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Electron-phonon interactions theory

Electron-phonon interactions from statistical sampling

The probability distribution of finding an atom within the coordinates ${\displaystyle \kappa +d\kappa }$ (where ${\displaystyle \kappa }$ denotes the Cartesian coordinates as well as the atom number) at temperature ${\displaystyle T}$ in the harmonic approximation is given by the following expression[1][2]

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

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

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

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

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

Full Monte-Carlo sampling

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

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

with the displacement

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

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

ZG configuration (one-shot method)

Motivated by the empirical observation that for increasing super-cell sizes the number of required structures in the MC method can be decreased, M. Zacharias and F. Giustino[3] proposed a one-shot method where only a single set of displacements is used

${\displaystyle \Delta \tau ^{\textrm {OS}}={\sqrt {\frac {1}{M_{\kappa }}}}\sum \limits _{\nu }^{3(N-1)}(-1)^{\nu -1}\varepsilon _{\kappa ,\nu }\sigma _{\nu ,T},}$

where the summation over the eigenmodes runs in an ascending order with respect to the values of the eigenfrequencies, and the magnitude of each displacement is given by

${\displaystyle \sigma _{\nu ,T}={\sqrt {(2n_{\nu ,T}+1){\frac {\hbar }{2\omega _{\nu }}}}}.}$

Here ${\displaystyle n_{\nu ,T}=[\mathrm {exp} (\hbar \omega _{\nu }/k_{B}T)-1]^{-1}}$ denotes the Bose-Einstein occupation number. In this way the sum for the observable ${\displaystyle \langle O(T)\rangle }$ is reduced to a single calculation. In Ref. [3] it was shown that for super-cell sizes ${\displaystyle N\rightarrow \infty }$ the structural configuration obtained using the ZG configuration should lead to equivalent results as fully converged MC calculations. In practice, it was shown that already relatively small super-cell sizes are sufficient to achieve good accuracy, but the convergence with respect to the cell size can vary between different materials. In Ref. [4] we have also used a slightly modified approach, in which the signs of the displacements are chosen randomly instead of ${\displaystyle \pm 1}$. This was only necessary, when calculating volume dependent ZPR, since the modes sometimes change the order as the volume changes. Using alternating signs for the displacement then causes small discontinuities in the ZPR volume curve of the order of 5 meV for carbon diamond. By averaging over many random phases this problem can be eliminated. Nevertheless 5 meV difference between the ZG configuration and full MC is considered as accurate enough in most calculations.