Introduction

Topological non-trivial magnetic structures such as chiral domain walls1, merons2, bimerons3,4, and skyrmions5,6 have attracted great research interest due to their rich physical properties and widespread application prospects in spintronic devices. Among these spin textures, magnetic skyrmions have been extensively studied due to their small size, low energy consumption, and low driving current density7,8. Magnetic skyrmions with non-collinear spin configurations can be induced by antisymmetric exchange coupling-Dzyaloshinskii-Moriya interaction (DMI) in structures with inversion symmetry breaking and spin-orbit coupling (SOC)9,10. The antisymmetric exchange interaction can be written as EDMI = D·(Si × Sj), where D is the DMI vector and Si and Sj represent the spins of sites i and j. Notably, magnetic skyrmions have been observed in non-centrosymmetric B20 bulk materials such as MnSi, FeGe, and FeCoSi6,11,12,13,14 and interfacial systems of multilayers such as Ir(111)/Fe, Ta/CoFeB, and Pt/Co15,16,17,18,19,20,21 with Cnv crystal symmetry. On the other hand, anisotropic DMI are reported in ultrathin epitaxial Au/Co/W(110) with C2v crystal symmetry22, and antiskyrmions with anisotropic DMI are reported in acentric tetragonal Heusler compounds with D2d crystal symmetry23,24 and non-centrosymmetric tetragonal structure with S4 crystal symmetry25. In parallel with the development of the hot study of skyrmions in these traditional bulk and multilayer thin films, 2D magnets, e.g., Fe3GeTe226, CrI327, CrGeTe328, MnSe229, and VSe230 with long-range magnetic orderings have been extensively reported in the last few years, which have been providing an ideal platform to study fundamental properties of magnetism such as magneto-optical and magnetoelectric effect for ultracompact spintronics in reduced dimensions. Moreover, recent works have proposed that Néel-type skyrmions with isotropic DMI can be realized in 2D Janus magnets, e.g., MnXY31, CrXY32, and multiferroics structures, e.g., CrN33, BaTiO3/SrRuO334, In2Se3/MnBi2Se2Te235. However, it is worth noting that anisotropic DMI has not been reported yet in 2D magnets. Different from previous materials with isotropic DMI vectors along with x and y directions, MCuX2 with special crystal symmetry has an anisotropic DMI vector. Skyrmion Hall effect will cause FM skyrmions with opposite topological charges to propagate in opposite directions, instead of moving parrel to injected current. Antiskyrmions Hall angle is strongly dependent on the direction of applied current related to the internal spin texture of antiskyrmion. When applying spin-polarized current to drive an antiskyrmion, the propagation direction of the antiskyrmion will follow the current direction without topological skyrmion Hall effect36,37,38,39,40,41. Therefore, it is possible to achieve the zero antiskyrmion Hall angle in a critical current direction.

In experiments, in order to discover 2D materials, a lot of efforts have been devoted to finding materials with characteristics of weak interlayer bonds, which allow their exfoliation down to a single layer by mechanical or liquid-phase approach42,43 peeling off the three-dimensional layered van der Waals materials, such as CrGeTe328, CrI327, and Fe3GeTe226, etc. Bulk FeCuTe2 is a layered magnetic material stacked by weak interlayer van der Waals interaction, and it has been reported experimentally44,45,46. It is reported bulk FeCuTe2 has a layered structure, and the unit cell parameters are as follows: a = 3.93 Å, c = 6.078 Å44. The layered compound FeCuTe2 presents an antiferromagnetically ordered state below TN = 254 K47. Based on first-principles calculations, we further determine the magnetic properties of bulk FeCuTe2. The calculated results show that the unit cell parameters a and c of bulk FeCuTe2 are 3.964 and 6.176 Å, respectively. Moreover, we also compare energies with different magnetic orderings as shown in Supplementary Fig. 10. Bulk FeCuTe2 prefers AFM V in the ground state. From these results, one can see that theoretical results are consistent with the experimental report. Before studying the properties of monolayer FeCuTe2, we theoretically simulate the exfoliation energy when blocks were gradually peeled into a two-dimensional structure as shown in Supplementary Fig. 1. The calculated cleavage energy is 0.46 J/m2, which is comparable to that of graphene or phosphene48,49, indicating the high possibility that layered 2D FeCuTe2 can be exfoliated from bulk. Interestingly, FeCuTe2 with D2d crystal symmetry matches the Moriya symmetry rules10 to achieve anisotropic DMI. Moreover, we predict a series of 2D MCuX2 structures with D2d crystal symmetry, where M and X represent the 3d transition metal (TM) and group VIA elements, respectively. Finally, we realize a series of FM antiskyrmions, FM bimerons, AFM antiskyrmions and AFM bimerons structures.

