Electron-phonon interactions theory - Revision history

Liebetreu: Reverted edits by Liebetreu (talk) to last revision by Csheldon

2026-04-15T12:38:54Z

Reverted edits by Liebetreu (talk) to last revision by Csheldon

← Older revision		Revision as of 12:38, 15 April 2026
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	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}^{\~~mathrm~~{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>
	\langle O(T)\rangle = \frac{1}{n} \sum\limits_{i=1}^{n} O(x_{T}^{\~~mathrm~~{MC,i}}).		\langle O(T)\rangle = \frac{1}{n} \sum\limits_{i=1}^{n} O(x_{T}^{\textrm{MC,i}}).
	</math>		</math>

	=== Full Monte Carlo sampling ===		=== Full Monte Carlo sampling ===

	Each set <math>i</math> is obtained from the equilibrium atomic positions <math>x_{\~~mathrm~~{eq}}</math> as		Each set <math>i</math> is obtained from the equilibrium atomic positions <math>x_{\textrm{eq}}</math> as

	<math>		<math>
	x_{T}^{\~~mathrm~~{MC,i}} = x_{\~~mathrm~~{eq}} + \Delta \tau^{\~~mathrm~~{MC,i}}		x_{T}^{\textrm{MC,i}} = x_{\textrm{eq}} + \Delta \tau^{\textrm{MC,i}}
	</math>		</math>

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	<math>		<math>
	\Delta \tau^{\~~mathrm~~{MC,i}} = \sqrt{\frac{1}{M_{\kappa}}} \sum\limits_{\nu}^{3(N-1)} \varepsilon_{\kappa,\nu} \mathcal{N}.		\Delta \tau^{\textrm{MC,i}} = \sqrt{\frac{1}{M_{\kappa}}} \sum\limits_{\nu}^{3(N-1)} \varepsilon_{\kappa,\nu} \mathcal{N}.
	</math>		</math>

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	<math>		<math>
	\Delta \tau^{\~~mathrm~~{OS}} = \sqrt{\frac{1}{M_{\kappa}}} \sum\limits_{\nu}^{3(N-1)} (-1)^{\nu-1} \varepsilon_{\kappa,\nu} \sigma_{\nu,T},		\Delta \tau^{\textrm{OS}} = \sqrt{\frac{1}{M_{\kappa}}} \sum\limits_{\nu}^{3(N-1)} (-1)^{\nu-1} \varepsilon_{\kappa,\nu} \sigma_{\nu,T},
	</math>		</math>

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

2026-04-14T11:34:34Z

Electron-phonon interactions from statistical sampling

← Older revision		Revision as of 11:34, 14 April 2026
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	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}^{\mathrm{MC},i}</math> with sample size <math>n</math> is taken

	<math>		<math>
	\langle O(T)\rangle = \frac{1}{n} \sum\limits_{i=1}^{n} O(x_{T}^{\~~textrm~~{MC,i}}).		\langle O(T)\rangle = \frac{1}{n} \sum\limits_{i=1}^{n} O(x_{T}^{\mathrm{MC,i}}).
	</math>		</math>

Liebetreu: /* Full Monte Carlo sampling */

2026-04-14T11:34:06Z

Full Monte Carlo sampling

← Older revision		Revision as of 11:34, 14 April 2026
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	<math>		<math>
	x_{T}^{\mathrm{MC,i}} = x_{\~~textrm~~{eq}} + \Delta \tau^{\mathrm{MC,i}}		x_{T}^{\mathrm{MC,i}} = x_{\mathrm{eq}} + \Delta \tau^{\mathrm{MC,i}}
	</math>		</math>

Liebetreu: /* Full Monte Carlo sampling */

2026-04-14T11:33:53Z

Full Monte Carlo sampling

← Older revision		Revision as of 11:33, 14 April 2026
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	=== Full Monte Carlo sampling ===		=== Full Monte Carlo sampling ===

	Each set <math>i</math> is obtained from the equilibrium atomic positions <math>x_{\~~textrm~~{eq}}</math> as		Each set <math>i</math> is obtained from the equilibrium atomic positions <math>x_{\mathrm{eq}}</math> as

	<math>		<math>
	x_{T}^{\~~textrm~~{MC,i}} = x_{\textrm{eq}} + \Delta \tau^{\~~textrm~~{MC,i}}		x_{T}^{\mathrm{MC,i}} = x_{\textrm{eq}} + \Delta \tau^{\mathrm{MC,i}}
	</math>		</math>

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	<math>		<math>
	\Delta \tau^{\~~textrm~~{MC,i}} = \sqrt{\frac{1}{M_{\kappa}}} \sum\limits_{\nu}^{3(N-1)} \varepsilon_{\kappa,\nu} \mathcal{N}.		\Delta \tau^{\mathrm{MC,i}} = \sqrt{\frac{1}{M_{\kappa}}} \sum\limits_{\nu}^{3(N-1)} \varepsilon_{\kappa,\nu} \mathcal{N}.
	</math>		</math>

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

2026-04-14T11:33:23Z

ZG configuration (one-shot method)

