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

TL;DR

This work develops a field-theoretic framework for ferromagnet-topological insulator heterostructures, where TI surface Dirac fermions mediate generalized RKKY-like spin interactions in the FM. By integrating out the TI degrees of freedom and expanding to second order in spins, the authors derive an anisotropic exchange tensor with Dzyaloshinskii–Moriya-type terms and a new off-diagonal, curl-coupled contribution to the Landau-Lifshitz-Gilbert dynamics. In the static limit, the TI-induced interactions exhibit RKKY-like oscillations with a period set by the Fermi momentum and an amplitude tunable by the chemical potential $\mu$ and interfacial exchange $\bar{J}$; the DM terms are zero inside the TI gap and become finite outside it, with angular dependence set by the SOC form. Dynamically, the TI proximity renormalizes FM parameters and introduces a curl-related term that affects skyrmion breathing and magnon spectra, including a tunable magnon gap and a softening inertial mode, which together enable controllable, topologically engineered spin textures at finite temperature.

Abstract

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.

Figures (3)

• Figure 1: Summary of the static spin theory in real space $\Delta {\mathbf{r}}$, containing induced anisotropic Heisenberg exchange interactions $J_\alpha \delta_{\alpha \beta}(\Delta {\mathbf{r}})$, Dzyaloshinskii–Moriya interactions $D_\gamma(\Delta {\mathbf{r}}) \epsilon_{\gamma \alpha \beta}$, and an off-diagonal spin-symmetric interaction $T_{\alpha \beta}^\text{sym}(\Delta {\mathbf{r}})(1-\delta_{\alpha \beta}).$ The displacement vector $\Delta {\mathbf{r}}$ connects two spin components, $S_\alpha({\mathbf{r}})$ and $S_\beta({\mathbf{r}} + \Delta {\mathbf{r}})$, and is given in units of the lattice constant $a$. The numerical values shown are obtained at chemical potential $\mu=300m eV,$ the Fermi velocity given by $\hbar v_{\mathrm{F}}/a=200m eV$, and interfacial exchange coupling $\bar{J}=75m eV$ using the choice of spin-orbit coupling (SOC) $\vb{d}_1$. The other choice $\vb{d}_2$ can be obtained by shifting the polar coordinate $\phi\to \phi+\pi/2$. The amplitudes remain the same for both kinds of SOC.
• Figure 2: Spin-spin interaction coefficients appearing in Eq. \ref{['eq:H_eff_prime_form']} in the main text as a function of chemical potential $\mu$. All coefficients are constant for $\mu$ in the gap, $m_0=2\bar{J}$, given by the interfacial exchange interaction $\bar{J}=35m eV.$ The DMI term $D^{(0,1)}$ and the skyrmion-dynamics term $T^{(1,1)}$ display a switching effect, where they suddenly become zero whenever $\abs{\mu} < m_0.$
• Figure 3: Spin-wave dispersion relations for the precession modes $\omega_-$ and the nutation modes $\omega_+$ using chemical potential $\mu=80m eV$ for the outside-the-gap curves, interfacial exchange $\bar{J}=35m eV$, and Fermi velocity $\hbar v_{\mathrm{F}}/a=500m eV$. The FM's intrinsic easy-axis anisotropy was set to $K=1m eV$, and the intrinsic exchange interaction to $J_\text{ex}=150m eV$. Further analysis and interpretations of negative frequencies are explained in the main text.