LaBr₂ bilayer multiferroic moiré superlattice with robust magnetoelectric coupling and magnetic bimerons

Abstract

Two-dimensional (2D) van der Waals (vdW) materials provide the versatile playground to stack two or more vdW layers for creation of superior materials with desired properties. Here we theoretically adopt a twisted stack-engineering of two LaBr₂ monolayers to break space inversion symmetry for ferroelectricity and ultimately multiferroism. The enhancement and reversal of electric polarization are accompanied with the transition from interlayer ferromagnetic and antiferromagnetic orderings, demonstrating an effective magnetoelectric coupling effect with a mechanism dissimilar to that of the conventional multiferroics. Magnetization dynamics simulations show that such magnetic phase transition can excite topologically protected bimeron, and the skyrmion Hall effect can be suppressed by bilayer-bimeron stabilized in both ferromagnetic and antiferromagnetic configurations. Moreover, in the small-angle twisted moiré superlattice, the uniform polarization will evolve into a staggered domain structure, accompanied with the appearance of bimeron, which forms a significant discrepancy with the non-twisted stack-engineered multiferroic LaBr₂ bilayer. This work provides a strategy for 2D multiferroic materials by twisted stack engineering of magnetic single layers.

Introduction

Multiferroic materials with robust magnetoelectric coupling is highly attractive with significant applications prospect in the field of spintronics and electronics such as high density and low energy consuming nonvolatile memories and high sensitivity sensors^1,2. However, the basic contradiction between the required empty d-electron configuration of the conventional ferroelectric (FE) phase and the partially filled d orbit demanded for the traditional magnetic phase, results in the scarcity of single phase multiferroic material with coexistence of the two different ferric properties^3,4.

The successful exfoliation of graphene⁵, has led to revolution in discovering two-dimensional (2D) van der Waals (vdW) materials with superior physical properties as well as revolution in materials engineering^{6,7,8,9,10,11}. This allows multiferroics to be achieved by simply stacking of ferromagnetic (FM) and FE monolayers to form heterostructure (HS). Such 2D multiferroic HS has shown strong magnetoelectric coupling, i.e, nonvolatile control of long-range magnetic ordering as well as topologically nontrivial spin textures by FE polarization^{12,13,14,15,16}. However, the limited availability of individual 2D FM and 2D FE materials hinder the stacking of 2D FM and 2D FE for 2D multiferroic HS. Furthermore, the asymmetric interfacial coupling in this type of 2D HS may seriously distort the FE double-well potential, thus destabilizing the ferroelectricity and causing the failure of stacking strategy¹⁷.

The recent vdW stack engineering of simple non-polar single layers to break the space inversion symmetry through the relative twisting between layers to induce out-of-plane polarization, has revolutionized the design 2D FE HS materials¹⁸. For example, the spontaneous out-of-plane ferroelectric polarization in stacked WTe₂ bilayer and BN bilayer can be reversed upon interlayer sliding of an in-plane translation by an external electric field^19,20,21. As the polarization in the stacked bilayers is caused by interlayer stacking, the rule for empty d for ferroelectricity are not applied, which can eliminate the contradiction for the FM and FE coexisting. The stacking induced ferroelectric polarization can be applied to construct 2D vdW multiferroic materials²². By twisted stacking 2D vdW magnetic bilayer, ferroelectric polarization can be induced, while the system will remain magnetic due to the existing of transition metal elements. The expected coexisting of ferroelectric polarization and magnetism will enable the system multiferroic. In addition, stacking of the identical materials can effectively address the potential ferroelectric damage due to asymmetric interface coupling in FM/FE HS.

The topological magnetic state with whirling spin configuration has attracted great interest in 2D materials. The topology protection and transport properties endow those topological magnetic states with great advantages in serving as information carriers for next-generation information storage and processing devices^23,24,25,26. Those topological spin configurations can be strongly affected by the magnetic anisotropy in 2D magnets. Particularly, magnetic skyrmion emerges in easy-axis magnets while bimeron in easy-plane magnets^{27,28,29,30,31,32}. Furthermore, by integrating with 2D vdW multiferroic materials, topological spin textures can be intriguingly tuned by polarization reversal, allowing skyrmion-based devices in atom-thick level with extreme low energy consumption^15,16.

