ELPH TRANSPORT DRIVER: Difference between revisions
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They are defined as | They are defined as | ||
:<math> | :<math> | ||
L_{ij} = \int d\epsilon \, \ | L_{ij} = \int d\epsilon \, \mathcal{T}(\epsilon) \, | ||
(\epsilon-\mu)^{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>\ | where <math>\mathcal{T}(\epsilon)</math> is the [[Electronic_transport_coefficients#Transport_distribution_function | transport distribution function]], | ||
<math>\mu</math> the chemical potential, and <math>f^0</math> the Fermi–Dirac distribution. | <math>\mu</math> the chemical potential, and <math>f^0</math> the Fermi–Dirac distribution. | ||
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::<math> | ::<math> | ||
L_{ij} \;\approx\; \sum_{k=1}^{N} w_k \; | L_{ij} \;\approx\; \sum_{k=1}^{N} w_k \; | ||
\ | \mathcal{T}(\epsilon_k)\; | ||
(\epsilon_k - \ | (\epsilon_k - \mu)^{\,i+j-2}\; | ||
\left( -\frac{\partial f^0}{\partial \epsilon} \right). | \left( -\frac{\partial f^0}{\partial \epsilon} \right). | ||
</math> | </math> | ||
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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(\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 abcissae of the Gauss-Legendre quadrature rule. | :with <math>w_k</math> and <math>x_k</math> the weights and abcissae of the Gauss-Legendre quadrature rule. | ||
Revision as of 14:17, 23 October 2025
ELPH_TRANSPORT_DRIVER = [integer]
Default: ELPH_TRANSPORT_DRIVER = 2
Description: choose method to compute the Onsager coefficients, which are then used to compute the transport coefficients.
| Mind: Available as of VASP 6.5.0 |
The Onsager coefficients can be computed using either of the options bellow, each with its own advantages and disadvantages. They are defined as
- [math]\displaystyle{ L_{ij} = \int d\epsilon \, \mathcal{T}(\epsilon) \, (\epsilon-\mu)^{i+j-2} \left( -\frac{\partial f^0}{\partial \epsilon} \right), }[/math]
where [math]\displaystyle{ \mathcal{T}(\epsilon) }[/math] is the transport distribution function, [math]\displaystyle{ \mu }[/math] the chemical potential, and [math]\displaystyle{ f^0 }[/math] the Fermi–Dirac distribution.
ELPH_TRANSPORT_DRIVER = 1- The discretized Onsager coefficient is evaluated as
- [math]\displaystyle{ L_{ij} \;\approx\; \sum_{k=1}^{N} w_k \; \mathcal{T}(\epsilon_k)\; (\epsilon_k - \mu)^{\,i+j-2}\; \left( -\frac{\partial f^0}{\partial \epsilon} \right). }[/math]
- with [math]\displaystyle{ \epsilon_k = \epsilon_\text{min}+(k-1)\Delta \epsilon,\;\; k=1,\dots,N }[/math] and [math]\displaystyle{ \Delta \epsilon = \tfrac{\epsilon_\text{max}-\epsilon_\text{min}}{N-1} }[/math] and [math]\displaystyle{ \epsilon_\text{min} }[/math]=ELPH_TRANSPORT_EMIN and [math]\displaystyle{ \epsilon_\text{max} }[/math]=ELPH_TRANSPORT_EMAX or alternatively both [math]\displaystyle{ \epsilon_\text{min} }[/math] and [math]\displaystyle{ \epsilon_\text{max} }[/math] are set by ELPH_TRANSPORT_DFERMI_TOL, [math]\displaystyle{ w_k }[/math] the weights due to the Simpson integration rule and N=TRANSPORT_NEDOS.
ELPH_TRANSPORT_DRIVER = 2- Use Gauss-Legendre integration to evaluate the Onsager coefficients. The convergence of the integral can be checked by performing a convergence study with respect to N=TRANSPORT_NEDOS alone. In this case the Onsager coefficients are evaluated using the following discretization
- [math]\displaystyle{ L_{ij} \;\approx\; \tfrac{1}{2} \sum_{k=1}^N w_k \, \left( \frac{k_B T}{-e} \ln \frac{1+x_k}{1-x_k} \right)^{i+j-2} \mathcal{T}\!\left(\mu + k_B T \ln\frac{1+x_k}{1-x_k}\right), }[/math]
- with [math]\displaystyle{ w_k }[/math] and [math]\displaystyle{ x_k }[/math] the weights and abcissae of the Gauss-Legendre quadrature rule.