Twisted Bilayer NiI2: Spin-Lattice Multiferroicity

• TBN is a spin-lattice multiferroic characterized by moiré geometry-driven coupling of lattice relaxation and noncollinear spin order to yield distinct ionic and electronic polarization channels.
• A high-accuracy SpinGNN++ model, trained on 5,981 DFT data points, captures self-consistent lattice and spin interactions in supercells containing up to 10^5 atoms with minimal energy error.
• Angle-dependent regimes reveal a moiré-locked spin spiral at small twist angles and a near-60° anti-aligned polarization state, highlighting the critical role of structural reconstruction in emergent topological textures.

Twisted bilayer NiI$_2$ (TBN) is a twisted magnetic van der Waals bilayer in which moiré geometry, structural relaxation, and noncollinear spin order become directly coupled to ferroelectric response. In relaxed moiré superlattices, TBN exhibits cooperative ionic and spin-driven ferroelectricity: ionic out-of-plane dipoles coexist with purely electronic in-plane polarization, and the resulting textures depend sensitively on twist angle and on whether lattice relaxation is included. Within this framework, TBN is described as a “spin-lattice multiferroic,” with distinct small-angle and near-$60^\circ$ regimes supporting different polar-magnetic topologies (Zhua et al., 18 Jul 2025).

1. Moiré geometry and structural relaxation

For a commensurate twist angle $\theta$, the moiré wavelength is

$L_m \simeq \frac{a}{2\sin(\theta/2)}.$

At $\theta=2.13^\circ$, this gives

$$L_m \simeq \frac{3.97~\text{\AA}}{2\sin 1.065^\circ} \simeq 1.07\times 10^2~\text{\AA},$$

or approximately $10.7~\text{nm}$. This moiré scale sets the spatial period over which local stackings interpolate through the AA$\rightarrow$AB$\rightarrow$AB$^\prime$ pattern (Zhua et al., 18 Jul 2025).

Structural relaxation generates moiré-periodic “bumps” that modulate the interlayer spacing and sharpen the real-space domain structure. At $60^\circ$0, the local Ni–Ni interlayer distance varies from $60^\circ$1 to $60^\circ$2, corresponding to an amplitude $60^\circ$3. In the high-energy AB$60^\circ$4 domains, the top-layer Ni ions shift in plane by up to $60^\circ$5, with an opposite shift of the same magnitude in the bottom layer. The reported trend is that these local in-plane strains sharpen domain walls as $60^\circ$6.

The structural reconstruction is not a secondary correction. It is the source of the spatially varying interlayer registry that later controls both the magnetic texture and the polarization texture. A plausible implication is that the moiré lattice in TBN is best viewed not as a rigid geometric overlay but as a spin-lattice-coupled reconstruction field.

2. SpinGNN++ and the spin-lattice interatomic model

Large moiré supercells in twisted magnetic bilayers are challenging because the ionic and spin degrees of freedom must be treated self-consistently over length scales far beyond standard direct first-principles simulations. TBN was modeled with an $60^\circ$7-equivariant graph neural network, “SpinGNN++,” trained on $60^\circ$8 DFT total energies, forces, and torques for bilayer NiI$60^\circ$9 in both aligned and anti-aligned stackings. The model attains a mean absolute energy error of $\theta$0 and $\theta$1, and it is designed so that lattice displacements enter the atomic-representation layers while also modulating all pairwise exchange interactions. This enables self-consistent relaxation of ionic and spin degrees of freedom in moiré supercells containing approximately $\theta$2–$\theta$3 atoms (Zhua et al., 18 Jul 2025).

For bilayer AB stacking, the model reproduces the key spin-interaction terms listed below.