Atomically thin layers of 2D materials can be assembled in vertical stacking, where their strong covalent bonds provide the in-plane stability, and the relatively weak vdW-like forces hold the advantage of easy-stacking^8,9, enabling coupling between monolayer crystals with incommensurate lattices and arbitrary mutual rotation. When single layers of 2D materials are stacked on top of one another with a small twist in orientation, the way of layers stacked will change periodically and smoothly over long range, and the resulting periodic moiré superlattices defines a different length and energy scale^33,34,35, which offers an additional degree of freedom for tuning material properties. For example, exotic and relevant physical phenomena, such as Mott insulators and unconventional superconductivity, have recently been observed in twisted graphene with small magic angles^36,37,38. For 2D bilayer magnets, such a long-period lateral modulation of spatial change of the stacking order can be available for controlling of interlayer exchange coupling and spatial symmetry^39,40,41, which gives rise to strong coupling between magnetism and polarization through interlayer interactions.

In this work, we investigated non-twisted (3 R stacking) and 180° twisted (2H stacking) stacking configurations of magnetic LaBr₂ bilayers, which were subsequently extended to the entire moiré pattern with smooth variations in local stacking at small twisting angles. Through our density functional theory (DFT) calculations, we demonstrate that the different stacking types of 2H and 3 R exhibit antiferromagnetic (AFM) and FM states, respectively (as shown in Fig. 1a, b). In addition, 3 R stacking induces an out-of-plane spontaneous polarization, and a strong magnetoelectric coupling effect occurs in this type, we designate it as stacking-engineered multiferroics. In 3 R type LaBr₂ bilayer, the enhancement and reversal of ferroelectric polarization will trigger magnetic phase transition. Spin dynamics simulations show that this can excite topologically protected bimeron from long-range magnetic orderings, and the skyrmion Hall effect can be suppressed by bilayer-bimeron stabilized in FM or AFM configurations. In addition, for twisting 3R-type LaBr₂ bilayer at small angles, 3R-type stacking configurations with opposite polarization exist simultaneously in the local region of the moiré superlattice, thus leading to staggered polarization domains and magnetic bimeron stabilized in domains. This indicates that there is a significant discrepancy between non-twisted and twisted stacking-engineered 2D multiferroics. Our work paves the way for the design of 2D HS multiferroics by twisted stacking 2D VDW magnetic single layers with strong ME coupling as well as skyrmion devices.

Fig. 1: The lattice structure of bilayer LaBr₂.

a, b show both the side and top views of 2H- and 3R-type stacking, respectively. S_La represents the spin direction of La ions in upper and lower layer. c The splitting of 5d orbitals for La²⁺ in the trigonal prismatic crystal field. d Spin density for 3 R LaBr₂ bilayer. e The maximally localized Wannier of La-5d_z² states for 2H and 3 R type. Yellow and cyan colors represent positive and negative, respectively, isovalue surfaces of the Wannier function. The red and blue arrows represent FM and AFM superexchange paths between different atoms, respectively.

Result

3 R stack-engineering induced multiferroicity and magnetoelectric coupling

Bulk LaBr₂ is a hexagonal layered crystal with a space group of P6₃/mmc⁴². Recently, a monolayered LaBr₂ has been proposed as an ideal FM semiconductor^43,44. As shown in Fig. 1c, each La atom is trigonal prismatic coordinated to six Br atoms. Under such a condition, the 5d orbitals of La²⁺ split into three groups, where the lowest energy d_Z² orbitals are filled with d¹ electrons, and giving rise to a finite magnetic moment of 1 μ_B on each La atom. The spin density distributions of 3 R stacking in Fig. 1d present that one part of spin density is distributed in the d_z² orbitals and the other is delocalized in the intralayer hexagonal cavity. The existence of such a group of La-d_z² orbitals, due to the weak electronegativity of La ions, enables the electrons in LaBr₂ more favorable to involve in interlayer coupling, serving as an ideal physical platform for the study of stacking-engineered vdW layered materials. Here, we consider two types of stacking orderings for the bilayer LaBr₂: 3 R and 2H stacking, see Fig. 1a, b. 2H type is a stacked structure consistent with the bulk that rotates the lower layer (relative to the upper layer) by 180°, which maintains the spatial inverse symmetry of the structure. In contrast, the 3 R type is a stacking of two LaBr₂ monolayers without rotation, thereby enabling the design of synthetic stacking-engineered layered multiferroic materials due to the broken space inversion symmetry. In addition, the results by DFT calculations show that the difference in stacking energy between the 2H and 3 R stacking is 0.16 meV per unit, and the very small energy difference allows both types of the LaBr₂ bilayer to be stable.