Results and discussion

Crystal structure of monolayer MCuX 2

Top and side views of the crystal structure of single-layer MCuX2 (M:3d TM; X:group VIA) are represented in Fig. 1a–c. It exhibits a tetrahedral structure with M-Xtop-M and M-Xbot-M configurations along x and y directions, and the face-centered coordinates are occupied by Cu atoms in the middle layer. The MCuX2 structure belonging to D2d (No.115) crystal symmetry has the symmetry generators: identity operation E: (x, y, z) → (x, y, z); mirror plane inversion M2: (x, y, z) → (-x, -y, z); fourfold roto-inversion IC4: (x, y, z) → (y, -x, -z); and twofold axis C2: (x, y, z) → (x, -y, z), which leads to its feature with intrinsic inversion symmetry broken. According to the Moriya rules10, the induced DMI sign should be opposite along x and y directions.

Fig. 1: Crystal structure of MCuX2 monolayer, where M and X represent 3d TM and group VIA elements, respectively.
figure 1

Top (a) and side (b, c) views of MCuX2 monolayer. The dashed lines in (a–c) represent the unit cell. The red and black arrows in (a) indicate the in-plane component of DMI with opposite chirality.

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Via first-principles calculations, we obtain the basic magnetic parameters of MCuX2 ternary compounds as shown in Table 1 (Computational details are presented in the experimental method). The J1 and J2 indicate the exchange coupling constants between NN and NNN atoms. In most systems, J1 is orders larger than J2. There are frustration coming from competing exchange interactions between J1 and J2 in the classical Heisenberg model on the tetrahedra structure except for VCuS2 and VCuSe2. Interestingly, for MCuX2, when M changes from V to Mn and from Fe to Ni, the systems are tuned from FM to G-type AFM, respectively. In tetrahedra crystal, the five d orbitals of transition metal ions split into two groups lower e (\(d_{x^2 - y^2}\), \(d_{z^2}\)) and higher t2 (dxy, dxz, dyz) levels due to the influence of the crystal field. The t2 ↔p↔e super-exchange interaction favors the appearance of FM coupling, while the e↔e direct exchange and t↔t direct exchange prefer AFM coupling. When the d orbit is more than half-filled with electrons, the AFM coupling mainly benefits from t↔t direct exchange. Thus, in the MCuX2 family with d orbit no less than half-filled, the competitive AFM coupling is stronger. Similar results are also depicted in the zinc-blende binary transition metal compounds50. Collecting all 3d TM atoms magnetic moments in MCuX2 monolayer [see Table 1]. We can find that Mn atoms have the highest magnetic moments and they monotonically decrease on both sides of Mn. It is obvious that the overall trend across the 3d TM row obeys Hund’s rule51.

Table 1 The calculated magnetic parameters of lattice constants a, NN, and NNN exchange coupling J1 and J2, magnetic anisotropy energy K and magnetic moment μM of M atom, and magnetic characteristic parameter R in MCuX2 family.
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DMI of monolayer MCuX 2

Figure 2 shows the calculated NN-DMI of MCuX2 structures based on the chirality-dependent total energy difference approach52. It is found that all systems have anisotropic DMI and dx and dy have opposite signs along x and y directions, which is consistent with the DMI analysis at the beginning. Besides, the DMI strength varies from 0 to 15 meV/atom. These DMIs are very large compared to many state-of-the-art FM/HM heterostructures and 2D Janus structures, e.g., Co/Pt (∼3.0 meV)52 and Fe/Ir(111) (∼1.7 meV)53 thin films and 2D MnSTe (~2.63 meV)31. In addition, it is worth noting that the DMI of VCuTe2 reaches up to 15.2 meV/atom. In order to verify the correctness of the DMI, we calculated the variation of DMI when the U values of the V atom were 2, 3, and 4 eV, which were 13.2, 15.2, and 11.6 meV/atom, respectively. Notably, CoCuTe2 is different from other antiferromagnetic structures. We find that this structure tends to the Strip-AFM structure with a different lattice constant in x and y direction. Meantime, CoCuTe2 also has the kind of anisotropic DMI in Fig. 2, which is −0.37 and 4.2 meV/atom along x and y directions, respectively.

