Electron-phonon interactions theory - Revision history

Huebsch at 14:44, 20 February 2024

2024-02-20T14:44:10Z

← Older revision		Revision as of 14:44, 20 February 2024
Line 60:		Line 60:

	----		----
	[[Category:Phonons]][[Category:Electron-phonon interactions]][[Category:Theory~~]][[Category:VASP6~~]]		[[Category:Phonons]][[Category:Electron-phonon interactions]][[Category:Theory]]

Miranda.henrique at 09:58, 19 July 2022

2022-07-19T09:58:58Z

← Older revision		Revision as of 09:58, 19 July 2022
Line 60:		Line 60:

	----		----
	[[Category:~~Lattice Vibrations~~]][[Category:Electron-phonon interactions]][[Category:Theory]][[Category:VASP6]]		[[Category:Phonons]][[Category:Electron-phonon interactions]][[Category:Theory]][[Category:VASP6]]

Schlipf at 08:17, 10 April 2019

2019-04-10T08:17:38Z

← Older revision		Revision as of 08:17, 10 April 2019
Line 15:		Line 15:
	Here <math>M_{\kappa}</math>, <math>\nu</math> and <math>\omega_{\nu}</math> denote the mass, phonon eigenmode and phonon eigenfrequency, respectively.		Here <math>M_{\kappa}</math>, <math>\nu</math> and <math>\omega_{\nu}</math> denote the mass, phonon eigenmode and phonon eigenfrequency, respectively.
	The equation for <math>dW</math> is valid at any temperature and the high (Maxwell--Boltzmann distribution) and low temperature limits are easily regained.		The equation for <math>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>O(T)</math> at a given temperature <math>T</math, the average of the observable sampled at different coordinate sets <math>x_{T}^{\textrm{MC},i}</math> with sample size <math>n</math> is taken		In order to obtain an observable <math>O(T)</math> at a given temperature <math>T</math>, the average of the observable sampled at different coordinate sets <math>x_{T}^{\textrm{MC},i}</math> with sample size <math>n</math> is taken

	<math>		<math>

Karsai: /* ZG configuration (one-shot method) */

2019-04-09T14:45:53Z

ZG configuration (one-shot method)

← Older revision		Revision as of 14:45, 9 April 2019
Line 54:		Line 54:

	Here <math>n_{\nu,T}=[\mathrm{exp}(\hbar \omega_{\nu} /k_{B}T)-1]^{-1}</math> denotes the Bose-Einstein occupation number.		Here <math>n_{\nu,T}=[\mathrm{exp}(\hbar \omega_{\nu} /k_{B}T)-1]^{-1}</math> denotes the Bose-Einstein occupation number.
	In this way the sum for the observable <math>\langle O(T)\rangle</math> is reduced to a single calculation. In Ref. {{cite\|zacharias:prb:2016}} it was shown that for super-cell sizes <math>N\rightarrow \infty</math> 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. {{cite\|karsai:njp:2018}} we have also used a slightly modified approach, in which the signs of the displacements are chosen randomly instead of <math>\pm 1</math>. 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 ~~changes in most calculations are~~ considered as accurate enough.		In this way the sum for the observable <math>\langle O(T)\rangle</math> is reduced to a single calculation. In Ref. {{cite\|zacharias:prb:2016}} it was shown that for super-cell sizes <math>N\rightarrow \infty</math> 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. {{cite\|karsai:njp:2018}} we have also used a slightly modified approach, in which the signs of the displacements are chosen randomly instead of <math>\pm 1</math>. 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 ==		== References ==

Karsai: /* ZG configuration (one-shot method) */

2019-04-09T14:44:55Z

ZG configuration (one-shot method)

← Older revision		Revision as of 14:44, 9 April 2019
Line 54:		Line 54:

