Capabilities (incomplete list)
Note : This is an incomplete list. Please refer to Documentation for details.
Integration
The code may be used to evaluate the following quantities, represented as Brillouin zone integrals.
Static (frequency-independent) quantities
'ahc': intrinsic anomalous Hall conductivity \(\sigma_{\alpha\beta}^{\rm AHE}\) (Nagaosa et al. 2010) via\[\sigma_{\alpha\beta}^{\rm AHE}=-\frac{e^2}{\hbar}\epsilon_{\alpha\beta\gamma}\int \frac{d{\bf k}}{(2\pi)^3}\Omega_\gamma({\bf k}).\]Anomalous Nernst conductivity (Xiao et al. 2006) \(\alpha_{\alpha\beta}^{\rm ANE}\) may be obtained from \(\sigma_{\alpha\beta}(\epsilon)^{\rm AHE}\) evaluated over a dense grid of Fermi levels \(\epsilon\)
(1)\[\alpha_{\alpha\beta}^{\rm ANE}=-\frac{1}{e}\int d\varepsilon \frac{\partial f}{\partial\varepsilon}\sigma_{\alpha\beta}(\varepsilon)\frac{\varepsilon-\mu}{T}, \label{eq:ANE}\]where \(f(\varepsilon)=1/\left(1+e^\frac{\varepsilon-\mu}{k_{\rm B}T}\right)\);
'Morb': orbital magnetization (Lopez et al. 2012)\[M^\gamma_n({\bf k})=\frac{e}{2\hbar}{\rm Im\,}\epsilon_{\alpha\beta\gamma}\int[d{\bf k}]\sum_n^{\rm occ}\Bigl[ \langle\partial_a u_{n{\bf k}}\vert H_{\bf k}+E_{n{\bf k}}-2E_F\vert\partial_b u_{n{\bf k}}\rangle\Bigr];\]'berry_dipole'and'berry_dipole_fsurf': berry curvature dipole\[D_{\alpha\beta}(\mu)=\int[d{\bf k}]\sum_n^{\rm occ} \partial_\alpha \Omega_n^{\beta}= \int[d{\bf k}]\sum_n^{\rm occ} \partial_\alpha E_{n\mathbf{k}} \Omega_n^{\beta} \delta(E_{n\mathbf{k}}-\mu)\]which describes nonlinear Hall effect (Sodemann and Fu 2015);
'gyrotropic_Korb'and'gyrotropic_Kspin' :gyrotropic magnetoelectric effect (GME) (Zhong, Moore, and Souza 2016) tensor (orbital and spin contributions) in the Fermi-sea formulation:\[K_{\alpha\beta}(\mu)=\int[d{\bf k}]\sum_n^{\rm occ} \partial_\alpha m_n^{\beta} ; \label{eq:gyro-K}\]'gyrotropic_Korb_fsurf'and'gyrotropic_Kspin_fsurf': gyrotropic magnetoelectric effect (GME) (Zhong, Moore, and Souza 2016) tensor (orbital and spin contributions) in the Fermi-surface formulation:\[K_{\alpha\beta}(\mu)=\int[d{\bf k}]\sum_n^{\rm occ} \partial_\alpha E_{n\mathbf{k}} m_n^{\beta} \delta (E_{n{\bf k}}-\mu)\]'conductivity_Ohmic'and'conductivity_Ohmic_fsurf'ohmic conductivity within the Boltzmann transport theory in constant relaxation