Fig. 2: The calculated anisotropic DMI of monolayer MCuX2.
figure 2

Yellow and blue bars indicate the DMI components of NN magnetic atoms along x and y direction, respectively. Here, d > 0 (d < 0) represents the anticlockwise (clockwise) chirality.

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To elucidate the microscopic energy source of DMI, we calculate the atomic resolved SOC energy difference ΔΕSOC with opposite chirality associated with DMI. Only the ΔΕSOC are presented along the x-direction in Fig. 3. We can see that M atom and Cu atom contribute a relatively large DMI when the X is the light element S with weak SOC. As the X varies from S to Te, the dominant contribution of the X element to DMI gradually improves in all MCuX2 monolayers due to the increase in SOC strength. Similar to the interface of HM/FM52,54, in which ΔΕsoc is contributed mainly by heavy 5d metal elements of the interfacial location. In our systems, the Fert–Levy mechanism of DMI can be understood that the heavy chalcogen element plays a significant role in inducing the spin-orbit scattering between two magnetic atoms. In addition, we noticed that the ΔΕSOC of NiCuTe2 is close to zero. From optimization, we also identify the equilibrium lattice constant values for FM configuration of MnCuTe2 and G-AFM configuration of NiCuTe2, the calculated results are obtained about 4.100 and 7.265 Å, as shown in Supplementary Fig. 3. The main reason is that the magnetic moment of Ni atoms in the system is very small, which leads to a small contribution to DMI. Of course, we also check the different U value from 2 to 4 eV, very small magnetic moments are obtained.

Fig. 3: Anatomy of DMI for MCuX2 monolayers.
figure 3

a–c Atomic resolved localization of SOC energy difference between clockwise and anticlockwise chirality for iron/cobalt/nickel (a), chromium/manganese (b), and vanadium ternary compounds.

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Next, according to Moriya’s rules10 and the structural symmetry analysis above, the DMI vector for each pair of NNN M atoms is parallel to their bonds because the twofold rotation axes are along the directions between two NNN magnetic atoms [see Supplementary Fig. 2]. In our calculated structure with D2d crystal symmetry, the staggered spin vector will rotate in the plane perpendicular to the propagation direction <110>. However, we ignore the NNN DMI in our theoretical calculations, because we find that the DMI between NN and NNN atoms differs by about two orders of magnitude by using the four-state energy-mapping analysis55, e.g., VCuSe2 and MnCuSe2 (NNN DMI is −0.082 and −0.075 meV). Although the NNN DMI is neglected, we still observe the magnetic structure of Bloch-type helicoid from the results of the micromagnetic simulation.

Chiral spin textures of monolayer MCuX 2

Furthermore, we perform the atomistic micromagnetic simulation based on first-principles calculated materials parameters as shown in Table 1 by using the VAMPIRE software package56. To get the dynamics of magnetization, the Landau–Lifshitz–Gilbert (LLG) equation was used with the Langevin dynamics as follows:

$$\frac{{\partial {{{\mathbf{S}}}}_i}}{{\partial t}} = - \frac{\gamma }{{\left( {1 + \lambda ^2} \right)}}\left[ {{{{\mathbf{S}}}}_i \times H_{{\mathrm{eff}}}^i + \lambda {{{\mathbf{S}}}}_i \times \left( {{{{\mathbf{S}}}}_i \times H_{{\mathrm{eff}}}^i} \right)} \right]$$
(1)

