First Principles Evaluations of the Spin-Orbit Couplings in Crystals*

The spin-orbit coupling (SOC) plays a crucial role in many spin-related phenomena such as anomalous Hall effect and spin Hall effect. However, the strength of the SOC in solids and its relation to the isolated limit have not been well understood. Here, we quantify the atomic SOC in solids by constructing the tight-binding Hamiltonian from the Wannier functions obtained by the first-principles calculation. By calculating the atomic SOC in monatomic crystals and binary compounds systematically, we find that the strength of the atomic SOC is not the atom-specific property but depends much on the crystal structure and chemical composition. We show that its crystal dependence is well explained by the spread of the Wannier function.