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Exchange tensors, generalized RKKY interactions, and magnetization dynamics in heterostructures of ferromagnets and topological insulators

We present a comprehensive theoretical analysis of magnetic heterostructures composed of ferromagnetic (FM) layers interfaced with three-dimensional topological insulators (TIs). Integrating out the topological surface states and computing the spin determinant to second order in spins, we derive the effective generalized Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interactions mediated by topological surface states. These interactions inherently incorporate spin-momentum locking and anisotropic spin susceptibilities stemming from the Dirac-like dispersion of the TI surface electrons. The analysis reveals that the interplay between the spin-orbit coupling intrinsic to the TI and the magnetization in the FM layer induces highly nonlocal and retarded, chiral, and Dzyaloshinskii-Moriya (DM)-like contributions to the effective spin Hamiltonian. Furthermore, the spin dynamics is studied through a derivation of the LLG equation for this problem. The induced interactions renormalize many of the FM's intrinsic properties, but a new term in the LLG equation is induced that is related to the rate of change of the magnetization's curl, which is relevant to skyrmion dynamics. The magnon dispersion exhibits modifications due to the TI-mediated interactions, including a softened inertial spin-wave mode and tunable magnon gaps, sensitive to a tunable chemical potential and interfacial exchange coupling strength. The results also apply to finite temperatures. They elucidate topologically induced magnetic phenomena and pave the way for engineering exotic spin textures, such as skyrmions and chiral domain walls, in TI-FM hybrid systems with tunable interactions.