Interaction	Value / characterization
$\theta$4	$\theta$5, ferromagnetic intralayer
$\theta$6	$\theta$7
$\theta$8	$\theta$9
$L_m \simeq \frac{a}{2\sin(\theta/2)}.$0	$L_m \simeq \frac{a}{2\sin(\theta/2)}.$1
$L_m \simeq \frac{a}{2\sin(\theta/2)}.$2	$L_m \simeq \frac{a}{2\sin(\theta/2)}.$3
$L_m \simeq \frac{a}{2\sin(\theta/2)}.$4	$L_m \simeq \frac{a}{2\sin(\theta/2)}.$5, AFM preferred
$L_m \simeq \frac{a}{2\sin(\theta/2)}.$6	$L_m \simeq \frac{a}{2\sin(\theta/2)}.$7, easy-plane

These energy scales define the microscopic competition underlying the observed noncollinear textures. The same summary also reports a rigid stacking-energy difference of AB$L_m \simeq \frac{a}{2\sin(\theta/2)}.$8–AB $L_m \simeq \frac{a}{2\sin(\theta/2)}.$9, indicating that the moiré pattern samples locally inequivalent registries with appreciably different energetic preferences.

3. Electronic and ionic polarization channels

The electronic, spin-driven contribution is evaluated through a generalized KNB mechanism. For each intralayer Ni–Ni bond $\theta=2.13^\circ$0,

$\theta=2.13^\circ$1

or, when only the antisymmetric part matters,

$\theta=2.13^\circ$2

For the first layer, the dominant coupling-tensor elements are $\theta=2.13^\circ$3 and $\theta=2.13^\circ$4, while all other $\theta=2.13^\circ$5 (Zhua et al., 18 Jul 2025).

The ionic contribution is written as

$\theta=2.13^\circ$6

where each atom $\theta=2.13^\circ$7 with effective charge $\theta=2.13^\circ$8 is displaced by $\theta=2.13^\circ$9. For the relaxed moiré bilayer at $$L_m \simeq \frac{3.97~\text{\AA}}{2\sin 1.065^\circ} \simeq 1.07\times 10^2~\text{\AA},$$0, this produces a net out-of-plane dipole $$L_m \simeq \frac{3.97~\text{\AA}}{2\sin 1.065^\circ} \simeq 1.07\times 10^2~\text{\AA},$$1.

The key point is that TBN contains two distinct polarization channels with different microscopic origins and different spatial localization. The ionic channel is tied to structural reconstruction and interlayer-spacing modulation, whereas the electronic channel is tied to noncollinear spin texture through bond-resolved spin chirality. Their coexistence is the basis for the reported magnetoelectric textures.

4. Small-angle moiré-locked regime

For twist angles in the interval $$L_m \simeq \frac{3.97~\text{\AA}}{2\sin 1.065^\circ} \simeq 1.07\times 10^2~\text{\AA},$$2, the relaxed bilayer develops a “moiré-locked” spin spiral pattern whose local wavevector $$L_m \simeq \frac{3.97~\text{\AA}}{2\sin 1.065^\circ} \simeq 1.07\times 10^2~\text{\AA},$$3 rotates so as to follow the triangular AA–AB–AB$$L_m \simeq \frac{3.97~\text{\AA}}{2\sin 1.065^\circ} \simeq 1.07\times 10^2~\text{\AA},$$4 stacking map. This is the regime in which both ionic and spin-driven polarization mechanisms become prominent, and it is the central small-angle multiferroic regime identified for TBN (Zhua et al., 18 Jul 2025).

The polarization texture decomposes into complementary components. The out-of-plane ionic dipoles $$L_m \simeq \frac{3.97~\text{\AA}}{2\sin 1.065^\circ} \simeq 1.07\times 10^2~\text{\AA},$$5 reach peaks of $$L_m \simeq \frac{3.97~\text{\AA}}{2\sin 1.065^\circ} \simeq 1.07\times 10^2~\text{\AA},$$6 and are concentrated in the AB$$L_m \simeq \frac{3.97~\text{\AA}}{2\sin 1.065^\circ} \simeq 1.07\times 10^2~\text{\AA},$$7 domains, where the interlayer spacing is largest. The in-plane electronic polarization $$L_m \simeq \frac{3.97~\text{\AA}}{2\sin 1.065^\circ} \simeq 1.07\times 10^2~\text{\AA},$$8 in the $$L_m \simeq \frac{3.97~\text{\AA}}{2\sin 1.065^\circ} \simeq 1.07\times 10^2~\text{\AA},$$9 plane reaches peaks of $10.7~\text{nm}$0 and appears in the AA/AB regions, where a single spiral $10.7~\text{nm}$1 rotation is un-frustrated.