The stacking of vdW magnetic layers can alter heterostructure symmetry, accompanied with the further change of interlayer coupling and even the overall magnetic properties, which is exemplified by the AFM and FM ordering of CrI₃ bilayer that correspond to a high-temperature and a low-temperature structure caused by interlayer sliding^22,45,46, respectively. For LaBr₂ bilayer, the related magnetism parameters, such as J and K were obtained by considering the following classical Heisenberg model for local spins:

$$\begin{array}{l}H = - K\mathop {\sum}\limits_i {\left( {{{{\mathbf{m}}}}_i \cdot {{{\hat{\mathbf z}}}}} \right)^2 - J_{{{\mathrm{L}}}}\mathop {\sum}\limits_{i,j} {{{{\mathbf{m}}}}_i \cdot {{{\mathbf{m}}}}_j - J_{{{{\mathrm{t}}}}1}\mathop {\sum}\limits_{i,j} {{{{\mathbf{m}}}}_i \cdot {{{\mathbf{m}}}}_j } }} \\ \qquad\,\,-\, J_{{\mathrm{t}}2}\mathop {\sum}\limits_{i,j} {{{{\mathbf{m}}}}_i \cdot {{{\mathbf{m}}}}_j - J_{{{{\mathrm{b}}}}1}\mathop {\sum}\limits_{i,j} {{{{\mathbf{m}}}}_i \cdot {{{\mathbf{m}}}}_j - J_{{{{\mathrm{b}}}}2}\mathop {\sum}\limits_{i,j} {{{{\mathbf{m}}}}_i \cdot {{{\mathbf{m}}}}_j} } } \end{array}$$

(1)

where, |m| = 1 μ_B is the magnetic moment of La²⁺, K is the single-ion perpendicular magnetocrystalline anisotropy arising from spin–orbital coupling, J_L is the interlayer exchange interaction, J_t1, J_b1, J_t2, and J_b2 represent the nearest and the next-nearest intralayer exchange interactions in upper and lower layer respectively, as shown in Fig. 1a, b. We performed the DFT calculations of interlayer couplings for 2H and 3 R stacking phases, which are summarized in Table 1. The 2H type maintains the interlayer AFM exchange while the 3 R type possesses weak interlayer FM exchange. Through the analysis of maximally localized Wannier orbitals⁴⁷, we further show, in Fig. 1e, that the main interaction path for interlayer exchange interaction J_L in our structure is the electron hopping between La-d_z² and La’-d_z² through the La-Br’-La’ or La’-Br-La bondings. Taking La-Br’-La’ superexchange interaction as an example, the La-d_z² orbital forming a σ-bonding with the Br’-p_z orbital, electron can transfer from Br’-p_z to La-d_z² orbital and the semicovalent exchange is achieved during this process. Such a covalent component extends the cation wavefunction out over the anions to give an orbital overlapping for the superexchange electron transfer. In addition, a π-bonding between Br’-p_z and the La’-d_z² orbitals enable a virtual electron transfer. According to Goodenough–Kanamori–Anderson rule^48,49,50, La-Br’-La’ exhibits AFM superexchange. The lattice symmetry determines that the other half of the loops La’-Br-La are also AFM superexchange, resulting in a strong antiferromagnetism for 2H-type stacking. For 3 R type, the AFM superexchange of La-Br’-La’ is retained, while the broken inversion symmetry of this stacking reduces the wave function mixing of La’-Br-La exchange interaction, making La’-Br-La exhibit FM superexchange interaction. Therefore, a competition occurs in 3 R type between interlayer FM and AFM superexchange, which eventually manifests as weak FM coupling. The calculation of the magnetic anisotropy was also carried out for 2H and 3 R types. They both show in-plane anisotropy with a value of −24 μeV, suggesting that the interlayer configuration has little effect on the magnetic anisotropy. In addition, we performed Monte Carlo simulations based on the Heisenberg model from Eq. 1 to evaluate the phase transition temperature for 2H and 3 R type, which are 66 K and 52 K (shown in Supplementary Fig. 1), respectively, higher than that of the experimentally known phase transition temperature of CrI₃ monolayer (45 K)⁵¹.

Table 1 Exchange interaction (J, in meV) and magnetocrystalline anisotropy (K, in meV) constant of 2H- and 3R-type LaBr₂ bilayer.

