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# Electron-phonon interactions theory

## Electron-phonon interactions from statistical sampling

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

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

Here , and denote the mass, phonon eigenmode and phonon eigenfrequency, respectively. The equation for is valid at any temperature and the high (Maxwell--Boltzmann distribution) and low temperature limits are easily regained. In order to obtain an observable at a given temperature , the average of the observable sampled at different coordinate sets with sample size is taken

### Full Monte-Carlo sampling

Each set is obtained from the equilibrium atomic positions as

with the displacement

Here denotes the unit vector of eigenmode on atom . The magnitude of the displacement in each Cartesian direction is obtained from the normal-distributed random variable with a probability distribution according to .

### 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

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

Here denotes the Bose-Einstein occupation number.
In this way the sum for the observable is reduced to a single calculation. In Ref. ^{[3]} it was shown that for super-cell sizes 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 . 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.

## References

- ↑ F. Bloch, Zeitschrift fuer Physik. 74, 295 (1932).
- ↑ [ L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics: Statistical Physics vol 5, (London: Pergamon), (1959).]
- ↑
^{a}^{b}M. Zacharias and F. Giustino, Phys. Rev. B 94, 075125 (2016). - ↑ F. Karsai, M. Engel, E. Flage-Larssen and G. Kresse, New J. of Phys. 20, 123008 (2018).