Abstract
Magnetic skyrmions, topologically protected chiral spin swirling quasiparticles, have attracted great attention in fundamental physics and applications. Recently, the discovery of twodimensional (2D) van der Waals (vdW) magnets have aroused great interest due to their appealing physical properties. Moreover, both experimental and theoretical works have revealed that isotropic Dzyaloshinskii–Moriya interaction (DMI) can be achieved in 2D magnets or ferromagnetbased heterostructures. However, 2D magnets with anisotropic DMI haven’t been reported yet. Here, via using firstprinciples calculations, we unveil that anisotropic DMI protected by D_{2d} crystal symmetry can exist in 2D ternary compounds MCuX_{2} (M: 3d transition metal (TM), X: group VIA). Interestingly, by using micromagnetic simulations, we demonstrate that ferromagnetic (FM) antiskyrmions, FM bimerons, antiferromagnetic (AFM) antiskyrmions, and AFM bimerons can be realized in the MCuX_{2} family. Our discovery opens up an avenue to creating antiskyrmions and bimerons with anisotropic DMI protected by D_{2d} crystal symmetry in 2D magnets.
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Introduction
Topological nontrivial magnetic structures such as chiral domain walls^{1}, merons^{2}, bimerons^{3,4}, and skyrmions^{5,6} 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 density^{7,8}. Magnetic skyrmions with noncollinear spin configurations can be induced by antisymmetric exchange couplingDzyaloshinskiiMoriya interaction (DMI) in structures with inversion symmetry breaking and spinorbit coupling (SOC)^{9,10}. The antisymmetric exchange interaction can be written as E_{DMI} = D·(S_{i} × S_{j}), where D is the DMI vector and S_{i} and S_{j} represent the spins of sites i and j. Notably, magnetic skyrmions have been observed in noncentrosymmetric B20 bulk materials such as MnSi, FeGe, and FeCoSi^{6,11,12,13,14} and interfacial systems of multilayers such as Ir(111)/Fe, Ta/CoFeB, and Pt/Co^{15,16,17,18,19,20,21} with C_{nv} crystal symmetry. On the other hand, anisotropic DMI are reported in ultrathin epitaxial Au/Co/W(110) with C_{2v} crystal symmetry^{22}, and antiskyrmions with anisotropic DMI are reported in acentric tetragonal Heusler compounds with D_{2d} crystal symmetry^{23,24} and noncentrosymmetric tetragonal structure with S_{4} crystal symmetry^{25}. In parallel with the development of the hot study of skyrmions in these traditional bulk and multilayer thin films, 2D magnets, e.g., Fe_{3}GeTe_{2}^{26}, CrI_{3}^{27}, CrGeTe_{3}^{28}, MnSe_{2}^{29}, and VSe_{2}^{30} with longrange 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 magnetooptical and magnetoelectric effect for ultracompact spintronics in reduced dimensions. Moreover, recent works have proposed that Néeltype skyrmions with isotropic DMI can be realized in 2D Janus magnets, e.g., MnXY^{31}, CrXY^{32}, and multiferroics structures, e.g., CrN^{33}, BaTiO_{3}/SrRuO_{3}^{34}, In_{2}Se_{3}/MnBi_{2}Se_{2}Te_{2}^{35}. 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, MCuX_{2} 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 spinpolarized current to drive an antiskyrmion, the propagation direction of the antiskyrmion will follow the current direction without topological skyrmion Hall effect^{36,37,38,39,40,41}. 