Electron-phonon interactions theory

Electron-phonon interactions from statistical sampling

The probability distribution of finding an atom within the coordinates $\kappa+d\kappa$ (where $\kappa$ denotes the Cartesian coordinates as well as the atom number) at temperature $T$ in the harmonic approximation is given by the following expression

$dW_{\nu}(\kappa,T)=\frac{1}{2\pi \langle u^{2}_{\nu \kappa}\rangle} e^{-\kappa^{2}/(2 \langle u^{2}_{\nu \kappa}\rangle)} d\kappa,$

where the mean-square displacement of the harmonic oscillator is given as

$\langle u^{2}_{\nu \kappa}\rangle = \frac{\hbar}{2 M_{\kappa} \omega_{\nu}} \coth{\frac{\hbar \omega_{\nu}}{2 k_{B}T}}.$

Here $M_{\kappa}$, $\nu$ and $\omega_{\nu}$ denote the mass, phonon eigenmode and phonon eigenfrequency, respectively. The equation for $dW$ is valid at any temperature and the high (Maxwell--Boltzmann distribution) and low temperature limits are easily regained. In order to obtain an observable $O(T)$ at a given temperature $T</math, the average of the observable sampled at different coordinate sets <math>x_{T}^{\textrm{MC},i}$ with sample size $n$ is taken

$\langle O(T)\rangle = \frac{1}{n} \sum\limits_{i=1}^{n} O(x_{T}^{\textrm{MC,i}}).$

Each set $i$ is obtained from the equilibrium atomic positions $x_{\textrm{eq}}$ as

$x_{T}^{\textrm{MC,i}} = x_{\textrm{eq}} + \Delta \tau^{\textrm{MC,i}}$

with the displacement

$\Delta \tau^{\textrm{MC,i}} = \sqrt{\frac{1}{M_{\kappa}}} \sum\limits_{\nu}^{3(N-1)} \varepsilon_{\kappa,\nu} \mathcal{N}.$

Here $\varepsilon_{\kappa,\nu}$ denotes the unit vector of eigenmode $\nu$ on atom $\kappa$. The magnitude of the displacement in each Cartesian direction is obtained from the normal-distributed random variable $\mathcal{N}$ with a probability distribution according to $dW$.

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