where Si is the normal unit vector of ith magnetic atom, γ is the gyromagnetic ratio and λ is the gilbert damping constant. The magnitude of the effective field is obtained by the equation:\(H_{{\mathrm{eff}}}^i = - \frac{1}{{\mu _i}}\frac{{\partial H}}{{\partial {{{\mathbf{S}}}}_i}}\), in which μI represents a magnetic moment of the site i and H is the Hamiltonian of the system. In all micromagnetic simulations, we relax a random state of 100-nm-wide with periodic boundary conditions as the initial state to get the ground states without an external magnetic field. For all FM systems, uniform FM and FM Néel chiral structures are observed from CrCuS2 to MnCuTe2 in Fig. 4 due to the enhancement of ferromagnetic exchange coupling. Besides, in the VCuSe2 monolayer, antiskyrmions with a diameter of 31 nm emerge on the large ferromagnetic domain. It is also important to know the topological charge Q. Here, we apply the lattice-based approach to calculate the topological charge. First, we perform the atomic spin model simulations to obtain the spin vector S on each lattice point. Furthermore, we use the formula: S · (∂xS × ∂yS) to calculate the topological charge density of each lattice point. Finally, we integrate the charge density of the zone holding the topological magnetic quasiparticle to obtain the final topological charge. The topological charge is +1 in the zoomed antiskyrmion in Fig. 4d. More interestingly, we also realize the FM anti-bimerons arising from strong in-plane magnetic anisotropy in MnCuTe2. In all AFM systems, the topological charge of each AFM antiskyrmion with antiparallel NN spin alignment is 0. It can be decomposed into two identical FM antiskyrmion sublattices, where the FM sublattice pairs have opposite topological charges +1 and −1. In FeCuSe2, AFM antiskyrmion and AFM antiskyrmionium are presented in Fig. 4i, j. It is well known that lots of topologically magnetic textures can be induced by exchange frustration57,58. Notably, previous work have demonstrated that DMI and exchange frustration can stabilize skyrmionium in CrGe(Se,Te)3 Janus monolayer when J1/J2 is within a certain range59. Similar to AFM antiskyrmion, AFM antiskyrmionioum with zero topological charge refers to a magnetic texture that can be view as two nested AFM antiskyrmions due to the frustration caused by opposite sign exchange coupling J1 and J2. In addition, we note that isolated bimerons and multiple bimerons are observed in CoCuS2 and MnCuTe2, respectively. The main reason is that FM exchange coupling can be consistent with an effective FM external field. When we reduce the J1 of CoCuS2, multiple bimerons appear, as is shown in Supplementary Fig. 9. Similar to that of CoCuS2, when we increase J1 or external magnetic field in the VCuS2 system, respectively, one can observe that isolated antiskyrmions emerge, as is shown in Supplementary Fig. 11. Meantime, we also adopt open boundary conditions to simulate the spin textures of FM MnCuSe2 and AFM CoCuSe2, as shown in Supplementary Fig. 6. One can see that the simulated FM antiskyrmions in MnCuSe2 and AFM antiskyrmions in CoCuSe2 under open boundary conditions have almost the same size and topological properties with periodic boundary conditions. Furthermore, we also take into account the effect of dipolar interaction on chiral magnetic textures. For antiferromagnetic systems, magnetic dipolar interaction diminishes due to the cancellation of the magnetic moment of coupled sublattices60. Thus, we simulated VCuS2 and MnCuSe2 monolayers based on first-principles calculated magnetic parameters, one can see that the calculated spin configurations of VCuS2 and MnCuTe2 are consistent with previous results as shown in the Supplementary Fig. 7.

Fig. 4: Spin configurations of ground states of monolayers MCuX2 in real-space.
figure 4

Spin configuration of a 100 nm × 100 nm square and zoom of isolated chiral magnetic profile of VCuS2 (a, b); VCuSe2 (c, d); MnCuS2 (e, f); MnCuTe2 (g, h); FeCuSe2 (i, j); NiCuSe2 (k, l); CoCuS2 (m, n) and CoCuSe2 (o, p). The color map indicates the out-of-plane spin component.

