Introduction

Notable attention has been paid to multiferroic materials since they possess intrinsic magnetoelectric coupling effects which leads to extraordinary advantages for building multifunctional and multi-field-driven spintronic devices. Type-II multiferroics, where ferroelectricity originated from spin-driven, are more interesting due to the spontaneous polarization varying with magnetic phase transitions (MPTs)1,2,3,4,5. Recently, type-II multiferroicity has been experimentally realized in van der Waals NiI2 as indicated by strong intensity of second-harmonic generation imaging6,7,8. The canted proper screw spin state and ferroelectric domains of NiI2 have also been successfully reproduced by a realistic spin model9, revealing that the multiferroicity of NiI2 is closely related to the strong Kitaev interaction10,11,12. However, apart from the monolayer NiI2, experimentally observed low-dimensional multiferroics are very rare. Besides, the inversion symmetric NiI2 prohibits the antisymmetric exchange, i.e., Dzyaloshinskii–Moriya interaction (DMI)13, which plays a key role in determining the spin textures. Therefore, despite the significant breakthrough, there are still two remaining challenges in studying multiferroics: (i) more van der Waals type-II multiferroic materials are desired to realize, and (ii) how the DMI contributes to the magnetic/ferroelectric ground states still needs further investigation.

Two-dimensional (2D) materials with Janus configurations introduce a diverse range of chemical compositions and are characterized by the out-of-plane asymmetry, which leads to the emergence of new physical phenomena and enhanced functionalities14,15,16,17,18. Janus magnets show extensive research value since topologically nontrivial spin textures tend to stabilize with DMI preserved and can be easily manipulated by external magnetic or electric fields19,20,21,22,23. Recent studies have demonstrated that the interplay between DMI and magnetic frustration would modulate the topological spin textures in monolayer Janus magnets, further determining the presence and helicity of skyrmions and driving MPTs as a dependence of DMI strength24,25,26. Consequently, materials with Janus configurations provide an ideal platform to study type-II multiferroic mechanism, especially the magnetic and ferroelectric orderings led by DMI manipulation in 2D terminals.

In this work, we build effective spin Hamiltonians for multiferroic Janus NiXY monolayers (X, Y = I, Br, Cl) using a symmetry-adapted cluster expansion method and investigate the inherent magnetoelectric coupling effects. Magnetic coefficients are fitted based on first-principles calculation results through a machine learning algorithm. Through parallel tempering Monte Carlo (MC) simulations, we predict an inclined cycloid helical ground state with in-plane 〈110〉 propagation induced by DMI and Kitaev interaction. For the first time we systematically reveal the correlations between DMI and the inclination of the spin rotation plane within coplanar antiferromagnetic spin spiral systems, which further indicates the tunable ferroelectricity through DMI manipulation attributed to the type-II multiferroicity mechanism, i.e., the horizontal component of polarization is determined by spin helicity. Considering this in-plane polarization intertwined with spin spiral arrangement is directly relate to the inherent DMI, we thus propose a feasible method for tailoring the magnetic and ferroelectric ground states in 2D multiferroics by manipulating the DMI through external means.

Results and discussion

Spin Hamiltonian for Janus NiXY monolayers

Janus NiXY crystallizes in a triangular lattice of transition-metal ions sandwiched by two different types of halogens (Fig. 1a). Therefore, NiXY monolayers belong to the P3m1 space group (C3v point group), featuring a threefold rotational axis and three mirror planes at 120° to each other. Due to the broken out-of-plane symmetry introduced by Janus configurations, inversion centers and two-fold axes no longer exist, accordingly leading to the preservation of DMI. To confirm the multiferroic ground states of Janus NiXY monolayers, we employ a symmetry-adapted cluster expansion method, as implemented in the PASP software27, to build the first-principle-based spin Hamiltonians. The invariants of the Janus NiXY, i.e., all possible combinations of spin components protected by crystal symmetry, are initially proposed. After carefully selecting important magnetic interactions with obvious magnetic frustration and strong spin-orbit coupling (SOC) effects considered in 1T-prototype candidates, we propose the spin Hamiltonian:

