Phonons

A three dimensional Einstein oscillator (for atom $n$) in a solid can be described by the following Hamiltonian

  $$\hat{\mathcal{H}}_{P1}(n) = \frac{a_0^2\hat{\mathbf{w}}_n^2}{... ...t{u}_{\beta}^n - \sum_{\alpha=1,2,3} F_{\alpha}(n) \hat{u}_{\alpha}^n$$ (19)

Here $\hat{\mathbf{u}}$ is the dimensionless displacement vector $\hat{\mathbf{u}} = \hat{\mathbf{P}}_n / a_0 = \Delta \hat{\mathbf{r}}_n / a_0$, with the Bohr radius $a_0 = 0.5219$ Angstrom), $m_n$ the mass of the atom $n$, $\hat{\mathbf{w}}_n = d\hat{\mathbf{u}}_n/dt = \mathbf{p}_n / a_0$ the conjugate momentum to $\hat{\mathbf{u}}_n$ and $\overline{K}(nn)$ the matrix describing the restoring force.

External force $\mathbf{F}(n)$ can correspond to the electric field $\mathbf{E}_{{\rm el}}$, i. e. $\mathbf{F}(n) = q_n \mathbf{E}_{{\rm el}} a_0$, where Bohr radius $a_0$ is included in order to yield $\mathbf{F}_{{\rm el}}(n)$ in units of meV.

Coupling such oscillators leads to the Hamiltonian

  $$\hat{\mathcal{H}}_{P2} = - \frac{1}{2} \sum_{n\ne n^{\pri... ...t{u}_{\alpha}^n K_{\alpha\beta}(nn^{\prime}) \hat{u}_{\beta}^{n^{\prime}}$$ (20)

where operators are defined as $\hat{\mathcal{I}}_{\alpha}^n \leftrightarrow \hat{\mathbf{u}}^n_{\alpha}$ and interaction parameters are $\mathcal{J}_{\alpha,\beta}(\mathbf{R}_{n^{\prime}} - \mathbf{R}_{n}) \leftrightarrow K_{\alpha\beta}(nn^{\prime})$.