Long-Range Spin-Orbit-Coupled Magnetoelectricity in Type-II Multiferroic NiI$_2$

Type-II multiferroics, where spin order induces ferroelectricity, exhibit strong magnetoelectric coupling. However, for the typical 2D type-II multiferroic NiI$_2$, the underlying magnetoelectric mechanism remains unclear. Here, applying generalized spin-current model, together with first-principles calculations and a tight-binding approach, we build a comprehensive magnetoelectric model for spin-induced polarization. Such model reveals that the spin-orbit coupling extends its influence to the third-nearest neighbors, whose contribution to polarization rivals that of the first-nearest neighbors. By analyzing the orbital-resolved contributions to polarization, our tight-binding model reveals that the long-range magnetoelectric coupling is enabled by the strong $e_g$-$p$ hopping of NiI$_2$. Monte Carlo simulations further predict a Bloch-type magnetic skyrmion lattice at moderate magnetic fields, accompanied by polar vortex arrays. These findings can guide the discovery and design of strongly magnetoelectric multiferroics.