Electronic transport coefficients - Revision history

Csheldon at 11:14, 9 February 2026

2026-02-09T11:14:08Z

← Older revision		Revision as of 11:14, 9 February 2026
Line 1:		Line 1:
	The theoretical framework is based on the [[Electron-phonon interactions theory#Transport coefficients \| linearized Boltzmann transport equation (BTE)]] within the relaxation time approximation (RTA).		The theoretical framework is based on the [[Electron-phonon interactions theory#Transport coefficients \| linearized Boltzmann transport equation (BTE)]] 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 goal of this page is to explain how to compute the electronic lifetimes, scattering rates, and transport coefficients such as the electrical conductivity, Seebeck coefficient, and the electronic thermal conductivity. If you would like to plot the scattering lifetimes, see the [https://www.vasp.at/tutorials/latest/electron-phonon/part4/#electron-phonon-e12 end of Exercise 12 of the electron-phonon tutorials].		The goal of this page is to explain how to compute the electronic lifetimes, scattering rates, and transport coefficients such as the electrical conductivity, Seebeck coefficient, and the electronic thermal conductivity. If you would like to plot the scattering lifetimes or the electron-phonon matrix elements, see the [https://www.vasp.at/tutorials/latest/electron-phonon/part4/#electron-phonon-e12 end of Exercise 12] and [https://www.vasp.at/tutorials/latest/electron-phonon/part3/#electron-phonon-e08 Exercise 9] of the electron-phonon tutorials, respectively.

	== Electron–phonon coupling matrix elements ==		== Electron–phonon coupling matrix elements ==

Csheldon at 11:07, 9 February 2026

2026-02-09T11:07:09Z

← Older revision		Revision as of 11:07, 9 February 2026
Line 1:		Line 1:
	The theoretical framework is based on the [[Electron-phonon interactions theory#Transport coefficients \| linearized Boltzmann transport equation (BTE)]] within the relaxation time approximation (RTA).		The theoretical framework is based on the [[Electron-phonon interactions theory#Transport coefficients \| linearized Boltzmann transport equation (BTE)]] 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 goal of this page is to explain how to compute the electronic lifetimes, scattering rates, and transport coefficients such as the electrical conductivity, Seebeck coefficient, and the electronic thermal conductivity. If you would like to plot the scattering ~~rates~~, see the [https://www.vasp.at/tutorials/latest/electron-phonon/part4/#electron-phonon-e12 end of Exercise 12 of the electron-phonon tutorials].		The goal of this page is to explain how to compute the electronic lifetimes, scattering rates, and transport coefficients such as the electrical conductivity, Seebeck coefficient, and the electronic thermal conductivity. If you would like to plot the scattering lifetimes, see the [https://www.vasp.at/tutorials/latest/electron-phonon/part4/#electron-phonon-e12 end of Exercise 12 of the electron-phonon tutorials].

	== Electron–phonon coupling matrix elements ==		== Electron–phonon coupling matrix elements ==

Csheldon at 11:06, 9 February 2026

2026-02-09T11:06:42Z

← Older revision		Revision as of 11:06, 9 February 2026
Line 1:		Line 1:
	The theoretical framework is based on the [[Electron-phonon interactions theory#Transport coefficients \| linearized Boltzmann transport equation (BTE)]] within the relaxation time approximation (RTA).		The theoretical framework is based on the [[Electron-phonon interactions theory#Transport coefficients \| linearized Boltzmann transport equation (BTE)]] 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 goal of this page is to explain how to compute the electronic lifetimes, scattering rates, and transport coefficients such as the electrical conductivity, Seebeck coefficient, and the electronic thermal conductivity.		The goal of this page is to explain how to compute the electronic lifetimes, scattering rates, and transport coefficients such as the electrical conductivity, Seebeck coefficient, and the electronic thermal conductivity. If you would like to plot the scattering rates, see the [https://www.vasp.at/tutorials/latest/electron-phonon/part4/#electron-phonon-e12 end of Exercise 12 of the electron-phonon tutorials].