$$\begin{array}{l}{\boldsymbol{\mathcal{H}}}=\mathop{\sum}\limits_{{\left\langle i,j\right\rangle}_{1}}\left\{J\left({\mathbf{S}}_{i} \cdot {\mathbf{S}}_{j}\right)+ {\boldsymbol{D}}_{{ij}} \cdot \left({\mathbf{S}}_{i} \times {\mathbf{S}}_{j}\right)+ B{\left({\mathbf{S}}_{i} \cdot {\mathbf{S}}_{j}\right)}^{2} \right. \\\qquad\;\; \left.+\, K\left(S_{i}^{\gamma } S_{j}^{\gamma }\right)+ {h}_{1} + {h}_{2}\right\}+ \mathop{\sum }\limits_{{\left\langle i,j\right\rangle }_{n}} {J}_{n} \left({\mathbf{S}}_{i} \cdot {\mathbf{S}}_{j}\right).\end{array}$$
(1)
Fig. 1: Structural and magnetic properties of Janus NiXY monolayers (X, Y=Cl, Br, I).
figure 1

a Top and side view of the Janus NiXY monolayers. Gray, purple, and orange balls denote Ni, X, and Y atoms, respectively. b The schematic of magnetic interactions in monolayer NiXY. Dashed purple lines denote the chemical bonds between Ni and ligands. Orange arrows denote the directions of the horizontal component of DMI between each nearest Ni-Ni pair, with the vertical components of DMI symbolled beside each arrow. Light purple lines indicate the mirrors.

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Here, isotropic Heisenberg interactions are counted ranging from the nearest-neighbors (NN) to the third-NN. All parameters including antisymmetric exchange (reflected as the DMI vector Dij), anisotropic symmetry exchange (reflected as the Kitaev interaction K), as well as the isotropic Heisenberg exchange J, can be considered as the components of magnetic-exchange-couplings matrix (\({\mathcal{J}}\) matrix) of the NN, where DMI and Kitaev interaction originate from SOC effects. According to Mariya’s rule, the DMI vector Dij always lies in the mirror plane between each pair as shown in (Fig. 1b)13. Biquadratic interaction B is also considerable in Janus NiXY monolayers, which is included to describe collinear spin alignment better28,29. In the spin model, \({h}_{1}\) and \({h}_{2}\) are respectively \({\Gamma }_{1}({S}_{i}^{\alpha }{S}_{j}^{\beta }+{S}_{i}^{\beta }{S}_{j}^{\alpha })\) and \({\Gamma }_{2}({S}_{i}^{\alpha }{S}_{j}^{\gamma }+{S}_{i}^{\beta }{S}_{j}^{\gamma }+{S}_{i}^{\gamma }{S}_{j}^{\beta }+{S}_{i}^{\gamma }{S}_{j}^{\alpha })\). Here, {α, β, γ} = {Ky, Kz, Kx}, {Kz, Kx, Ky}, and {Kx, Ky, Kz}, where Kx, Ky, and Kz correspond to axes of the Kitaev basis as shown in Supplementary Fig. S1a. Both Γ1 and Γ2 are off-diagonal exchanges beyond the “\({\boldsymbol{\mathcal{J}}}\) matrix under Kitaev basis” (\({\boldsymbol{\mathcal{J}}}_{{\bf{K}}}\)), which indicate a unique magnetic anisotropy or induce long-range magnetic order30,31.

We firstly focus on the spin ground states of three Janus NiXY magnets. By fitting the total energies of random spin structures calculated within HSE functionals through a machine learning method (see Supplementary Material and Supplementary Fig. S2 for details), we respectively obtain magnetic coefficients for Janus NiIBr, NiICl, and NiBrCl, as illustrated in Table 1. Fitting mean average errors for three materials are 0.042, 0.045, and 0.018 meV per unit cell respectively, indicating high fitting precision and accuracy. All candidates feature a strong ferromagnetic (FM) J1, a negligible FM J2, and an antiferromagnetic J3 comparable in magnitude, which indicates strong magnetic frustration. The DMI and Kitaev interactions in NiBrCl are small in value, which is in accordance with weak SOC effects provided by light ligands. NiICl and NiIBr show larger D and K values due to heavy I ligand, meanwhile, D and K are larger in value of NiICl than that of NiIBr (D is used when describing the scalar value of Dij).

