RKKY interaction in three-dimensional electron gases with linear spin-orbit coupling

We theoretically study the impacts of linear spin-orbit coupling (SOC) on the Ruderman-Kittel-Kasuya-Yosida interaction between magnetic impurities in two kinds of three-dimensional noncentrosymmetric systems. It has been found that linear SOCs lead to the Dzyaloshinskii-Moriya interaction and the Ising interaction, in addition to the conventional Heisenberg interaction. These interactions possess distinct range functions from three dimensional electron gases and Dirac/Weyl semimetals. In the weak SOC limit, the Heisenberg interaction dominates over the other two interactions in a moderately large region of parameters. Sufficiently strong Rashba SOC makes the Dzyaloshinskii-Moriya interaction or the Ising interaction dominate over the Heisenberg interaction in some regions. The change in topology of the Fermi surface leads to some quantitative changes in periods of oscillations of range functions. The anisotropy of Ruderman-Kittel-Kasuya-Yosida interaction in bismuth tellurohalides family BiTe$X$ ($X$ = Br, Cl, and I) originates from both the specific form of Rashba SOC and the anisotropic effective mass. Our work provides some insights into understanding observed spin textures and the application of these materials in spintronics.