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In addition, we also calculate magnetic parameters J1, J2, K, and D of monolayer VCuSe2 when the tensile strain increases from 1 to 5%, as shown in Table 2. We find that NN and NNN FM exchange coupling strengths decrease a lot while DMI changes slightly, resulting in a large ratio of D/J. Therefore, antiskyrmion with a smaller diameter is achieved under tensile strain7. The phase diagram of the VCuSe2 monolayer under different stress and temperatures is shown in Fig. 5. For pristine VCuSe2, the FM antiskyrmions embedded in the background of the large-size domain are observed [Fig. 4c]. It is found that stable chiral domain and magnetic antiskyrmions appear slowly when the temperature varies from 300 to 250 K. Furthermore, if we keep the temperature decreasing, one can see that the chiral magnetic structures can be basically stabilized at 250 K, and these chiral structures also undergo some small changes as the temperature continues to decrease to the finite temperature. Moreover, as the strain increases, the density of antiskyrmions increases, and the size of the domain becomes much smaller than that of the pristine state. Supplementary Fig. 8 presents the results of the micromagnetic simulations for spin spiral length as a function of strain. We can clearly observe the strain dependence of spin spiral length with the increasing D and decreasing exchange J, which is consistent with |D/J|.

Table 2 The calculated magnetic parameters of NN and NNN exchange coupling J1 and J2, magnetic anisotropy energy K, DMI parameters dy and magnetic moment μM of V atom when the tensile strain varies from 1 to 5% in monolayer VCuSe2.
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Fig. 5: Spin textures for VCuSe2 in real-space.
figure 5

Phase of spin configurations under increased tensile strain and decreased temperature for VCuSe2 monolayer.

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In summary, we discover a group of 2D layered ternary compounds MCuX2 (M:3d TM; X:group VIA) with anisotropic DMI protected by D2d crystal symmetry from first-principles calculations. We show that the anisotropic DMI can vary from 0 to 15 meV/atom in 2D layered ternary compounds MCuX2, where M represents 3d transition metal, and X represents the VIA group element. Thanks to the large enough anisotropic DMI, we demonstrate that a series of FM (AFM) antiskyrmions, bimerons, and antiskyrmionioum can be realized without an external field in the MCuX2 family. The discovery will provide a platform to find various FM/AFM anti-topological spin textures with crystal symmetry-protected anisotropic DMI. In addition, our calculations show that magnetic parameters of VCuSe2 are sensible to strain, and the possibility of antiskyrmions formation up to a hundred Kelvin is demonstrated in 2D VCuSe2. Our work will benefit both fundamental research and applications in the fields of 2D van der Waals materials and spintronics.

Methods

DFT calculations

First-principles calculations are carried out based on density functional theory (DFT) implemented in Vienna ab-initio Simulation Package (VASP)61. We adopt Perdew–Burke–Ernzerhof (PBE) functionals of the generalized gradient approximation (GGA)62 as the exchange-correlation potential, and use projector augmented plane-wave (PAW) method63,64 to deal with the interaction between nuclear electrons and valence electrons. We set the cutoff energy of 520 eV for the plane-wave basis set, and 24 × 24 × 1 k-point with Γ-centered meshes for the Brillouin zone integration. Partially occupied d orbitals of transition metal atoms are treated by GGA+U65 with U = 3 eV for the 3d orbitals of M and Cu elements. We set a vacuum layer with a thickness of 25 Å in the z-direction to ensure that there is no interaction between the periodic images. The convergence criteria of the total energy in the ion relaxation process and the Hellmann–Feynman force between atoms were set to be 1 × 10−7 eV and 0.001 eV/Å, respectively. To describe our magnetic system, we adopt the following Hamiltonian model:

$$H = - J_1\mathop {\sum}\nolimits_{\left\langle {i{{{\mathrm{,}}}}j} \right\rangle } {{{{\mathbf{S}}}}_i \cdot {{{\mathbf{S}}}}_j - J_2} \mathop {\sum}\nolimits_{\left\langle {i{{{\mathrm{,}}}}j} \right\rangle } {{{{\mathbf{S}}}}_i \cdot {{{\mathbf{S}}}}_j - } \mathop {\sum}\nolimits_{\left\langle {i{{{\mathrm{,}}}}j} \right\rangle } {{{{\mathbf{D}}}}_{ij} \cdot \left( {{{{\mathbf{S}}}}_i \cdot {{{\mathbf{S}}}}_j} \right) - K} \mathop {\sum}\nolimits_i {\left( {{{{\mathbf{S}}}}_i^z} \right)^2}$$
(2)

where Si (Sj) is the normal spin vector of ith (jth) magnetic atom, the J1 and J2 represent exchange coupling constants between Nearest-Neighbor (NN) and Next-Nearest-Neighbor (NNN) atoms, respectively. K is the magnetic anisotropy constant and Dij is the DMI vectors. The methods to calculate J, K, and D is described in the experimental section.