Table 1 Magnetic parameters of the Eq. (1) fitting from HSE functionals of NiXY monolayers, in units of meV
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Magnetic ground states of Janus NiXY are determined by employing the Hamiltonian of Eq. (1) within Monte Carlo simulations and conjugate gradient (CG) methods. NiBrCl is an FM semiconductor, while NiIBr and NiICl yield a coplanar cycloid helical state that propagates along the in-plane 〈110〉 direction with distinct inclined spin rotation plane. Although a frustrated J3/J suggests a helical ground state, large biquadratic interaction (B/J = 0.18) and weak SOC effects finally lead NiBrCl to a collinear FM alignment rather than multiferroics. In contrast, strong SOC effects, especially intrinsic DMI, determine the spiral ground states in both NiIBr and NiICl. To confirm the ground states of Janus NiXY monolayers, large-scale spin simulations are performed based on the spin Hamiltonians as implemented in a 25 nm × 43 nm supercell. Spin patterns of Janus NiXY magnets are obtained. Three magnetic domains are illustrated in both NiIBr (Fig. 2) and NiICl (Supplementary Fig. S3a), covering most of the area and propagating along 〈\(110\)〉 and its equivalent directions which is determined by the inherent C3v point group. The FM ground state is also confirmed in NiBrCl (Supplementary Fig. S3b). Topological defects such as bimerons are naturally formed at the domain walls in both spin helical magnets, while skyrmion is also found in the NiBrCl due to intrinsic DMI. Using the existing MC simulations, as shown in Supplementary Fig. S4, we estimate the critical temperatures of NiIBr, NiICl, and NiBrCl as 17 K, 14 K, and 26 K, respectively, which are comparable to the critical temperature of the monolayer NiI2 (~21 K)6.

Fig. 2: Large-scale spin patterns from MC simulation and a following CG optimization for Janus magnets NiIBr.
figure 2

Cones denote spin where red and blue colors indicate positive and negative values of the z component Sz. The background warm and cold colors indicate positive and negative topological charges. The 〈110〉 propagation direction and its equivalent are indicated as red dashed lines, with b1 and b2 denoting the reciprocal lattice.

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DMI manipulation and analyses

The intrinsic DMI plays a decisive role in determining the magnetic ground states in Janus NiXY monolayers. Given that both NiIBr and NiICl possess similar cycloid ground states (Supplementary Fig. S5a, b), we take NiIBr as an example to show how DMI works. We artificially manipulate the horizontal and vertical components of the DMI (namely, \({{\boldsymbol{D}}}_{{\parallel}}\) and \({{\boldsymbol{D}}}_{{\bf{z}}}\)) and propose a phase diagram to investigate whether DMI can give rise to an MPT. Note that DMI of the second- and third-NN are negligibly small and not included. As shown in Fig. 3a, all MC simulations consistently demonstrate that with the variations in | D||/J | and | Dz/J| (symbols D|| and Dz are used when describing their scalar values), the cycloid helical ground state with 〈\(110\)〉 propagation remains unchanged, thereby suggesting no apparent MPT occurs within DMI manipulation. However, both the inclination of the rotation plane (θ, refers to the canting angle with respect to the z-axis as depicted in Supplementary Fig. S6b) and the period of the spin spiral (n) are significantly affected by the strength of the D|| and Dz relatively. As shown in Fig. 3a, a stronger D|| precisely reduces θ, while a larger Dz usually leads to an increment of θ. Besides, Fig. 3b shows that n is highly dependent on both D|| and Dz, where a weaker D resulting in a longer spiral period.

Fig. 3: DMI manipulation in Janus NiIBr monolayer.
figure 3

a Phase diagram illustrating the magnetic ground state and inclination of the spin rotation plane as a dependence of | D||/J | and | Dz/J |. The purple background represents the 〈110〉 propagation direction and the gray contour indicates the inclination angles. The orange star denotes the model-predicted position for NiIBr in this phase diagram. Yellow and red stars denote the positions with distinct DMI strengths. b Period of cycloid helical states as a dependence of | D||/J | and | Dz/J |, with each term individually considered. c Comparison of energies obtained by mathematical model and obtained by effective Hamiltonian when each component of DMI is only considered as a function of inclination with n = 4. d Calculated total energy contributed from horizontal and vertical components of DMI vector as a function of period n. Here, values of D|| and Dz are set to be equal in panels c and d for simplification. The yellow dashed line (E = 0) is included in panels c and d for reference.