	== Electron–phonon coupling matrix elements ==		== Electron–phonon coupling matrix elements ==

Csheldon: /* Electron and hole mobilities in semiconductors */

2025-11-04T09:45:08Z

Electron and hole mobilities in semiconductors

← Older revision		Revision as of 09:45, 4 November 2025
Line 133:		Line 133:
	In semiconductors, the electrical conductivity can be separated into contributions from conduction-band electrons and valence-band holes.		In semiconductors, the electrical conductivity can be separated into contributions from conduction-band electrons and valence-band holes.
	This is only meaningful in materials with a finite band gap, where carriers can be clearly identified as either electrons in the conduction band (CB) or holes in the valence band (VB).		This is only meaningful in materials with a finite band gap, where carriers can be clearly identified as either electrons in the conduction band (CB) or holes in the valence band (VB).
			{{NB\|important\|Be careful not to mistake the [[Chemical potential in electron-phonon interactions \| chemical potential]] <math>\mu</math> for the mobility <math>\mu_e</math> or <math>\mu_h</math>.}}

	{\| class="wikitable" style="text-align:center"		{\| class="wikitable" style="text-align:center"
Line 154:		Line 155:
	* <math>N_\mathbf{k}</math> is the total number of k-points.		* <math>N_\mathbf{k}</math> is the total number of k-points.
	* <math>\mu</math> is the [[Chemical potential in electron-phonon interactions \| chemical potential]] at the given temperature.		* <math>\mu</math> is the [[Chemical potential in electron-phonon interactions \| chemical potential]] at the given temperature.
	~~{{NB\|mind\|Be careful not to mistake the [[Chemical potential in electron-phonon interactions \| chemical potential]] <math>\mu</math> for the mobility <math>\mu_e</math> or <math>\mu_h</math>.}}~~

	== Approximations and methods ==		== Approximations and methods ==

Csheldon at 09:42, 4 November 2025

2025-11-04T09:42:55Z

← Older revision		Revision as of 09:42, 4 November 2025
Line 141:		Line 141:
	\| Electron mobility <math>\mu_e</math>		\| Electron mobility <math>\mu_e</math>
	\| <math>\mu_e = \tfrac{\sigma_{n \in \text{CB}}}{n_e}</math>		\| <math>\mu_e = \tfrac{\sigma_{n \in \text{CB}}}{n_e}</math>
	\| <math>n_e = \frac{1}{\Omega N_\mathbf{k}}\sum_{\mathbf{k}n \in \text{CB}} f(\varepsilon_{\mathbf{k}n}, T, \~~eta~~)</math>		\| <math>n_e = \frac{1}{\Omega N_\mathbf{k}}\sum_{\mathbf{k}n \in \text{CB}} f(\varepsilon_{\mathbf{k}n}, T, \mu)</math>
	\|-		\|-
	\| Hole mobility <math>\mu_h</math>		\| Hole mobility <math>\mu_h</math>
	\| <math>\mu_h = \tfrac{\sigma_{n \in \text{VB}}}{n_h}</math>		\| <math>\mu_h = \tfrac{\sigma_{n \in \text{VB}}}{n_h}</math>
	\| <math>n_h = \frac{1}{\Omega N_\mathbf{k}}\sum_{\mathbf{k}n \in \text{VB}} \big[1 - f(\varepsilon_{\mathbf{k}n}, T, \~~eta~~)\big]</math>		\| <math>n_h = \frac{1}{\Omega N_\mathbf{k}}\sum_{\mathbf{k}n \in \text{VB}} \big[1 - f(\varepsilon_{\mathbf{k}n}, T, \mu)\big]</math>
	\|}		\|}

