Systematic first-principles study of the on-site spin-orbit coupling in crystals

Spin-orbit coupling (SOC) provides a lot of interesting phenomena in condensed-matter physics, and the strength of the SOC is one of the key factors to control such physical properties. In this paper, we present a systematic first-principles study of the strength of SOC in crystals by using the on-site SOC, namely, the atomic SOC defined by Wannier functions of the corresponding atomic orbitals. By calculating the on-site SOC in isolated atoms, monatomic crystals, and binary compounds, we find that the valence dependence of the atomic SOC in isolated atoms cannot explain the strength of the on-site SOC in crystals. Instead, it is shown that the spread of the Wannier function well describes the material dependence and pressure dependence of the on-site SOC in crystals. The material dependence is found to be substantially larger than the valence dependence for isolated atoms. Particularly, for Bi, the strength of the on-site SOC for the Bi $6p$ orbitals changes from 1.0 eV (${\mathrm{Cs}}₃\mathrm{Bi}$) to 2.5 eV (${\mathrm{BiF}}₃$).