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Therefore, one can manipulate the spin rotation plane through regulating DMI in a spin frustration system. For instance, the orange star in Fig. 3a corresponds to the initial magnetic ground state of NiIBr where | D||/J| = 0.093 and | Dz/J| = 0.068. The period and inclination of the NiIBr spin ground state are n = 7.25a and θ = 27° (a denotes the in-plane lattice parameter of the unit cell). Keeping other magnetic coefficients steady while artificially altering the DMI, as D|| increases to | D||/J| = 0.17 and Dz decreases to | Dz/J| = 0.010, an approximately out-of-plane cycloid state with n = 6.75a and θ = 9° is obtained symbol as a yellow star in Fig. 3a. In contrast, when the DMI value renders | D||/J| = 0.026 and | Dz/J| = 0.13 (denoted by red star in Fig. 3a), the system exhibits a longer period n = 8a and an enlarged inclination angle θ = 67°, indicating nearly in-plane cycloid spin orientations. Spin ground states after DMI manipulations are illustrated in Supplementary Fig. S5c, d.

We now develop a numerical model to attribute why Dij affects the inclination θ and the period n. We initially construct a coplanar \({{CY}}^{\left\langle 110\right\rangle }\) state, i.e., a cycloid helical state propagating along 〈\(110\)〉, and only adopt the DMI. The total energy contributed by the horizontal (E1) and vertical (E2) components of the DMI vector are expressed as:

$${\rm{E}}_{1}=-2{D}_{\parallel }\cdot \cos \theta \cdot (\sin 2\varphi +\sin \varphi ),$$
(2)
$${\rm{E}}_{2}=2{D}_{\rm{z}}\cdot \sin \theta \cdot (\sin 2\varphi -2\sin \varphi).$$
(3)

Here, θ is the inclination as mentioned before, and φ is the interval angle between neighboring spins (\(n=\pi /\varphi\)) as depicted in Supplementary Fig. S6c. It is found that: (i) E1 has its overall minimum at θ = 0° while E2 has its overall minimum at θ = 90° as shown in Fig. 3c, which is precisely consistent with the tunable trend of θ depending on D|| and Dz in the phase diagram and also coordinates with effective Hamiltonian predictions; and (ii) a shorter period indicates a larger φ (when n > 4), which further helps to reduce the total energy (Fig. 3d). The expressions explain why Dz terms play a limited role in deciding spin orientations compared with D|| terms and demonstrate that Janus NiXY only exhibits a single chirality along each propagating direction. (see Supplementary Material for details)