	Here <math>n_{\nu,T}=[\mathrm{exp}(\hbar \omega_{\nu} /k_{B}T)-1]^{-1}</math> denotes the Bose-Einstein occupation number.		Here <math>n_{\nu,T}=[\mathrm{exp}(\hbar \omega_{\nu} /k_{B}T)-1]^{-1}</math> denotes the Bose-Einstein occupation number.
	In this way the sum for the observable <math>\langle O(T)\rangle</math> is reduced to a single calculation. In Ref. {{cite\|zacharias:prb:2016}} it was shown that for super-cell sizes <math>N\rightarrow \infty</math> 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. {{cite\|karsai:njp:2018}} have also used a slightly modified approach, in which the signs of the displacements are chosen randomly instead of <math>\pm 1</math>. 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 changes in most calculations are considered as accurate enough.		In this way the sum for the observable <math>\langle O(T)\rangle</math> is reduced to a single calculation. In Ref. {{cite\|zacharias:prb:2016}} it was shown that for super-cell sizes <math>N\rightarrow \infty</math> 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. {{cite\|karsai:njp:2018}} we have also used a slightly modified approach, in which the signs of the displacements are chosen randomly instead of <math>\pm 1</math>. 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 changes in most calculations are considered as accurate enough.

	== References ==		== References ==

Karsai: /* ZG configuration (one-shot method) */

2019-04-09T14:44:36Z

ZG configuration (one-shot method)

← Older revision		Revision as of 14:44, 9 April 2019
Line 54:		Line 54:

	Here <math>n_{\nu,T}=[\mathrm{exp}(\hbar \omega_{\nu} /k_{B}T)-1]^{-1}</math> denotes the Bose-Einstein occupation number.		Here <math>n_{\nu,T}=[\mathrm{exp}(\hbar \omega_{\nu} /k_{B}T)-1]^{-1}</math> denotes the Bose-Einstein occupation number.
	In this way the sum for the observable <math>\langle O(T)\rangle</math> is reduced to a single calculation. In Ref. {{~~TAG~~\|zacharias:prb:2016}} it was shown that for super-cell sizes <math>N\rightarrow \infty</math> 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. {{cite\|karsai:njp:2018}} have also used a slightly modified approach, in which the signs of the displacements are chosen randomly instead of <math>\pm 1</math>. 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 changes in most calculations are considered as accurate enough.		In this way the sum for the observable <math>\langle O(T)\rangle</math> is reduced to a single calculation. In Ref. {{cite\|zacharias:prb:2016}} it was shown that for super-cell sizes <math>N\rightarrow \infty</math> 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. {{cite\|karsai:njp:2018}} have also used a slightly modified approach, in which the signs of the displacements are chosen randomly instead of <math>\pm 1</math>. 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 changes in most calculations are considered as accurate enough.


	== References ==		== References ==

Karsai at 14:44, 9 April 2019

2019-04-09T14:44:19Z

← Older revision		Revision as of 14:44, 9 April 2019
Line 55:		Line 55:
	Here <math>n_{\nu,T}=[\mathrm{exp}(\hbar \omega_{\nu} /k_{B}T)-1]^{-1}</math> denotes the Bose-Einstein occupation number.		Here <math>n_{\nu,T}=[\mathrm{exp}(\hbar \omega_{\nu} /k_{B}T)-1]^{-1}</math> denotes the Bose-Einstein occupation number.
	In this way the sum for the observable <math>\langle O(T)\rangle</math> is reduced to a single calculation. In Ref. {{TAG\|zacharias:prb:2016}} it was shown that for super-cell sizes <math>N\rightarrow \infty</math> 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. {{cite\|karsai:njp:2018}} have also used a slightly modified approach, in which the signs of the displacements are chosen randomly instead of <math>\pm 1</math>. 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 changes in most calculations are considered as accurate enough.		In this way the sum for the observable <math>\langle O(T)\rangle</math> is reduced to a single calculation. In Ref. {{TAG\|zacharias:prb:2016}} it was shown that for super-cell sizes <math>N\rightarrow \infty</math> 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. {{cite\|karsai:njp:2018}} have also used a slightly modified approach, in which the signs of the displacements are chosen randomly instead of <math>\pm 1</math>. 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 changes in most calculations are considered as accurate enough.