Calculations based on the Berry phase method, the FE polarization value P of 3 R stacked magnetic bilayer LaBr₂ is 1.16 × 10⁻⁴ C m⁻², which is comparable to the experimentally measured magnitude of nonmagnetic WTe₂ (3.204 × 10⁻⁴ C m^{−2 19}) and much higher than experimental value of BN (2.25 × 10⁻¹² C m^{−2 18}). Similar to WTe₂ and BN bilayer, where the polarization switching by applied electric field has been experimentally achieved^18,19, the out-of-plane FE polarization of magnetic bilayer LaBr₂ can also be reversed by a slight interlayer sliding, as displayed by the blue arrow in Fig. 1b. By using the climbing image nudged elastic band (CI-NEB) calculations⁵², we found that the FE switching should overcome an energy barrier about 14.81 meV per unit cell, as shown in Fig. 2a. In addition, interlayer sliding of an in-plane translation for polarization reversion induces an interlayer FM-AFM-FM phase transition, as shown in Fig. 2b. Such a magnetic phase transition was a result of competition of FM and AFM superexchange, which enables the manipulation of magnetism through manipulation of overall the interlayer exchange coupling. Based on the fact that both magnetism and polarization are sensitive to the variations of interlayer spacing, we investigated the compressive vertical strains effect on magnetization and polarization by changing the interlayer distance of two LaBr₂ layers in 3 R stacking, as displayed in Fig. 2c–e. The changing of the interlayer distance δ = d – d₀, where d and d₀ are the strained and equilibrium interlayer distances, will drive the bilayer LaBr₂ transition from FM to AFM accompanied by a significant enhancement in out-of-plane polarization. In contrast, the compressive vertical strains applied on 2H-type stacking will not change the AFM ordering, demonstrating that the stable AFM exchange of 2H type, as shown in Supplementary Fig. 2.

Fig. 2: Ferroelectricity and magnetism of 3 R LaBr₂ bilayer.

a FE switching pathway of 3 R LaBr₂ bilayer, the red arrows denote the polarization direction. b Total energy difference between AFM and FM in FE switching pathway. c defines the distance d between two LaBr₂ layers. d, e Energy difference (E_AFM-E_FM) and polarization (P) are as a function of δ, respectively.

It is imperative to reveal the coupling mechanism between ferromagnetism and ferroelectricity in 3R-type LaBr₂ bilayer, which is the key of instruction for design superior 2D multiferroic system with robust magnetoelectric coupling. As the charge accumulation after stacking those two LaBr₂ layers in Fig. 3a shows, in 3 R stacking, the La The above results indicates atoms in the upper layer sit above the Br’ atoms in the lower layer, while the Br atoms in the upper layer lay above the empty site at the centre of the triangle in the lower layer. Such the atomic arrangement allows La-d_z² and Br’-p_z orbitals to overlapping, and the accumulated electrons surrounded Br’ atoms, thereby generating an electric dipole moment along the positive z direction. Due to the reduce of layer distance, the orbitals overlapping and electric dipole moment increase, as shown in Fig. 3b, c. Note that charge transfer also exists in 2H type stacking (see Fig. 3a), but the resulted dipoles cancel each other out due to structural inversion symmetry. In addition, strain-enhanced orbital overlapping in 3R-type LaBr₂ bilayer allows the Br’-mediated La’-Br’-La AFM superexchange to dominate interlayer exchange, then leading to the FM-AFM phase transition. Clearly, in the present stacking-engineered multiferroic system, the strong magnetoelectric coupling effect occurs due to the synchronized changes of magnetism and polarization under compressive vertical strains, which are different from the conventional type-I and type-II multiferroics.

Fig. 3: The electron transfer of bilayer LaBr₂.

a Charge accumulation after stacking those two LaBr₂ layers in 3 R and 2H stacking. The total differential electron along the c-axis (b) and interlayer transferred charge number (c) are calculated by integrating the differential charge at different δ values in 3 R stacking.

Spin dynamics of frustrated bimerons in stacking engineered multiferroic LaBr₂

Besides the tuning in the long-range magnetic ordering, controlling of short-range spin texture is a superior form of magnetoelectric coupling. Since the value of in-plane nearest and next-nearest exchange interaction parameters in both 2H and 3 R stacked LaBr₂ bilayers are ferromagnetic (positive) and antiferromagnetic (negative) coupling, respectively, as shown in Table 1, indicating that topological spin textures may be induced by frustrated exchange interactions^53,54. We performed atomically spin dynamics simulation based on Landau-Lifshitz-Gilbert equation on a 100 × 100 supercell with periodic boundary condition. Figure 4a–c show the spin textures under the layer spacing, which is generated by relaxing the spin configurations from random states. All magnetic frustrated samples exhibit topologically protected bimerons with topological charge Q = ±1. When δ ranges from −0.4 to 0 Å, the interlayer coupling J_L is relatively weak. The isolated bimerons are distributed randomly in the presence of upper or lower layers and then lead to a nonuniform magnetization instead of a bimeron in the adjacent layer, as shown in Fig. 4a. When the δ decreasing from −0.4 to −0.6 Å and the interlayer exchange coupling transits from FM to AFM in Fig. 4b, this also results in isolated bimerons and a nonuniform magnetization. However, when δ decreases to −0.8 Å, due to the further enhanced J_L, a pair of antiferromagnetic coupling bimerons appear in the both upper and lower layers (so called bilayer-bimeron), as shown in Fig. 4c. In addition, for the 2H stacking phase, we found that the spin texture is essentially identical with that of the 3R-type with δ = −0.8 Å, since a strong J_L always exists in 2H stacking at the range of δ = −0.8 ~ 0 Å. The spin textures with different δ values in 2H and 3 R stacking are shown in Supplementary Fig. 3.