Line 153:		Line 153:
	* <math>\Omega</math> is the volume of the unit cell.		* <math>\Omega</math> is the volume of the unit cell.
	* <math>N_\mathbf{k}</math> is the total number of k-points.		* <math>N_\mathbf{k}</math> is the total number of k-points.
	* <math>\~~eta~~</math> is the [[Chemical potential in electron-phonon interactions \| chemical potential]] at the given temperature.		* <math>\mu</math> is the [[Chemical potential in electron-phonon interactions \| chemical potential]] at the given temperature.
	{{NB\|~~important~~\|~~The~~ [[Chemical potential in electron-phonon interactions \| chemical potential]] ~~is usually written as~~ <math>\mu</math> or <math>\~~varepsilon_F~~</math>~~. To avoid confusion with the mobility, we use the notation~~ <math>\~~eta~~</math>.}}		{{NB\|mind\|Be careful not to mistake the [[Chemical potential in electron-phonon interactions \| chemical potential]] <math>\mu</math> for the mobility <math>\mu_e</math> or <math>\mu_h</math>.}}

	== Approximations and methods ==		== Approximations and methods ==

Csheldon: /* Onsager coefficients */

2025-11-04T09:41:15Z

Onsager coefficients

← Older revision		Revision as of 09:41, 4 November 2025
Line 67:		Line 67:
	where <math>\mathcal{T}(\epsilon)</math> is the transport distribution function,		where <math>\mathcal{T}(\epsilon)</math> is the transport distribution function,
	<math>\mu</math> the [[Chemical potential in electron-phonon interactions \| chemical potential]], and <math>f^0</math> the Fermi–Dirac distribution.		<math>\mu</math> the [[Chemical potential in electron-phonon interactions \| chemical potential]], and <math>f^0</math> the Fermi–Dirac distribution.
	{{NB\|important\|The [[Chemical potential in electron-phonon interactions \| chemical potential]] is written as <math>\mu</math>. The mobility is <math>\mu_e</math> or <math>\mu_h</math>, and should not be ~~mixed up with~~ it.}}		{{NB\|important\|The [[Chemical potential in electron-phonon interactions \| chemical potential]] is written as <math>\mu</math>. The mobility is <math>\mu_e</math> or <math>\mu_h</math>, and should not be mistaken for it.}}

	In practice, this integral can be evaluated in one of two ways determined by {{TAG\|ELPH_TRANSPORT_DRIVER}}		In practice, this integral can be evaluated in one of two ways determined by {{TAG\|ELPH_TRANSPORT_DRIVER}}

Csheldon: /* Onsager coefficients */

2025-11-04T09:41:00Z

Onsager coefficients

← Older revision		Revision as of 09:41, 4 November 2025
Line 56:		Line 56:
	* <math>\mathbf{J}_e</math> = electrical current density,		* <math>\mathbf{J}_e</math> = electrical current density,
	* <math>\mathbf{J}_q</math> = heat current density carried by the electrons,		* <math>\mathbf{J}_q</math> = heat current density carried by the electrons,
	* <math>\mathbf{\mathcal{E}} = E + \frac{\nabla\~~eta~~}{e}</math> electrochemical potential with <math>\~~eta~~</math> the [[Chemical potential in electron-phonon interactions \| chemical potential]] ~~usually written as <math>\mu</math> or <math>\epsilon_F</math>)~~		* <math>\mathbf{\mathcal{E}} = E + \frac{\nabla\mu}{e}</math> electrochemical potential with <math>\mu</math> the [[Chemical potential in electron-phonon interactions \| chemical potential]]
	* <math>T</math> = temperature.		* <math>T</math> = temperature.