time (\(\tau\)) - Femi-sea and Fermi-surface formula approximation:\[\sigma_{\alpha\beta}^{\rm Ohm}(\mu) =\tau\int[d{\bf k}]\sum_n^{E_{n{\bf k}}<\mu} \partial^2_{\alpha\beta} E_{n{\bf k}} =\tau\int[d{\bf k}]\sum_n^{\rm occ} \partial_\alpha E_{n{\bf k}}\partial_\beta E_{n{\bf k}} \delta(E_{n{\bf k}}-\mu) ; \label{eq:ohmic}\]'dos': density of states \(n(E)\)'cumdos': cumulative density of states\[N(E) = \int\limits_{-\infty}^En(\epsilon)d\epsilon. \label{eq:cDOS}\]'shc_static_ryoo'and'shc_static_qiao': Kubo-Greenwood formula for static spin Hall conductivity (SHC) (Ryoo, Park, and Souza 2019) or (Qiao, Zhou, Yuan, and Zhao 2018). Equivalent to setting \(\omega=0\) in'opt_SHCryoo'and'opt_SHCqiao'.\[\sigma^{\gamma}_{\alpha\beta}(\mu) = \frac{e\hbar}{N_k\Omega_c} \sum_{\bf k} \sum_n^{\rm occ} \Omega^{{\rm spin};\,\gamma}_{\alpha\beta, n}({\bf k}),\]where
\[\Omega^{{\rm spin};\,\gamma}_{\alpha\beta, n}({\bf k}) = -2 {\rm Im} \sum_l^{\rm unocc} \frac{\langle\psi_{n{\bf k}}\vert \frac{1}{2} \{ s^{\gamma}, v_\alpha \} \vert\psi_{l{\bf k}}\rangle \langle\psi_{l{\bf k}}\vert v_\beta\vert\psi_{n{\bf k}}\rangle} {(\varepsilon_{n{\bf k}}-\varepsilon_{l{\bf k}})^2}.\]
Dynamic (frequency-dependent) quantities
'opt_conductivity': Kubo-greenwood formula for optical conductivity (example)(2)\[\sigma_{\alpha\beta}(\hbar\omega)=\frac{ie^2\hbar}{N_k\Omega_c} \sum_{\bf k}\sum_{n,m} \frac{f_{m{\bf k}}-f_{n{\bf k}}} {\varepsilon_{m{\bf k}}-\varepsilon_{n{\bf k}}} \frac{\langle\psi_{n{\bf k}}\vert v_\alpha\vert\psi_{m{\bf k}}\rangle \langle\psi_{m{\bf k}}\vert v_\beta\vert\psi_{n{\bf k}}\rangle} {\varepsilon_{m{\bf k}}-\varepsilon_{n{\bf k}}-(\hbar\omega+i\eta)}.\]'opt_shiftcurrent': shift photocurrent (PRB 2018)(3)\[\sigma^{abc}(0;\omega,-\omega) = -\frac{i\pi e^3}{4\hbar^2} \sum_{\bf k}\sum_{n,m}\left( f_{n{\bf k}}-f_{m{\bf k}} \right) \left(I^{abc}_{mn}+I^{acb}_{mn}\right) \times \left[\delta(\omega_{mn}-\omega)+\delta(\omega_{nm}-\omega)\right].\]where \(I^{abc}_{mn}=r^b_{mn}r^{c;a}_{nm}\); \(r^a_{\mathbf{k}nm}=(1-\delta_{nm})A^a_{\mathbf{k} nm}\); \(r^{a;b}_{\mathbf{k} nm}=\partial_b r^a_{\mathbf{k} nm} -i\left(A^b_{\mathbf{k}nn}-A^b_{\mathbf{k} mm}\right)r^a_{\mathbf{k} nm}\); \(A^a_{\mathbf{k} nm}=i\langle{u_{\mathbf{k} n}}|{\partial_a u_{\mathbf{k} m}}\rangle\).