Fig. 4: Frustrated bimeron texture in 3 R LaBr₂ bilayer.

a–c represent the spin texture of δ = −0.4~0, δ = −0.6, and δ = −0.8 Å in LaBr₂ with polarization along positive z direction, respectively. d is the relative energy of FM, bimeron, and bilayer-bimeron states. e–g and h–i are spin texture evolution from FM initial state caused by polarization inversion and enhancement, respectively.

As already discussed, enhanced and reversed polarization will modify interlayer exchange coupling J_L, thus, the bilayer system cannot retain the previous magnetic state and subsequent spin rearrangement will be triggered during the energy dissipative process of spin relaxation. As shown in Fig. 4d, we calculated the total energy of the FM state, isolated bimeron with FM background, and the bilayer-bimeron with FM background, and the results show that the FM state is the most stable, while the FM backgrounds with isolated bimeron and bilayer-bimeron states are metastable. Here, we use the FM ordering as the initial magnetic state for spin relaxation of system. Notably, unlike the scenario that the accessible topological spin texture was obtained by the relaxation of the random state, we consider the FM state as the initial magnetic state to investigate the effect of polarization on the spin texture. Figure 4e–g show that the FM-AFM-FM phase transition are caused by polarization reversal. When the ferroelectric polarization transits to the intermediate state with the J_L sign becomes negative, due to the same initial FM orderings of the upper and lower layers, the spin spiral configurations occur in the same position of two layers during the spin relaxed process, thus leading to bimeron in one layer or in the identical positions of two layers as the bilayer-bimeron form. Since the bilayer-bimeron endows with higher energy, the bimeron tends to appear only in one layer of the LaBr₂ HS while the other layer remains in FM state, as shown in Fig. 4f. Subsequently, J_L sign returns to a positive value when the polarization switches to another steady state, the bimeron texture breaks the consistency of the spin in the upper and lower layers during the relaxed process, allowing it to appear in anywhere of the bilayer LaBr₂, as shown in Fig. 4g. In addition, we also corroborate that when polarization is enhanced caused by the vertical strain, and the result is similar to that of the polarization reversed, as shown in Fig. 4h, i. The above results show that the reversal or enhancement of polarization can excite frustrated bimeron from long-range magnetic ordering, exhibiting a strong coupling effect of mechanical force, electric polarization, and magnetism.

Evolution of polarization texture and spin texture of twisted LaBr₂ bilayer moiré superlattice

With identifying the 3R-type LaBr₂ multiferroics and the understanding of the stacking dependence of interlayer interactions in the bilayers, we now turn to the study of possible spin and polar textures in LaBr₂ bilayers with small twist angles. For 3R-type LaBr₂ bilayers, a long-period moiré pattern can be created by a small twisting or strain between the layers, as seen in Fig. 5a. The moiré periodicity is approximately $A \approx a/\sqrt {\sigma ^2 + \theta ^2}$ for a small lattice mismatch σ and/or twisting angle θ, where a is the lattice constant of the monolayer. The relevance illustrates that moiré periodicity A increases as the relative twisting or strain between the layers decreases. For small twist angles, the stacking order at any local region in the moiré pattern can be obtained by translating the upper layer of the LaBr₂ bilayer by the vector r = ηa₁ + νa₂, where η, ν ∈ [0, 1], and a₁ and a₂ are the lattice vectors of the unit cell. In one moiré cycle, the local regions R₀, R₁, R₂ stacking configuration corresponds to the upper lateral shift $r = \left\{ {a_1 + a_2,\frac{1}{3}a_1 + \frac{1}{3}a_2,\frac{2}{3}a_1 + \frac{2}{3}a_2} \right\}$ with respect to lower layer, among which the R₀ configuration possesses an inversion symmetry without polarization value, while the R₁ and R₂ configurations are two different polarization states of the 3 R stacking above, respectively, as shown in Fig. 5a. The moiré field is extracted from the stacking-dependent interlayer interaction in these bilayers^39,40,41, which can be achieved by performing DFT calculations (see Supplementary Materials).

Fig. 5: The moiré pattern of LaBr₂ bilayer at small twisting angle.

a The moiré superlattice of 3R-type LaBr₂ bilayer from twisting. Three local regions R₀, R₁, R₂ stacking configuration are amplified to show their different interlayer configurations in the moiré superlattice. b, c represent polarization and interlayer exchange interaction textures in 3R-type LaBr₂ moiré superlattice, respectively. d is the magnetic texture at the twisting angle θ = 0.57°. e is the local regions R₀, R₁, R₂ stacking configuration for 2H-type moiré superlattice, and the polarization (f) and interlayer exchange (g) texture formed in the 2H-type moiré superlattice.

