ELPH SCATTERING APPROX

ELPH_SCATTERING_APPROX = [string]
Default: ELPH_SCATTERING_APPROX = SERTA MRTA_LAMBDA 

Description: Select which type of approximation is used to compute the electron scattering lifetimes due to electron-phonon coupling

Mind: Available as of VASP 6.5.0

---

There are different approximations to compute the electronic lifetimes due to electron-phonon scattering. Each of these can lead to significantly different transport coefficients. It is possible to select more than one approximation in ELPH_SCATTERING_APPROX. In this case, additional electron-phonon accumulators are created for each scattering approximation.

Options to select

ELPH_SCATTERING_APPROX = CRTA - Constant Relaxation-Time Approximation

The relaxation time is assumed constant. It needs to be specified via TRANSPORT_RELAXATION_TIME. In this case, the computation of electron-phonon matrix elements is skipped entirely, which is a huge performance boost compared to the other relaxation-time approximations.

Warning: While the CRTA can be a reasonable approximation for metals, it will generally fail for insulators.

ELPH_SCATTERING_APPROX = SERTA - Self-Energy Relaxation-Time Approximation

Computes the relaxation time from the imaginary part of the Fan self-energy, evaluated on the electronic eigenenergy:

[math]\displaystyle{ \frac{1}{\tau^{\mathrm{SERTA}}_{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] }[/math]

where [math]\displaystyle{ {\tau^{\mathrm{SERTA}}_{n\mathbf{k}}} }[/math] is the relaxation time (or scattering time, or lifetime) for state [math]\displaystyle{ (n,\mathbf{k}) }[/math], [math]\displaystyle{ w_{n\mathbf{k},n'\mathbf{k}'} }[/math] is the scattering weight, [math]\displaystyle{ g^{\nu}_{n\mathbf{k},n'\mathbf{k}'} }[/math] is the electron-phonon coupling matrix element, [math]\displaystyle{ f_{n\mathbf{k}} }[/math] is the population of the electronic state (Fermi-Dirac distribution), [math]\displaystyle{ n_{\nu\mathbf{q}} }[/math] is the population of the phononic state (Bose-Einstein distribution), [math]\displaystyle{ \varepsilon_{n\mathbf{k}} }[/math] is the energy of an electron band, [math]\displaystyle{ \omega_{\nu\mathbf{q}} }[/math] is the phonon frequency, and [math]\displaystyle{ \delta }[/math] is the Dirac delta function.

For SERTA, the scattering weight is:

[math]\displaystyle{ w_{n\mathbf{k},n'\mathbf{k}'} = 1 }[/math]

ELPH_SCATTERING_APPROX = ERTA_LAMDBA - Energy Relaxation-Time Approximation (mean-free path approximation)

Applies an energy-projected weight scaled by mean-free path (where [math]\displaystyle{ \mu }[/math] is the chemical potential):

[math]\displaystyle{ w_{n\mathbf{k},n'\mathbf{k}'} = \left(1 - \frac{\mathbf{v}_{n\mathbf{k}} \cdot \mathbf{v}_{n'\mathbf{k}'}}{|\mathbf{v}_{n\mathbf{k}}| |\mathbf{v}_{n'\mathbf{k}'}|} \left| \frac{\varepsilon_{n'\mathbf{k}'} - \mu}{\varepsilon_{n\mathbf{k}} - \mu} \right|\right) }[/math]

Warning: The formula above is correct and used from the next release of VASP onwards. In VASP 6.5.0 and 6.5.1, the following formula is used: [math]\displaystyle{ w_{n\mathbf{k},n'\mathbf{k}'} = \left(1 - \frac{\mathbf{v}_{n\mathbf{k}} \cdot \mathbf{v}_{n'\mathbf{k}'}}{|\mathbf{v}_{n\mathbf{k}}| |\mathbf{v}_{n'\mathbf{k}'}|}\right) \left| \frac{\varepsilon_{n'\mathbf{k}'} - \mu}{\varepsilon_{n\mathbf{k}} - \mu} \right| }[/math]

ELPH_SCATTERING_APPROX = ERTA_TAU - Energy Relaxation-Time Approximation (lifetime approximation)

[math]\displaystyle{ w_{n\mathbf{k},n'\mathbf{k}'} = \left(1 - \frac{\mathbf{v}_{n\mathbf{k}} \cdot \mathbf{v}_{n'\mathbf{k}'}}{|\mathbf{v}_{n\mathbf{k}}|^2} \left| \frac{\varepsilon_{n'\mathbf{k}'} - \mu}{\varepsilon_{n\mathbf{k}} - \mu} \right|\right) }[/math]

Warning: The formula above is correct and used from the next release of VASP onwards. In VASP 6.5.0 and 6.5.1, the following formula is used: [math]\displaystyle{ w_{n\mathbf{k},n'\mathbf{k}'} = \left(1 - \frac{\mathbf{v}_{n\mathbf{k}} \cdot \mathbf{v}_{n'\mathbf{k}'}}{|\mathbf{v}_{n\mathbf{k}}|^2}\right) \left| \frac{\varepsilon_{n'\mathbf{k}'} - \mu}{\varepsilon_{n\mathbf{k}} - \mu} \right| }[/math]

ELPH_SCATTERING_APPROX = MRTA_LAMDBA - Momentum Relaxation-Time Approximation (mean-free path approximation)

[math]\displaystyle{ w_{n\mathbf{k},n'\mathbf{k}'} = \left(1 - \frac{\mathbf{v}_{n\mathbf{k}} \cdot \mathbf{v}_{n'\mathbf{k}'}}{|\mathbf{v}_{n\mathbf{k}}| |\mathbf{v}_{n'\mathbf{k}'}|}\right) }[/math]

ELPH_SCATTERING_APPROX = MRTA_TAU - Momentum Relaxation-Time Approximation (lifetime approximation)

[math]\displaystyle{ w_{n\mathbf{k},n'\mathbf{k}'} = \left(1 - \frac{\mathbf{v}_{n\mathbf{k}} \cdot \mathbf{v}_{n'\mathbf{k}'}}{|\mathbf{v}_{n\mathbf{k}}|^2}\right) }[/math]

Related tags and articles

• Transport calculations
• Electronic transport coefficients
• Electron-phonon accumulators
• ELPH_RUN
• ELPH_TRANSPORT
• ELPH_TRANSPORT_DRIVER
• TRANSPORT_RELAXATION_TIME