'opt_SHCryoo'and'opt_SHCqiao': Kubo-Greenwood formula for spin Hall conductivity (SHC) under time-reversal symmetry (Ryoo, Park, and Souza 2019) or (Qiao, Zhou, Yuan, and Zhao 2018)\[\sigma^{\gamma}_{\alpha\beta}(\hbar\omega)=\frac{-e\hbar}{N_k\Omega_c} \sum_{\bf k}\sum_{n,m} \left(f_{n{\bf k}}-f_{m{\bf k}}\right) \frac{\textrm{Im}\left[\langle\psi_{n{\bf k}}\vert \frac{1}{2}\{ s^{\gamma}, v_\alpha \} \vert\psi_{m{\bf k}}\rangle \langle\psi_{m{\bf k}}\vert v_\beta\vert\psi_{n{\bf k}}\rangle\right]} {(\varepsilon_{n{\bf k}}-\varepsilon_{m{\bf k}})^2-(\hbar\omega+i\eta)^2}.\]
Tabulating
Fig. 1 Fermi surface of bcc iron, colored by the Berry curvature \(\Omega_z\). Figure produced using FermiSurfer.
WannerBerri can also tabulate certain band-resolved quantities over the
Brillouin zone producing files Fe_berry-?.frmsf, containing the Energies
and Berry curvature of bands 4-9 (band counting starts from zero).
The format of the files allows to be directly passed to the
FermiSurfer visualization tool (Kawamura 2019) which can produce a
plot like Fig. 1. Transformation of files to other
visualization software is straightforward.
Some of the quantites that are available to tabulate are:
'berry': Berry curvature [Å2]\[\Omega^\gamma_n({\bf k})=-\epsilon_{\alpha\beta\gamma}{\rm Im\,}\langle\partial_\alpha u_{n{\bf k}}\vert\partial_\beta u_{n{\bf k}}\rangle;\]'morb': orbital moment of Bloch states [eV·Å2]\[m^\gamma_n({\bf k})=\frac{e}{2\hbar}\epsilon_{\alpha\beta\gamma}{\rm Im\,}\langle\partial_\alpha u_{n{\bf k}}\vert H_{\bf k}-E_{n{\bf k}}\vert\partial_\beta u_{n{\bf k}}\rangle;\]'spin': the expectation value of the Pauli operator [ħ]\[\mathbf{s}_n({\bf k})=\langle u_{n{\bf k}}\vert\hat{\bf \sigma}\vert u_{n{\bf k}}\rangle;\]'V': the band gradients [eV·Å] \(\nabla_{\bf k}E_{n{\bf k}}\).'spin_berry': Spin Berry curvature [ħ·Å2]. Requires an additional parameterspin_current_typewhich can be"ryoo"or"qiao".\[\begin{split}\Omega^{{\rm spin};\,\gamma}_{\alpha\beta, n}({\bf k}) = -2 {\rm Im} \sum_{\substack{l \\ \varepsilon_{l{\bf k}} \neq \varepsilon_{n{\bf k}}}} \frac{\langle\psi_{n{\bf k}}\vert \frac{1}{2} \{ s^{\gamma}, v_\alpha \} \vert\psi_{l{\bf k}}\rangle \langle\psi_{l{\bf k}}\vert v_\beta\vert\psi_{n{\bf k}}\rangle} {(\varepsilon_{n{\bf k}}-\varepsilon_{l{\bf k}})^2}.\end{split}\]
Evaluation of additional matrix elements
In order to produce the matrix elements that are not evaluated by a particular ab initio code, the following interfaces have been developed:
mmn2uHu
The wannierberri.utils.mmn2uHu module evaluates the (.uHu file) containing the matrix elements needed for orbital moment calculations
on the basis of the .mmn and .eig files by means of the sum-over-states formula
and the (.sHu and .sIu file) containing the matrix elements needed for Ryoo’s spin current calculations(Ryoo, Park, and Souza 2019)
on the basis of the .mmn, .spn and .eig files by means of the sum-over-states formula
see sec-mmn2uHu for more details
vaspspn
The wannierberri.utils.vaspspn computes the spin matrix
based on the normalized pseudo-wavefunction read from the WAVECAR file written by
VASP
see sec-vaspspn for more details
The wannierberri.utils.mmn2uHu and wannierberri.utils.vaspspn modules were initially developed and
used in (Tsirkin, Puente, and Souza 2018) as separate scripts, but were
not published so far. Now they are included in the WannierBerri
package with a hope of being useful for the community.