Noncollinear interlayer exchange coupling caused by interface spin-orbit interaction

The interlayer exchange couplings between neighboring ferromagnetic layers in ferromagnetic-nonmagnetic multilayer structures are derived analytically in the frame of the extended Anderson s-d mixing model with spin-orbit interaction. After transforming the extended Anderson mixing model into an s-d exchange model and taking into account the spin frustration at interfaces, the second-order perturbation calculation naturally gives rise to both the noncollinear Dzyaloshinski-Moriya (DM) exchange coupling as well as the usual isotropic Ruderman-Kittel-Kasuya-Yosida exchange coupling between the neighboring ferromagnetic layers. The isotropic and anisotropic exchange couplings have a decaying oscillatory behavior as a function of spacer layer thickness, but they differ by a phase factor of pi/2. While the exact value of the DM coupling depends on the material parameters as well as the interface s-d mixing effect, a rough estimate suggests that it is significant. Thus our result offers an alternative explanation to the noncollinear exchange coupling as was observed in the experiments.