Figure 5b shows the polar texture of moiré superlattices evolved from 3R-type stacking. The domains with staggered polarization are formed due to the simultaneous generation of stacking configurations with opposite polarization states upon small twisting angles. According to the relationship between moiré periodicity and twisting angles, the polarization domain density enhances as the twisting angle increase. We also calculated the moiré lattice of 2H-type LaBr₂ (see Supplementary Fig. 4), however, our results proving that the polarization cannot be induced by a small twisting angle, since any local region of the system remains its central inversion symmetry, as shown in Fig. 5e, f.

According to previous discussion, different kinds of stacking configurations have little effect on the intralayer exchange interaction in terms of the magnetic exchange coupling, while it is particularly sensitive to the interlayer exchange coupling. This is because that the relative twisting between LaBr₂ bilayers shifts the interlayer double exchange path, enabling it to locally display a strong AFM coupling or weak ferromagnetic coupling as shown in Fig. 1e. Therefore, this offers the possibility of uneven magnetization textures in the system. Here, we calculated the interlayer coupling constants of 3R- and 2H-type LaBr₂ moiré superlattices, as shown in Fig. 5c, f. Among which, for 3R-type moiré superlattice, the interlayer coupling near the R₁, R₂ configurations exhibits weak ferromagnetism, while in R₀ region, it is antiferromagnetic coupling. We performed micromagnetic simulations for the moiré superlattices with a twisting angle θ = 0.57°, as shown in Fig. 5d. For the non-polar R₀ region, the magnetic configuration is a unified interlayer AFM ordering. In contrast, near the R₁ and R₂ configurations, the density of magnetic bimerons increases, owing to the strong interlayer magnetic coupling in the R₀ region where it is not conducive to the survival of bimeron, thus, bimerons can only aggregate to the weakly interlayer coupled R₁ and R₂ regions under the influence of the moiré potential. This leads to the formation of multimeron states^20,54,55, corresponding to ferroelectric domains. Notably, unlike the case where the polarization domain size varies with the size of superlattice, bimeron size (about 8 nm) is immune to the size of superlattice, but depends on local exchange interactions and can be easily affected by the adjacent spin configuration (shown in Supplementary Fig. 3). The above results indicate that there is also a strong magnetoelectric coupling in the moiré superlattice. On the other hand, for 2H-type, the interlayer coupling is reduced near the R₁ and R₂, but the overall AFM coupling still remains. It should be noted that although the bimeron can also be stabilized around the R₁, R₂ region, the moiré lattice structures of 2H-type are completely different with the 3R-type. Polarization domains and magnetic textures underline the significant discrepancy in stack-engineered multiferroics between twisted and non-twisted systems, providing theoretical guidance for twist multiferroics engineering of two-dimensional vdW systems.

Spin dynamics of bilayer bimeron

Topological spin texture can be used as information carriers in future advanced memories. However, there exists an obstacle known as the skyrmion Hall effect, that is, the bimeron trajectories deviate away from the driving current direction due to the Magnus force²⁶. One solution for this problem is to fabricate synthetic antiferromagnets, formed by considering two perpendicularly magnetized FM sublayers strongly coupled via the AFM exchange interaction separated by a metallic layer, where skyrmion Hall effect can be completely suppressed^56,57,58. Intriguingly, our bilayer LaBr₂ can naturally eliminate the effect of the skyrmion Hall effect without the need to construct artificial synthetic antiferromagnets consisting a nonmagnetic layer sandwiched by two magnetic layers. The calculated results show that bilayer-bimeron has four forms when interlayer FM (J_L > 0) and AFM (J_L < 0) exchange are considered, as shown in Fig. 6a–d. Although the out-of-plane components of the magnetic moments m_z of the upper and lower layers can be aligned in the same or opposite direction and not need to satisfy the most energetically favourable configurations in Fig. 6b, d, while bilayer-bimeron can be stabilized in LaBr₂ due to the in-plane magnetic component satisfying of J_L.

Fig. 6: The bilayer-bimeron texture in 3 R LaBr₂ bilayer.

a–d is the four combination forms of bilayer-bimeron. e, f represent the top view of the system with δ = 0 (J_L > 0) and δ = −0.6 (J_L < 0), respectively. The solid and dotted line represent the trajectory of bilayer bimeron with Q = 0 and Q = 2 respectively.

