Robust Wannierization including magnetization and spin-orbit coupling via projectability disentanglement

TL;DR

The paper tackles robust Wannierization in magnetic and spin-orbit coupled materials by extending PDWFs to include SOC and magnetization, and by automatically expanding the projection space with hydrogenic atomic orbitals. It introduces a fully automated workflow that handles $j$-dependent projections and orthogonalization, enabling high-throughput generation of accurate MLWFs up to $E_F+2$ eV across diverse chemistries. Benchmark results on 200 materials (including SOC) show dramatically improvedband interpolation quality, with $\eta_2$ typically below a few meV and a 100% success rate when external hydrogenic projectors are used. The work reduces pseudopotential dependence, supports collinear and non-collinear magnetism, and provides open-source, automated tools (AiiDA workflows) to widely enable PDWF-based tight-binding models for magnetic and SOC systems in materials discovery and property calculations.

Abstract

Maximally-localized Wannier functions (MLWFs) are widely employed as an essential tool for calculating the physical properties of materials due to their localized nature and computational efficiency. Projectability-disentangled Wannier functions (PDWFs) have recently emerged as a reliable and efficient approach for automatically constructing MLWFs that span both occupied and lowest unoccupied bands. Here, we extend the applicability of PDWFs to magnetic systems and/or those including spin-orbit coupling, and implement such extensions in automated workflows. Furthermore, we enhance the robustness and reliability of constructing PDWFs by defining an extended protocol that automatically expands the projectors manifold, when required, by introducing additional appropriate hydrogenic atomic orbitals. We benchmark our extended protocol on a set of 200 chemically diverse materials, as well as on the 40 systems with the largest band distance obtained with the standard PDWF approach, showing that on our test set the present approach delivers a 100% success rate in obtaining accurate Wannier-function interpolations, i.e., an average band distance below 15 meV between the DFT and Wannier-interpolated bands, up to 2 eV above the Fermi level.

Figures (6)

• Figure 1: Electronic band structure of BCC tungsten. The gray and black solid lines are the energy bands obtained directly from first-principles DFT calculations without and with SOC, respectively. The red dashed lines are the energy bands obtained from Wannier interpolation with SOC using our extended PDWF method. The Fermi level is marked as a horizontal blue dashed line.
• Figure 2: Band distance $\eta_2$ of 200 systems with different pseudopotentials, including SOC effects. Histogram (red) and cumulative histogram (blue) of the band distance $\eta_2$ of 200 spin-orbit coupling systems with different pseudopotentials sets: (a) the PseudoDojo library, and (b) the modified-pslibrary set (see main text). External hydrogenic atomic orbitals are introduced to the projectors to enhance the robustness of the PDWF method, making results obtained with the two pseudopotential libraries qualitatively very similar. As a comparison, the blue dashed lines are the cumulative histogram of $\eta_2$ without introducing such external projectors, exhibiting a lower success rate and a strong dependence on the pseudopotential library. The orange (green) vertical line is the mean (median) band distance $\eta_2$ of the 200 structures with external projectors; their values are shown in the legend of each panel. All 200 structures can be interpolated with a resulting $\eta_2\le 15$ meV for both pseudopotentials libraries, once hydrogenic atomic orbitals are introduced.
• Figure 3: Effect of the introduction of additional hydrogenic projectors on the band structure and projectability of AlCo. (a) DFT bands (black solid lines) compared with Wannier-interpolated bands (red dashed lines) along the R--M path for AlCo without additional hydrogenic projectors, including only the orbitals from the pseudopotential files in the PseudoDojo library VanSetten2018: $3s$, $3p$, $3d$, $4s$ orbitals for cobalt, and $3s$, $3p$ orbitals for aluminum. (b) Projectability for all $k$-points for the system of panel (a). The projectability starts to decrease rapidly when the energy is larger than the Fermi level. (c) DFT bands compared with Wannier-interpolated bands when additional hydrogenic $4p$ orbitals are included for cobalt. (d) Projectability for all $k$-points for the system of panel (c). With the help of hydrogenic AOs, the projectability remains close to one up to a few eV above $E_F$.
• Figure 4: Results of recalculating the 40 systems with largest band distance from Ref. Qiao2023. The red crosses are the original data directly from Ref. Qiao2023, whose projectors are PAOs from pseudopotentials in SSSP PBEsol Efficiency v1.1. The blue circles are the recalculated data where the same PAOs are complemented by additional hydrogenic AOs. The blue dashed rectangle area (which includes all blue circles) is zoomed in the inset, showing that after adding hydrogenic AOs, all 40 materials have a $\eta_2\le 10$ meV, thus demonstrating the effectiveness of our algorithm even in extremely challenging cases.
• Figure 5: Summary of the effect of several ingredients of the extended PDWF algorithm on the quality of Wannier interpolation. Histogram (red) and cumulative histogram (blue) of band distance $\eta_2$ for our test set of 200 structures, using different algorithms and projectors (all without SOC). All sets of calculations use the same number of projectors and same configurations of semi-core states. All frozen windows were set to $E_F + 2$ eV. (a) ED with the default all-hydrogenic AOs; (b) ED with the corrected all-hydrogenic AOs (see Table \ref{['text:tab:hydrogenic_radial_function']}); (c) PDWF with the corrected all-hydrogenic AOs; (d) ED with PAOs and external hydrogenic AOs; and (e) PDWF with PAOs and external hydrogenic AOs. The orange (green) vertical lines are the mean (median) band distance $\eta_2$, whose values are shown in the legend of each panel. The difference and the relation between each set of data are shown in the top right panel. Additional comparisons with further combinations of disentanglement method and starting projectors can be found in Supplementary Section \ref{['sm-sec:additional_compare']}.