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 liquidphase approach^{42,43} peeling off the threedimensional layered van der Waals materials, such as CrGeTe_{3}^{28}, CrI_{3}^{27}, and Fe_{3}GeTe_{2}^{26}, etc. Bulk FeCuTe_{2} is a layered magnetic material stacked by weak interlayer van der Waals interaction, and it has been reported experimentally^{44,45,46}. It is reported bulk FeCuTe_{2} has a layered structure, and the unit cell parameters are as follows: a = 3.93 Å, c = 6.078 Å^{44}. The layered compound FeCuTe_{2} presents an antiferromagnetically ordered state below T_{N} = 254 K^{47}. Based on firstprinciples calculations, we further determine the magnetic properties of bulk FeCuTe_{2}. The calculated results show that the unit cell parameters a and c of bulk FeCuTe_{2} are 3.964 and 6.176 Å, respectively. Moreover, we also compare energies with different magnetic orderings as shown in Supplementary Fig. 10. Bulk FeCuTe_{2} 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 FeCuTe_{2}, we theoretically simulate the exfoliation energy when blocks were gradually peeled into a twodimensional structure as shown in Supplementary Fig. 1. The calculated cleavage energy is 0.46 J/m^{2}, which is comparable to that of graphene or phosphene^{48,49}, indicating the high possibility that layered 2D FeCuTe_{2} can be exfoliated from bulk. Interestingly, FeCuTe_{2} with D_{2d} crystal symmetry matches the Moriya symmetry rules^{10} to achieve anisotropic DMI. Moreover, we predict a series of 2D MCuX_{2} structures with D_{2d} 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 singlelayer MCuX_{2} (M:3d TM; X:group VIA) are represented in Fig. 1a–c. It exhibits a tetrahedral structure with MX_{top}M and MX_{bot}M configurations along x and y directions, and the facecentered coordinates are occupied by Cu atoms in the middle layer. The MCuX_{2} structure belonging to D_{2d} (No.115) crystal symmetry has the symmetry generators: identity operation E: (x, y, z) → (x, y, z); mirror plane inversion M_{2}: (x, y, z) → (x, y, z); fourfold rotoinversion IC_{4}: (x, y, z) → (y, x, z); and twofold axis C_{2}: (x, y, z) → (x, y, z), which leads to its feature with intrinsic inversion symmetry broken. According to the Moriya rules^{10}, the induced DMI sign should be opposite along x and y directions.
Via firstprinciples calculations, we obtain the basic magnetic parameters of MCuX_{2} ternary compounds as shown in Table 1 (Computational details are presented in the experimental method). The J_{1} and J_{2} indicate the exchange coupling constants between NN and NNN atoms. In most systems, J_{1} is orders larger than J_{2}. There are frustration coming from competing exchange interactions between J_{1} and J_{2} in the classical Heisenberg model on the tetrahedra structure except for VCuS_{2} and VCuSe_{2}. Interestingly, for MCuX_{2}, when M changes from V to Mn and from Fe to Ni, the systems are tuned from FM to Gtype 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 t_{2} (d_{xy}, d_{xz}, d_{yz}) levels due to the influence of the crystal field. The t_{2} ↔p↔e superexchange 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 halffilled with electrons, the AFM coupling mainly benefits from t↔t direct exchange. Thus, in the MCuX_{2} family with d orbit no less than halffilled, the competitive AFM coupling is stronger. Similar results are also depicted in the zincblende binary transition metal compounds^{50}. Collecting all 3d TM atoms magnetic moments in MCuX_{2} 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 rule^{51}.
DMI of monolayer MCuX _{2}
Figure 2 shows the calculated NNDMI of MCuX_{2} structures based on the chiralitydependent total energy difference approach^{52}. It is found that all systems have anisotropic DMI and d_{x} and d_{y} 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 stateoftheart 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 VCuTe_{2} 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, CoCuTe_{2} is different from other antiferromagnetic structures. We find that this structure tends to the StripAFM structure with a different lattice constant in x and y direction. Meantime, CoCuTe_{2} 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.
