Effects of non-local emergent electromagnetic fields driven by Rashba spin-orbit coupling

We theoretically study effects of emergent electromagnetic fields induced by the Rashba spin-orbit coupling (RSOC) at a distance from the exchange coupling (ExC), considering, for example, the tri-layer metallic structure of ferromagnetic (FM), normal (NM), and heavy (HM) metals, where ExC is only in FM, and RSOC is near the interface between NM and HM. In such a structure, the magnetization dynamics in FM may give rise to the spin-motive force at the interface between NM and HM due to RSOC. We show that the motive force can be understood as that driven by a non-local emergent electric field, in which RSOC connected to ExC through the diffusion of electrons. We also find that a magnetic field as the counterpart of the electric field does not propagate in equilibrium but propagates when the electric current parallel to the layers flows. The Rashba-Edelstein magnetoresistance can be understood as a normal magnetoresistance due to the non-local emergent magnetic field. The calculations are done using the Green function and the Feynman diagram without setting specific geometries like the above tri-layer structure.