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

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

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

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

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 $T</math,theaverageoftheobservablesampledatdifferentcoordinatesets<math>x_{T}^{{\textrm {MC}},i}$ 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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