To elucidate the microscopic energy source of DMI, we calculate the atomic resolved SOC energy difference ΔΕ_{SOC} with opposite chirality associated with DMI. Only the ΔΕ_{SOC} are presented along the xdirection 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 MCuX_{2} monolayers due to the increase in SOC strength. Similar to the interface of HM/FM^{52,54}, in which ΔΕ_{soc} 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 spinorbit scattering between two magnetic atoms. In addition, we noticed that the ΔΕ_{SOC} of NiCuTe_{2} is close to zero. From optimization, we also identify the equilibrium lattice constant values for FM configuration of MnCuTe_{2} and GAFM configuration of NiCuTe_{2}, 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.
Next, according to Moriya’s rules^{10} 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 D_{2d} 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 fourstate energymapping analysis^{55}, e.g., VCuSe_{2} and MnCuSe_{2} (NNN DMI is −0.082 and −0.075 meV). Although the NNN DMI is neglected, we still observe the magnetic structure of Blochtype helicoid from the results of the micromagnetic simulation.
Chiral spin textures of monolayer MCuX _{2}
Furthermore, we perform the atomistic micromagnetic simulation based on firstprinciples calculated materials parameters as shown in Table 1 by using the VAMPIRE software package^{56}. To get the dynamics of magnetization, the Landau–Lifshitz–Gilbert (LLG) equation was used with the Langevin dynamics as follows:
where S_{i} 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 μ_{I} 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 100nmwide 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 CrCuS_{2} to MnCuTe_{2} in Fig. 4 due to the enhancement of ferromagnetic exchange coupling. Besides, in the VCuSe_{2} 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 latticebased 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 · (∂_{x}S × ∂_{y}S) 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 antibimerons arising from strong inplane magnetic anisotropy in MnCuTe_{2}. 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 FeCuSe_{2}, 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 frustration^{57,58}. Notably, previous work have demonstrated that DMI and exchange frustration can stabilize skyrmionium in CrGe(Se,Te)_{3} Janus monolayer when J_{1}/J_{2} is within a certain range^{59}. 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 J_{1} and J_{2}. In addition, we note that isolated bimerons and multiple bimerons are observed in CoCuS_{2} and MnCuTe_{2}, respectively. The main reason is that FM exchange coupling can be consistent with an effective FM external field. When we reduce the J_{1} of CoCuS_{2}, multiple bimerons appear, as is shown in Supplementary Fig. 9. Similar to that of CoCuS_{2}, when we increase J_{1} or external magnetic field in the VCuS_{2} 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 MnCuSe_{2} and AFM CoCuSe_{2}, as shown in Supplementary Fig. 6. One can see that the simulated FM antiskyrmions in MnCuSe_{2} and AFM antiskyrmions in CoCuSe_{2} 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 sublattices^{60}. Thus, we simulated VCuS_{2} and MnCuSe_{2} monolayers based on firstprinciples calculated magnetic parameters, one can see that the calculated spin configurations of VCuS_{2} and MnCuTe_{2} are consistent with previous results as shown in the Supplementary Fig. 7.
In addition, we also calculate magnetic parameters J_{1}, J_{2}, K, and D of monolayer VCuSe_{2} 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 strain^{7}. The phase diagram of the VCuSe_{2} monolayer under different stress and temperatures is shown in Fig. 5. For pristine VCuSe_{2}, the FM antiskyrmions embedded in the background of the largesize 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.
In summary, we discover a group of 2D layered ternary compounds MCuX_{2} (M:3d TM; X:group VIA) with anisotropic DMI protected by D_{2d} crystal symmetry from firstprinciples calculations. We show that the anisotropic DMI can vary from 0 to 15 meV/atom in 2D layered ternary compounds MCuX_{2}, 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 MCuX_{2} family. The discovery will provide a platform to find various FM/AFM antitopological spin textures with crystal symmetryprotected anisotropic DMI. In addition, our calculations show that magnetic parameters of VCuSe_{2} are sensible to strain, and the possibility of antiskyrmions formation up to a hundred Kelvin is demonstrated in 2D VCuSe_{2}. Our work will benefit both fundamental research and applications in the fields of 2D van der Waals materials and spintronics.