← Older revision		Revision as of 11:33, 14 April 2026
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	<math>		<math>
	\Delta \tau^{\~~textrm~~{OS}} = \sqrt{\frac{1}{M_{\kappa}}} \sum\limits_{\nu}^{3(N-1)} (-1)^{\nu-1} \varepsilon_{\kappa,\nu} \sigma_{\nu,T},		\Delta \tau^{\mathrm{OS}} = \sqrt{\frac{1}{M_{\kappa}}} \sum\limits_{\nu}^{3(N-1)} (-1)^{\nu-1} \varepsilon_{\kappa,\nu} \sigma_{\nu,T},
	</math>		</math>

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

2025-10-24T08:00:46Z

Electron-phonon interactions from statistical sampling

← Older revision		Revision as of 08:00, 24 October 2025
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	\langle O(T)\rangle = \frac{1}{n} \sum\limits_{i=1}^{n} O(x_{T}^{\textrm{MC,i}}).		\langle O(T)\rangle = \frac{1}{n} \sum\limits_{i=1}^{n} O(x_{T}^{\textrm{MC,i}}).
	</math>		</math>


	=== Full Monte Carlo sampling ===		=== Full Monte Carlo sampling ===

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

2025-10-24T08:00:40Z

Electron-phonon interactions from statistical sampling

← Older revision		Revision as of 08:00, 24 October 2025
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	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

Csheldon: /* Full Monte Carlo sampling */

2025-10-22T14:53:34Z

Full Monte Carlo sampling

← Older revision		Revision as of 14:53, 22 October 2025
Line 122:		Line 122:


	=== Full Monte-Carlo sampling ===		=== Full Monte Carlo sampling ===

	Each set <math>i</math> is obtained from the equilibrium atomic positions <math>x_{\textrm{eq}}</math> as		Each set <math>i</math> is obtained from the equilibrium atomic positions <math>x_{\textrm{eq}}</math> as

Csheldon: /* Full Monte Carlo sampling */

2025-10-22T14:51:22Z

Full Monte Carlo sampling

← Older revision		Revision as of 14:51, 22 October 2025
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	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>.		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>.


	=== ZG configuration (one-shot method) ===		=== ZG configuration (one-shot method) ===

Csheldon: /* Transport coefficients */

2025-10-22T14:18:18Z

Transport coefficients

← Older revision		Revision as of 14:18, 22 October 2025
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	==== Transport coefficients ====		==== Transport coefficients ====
	The [[Electronic transport coefficients \| transport coefficients]] describe how efficiently charge, heat, or another quantity is carried through the material. The electronic transport coefficients are derived via the linearized Boltzmann equation {{Cite\|giustino:rmp:2017}}:		The [[Electronic transport coefficients \| transport coefficients]] describe how efficiently charge, heat, or another quantity is carried through the material. The electronic transport coefficients are derived via the linearized Boltzmann equation within the relaxation time approximation (RTA) {{Cite\|giustino:rmp:2017}}:

	<math>		<math>
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	where <math>f^0_{n \mathbf{k} + \mathbf{q}}</math> is the electron occupation at band <math>n</math> at k-point <math>\mathbf{k}</math> and q-point <math>\mathbf{k}</math>, <math>\varepsilon_{n \mathbf{k}}</math> is the electron band energy, <math>\textbf{v}_{n \mathbf{k}}</math> is the carrier velocity, <math>\textbf{E}</math> is the electric field, <math>\Omega_{BZ}</math> is the cell volume, and <math>\Gamma_{mn \nu}(\mathbf{k}, \mathbf{q})</math> is a kernel describing the interaction between the electrons and phonons.		where <math>f^0_{n \mathbf{k} + \mathbf{q}}</math> is the electron occupation at band <math>n</math> at k-point <math>\mathbf{k}</math> and q-point <math>\mathbf{k}</math>, <math>\varepsilon_{n \mathbf{k}}</math> is the electron band energy, <math>\textbf{v}_{n \mathbf{k}}</math> is the carrier velocity, <math>\textbf{E}</math> is the electric field, <math>\Omega_{BZ}</math> is the cell volume, and <math>\Gamma_{mn \nu}(\mathbf{k}, \mathbf{q})</math> is a kernel describing the interaction between the electrons and phonons.

	~~within the relaxation time approximation (RTA).~~ We employ the frozen-band approximation, which assumes that the electronic potential and eigenvalues computed for the undoped system remain unchanged when electrons are added or removed.		We employ the frozen-band approximation, which assumes that the electronic potential and eigenvalues computed for the undoped system remain unchanged when electrons are added or removed.

	The current <math>J</math> can then be iteratively solved and the Onsager coefficients derived. From the Onsager coefficients, the electrical conductivity <math>\sigma</math>, the Seebeck coefficient <math>S</math>, Peltier coefficient <math>\Pi</math>, and electronic thermal conductivity <math>\kappa_e</math> can be calculated. The electronic transport coefficients are described in more detail in [[Electronic transport coefficients \| this theory page]].		The current <math>J</math> can then be iteratively solved and the Onsager coefficients derived. From the Onsager coefficients, the electrical conductivity <math>\sigma</math>, the Seebeck coefficient <math>S</math>, Peltier coefficient <math>\Pi</math>, and electronic thermal conductivity <math>\kappa_e</math> can be calculated. The electronic transport coefficients are described in more detail in [[Electronic transport coefficients \| this theory page]].