Line 62:		Line 62:
	:<math>		:<math>
	L_{ij} = \int d\epsilon \, \mathcal{T}(\epsilon) \,		L_{ij} = \int d\epsilon \, \mathcal{T}(\epsilon) \,
	(\epsilon-\~~eta~~)^{i+j-2}		(\epsilon-\mu)^{i+j-2}
	\left( -\frac{\partial f^0}{\partial \epsilon} \right),		\left( -\frac{\partial f^0}{\partial \epsilon} \right),
	</math>		</math>
	where <math>\mathcal{T}(\epsilon)</math> is the transport distribution function,		where <math>\mathcal{T}(\epsilon)</math> is the transport distribution function,
	<math>\~~eta~~</math> the [[Chemical potential in electron-phonon interactions \| chemical potential]], and <math>f^0</math> the Fermi–Dirac distribution.		<math>\mu</math> the [[Chemical potential in electron-phonon interactions \| chemical potential]], and <math>f^0</math> the Fermi–Dirac distribution.
	{{NB\|important\|The [[Chemical potential in electron-phonon interactions \| chemical potential]] is ~~usually~~ written as <math>\mu</math> or <math>\~~varepsilon_F~~</math>~~. To avoid confusion with the mobility, we use the notation~~ <math>\~~eta~~</math>.}}		{{NB\|important\|The [[Chemical potential in electron-phonon interactions \| chemical potential]] is written as <math>\mu</math>. The mobility is <math>\mu_e</math> or <math>\mu_h</math>, and should not be mixed up with it.}}

	In practice, this integral can be evaluated in one of two ways determined by {{TAG\|ELPH_TRANSPORT_DRIVER}}		In practice, this integral can be evaluated in one of two ways determined by {{TAG\|ELPH_TRANSPORT_DRIVER}}
Line 76:		Line 76:
	L_{ij} \;\approx\; \sum_{k=1}^{N} w_k \;		L_{ij} \;\approx\; \sum_{k=1}^{N} w_k \;
	\mathcal{T}(\epsilon_k)\;		\mathcal{T}(\epsilon_k)\;
	(\epsilon_k - \~~eta~~)^{\,i+j-2}\;		(\epsilon_k - \mu)^{\,i+j-2}\;
	\left( -\frac{\partial f^0}{\partial \epsilon} \right).		\left( -\frac{\partial f^0}{\partial \epsilon} \right).
	</math>		</math>
Line 89:		Line 89:
	Define		Define
	:<math>		:<math>
	x = 1-2f(\epsilon-\~~eta~~,T)		x = 1-2f(\epsilon-\mu,T)
	</math>		</math>
	so that <math>\epsilon = \~~eta~~ + k_B T \ln\frac{1+x}{1-x}</math>.		so that <math>\epsilon = \mu+ k_B T \ln\frac{1+x}{1-x}</math>.
	With this substitution, the derivative of the Fermi–Dirac distribution is absorbed into the Jacobian, and the Onsager coefficients take the form		With this substitution, the derivative of the Fermi–Dirac distribution is absorbed into the Jacobian, and the Onsager coefficients take the form
	:<math>		:<math>
Line 97:		Line 97:
	w_k \,		w_k \,
	\left( \frac{k_B T}{-e} \ln \frac{1+x_k}{1-x_k} \right)^{i+j-2}		\left( \frac{k_B T}{-e} \ln \frac{1+x_k}{1-x_k} \right)^{i+j-2}
	\mathcal{T}\!\left(\~~eta~~ + k_B T \ln\frac{1+x_k}{1-x_k}\right),		\mathcal{T}\!\left(\mu+ k_B T \ln\frac{1+x_k}{1-x_k}\right),
	</math>		</math>
	with <math>w_k</math> and <math>x_k</math> the weights and abscissae of the Gauss-Legendre quadrature rule.		with <math>w_k</math> and <math>x_k</math> the weights and abscissae of the Gauss-Legendre quadrature rule.

Miranda.henrique: /* Onsager coefficients */

2025-10-31T15:52:23Z

Onsager coefficients

← Older revision		Revision as of 15:52, 31 October 2025
Line 56:		Line 56:
	* <math>\mathbf{J}_e</math> = electrical current density,		* <math>\mathbf{J}_e</math> = electrical current density,
	* <math>\mathbf{J}_q</math> = heat current density carried by the electrons,		* <math>\mathbf{J}_q</math> = heat current density carried by the electrons,
	* <math>\mathbf{\mathcal{E}} = E + \frac{\nabla\eta}{e}</math> electrochemical potential with <math>\eta</math> the chemical potential usually written as <math>\mu</math> or <math>\epsilon_F</math>)		* <math>\mathbf{\mathcal{E}} = E + \frac{\nabla\eta}{e}</math> electrochemical potential with <math>\eta</math> the [[Chemical potential in electron-phonon interactions \| chemical potential]] usually written as <math>\mu</math> or <math>\epsilon_F</math>)
	* <math>T</math> = temperature.		* <math>T</math> = temperature.

