Validity of the on-site spin-orbit coupling approximation

Spin-orbit coupling (SOC) is generally understood as a highly localized interaction within each atom, whereby core electrons holding large J splittings transfer the SOC to the valence electrons of the same atom, while their direct impact on neighbor valence orbitals is usually small. Seivane and Ferrer [Phys. Rev. Lett. 99, 183401 (2007)PRLTAO0031-900710.1103/PhysRevLett.99.183401] proposed an approach within a tight-binding type ab initio framework assuming that the transfer of SOC from core to valence orbitals only takes place when both are on the same atom, leading to the so-called on-site approximation, which then has been successfully applied to a variety of systems. In this work we thoroughly test its general validity by confronting SOC related properties such as spin splittings, spin textures, or magnetic anisotropies calculated under the on-site approximation versus the more general approach where all the contributions to the SOC, including three-center integrals, are explicitly included. After considering a variety of systems with different dimensionalities, all presenting a strong SOC, we conclude that although the on-site approximation often provides accurate results, it breaks down in some systems where 5d electrons are close to the Fermi level due to their strong SOC and moderately large spatial extension. Furthermore, there are a few examples where subtle inaccuracies lead to qualitatively wrong conclusions, the most clear case being the doping of the topological surface state in Bi2Se3(0001). Finally, magnetic anisotropy energies calculated under this approximation tend to be underestimated.
This work was financially supported by the Spanish MINECO, MICIU, AEI, and EU FEDER (Grants No. MAT2015-66888-C3-1R, No. RTI2018-097895-B-C41, No. PGC2018-094783, No. PGC2018-096955-C43, and No. PGC2018-096955-C44), Generalitat de Catalunya (Grant No. 2017SGR1506), and the EU MaX Center of Excellence (EU-H2020 Grant No. 824143). ICN2 is supported by the Severo Ochoa program from Spanish MINECO (Grant No. SEV-2017-0706) and the CERCA Program of Generalitat de Catalunya. ICMAB is supported by the Spanish MICINN through the Severo Ochoa Centers of Excellence Program (Grant No. CEX2019-000917-S). R.C. acknowledges the funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodoswka-Curie Grant Agreement No. 665919.
With funding from the Spanish government through the ‘Severo Ochoa Centre of Excellence’ accreditation (CEX2019-000917-S).