Magnetic parameters

Exchange coupling constant

We construct a 2 × 2 × 1 supercell to study three different magnetic configurations, which are FM, G-type AFM where nearest-neighbor spins are aligned antiparallel and Stripe-type AFM where spins are ordered antiferromagnetically (ferromagnetically) along with x (y) axis (see Supplementary Fig. 4). Exchange coupling constant of nearest neighbor (NN) and next-nearest neighbor (NNN) magnetic atoms are obtained based on the following formula:

$$E_{{\mathrm{FM}}} = - \frac{1}{2} \ast 4\left( {4J_1 + 4J_2} \right) + E_0$$
(3)
$$E_{{\mathrm{AFM}}{{{\mathrm{I}}}}} = - \frac{1}{2} \ast 4\left( { - 4J_2} \right) + E_0$$
(4)
$$E_{{\mathrm{AFM}}{{{\mathrm{II}}}}} = - \frac{1}{2} \ast 4\left( { - 4J_1 + 4J_2} \right) + E_0$$
(5)
$$J_1 = \frac{{E_{{\mathrm{AFM}}{{{\mathrm{II}}}}} - E_{{\mathrm{FM}}}}}{{16}}$$
(6)
$$J_2 = \frac{{2 \ast E_{{\mathrm{AFM}}{{{\mathrm{I}}}}} - E_{{\mathrm{AFM}}{{{\mathrm{II}}}}} - E_{{\mathrm{FM}}}}}{{32}}$$
(7)

where the positive/negative value corresponds to FM/AFM coupling.

Magnetic anisotropy energy K

Magnetic anisotropy energy is defined as the energy difference between the in-plane magnetized [100] axis and the out-of-plane magnetized [001] axis:

$$K = E_{100} - E_{001}$$
(8)

NN Dzyaloshinskii–Moriya interaction (NN-DMI) D

We performed DMI calculations using the chirality-dependent total energy difference method52. First of all, a 4 × 1 × 1 supercell is constructed to obtain the charge distribution of the system’s ground state by solving the Kohn–Sham equations in the absence of spin-orbit coupling (SOC). Then, SOC is included and we set spin spirals to determine the self-consistent total energies in the clockwise and anticlockwise rotation. Finally, the energy difference between clockwise and anticlockwise rotation is calculated to obtain the anisotropic D. The DMI can be obtained by the following formula:

$$D = (E_{{\mathrm{cw}}} - E_{{\mathrm{acw}}})/8$$
(9)

NNN Dzyaloshinskii–Moriya interaction (NNN DMI)

We performed the four-state energy-mapping analysis55. First of all, a 4 × 4 × 1 supercell is constructed to set all the spin configuration in the y direction, then change the spins between the two NN atoms. The DMI between NN M spins were calculated based on four spin configurations: (i) S1 = (S, 0, 0), S2 = (0, 0, S); (ii) S1 = (S, 0, 0), S2 = (0, 0, -S); (iii) S1 = (-S, 0, 0), S2 = (0, 0, S); (iv) S1 = (−S, 0, 0), S2 = (0, 0, −S). Next, according to spin interaction energy based different spin configurations, we can solve the in-plane component Dy: Dy = (E1 + E4 – E2 – E3)/(4S2).

Phonon spectrum

In calculations, based on the PHONOPY code66,67, we use 3 × 3 × 1, \(3\sqrt 2 \times 3\sqrt 2 \times 1\) and 4 × 4 × 1 supercells with the frozen phonon approximation to calculate the phonon dispersions of single-layer MCuX2. Supplementary Fig. 5 shows that Γ point of some of the MCuX2 monolayers have very small imaginary frequencies within the entire wave-vector space, which can be attributed to a wavelength of particular mode68.