Miranda.henrique: /* Scattering rates and lifetimes */

2025-10-31T15:33:19Z

Scattering rates and lifetimes

← Older revision		Revision as of 15:33, 31 October 2025
Line 14:		Line 14:
	Within Fermi’s golden rule, the inverse lifetime (scattering rate) of an electron in state <math>(n,\mathbf{k})</math> is		Within Fermi’s golden rule, the inverse lifetime (scattering rate) of an electron in state <math>(n,\mathbf{k})</math> is
	:<math>		:<math>
	\frac{1}{\tau_{n\mathbf{k}}} = \frac{2\pi}{\hbar} \sum_{n'\nu\mathbf{k}'} w_{n\mathbf{k},n'\mathbf{k}'} \, \|g^{\nu}_{n\mathbf{k},n'\mathbf{k}'}\|^2 \left[ (n_{\nu\mathbf{q}} + 1 - f_{n'\mathbf{k}'}) \, \delta(\varepsilon_{n\mathbf{k}} - \varepsilon_{n'\mathbf{k}'} - \hbar\omega_{\nu\mathbf{q}}) + (n_{\nu\mathbf{q}} + f_{n'\mathbf{k}'}) \, \delta(\varepsilon_{n\mathbf{k}} - \varepsilon_{n'\mathbf{k}'} + \hbar\omega_{\nu\mathbf{q}}) \right]		\frac{1}{\tau_{n\mathbf{k}}} = \frac{2\pi}{\hbar} \sum_{n'\nu\mathbf{k}'} w_{n\mathbf{k},n'\mathbf{k}'} \, \|g^{\nu\mathbf{q}}_{n\mathbf{k},n'\mathbf{k}'}\|^2 \left[ (n_{\nu\mathbf{q}} + 1 - f_{n'\mathbf{k}'}) \, \delta(\varepsilon_{n\mathbf{k}} - \varepsilon_{n'\mathbf{k}'} - \hbar\omega_{\nu\mathbf{q}}) + (n_{\nu\mathbf{q}} + f_{n'\mathbf{k}'}) \, \delta(\varepsilon_{n\mathbf{k}} - \varepsilon_{n'\mathbf{k}'} + \hbar\omega_{\nu\mathbf{q}}) \right]
	</math>		</math>
	where:		where:

Miranda.henrique at 12:21, 31 October 2025

2025-10-31T12:21:59Z

← Older revision		Revision as of 12:21, 31 October 2025
Line 48:		Line 48:
	\end{pmatrix}		\end{pmatrix}
	\begin{pmatrix}		\begin{pmatrix}
	-\~~nabla~~ \~~eta~~ \\		\mathbf{\mathcal{E}} \\
	-\nabla T / T		-\nabla T / T
	\end{pmatrix},		\end{pmatrix},
Line 56:		Line 56:
	* <math>\mathbf{J}_e</math> = electrical current density,		* <math>\mathbf{J}_e</math> = electrical current density,
	* <math>\mathbf{J}_q</math> = heat current density carried by the electrons,		* <math>\mathbf{J}_q</math> = heat current density carried by the electrons,
	* <math>\eta</math> = chemical potential (usually written as <math>\mu</math> or <math>\epsilon_F</math>)		* <math>\mathbf{\mathcal{E}} = E + \frac{\nabla\eta}{e}</math> electrochemical potential with <math>\eta</math> the chemical potential usually written as <math>\mu</math> or <math>\epsilon_F</math>)
	* <math>T</math> = temperature.		* <math>T</math> = temperature.