Phonons: Theory

To understand them we start by looking at the Taylor expansion of 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 term in the expansion 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 to the second-order force-constants

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

We can define a variable that corresponds to the displacement of the atoms with respect to the equilibrium position $R_{I\alpha} = R^0_{I\alpha}+u_{I\alpha}$ which leads to

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

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

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

and the equation of motion

$M_I \ddot{u}^2_{I\alpha} = - C_{I\alpha J\beta} u_{J\beta}$

Using the following ansatz

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

where $\xi^{\mu }_{I\alpha}(\mathbf{q})$ are the phonon mode eigenvectors. Replacing 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

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

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 $\omega^\mu(\mathbf{q})^2$ at any arbitrary q point.