Electron-phonon interactions theory
Electron-phonon interactions from statistical sampling
The probability distribution of finding an atom within the coordinates [math]\displaystyle{ \kappa+d\kappa }[/math] (where [math]\displaystyle{ \kappa }[/math] denotes the Cartesian coordinates as well as the atom number) at temperature [math]\displaystyle{ T }[/math] in the harmonic approximation is given by the following expression[1][2]
[math]\displaystyle{ 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, }[/math]
where the mean-square displacement of the harmonic oscillator is given as
[math]\displaystyle{ \langle u^{2}_{\nu \kappa}\rangle = \frac{\hbar}{2 M_{\kappa} \omega_{\nu}} \coth{\frac{\hbar \omega_{\nu}}{2 k_{B}T}}. }[/math]
Here [math]\displaystyle{ M_{\kappa} }[/math], [math]\displaystyle{ \nu }[/math] and [math]\displaystyle{ \omega_{\nu} }[/math] denote the mass, phonon eigenmode and phonon eigenfrequency, respectively. The equation for [math]\displaystyle{ dW }[/math] is valid at any temperature and the high (Maxwell--Boltzmann distribution) and low temperature limits are easily regained. In order to obtain an observable [math]\displaystyle{ O(T) }[/math] at a given temperature [math]\displaystyle{ T\lt /math, the average of the observable sampled at different coordinate sets \lt math\gt x_{T}^{\textrm{MC},i} }[/math] with sample size [math]\displaystyle{ n }[/math] is taken
[math]\displaystyle{ \langle O(T)\rangle = \frac{1}{n} \sum\limits_{i=1}^{n} O(x_{T}^{\textrm{MC,i}}). }[/math]
Each set [math]\displaystyle{ i }[/math] is obtained from the equilibrium atomic positions [math]\displaystyle{ x_{\textrm{eq}} }[/math] as
[math]\displaystyle{ x_{T}^{\textrm{MC,i}} = x_{\textrm{eq}} + \Delta \tau^{\textrm{MC,i}} }[/math]
with the displacement
[math]\displaystyle{ \Delta \tau^{\textrm{MC,i}} = \sqrt{\frac{1}{M_{\kappa}}} \sum\limits_{\nu}^{3(N-1)} \varepsilon_{\kappa,\nu} \mathcal{N}. }[/math]
Here [math]\displaystyle{ \varepsilon_{\kappa,\nu} }[/math] denotes the unit vector of eigenmode [math]\displaystyle{ \nu }[/math] on atom [math]\displaystyle{ \kappa }[/math]. The magnitude of the displacement in each Cartesian direction is obtained from the normal-distributed random variable [math]\displaystyle{ \mathcal{N} }[/math] with a probability distribution according to [math]\displaystyle{ dW }[/math].