At $10.7~\text{nm}$2, these two components tile the moiré cell in a complementary, $10.7~\text{nm}$3-symmetric domain structure: ionic out-of-plane patches occupy the AB$10.7~\text{nm}$4 network, while electronic in-plane domains occupy the AA/AB network. This spatial complementarity is one of the defining features of the reported TBN state. A plausible implication is that electrical functionality in this regime could depend not only on polarization magnitude but also on how distinct polarization channels are spatially partitioned across the moiré cell.

5. Lattice relaxation, topological defects, and the absence of rigid-bilayer skyrmions

A central result is that twist alone does not generate the polar-magnetic topologies reported for TBN. In the rigid, unrelaxed bilayer at $10.7~\text{nm}$5—and more generally at small $10.7~\text{nm}$6—Monte Carlo/CG calculations using the same spin Hamiltonian yield uniform in-plane spin spirals nearly identical to those of untwisted bilayers. In that limit there is no spiral locking, no topological defects, and no skyrmions (Zhua et al., 18 Jul 2025).

By contrast, once ionic degrees of freedom are relaxed, spin-lattice couplings twist the spiral and generate converging $10.7~\text{nm}$7-vector patches described as precursors to topological spin defects. The conclusion drawn is explicit: interlayer ionic modulation is essential to break the residual symmetry that would otherwise forbid emergent moiré skyrmions.

This resolves a common simplification in moiré-magnet discussions, namely the assumption that geometric twisting by itself is sufficient to produce nontrivial topological textures. In TBN, the decisive ingredient is the relaxation-induced modulation of local stacking and spacing. The reported skyrmion-related behavior is therefore relaxation-enabled rather than twist-only.

6. Near-$10.7~\text{nm}$8 anti-alignment and the global angle-dependent phenomenology

Near $10.7~\text{nm}$9, TBN enters a qualitatively different regime. Under $\rightarrow$0 anti-alignment, sliding-induced ferroelectricity described by BSF theory produces stacking-dependent dipoles. The out-of-plane component satisfies $\rightarrow$1 at R-AB and R-AB$\rightarrow$2, while the in-plane component vanishes at R-AA and reaches $\rightarrow$3 at the midpoints between R-AB and R-AB$\rightarrow$4 (Zhua et al., 18 Jul 2025).

For a twisted mapping at $\rightarrow$5, this stacking-dependent polarization becomes a real-space field $\rightarrow$6 exhibiting vortex–antivortex, or meron–antimeron, textures around each R-AB/R-AB$\rightarrow$7 core. Each texture carries an integer winding number $\rightarrow$8 and obeys

$\rightarrow$9

In this near-$\rightarrow$0 regime, the magnetic state remains a uniform cycloid, but the polarization field is topological.

Across the full angle dependence, the reported regimes are:

• $\rightarrow$1 or $\rightarrow$2: nearly rigid-like behavior, with uniform in-plane spirals along $\rightarrow$3 and period $\rightarrow$4.
• $\rightarrow$5: moiré-locked spin spiral pattern with complementary ionic and electronic polarization.
• $\rightarrow$6: anti-aligned meron–antimeron polar network, with uniform magnetic cycloid and topological polarization.

The thresholds at $\rightarrow$7 and $\rightarrow$8 are identified as a spiral-locking transition. Taken together, these angle-dependent results define TBN as a platform where moiré geometry, structural relaxation, and noncollinear spin texture intertwine to produce ionic out-of-plane and purely electronic in-plane ferroelectric domains, while near $\rightarrow$9 the dominant topology shifts from the magnetic-polar coupling of the small-angle regime to a polarization-field meron–antimeron network.