J\beta}} \right|_{\mathbf{R} =\mathbf{R^0}} = - \left. \frac{\partial F_{I\alpha}(\{\mathbf{R}\})}{\partial R_{J\beta}} \right|_{\mathbf{R} =\mathbf{R^0}}.$

Changing variables in the Taylor expansion of the total energy with $u_{I\alpha} = R_{I\alpha}-R^0_{I\alpha}$ that corresponds to the displacement of the atoms with respect to their equilibrium position $R^0_{I\alpha}$ leads to

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

If we take the system to be in equilibrium, the forces on the atoms are zero and then the Hamiltonian of the system is

$H = \frac{1}{2} \sum_{I\alpha} M_I \dot{u}^2_{I\alpha} + \frac{1}{2} \sum_{I\alpha J\beta} \Phi_{I\alpha J\beta} u_{I\alpha} u_{J\beta},$

with $M_I$ the mass of the $I$-th nucleus. The equation of motion is then given by

$M_I \ddot{u}_{I\alpha} = - \Phi_{I\alpha J\beta} u_{J\beta}.$

We then look for solutions of the form of plane waves traveling parallel to the wave vector $\mathbf{q}$, i.e.

$\mathbf{u}^\mu_{I\alpha}(\mathbf{q},t) = \frac{1}{2} \frac{1}{\sqrt{M_I}} \left\{ A^\mu(\mathbf{q}) \xi^{\mu }_{I\alpha}(\mathbf{q}) e^{ i [\mathbf{q} \cdot \mathbf{\mathbf{R}}_I -\omega_\mu(\mathbf{q})t]}+ A^{\mu*}(\mathbf{q}) \xi^{\mu*}_{I\alpha}(\mathbf{q}) e^{-i [\mathbf{q} \cdot \mathbf{\mathbf{R}}_I -\omega_\mu(\mathbf{q})t]} \right\}$

where $\xi^{\mu }_{I\alpha}(\mathbf{q})$ are the phonon mode eigenvectors and $A^\mu(\mathbf{q})$ the amplitudes. Replacing it in the equation of motion we obtain the following eigenvalue problem

$\sum_{J\beta} D_{I\alpha J\beta} (\mathbf{q}) \xi^{\mu }_{J\beta}(\mathbf{q}) = \omega^\mu(\mathbf{q})^2 \xi^{\mu }_{I\alpha}(\mathbf{q})$

with

$D_{I\alpha J\beta} (\mathbf{q}) = \frac{1}{\sqrt{M_I M_J}} \Phi_{I\alpha J\beta} e^{i\mathbf{q} \cdot (\mathbf{R}_J-\mathbf{R}_I)}$

the dynamical matrix in the harmonic approximation. Now by solving the eigenvalue problem above we can obtain the phonon modes $\xi^{\mu }_{I\alpha}(\mathbf{q})$ and frequencies $\omega^\mu(\mathbf{q})$ at any arbitrary q point.

Finite differences

The second-order force constants are computed using finite differences of the forces when each ion is displaced in each independent direction. This is done by creating systems with finite ionic displacement of atom $a$ in direction $i$ with magnitude $\lambda$, computing the orbitals $\psi^{u_{I\alpha}}_{\lambda}$ and the forces for these systems. The second-order force constants are then computed using

$\Phi_{I\alpha J\beta}= \frac{\partial^2E}{\partial u_{I\alpha} \partial u_{J\beta}}= -\frac{\partial F_{I\alpha}}{\partial u_{J\beta}} \approx -\frac{ \left( \mathbf{F}[\{\psi^{u_{J\beta}}_{\lambda/2}\}]- \mathbf{F}[\{\psi^{u_{J\beta}}_{-\lambda/2}\}] \right)_{I\alpha}}{\lambda}, \quad {I=1,..,N_\text{atoms}} \quad {J=1,..,N_\text{atoms}} \quad {\alpha=x,y,z} \quad {\beta=x,y,z}$

where $u_{I\alpha}$ corresponds to the displacement of atom $I$ in the cartesian direction $\alpha$ and $\mathbf{F}[\psi]$ retrieves the set of forces acting on all the ions given the $\psi_{n\mathbf{k}}$ KS orbitals.