Methods
DFT calculations
Firstprinciples calculations are carried out based on density functional theory (DFT) implemented in Vienna abinitio Simulation Package (VASP)^{61}. We adopt Perdew–Burke–Ernzerhof (PBE) functionals of the generalized gradient approximation (GGA)^{62} as the exchangecorrelation potential, and use projector augmented planewave (PAW) method^{63,64} to deal with the interaction between nuclear electrons and valence electrons. We set the cutoff energy of 520 eV for the planewave basis set, and 24 × 24 × 1 kpoint with Γcentered meshes for the Brillouin zone integration. Partially occupied d orbitals of transition metal atoms are treated by GGA+U^{65} 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 zdirection 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:
where S_{i} (S_{j}) is the normal spin vector of ith (jth) magnetic atom, the J_{1} and J_{2} represent exchange coupling constants between NearestNeighbor (NN) and NextNearestNeighbor (NNN) atoms, respectively. K is the magnetic anisotropy constant and D_{ij} 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, Gtype AFM where nearestneighbor spins are aligned antiparallel and Stripetype AFM where spins are ordered antiferromagnetically (ferromagnetically) along with x (y) axis (see Supplementary Fig. 4). Exchange coupling constant of nearest neighbor (NN) and nextnearest neighbor (NNN) magnetic atoms are obtained based on the following formula:
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 inplane magnetized [100] axis and the outofplane magnetized [001] axis:
NN Dzyaloshinskii–Moriya interaction (NNDMI) D
We performed DMI calculations using the chiralitydependent total energy difference method^{52}. 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 spinorbit coupling (SOC). Then, SOC is included and we set spin spirals to determine the selfconsistent 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:
NNN Dzyaloshinskii–Moriya interaction (NNN DMI)
We performed the fourstate energymapping analysis^{55}. 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) S_{1} = (S, 0, 0), S_{2} = (0, 0, S); (ii) S_{1} = (S, 0, 0), S_{2} = (0, 0, S); (iii) S_{1} = (S, 0, 0), S_{2} = (0, 0, S); (iv) S_{1} = (−S, 0, 0), S_{2} = (0, 0, −S). Next, according to spin interaction energy based different spin configurations, we can solve the inplane component D_{y}: D_{y} = (E_{1} + E_{4} – E_{2} – E_{3})/(4S^{2}).
Phonon spectrum
In calculations, based on the PHONOPY code^{66,67}, 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 singlelayer MCuX_{2}. Supplementary Fig. 5 shows that Γ point of some of the MCuX_{2} monolayers have very small imaginary frequencies within the entire wavevector space, which can be attributed to a wavelength of particular mode^{68}.
Data availability
The datasets used in this article are available from the corresponding author upon request.
Code availability
Code that supports the findings of this study are available from the corresponding authors on reasonable request.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11874059 and 12174405); Key Research Program of Frontier Sciences, CAS (Grant NO. ZDBSLY7021); Pioneer and Leading Goose R&D Program of Zhejiang Province (Grant No. 2022C01053); Ningbo Key Scientific and Technological Project (Grant No. 2021000215); Zhejiang Provincial Natural Science Foundation (Grant No. LR19A040002); Beijing National Laboratory for Condensed Matter Physics (Grant No. 2021000123); and Ningbo 3315 project.
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H.Y. conceived the project. Y.G. performed the calculations. Y.G., Q.C., and H.Y. analyzed the results and wrote the manuscript. All the authors contributed to the discussion and the writing.
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Ga, Y., Cui, Q., Zhu, Y. et al. Anisotropic DzyaloshinskiiMoriya interaction protected by D_{2d} crystal symmetry in twodimensional ternary compounds. npj Comput Mater 8, 128 (2022). https://doi.org/10.1038/s41524022008094
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DOI: https://doi.org/10.1038/s41524022008094
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