The current-driven motion of a rigid bimeron is described by a modified Thiele’s equation^59,60:

$${{{\mathbf{G}}}} \times {{{\mathbf{v}}}} - \alpha {{{\mathbf{{{{\mathcal{D}}}}}}}} \cdot {{{\mathbf{v}}}} + 4\pi {{{\mathbf{{{{\mathcal{B}}}}}}}} \cdot {{{\mathbf{j}}}} = 0$$

(2)

where G = (0, 0, −4πQ) is the gyromagnetic coupling vector, α is the magnetic damping parameter and ${{{\mathbf{{{{\mathcal{D}}}}}}}}$ is the dissipative force tensor. ${{{\mathbf{{{{\mathcal{B}}}}}}}}$ is the tensor relating to the driving force. v = (v_x, v_y) is the propagation velocity along the x and y axis, respectively. Among them, the topological Hall effect is included in the Magnus force of the first term in Eq. 2, which is proportional to the topological charge Q. The Thiele equation yields $v_{{{\mathrm{x}}}} = \left[ { - \alpha {{{\mathcal{D}}}} / \left( {Q^2 + \alpha ^2{{{\mathcal{D}}}}^2} \right)} \right]{{{\mathcal{B}}}}j$ and $v_{{{\mathrm{y}}}} = \left[ {Q / \left( {Q^2 + \alpha ^2{{{\mathcal{D}}}}^2} \right)} \right]{{{\mathcal{B}}}}j$ for the velocity components of bimeron along x and y direction. In Fig. 6b, c, the opposite Q signs result in the opposite transverse motion v_y of bimeron in the upper and lower layers, leading the bilayer-bimeron moves horizontally. Particularly, the positive J_L value in Fig. 6b allows bilayer-bimeron to naturally suppress the skyrmion Hall effect in the FM background without resorting to any external fields. And for the bilayer-bimeron with topological number Q = 2 in Fig. 6a, c, the trajectory will be shifted under the influence of skyrmion Hall effect. We performed a simulation that the bilayer-bimeron was driven by current on the non-twisted 3 R LaBr₂ system. The systems with δ = 0 and δ = −0.6 are adopted to simulate the scenario of bilayer-bimeron configurations with J_L > 0 and J_L < 0, respectively. As expected, the bilayer-bimeron with opposite topological numbers can move parallelly in nanoribbon, as shown in Fig. 6e.

Discussion

In summary, using first-principle calculations, we designed a multiferroic system by using the twisted stack engineering of 2D magnetic layers. The 3 R stacked LaBr₂ create out-of-plane spontaneous ferroelectricity due to the broken inversion symmetry. Such a polarization can be reversed by a slight interlayer sliding, which is equivalent to an energy barrier of 14.81 meV per unit. The interlayer weak FM is highly sensitive to the stacking distance. Therefore, the enhancement and reversal of polarization will trigger magnetic phase transition, which makes the 3 R stacked LaBr₂ exhibit strong magnetoelectric coupling effects. Spin dynamics simulations show that such magnetoelectric coupling-induced magnetic phase transition can excite topologically protected bimeron from long-range magnetic orderings. Moreover, the topological Hall effect can be suppressed in ferromagnetically or antiferromagnetically exchange-coupled bilayer-bimeron system by counteracting the Magnus forces in the upper and lower layers. We also calculated the moiré pattern with small twisting angles of the 3 R bilayer LaBr₂, where the small twisting of the stacked multiferroic system results in staggered polarization domains and an uneven spatial distribution of magnetic topological textures. Our work points the direction for the design of superior multiferroics with strong magnetoelectric coupling.

Method

The DFT method

Magnetic parameters and electronic structure were calculated by performing first-principle simulations within the DFT^61,62, using the projected augmented wave pseudopotentials method as implemented in the Vienna Ab-initio Simulation Package (VASP)^63,64. The exchange correlation energy is calculated within the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof form⁶⁵. Due to that, the GGA algorithm will underestimate the bandgap of the f orbitals, an effective 5 eV Hubbard U_eff parameter is applied on La’s f orbitals using the Dudarev method in order to correct its bandgap and prevents f orbitals from participating in orbital hybridization near the Fermi level⁶⁶. The plane wave cutoff energy was set to 500 eV and the van der Waals interaction is included by the semiempirical DFT-D3 method⁶⁷. We utilized a centered 18 × 18 × 1 Monkhorst-Pack k-point mesh for calculating properties of LaBr₂. Utilizing the conjugate gradient method, the plane lattice constant and atomic coordinates are fully relaxed until the energy and force converge to 10^–5 eV and 10^–2 eV Å⁻¹, respectively. In order to eliminate the periodic boundary effect, a 15 Å thin vacuum layer is introduced along the z direction. Spin-orbit coupling is used to calculate magnetocrystalline anisotropy. In the LaBr₂ moiré superlattice, the interlayer exchange constant was parametrized by evaluating the energy on a grid of points for an LaBr₂ bilayer, the upper layer of LaBr₂ is shifted relative to the lower layer over a 6 × 6 grid in the unit cell to sample the entire moiré range. The in-plane positions of all atoms in the bilayer are fixed during optimization, but they are allowed to relax in the out-of-plane direction.