	== References ==		== References ==

Karsai at 14:44, 9 April 2019

2019-04-09T14:44:04Z

← Older revision		Revision as of 14:44, 9 April 2019
Line 54:		Line 54:

	Here <math>n_{\nu,T}=[\mathrm{exp}(\hbar \omega_{\nu} /k_{B}T)-1]^{-1}</math> denotes the Bose-Einstein occupation number.		Here <math>n_{\nu,T}=[\mathrm{exp}(\hbar \omega_{\nu} /k_{B}T)-1]^{-1}</math> denotes the Bose-Einstein occupation number.
	In this way the sum for the observable <math>\langle O(T)\rangle</math> is reduced to a single calculation. In Ref. {{TAG\|zacharias:prb:2016}} it was shown that for super-cell sizes <math>N\rightarrow \infty</math> 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. {{~~TAG~~\|karsai:njp:2018}} have also used a slightly modified approach, in which the signs of the displacements are chosen randomly instead of <math>\pm 1</math>. 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 changes in most calculations are considered as accurate enough.		In this way the sum for the observable <math>\langle O(T)\rangle</math> is reduced to a single calculation. In Ref. {{TAG\|zacharias:prb:2016}} it was shown that for super-cell sizes <math>N\rightarrow \infty</math> 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. {{cite\|karsai:njp:2018}} have also used a slightly modified approach, in which the signs of the displacements are chosen randomly instead of <math>\pm 1</math>. 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 changes in most calculations are considered as accurate enough.

			== References ==
			<references/>

	----		----
	[[Category:Lattice Vibrations]][[Category:Electron-phonon interactions]][[Category:Theory]][[Category:VASP6]]		[[Category:Lattice Vibrations]][[Category:Electron-phonon interactions]][[Category:Theory]][[Category:VASP6]]

Karsai at 14:43, 9 April 2019

2019-04-09T14:43:12Z

@@ Line 20: / Line 20: @@
 \langle O(T)\rangle = \frac{1}{n} \sum\limits_{i=1}^{n} O(x_{T}^{\textrm{MC,i}}).
 </math>
 Each set <math>i</math> is obtained from the equilibrium atomic positions <math>x_{\textrm{eq}}</math> as
@@ Line 34: / Line 37: @@
 Here <math>\varepsilon_{\kappa,\nu}</math> denotes the unit vector of eigenmode <math>\nu</math> on atom <math>\kappa</math>. The magnitude of the displacement in each Cartesian direction is obtained from the normal-distributed random variable <math>\mathcal{N}</math> with a probability distribution according to <math>dW</math>.
 ----
 [[Category:Lattice Vibrations]][[Category:Electron-phonon interactions]][[Category:Theory]][[Category:VASP6]]

Karsai: /* Electron-phonon interactions from statistical sampling */

2019-04-09T14:31:13Z

Electron-phonon interactions from statistical sampling

← Older revision		Revision as of 14:31, 9 April 2019
Line 1:		Line 1:
	== Electron-phonon interactions from statistical sampling ==		== Electron-phonon interactions from statistical sampling ==

	The probability distribution of finding an atom within the coordinates <math>\kappa+d\kappa</math> (where <math>\kappa</math> denotes the Cartesian coordinates as well as the atom number) at temperature <math>T</math> in the harmonic approximation is given by the following expression		The probability distribution of finding an atom within the coordinates <math>\kappa+d\kappa</math> (where <math>\kappa</math> denotes the Cartesian coordinates as well as the atom number) at temperature <math>T</math> in the harmonic approximation is given by the following expression{{cite\|bloch:zfp:1932}}{{cite\|landau:ctp:1959}}

	<math>		<math>