To further elucidate how each magnetic interaction in Eq. (1) determines the magnetic ground state, the role of each magnetic term in the effective Hamiltonian is analyzed and demonstrated. Three typical magnetic states are considered with their total energy decomposed: (i) the predicted ground state of NiIBr: \({{CY}}_{{\rm{cant}}}^{\left\langle 110\right\rangle }\), i.e., an inclined cycloid helical state propagating along 〈110〉 with n = 7.25a; (ii) \({{CY}}_{{\rm{in}}}^{\left\langle 110\right\rangle }\), an in-plane cycloid helical state propagating along 〈110〉 with n = 9a which approximately corresponds to the ground state of NiIBr if DMI is not considered; (iii) \({{PS}}_{{\rm{cant}}}^{\left\langle 1\bar{1}0\right\rangle }\), a canted proper screw state propagating along \(\left\langle 1\bar{1}0\right\rangle\) similar as NiI29. Decomposition means that we artificially remove each term in the effective Hamiltonian and obtain the energy of the remaining components. Firstly, as demonstrated in Supplementary Table S1, if isotropic Heisenberg interactions (J, J2, and J3) are considered only, the total energy of \({{CY}}_{{\rm{cant}}}^{\left\langle 110\right\rangle }\) is the lowest. Secondly, with B and K terms added, \({{PS}}_{{\rm{cant}}}^{\left\langle 1\bar{1}0\right\rangle }\) becomes the lowest, corresponding to the magnetic ground state of NiI2 dominated by the Kitaev interaction. However, we find no MPT occurs in Fig. 3a, suggesting that \({h}_{1}\) and \({h}_{2}\) in Eq. (1) make a difference. To confirm this, we artificially remove DMI as well as Γ1 and Γ2 terms from the \({\boldsymbol{\mathcal{J}}}\) matrix while keep other parameters in Supplementary Table S1 steady. As shown in Supplementary Fig. S1b, a series of MC simulations prove that a smaller K induces an MPT here. As Γ1 and Γ2 in \({\boldsymbol{\mathcal{J}}}_{{\bf{K}}}\) included, MPT will only occur after K > 1.0 meV in Supplementary Fig. S1c, indicating that the positive Γ1 and Γ2 stabilize the 〈110〉 propagation, which coordinates to the fact that \({{CY}}_{{\rm{in}}}^{\left\langle 110\right\rangle }\) becomes the energy minimum if only DMI is excluded. Thirdly, we find that DMI plays a decisive role in the inclined cycloid helical ground state because it reduces the energy of \({{CY}}_{{\rm{cant}}}^{\left\langle 110\right\rangle }\) but virtually contributes nothing to two others. Finally, we propose that \({{CY}}_{{\rm{cant}}}^{\left\langle 110\right\rangle }\) is robust with DMI considered, where MPT will not occur even when K is very large as indicated in Supplementary Fig. S1d.

Magnetoelectric coupling effects

The DMI in Janus NiXY monolayers can be externally modulated, leading to direct magnetoelectric coupling effects. Note that PBE functionals instead of HSE functionals are employed in this part due to the computation cost. Despite PBE overestimates J3 and K, the correct magnetic ground state, i.e., a cycloid helical state orientating 〈110〉 direction, is still obtained. Given that Janus NiXY monolayers are type-II multiferroics possessing both out-of-plane polarization originating from its intrinsic Janus configuration and in-plane polarization originating from the spin spiral, we propose: (i) on one hand, polarization along the vertical direction would naturally response to the external vertical electrostatic fields (Ez) with varying direction and strength; (ii) on the other hand, the applied Ez neither triggers great crystal deformation nor alters the invariants (combinations of spin components) of Janus NiXY monolayers but directly interferes with magnetic coefficients, especially the DMI. As a result, a clear monotonic variation of | D||/J | of the NiIBr is obtained and illustrated in Fig. 4a, b (note that | Dz/J | plays a limited role here). Interestingly, as previously discussed, the inclination of the spin rotation plane strongly depends on | D||/J |. Therefore, by manipulating DMI through external Ez, the magnetic ground state, more specifically, the inclination of the spin rotation plane can be continuously tuned.

Fig. 4: Direct magnetoelectric coupling effects in Janus NiIBr monolayer.
figure 4

a, b Magnetic coefficients, their ratio to J, and the inclination θ, as functions of the applied vertical electrostatic field Ez. c, d Spin-induced ferroelectric polarization as a function of inclination of the spin rotation plane. Panel d provides a zoomed-in view of the features shown in c, with a comparison to the theoretical gKNB model predictions.

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Type-II multiferroicity nature suggests adjustable averaged electric polarization Pij = (Px, Py, Pz) led by DMI manipulation. As shown in Fig. 4c, by artificially inclining the spin rotation plane while maintaining the cycloid order and period, the horizontal component of the Pij, specifically Py, exhibits a significant variation. This spin helicity derived Py is protected by the preserved joint mirror-spatial-symmetry and time-reversal-symmetry, which can also be theoretically estimated by the generalized Katsura-Nagaosa-Balatsky (gKNB) model:

$${\boldsymbol{P}}_{{ij}}\left({\mathbf{S}}_{i},{\mathbf{S}}_{j}\right)={\mathbf{S}}_{i}^{T}\cdot {\boldsymbol{\mathcal{P}}}_{{\rm{s}}}\cdot {\mathbf{S}}_{j}.$$
(4)