Similarly, the internal strain tensor is

$\Xi_{I\alpha l}=\frac{\partial^2 E}{\partial u_{I\alpha} \partial \eta_l}= \frac{\partial \sigma_l}{\partial u_{I\alpha}} \approx \frac{ \left( \mathbf{\sigma}[\{\psi^{u_{I\alpha}}_{\lambda/2}\}]- \mathbf{\sigma}[\{\psi^{u_{I\alpha}}_{-\lambda/2}\}] \right)_l }{\lambda} ,\qquad {l=xx, yy, zz, xy, yz, zx} \quad {\alpha=x,y,z} \quad {J=1,..,N_\text{atoms}}$

where $\mathbf{\sigma}[\psi_{n\mathbf{k}}]$ computes the strain tensor given the $\psi_{n\mathbf{k}}$ KS orbitals.

Density functional perturbation theory

Density-functional-perturbation theory provides a way to compute the second-order linear response to ionic displacement, strain, and electric fields. The equations are derived as follows.

In density-functional theory, we solve the Kohn-Sham (KS) equations

$H(\mathbf{k}) | \psi_{n\mathbf{k}} \rangle= e_{n\mathbf{k}}S(\mathbf{k}) | \psi_{n\mathbf{k}} \rangle,$

where $H(\mathbf{k})$ is the DFT Hamiltonian, $S(\mathbf{k})$ is the overlap operator and, $| \psi_{n\mathbf{k}} \rangle$ and $e_{n\mathbf{k}}$ are the KS eigenstates.

Taking the derivative with respect to the ionic displacements $u_{I\alpha}$, we obtain the Sternheimer equations

$\left[ H(\mathbf{k}) - e_{n\mathbf{k}}S(\mathbf{k}) \right] | \partial_{u_{I\alpha}}\psi_{n\mathbf{k}} \rangle = -\partial_{u_{I\alpha}} \left[ H(\mathbf{k}) - e_{n\mathbf{k}}S(\mathbf{k})\right] | \psi_{n\mathbf{k}} \rangle$

Once the derivative of the KS orbitals is computed, we can write

$| \psi^{u_{I\alpha}}_\lambda \rangle = | \psi \rangle + \lambda | \partial_{u_{I\alpha}}\psi \rangle.$

where $\lambda$ is a small numeric value to use in the finite differences formulas below.

The second-order response to ionic displacement, i.e., the force constants or Hessian matrix can be computed using the same equation used in the case of the finite differences approach

$\Phi_{I\alpha J\beta}= \frac{\partial^2E}{\partial u_{I\alpha} \partial u_{J\beta}}= -\frac{\partial F_{I\alpha}}{\partial u_{J\beta}} \approx -\frac{ \left( \mathbf{F}[\{\psi^{u_{J\beta}}_{\lambda/2}\}]- \mathbf{F}[\{\psi^{u_{J\beta}}_{-\lambda/2}\}] \right)_{I\alpha}}{\lambda}, \quad {I=1,..,N_\text{atoms}} \quad {J=1,..,N_\text{atoms}} \quad {\alpha=x,y,z} \quad {\beta=x,y,z}$

where again $\mathbf{F}[\{\psi\}]$ yields the forces for a given set of $\psi_{n\mathbf{k}}$ KS orbitals.

Similarly, the internal strain tensor is computed using

$\Xi_{I\alpha l}=\frac{\partial^2 E}{\partial u_{I\alpha} \partial \eta_l}= \frac{\partial \sigma_l}{\partial u_{I\alpha}} \approx \frac{ \left( \mathbf{\sigma}[\{\psi^{u_{I\alpha}}_{\lambda/2}\}]- \mathbf{\sigma}[\{\psi^{u_{I\alpha}}_{-\lambda/2}\}] \right)_l }{\lambda} ,\qquad {l=xx, yy, zz, xy, yz, zx} \quad {\alpha=x,y,z} \quad {J=1,..,N_\text{atoms}}$

where $\mathbf{\sigma}[\psi_{n\mathbf{k}}]$ computes the strain tensor given the $\psi_{n\mathbf{k}}$ KS orbitals. The Born effective charges are then computed using Eq. (42) of Ref. .

$Z^*_{I\alpha \gamma} = 2 \frac{\Omega_0}{(2\pi)^3} \int_\text{BZ} \sum_n \langle \partial_{u_{I\alpha}}\psi_{n\mathbf{k}} | (\vec{\beta}_{n\mathbf{k}})_\gamma \rangle d\mathbf{k}$

where $I$ is the atom index, $\alpha$ the direction of the displacement of the atom, $\gamma$ the polarization direction, and $| \vec{\beta}_{n\mathbf{k}} \rangle$ is the polarization vector defined in Eq. (30) in Ref. . The results should be equivalent to the ones obtained using LCALCEPS and LEPSILON.