Micromagnetic simulations

Using Spirit package to simulate the spin dynamics of LaBr₂ bilayer⁶⁸, the time evolution of the magnetization is described by the LLG equation^69,70, it can be written as:

$$\begin{array}{l}\frac{{\partial {{{\mathbf{m}}}}_i}}{{\partial t}} = - {\displaystyle{\frac{\gamma}{{\left( {1 + \alpha ^2} \right)\mu _i}}}}{{{\mathbf{m}}}}_i \times {{{\mathbf{B}}}}_i^{{{{\mathrm{eff}}}}}\left( t \right) - {\displaystyle{\frac{{\gamma \alpha }}{{\left( {1 + \alpha ^2} \right)\mu _i}}}}{{{\mathbf{m}}}}_i \times \left[ {{{{\mathbf{m}}}}_i \times {{{\mathbf{B}}}}_i^{{{{\mathrm{eff}}}}}\left( t \right)} \right] - {\displaystyle{\frac{{\alpha - \beta }}{{\left( {1 + \alpha ^2} \right)}}}}u{{{\mathbf{m}}}}_i\\ \qquad\quad\,\,\times \left( {{{{\hat{\mathbf j}}}} \cdot \nabla } \right){{{\mathbf{m}}}}_i + {\displaystyle{\frac{{1 + \beta \alpha }}{{\left( {1 + \alpha ^2} \right)}}}}u{{{\mathbf{m}}}}_i \times \left[ {{{{\mathbf{m}}}}_i \times \left( {{{{\hat{\mathbf j}}}} \cdot \nabla } \right){{{\mathbf{m}}}}_i} \right]\end{array}$$

(3)

where γ is the electron gyromagnetic ratio, α = 0.05 is the damping parameter, β = 0.01 is a nonadiabaticity parameter, $u = jPg\mu _B/\left( {2eM_s} \right)$ with current density j, P is the polarization of the current, e is the electron charge, M_s is the saturation magnetization. ${{{\hat{\mathbf j}}}} = {{{\boldsymbol{x}}}}$ denotes the electron current normal vector, and ∇ = ∂/∂r is the spatial gradient acting on the spin orientation.

For discrete lattice, the topological charge Q is calculated using the definition of Berg and Lüscher⁷¹, and arrive at the following expression:

$$Q = \frac{1}{{4\pi }}\mathop {\sum}\limits_l {A_l}$$

(4)

with

$$cos\left( {\frac{{A_l}}{2}} \right) = \frac{{1 + {{{\mathbf{m}}}}_i \cdot {{{\mathbf{m}}}}_j + {{{\mathbf{m}}}}_j \cdot {{{\mathbf{m}}}}_k + {{{\mathbf{m}}}}_k \cdot {{{\mathbf{m}}}}_i}}{{\sqrt {2\left( {1 + {{{\mathbf{m}}}}_i \cdot {{{\mathbf{m}}}}_j} \right)\left( {1 + {{{\mathbf{m}}}}_j \cdot {{{\mathbf{m}}}}_k} \right)\left( {1 + {{{\mathbf{m}}}}_k \cdot {{{\mathbf{m}}}}_i} \right)} }}$$

(5)

where l runs over all elementary triangles of any triangulated regular lattice and A_l is the solid angle formed by the three spin vectors m_i, m_j and m_k of the lth triangle. The sign of A_l is determined as ${{{\mathrm{sign}}}}\left( {A_l} \right) = {{{\mathrm{sign}}}}\left[ {{{{\mathbf{m}}}}_i \cdot \left( {{{{\mathbf{m}}}}_j \times {{{\mathbf{m}}}}_k} \right)} \right]$. Note that the triangles should cover all lattice area with no overlap and m_i, m_j and m_k should be anticlockwisely ordered.

Monte Carlo simulations

The magnetic phase transitions of LaBr₂ bilayer determined by performing Monte Carlo simulations with a 100 × 100 × 1 supercell based on the Heisenberg model of Eq. 1. For each temperature, the initial 10⁵ Monte Carlo steps were discarded for equilibrium consideration and additional 10⁵ Monte Carlo steps were retained for statistical averaging of the simulation. The critical temperature is estimated by the peak position of the specific heat (C), which is defined as

$$C = \frac{{\left( \langle{E^2}\rangle - \langle{E^2}\rangle \right)}}{{k_BT^2}}$$

(6)

where E is the thermal energy, k_B is Boltzmann constant, and T is temperature.