The polarization tensors of the nearest neighbors, \({\boldsymbol{\mathcal{P}}}_{{\rm{s}}}\), were obtained using the “four-states mapping” method based on a 4 × 4 × 1 superlattice of monolayer NiIBr (see Supplementary Material for details)32,33,34,35. As shown in Fig. 4b, d, within the tunable range of the electrostatic field (±5V/nm), the ground state of NiIBr remains a cycloid helical state with the inclination of the spin rotation plane gradually increasing. This variation leads to a monotonic increase in the magnitude of horizontal component of polarization, where a positive Ez enhances the Py, and a reversed Ez decreases the Py. The observed Py-Ez relationship indicates that type-II multiferroic Janus NiXY monolayers provide an ideal platform for studying tunable in-plane ferroelectric polarization driven by vertical electric fields through DMI manipulation, showing the advantage of direct magnetoelectric coupling effects.

In summary, based on the constructed effective spin Hamiltonian, our study predicts a cycloid helical ground state with a 〈110〉 propagating direction and an inclined rotation plane for both NiIBr and NiICl monolayers, as well as an FM state for NiBrCl monolayer. By employing spin Hamiltonian into Monte Carlo simulations, we decompose total energy for various spin configurations, revealing that the DMI vector precisely influences both the inclination and the period of the spin spiral, in agreement with mathematical models. Taken type-II multiferroic Janus monolayers as an example, we uncover a novel mechanism that the in-plane ferroelectric polarization can be continuously tuned by altering the inclination of the spin rotation plane, achievable through DMI manipulation induced by external forces. Therefore, the tunable horizontal component of polarization driven by vertical electric fields demonstrate strong correlations between DMI, spin ground states, and ferroelectric orders, further highlighting inherent magnetoelectric coupling effects in multiferroic Janus materials.

Methods

First-principles calculations

The density functional theory calculations are performed with Vienna Ab initio Simulation Package (VASP), where projector augmented-wave (PAW) potentials are employed with a 400 eV cutoff energy of plane wave basis36,37. Both the hybrid functionals HSE06 and the Perdew-Burke-Ernzerhof (PBE) functionals are used to calculate the total energy of Janus NiXY supercells of random spin configurations with a 2 × 2 × 1 Monkhorst-Pack k-point grid, where each supercell contains 4 × 4 unit cells38,39. All unit cell structures are fully relaxed until the Hellmann-Feynman forces on each atom are within 10−3 eV/Å−1 and free energy is less than 10−7 eV, with the Brillouin zone sampled as 10 × 10 × 1 k-points. The following electrons are treated as valence states: Ni 3p, 3d, and 4s; I 5s and 5p; Br 4s and 4p; and Cl 3s and 3p. Ferroelectric polarization is calculated through Berry phase methods40. The vacuum layer of about 30 Å is considered to avoid the interaction between periodic layers along z direction. The optimized lattice parameters for NiXY unit cells are: a = 3.871 Å for NiIBr; a = 3.795 Å for NiICl; and a = 3.622 Å for NiBrCl.

Monte Carlo simulations

The parallel tempering Monte Carlo (MC) simulations are conducted using the Hamiltonian of Eq. (1) as implemented in PASP software27. Different sizes and shapes of supercells are used in our study to validate the magnetic ground state. A 64a × 32b’ ×1c supercell (b’ = a + 2b, where a, b, c are lattice vectors of one unit cell), that contains 4096 NiXY unit cells is adopted for the results of Fig. 2 and Supplementary Fig. S3. The MC simulations last 120,000 sweeps with 32 parallel temperatures implemented. The conjugate gradient (CG) method is also applied to optimize the spin configuration. The directions of spins are described by independent variables (θi, φi), which are optimized locally by the CG method to minimize the force on each spin23,41. The MC simulations and CG optimizations guarantees the local energy minimum of all magnetic configurations. The energy convergence of this CG algorithm is set to be 10−6 eV. Moreover, the existing MC simulations are also used to estimate the critical temperature of the helical-to-paramagnetic transition within the temperature range of 4 K to 99 K, with an interval of about 3 K.