ELPH_DECOMPOSE
ELPH_DECOMPOSE = [string]
Default: ELPH_DECOMPOSE = VDPR
Description: Chooses which contributions to include in the computation of the electron-phonon matrix elements.
| Mind: Available as of VASP 6.5.0 |
The electron-phonon matrix element can be formulated in the projector-augmented-wave (PAW) method in terms of individual contributions[1]. Each contribution can be included by specifying the associated letter in ELPH_DECOMPOSE. We suggest two different combinations to define matrix elements:
ELPH_DECOMPOSE = VDPR- "All-electron" matrix element[1][2]
ELPH_DECOMPOSE = VDQ- "Pseudo" matrix element[1][3]
Available contributions
- V - Derivative of pseudopotential, [math]\displaystyle{ \tilde{v} }[/math]
- [math]\displaystyle{ g^{(\text{V})}_{m \mathbf{k}', n \mathbf{k}, a} \equiv \langle \tilde{\psi}_{m \mathbf{k}'} | \frac{\partial \tilde{v}}{\partial u_{a}} | \tilde{\psi}_{n \mathbf{k}} \rangle }[/math]
- This term is the pure plane-wave contribution to the total PAW matrix element. If the PAW augmentation region were vanishingly small, this would be the sole contribution.
- D - Derivative of PAW strength parameters, [math]\displaystyle{ D_{a, ij} }[/math]
- [math]\displaystyle{ g^{(\text{D})}_{m \mathbf{k}', n \mathbf{k}, a} \equiv \sum_{bij} \langle \tilde{\psi}_{m \mathbf{k}'} | \tilde{p}_{b i} \rangle \frac{\partial D_{b, ij}}{\partial u_{a}} \langle \tilde{p}_{b j} | \tilde{\psi}_{n \mathbf{k}} \rangle }[/math]
- This contribution stems from the PAW treatment of the electronic Hamiltonian. It is of the same nature as [math]\displaystyle{ g^{(\text{V})} }[/math] but is treated in the local basis inside the augmentation region. For a detailed discussion of the PAW strength parameters, we refer to Ref. [4].
- P - Derivative of PAW projectors, [math]\displaystyle{ |\tilde{p}_{ai}\rangle }[/math]
- [math]\displaystyle{ \begin{split} g^{(\text{P})}_{m \mathbf{k}', n \mathbf{k}, a} & \equiv \sum_{ij} \langle \tilde{\psi}_{m \mathbf{k}'} | \frac{\partial \tilde{p}_{a i}}{\partial u_{a}} \rangle ( D_{a, ij} - \varepsilon_{n \mathbf{k}} Q_{a, ij} ) \langle \tilde{p}_{a j} | \tilde{\psi}_{n \mathbf{k}} \rangle \\ & + \sum_{ij} \langle \tilde{\psi}_{m \mathbf{k}'} | \tilde{p}_{a i} \rangle ( D_{a, ij} - \varepsilon_{m \mathbf{k}'} Q_{a, ij} ) \langle \frac{\partial \tilde{p}_{a j}}{\partial u_{a}} | \tilde{\psi}_{n \mathbf{k}} \rangle \end{split} }[/math]
- R - Derivative of PAW partial waves, [math]\displaystyle{ |\phi_{ai}\rangle }[/math] and [math]\displaystyle{ |\tilde{\phi}_{ai}\rangle }[/math]
- [math]\displaystyle{ g^{(\text{R})}_{m \mathbf{k}', n \mathbf{k}, a} \equiv (\varepsilon_{n \mathbf{k}} - \varepsilon_{m \mathbf{k}'}) \sum_{ij} \langle \tilde{\psi}_{m \mathbf{k}'} | \tilde{p}_{a i} \rangle R_{a, ij} \langle \tilde{p}_{a j} | \tilde{\psi}_{n \mathbf{k}} \rangle }[/math]
- with [math]\displaystyle{ R_{a, ij} \equiv \langle \phi_{a i} | \frac{\partial \phi_{a j}}{\partial u_{a}} \rangle - \langle \tilde{\phi}_{a i} | \frac{\partial \tilde{\phi}_{a j}}{\partial u_{a}} \rangle }[/math]
- Q - Derivative of PAW projectors, [math]\displaystyle{ |\tilde{p}_{ai}\rangle }[/math] (different eigenvalues)
- [math]\displaystyle{ \begin{split} g^{(\text{Q})}_{m \mathbf{k}', n \mathbf{k}, a} & \equiv \sum_{ij} \langle \tilde{\psi}_{m \mathbf{k}'} | \frac{\partial \tilde{p}_{a i}}{\partial u_{a}} \rangle ( D_{a, ij} - \varepsilon_{n \mathbf{k}} Q_{a, ij} ) \langle \tilde{p}_{a j} | \tilde{\psi}_{n \mathbf{k}} \rangle \\ & + \sum_{ij} \langle \tilde{\psi}_{m \mathbf{k}'} | \tilde{p}_{a i} \rangle ( D_{a, ij} - \varepsilon_{n \mathbf{k}} Q_{a, ij} ) \langle \frac{\partial \tilde{p}_{a j}}{\partial u_{a}} | \tilde{\psi}_{n \mathbf{k}} \rangle \end{split} }[/math]
- This contribution is very similar to [math]\displaystyle{ g^{(\text{P})} }[/math]. The only difference is in the Kohn-Sham eigenvalues. While [math]\displaystyle{ g^{(\text{P})} }[/math] uses the eigenvalues of both the initial and final state (so [math]\displaystyle{ \varepsilon_{n \mathbf{k}} }[/math] and [math]\displaystyle{ \varepsilon_{m \mathbf{k}'} }[/math]), [math]\displaystyle{ g^{(\text{Q})} }[/math] only uses the eigenvalues of the initial state ([math]\displaystyle{ \varepsilon_{n \mathbf{k}} }[/math]).
Related tags and articles
References
- ↑ a b c M. Engel, H. Miranda, L. Chaput, A. Togo, C. Verdi, M. Marsman, and G. Kresse, Zero-point renormalization of the band gap of semiconductors and insulators using the projector augmented wave method, Phys. Rev. B 106, 094316 (2022).
- ↑ L. Chaput, A. Togo, and I. Tanaka, Finite-displacement computation of the electron-phonon interaction within the projector augmented-wave method, Phys. Rev. B 100, 174304 (2019).
- ↑ M. Engel, M. Marsman, C. Franchini, and G. Kresse, Electron-phonon interactions using the projector augmented-wave method and Wannier functions, Phys. Rev. B 101, 184302 (2020).
- ↑ I. G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).