Strain-Enabled Control of Chiral Magnetic Structures in MnSeTe Monolayer

Zhiwen Wang(王智文)$^{1,2}$, Jinghua Liang(梁敬华)$^{2}$, and Hongxin Yang(杨洪新)$^{1,2\hyperlink{s}{*}}$

doi:10.1088/0256-307X/40/1/017501

⇦Chinese Physics Letters, 2023, Vol. 40, No. 1, Article code 017501⇨ Strain-Enabled Control of Chiral Magnetic Structures in MnSeTe Monolayer Zhiwen Wang (王智文)1,2, Jinghua Liang (梁敬华)2, and Hongxin Yang (杨洪新)1,2* Affiliations 1National Laboratory of Solid State Microstructures, School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China 2Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo 315201, China Received 28 September 2022; accepted manuscript online 5 December 2022; published online 21 December 2022 ^*Corresponding author. Email: hongxin.yang@nju.edu.cn Citation Text: Wang Z W, Liang J H, and Yang H X 2023 Chin. Phys. Lett. 40 017501 Abstract Chiral magnetic states are promising for future spintronic applications. Recent progress of chiral spin textures in two-dimensional magnets, such as chiral domain walls, skyrmions, and bimerons, have been drawing extensive attention. Here, via first-principles calculations, we show that biaxial strain can effectively manipulate the magnetic parameters of the Janus MnSeTe monolayer. Interestingly, we find that both the magnitude and the sign of the magnetic constants of the Heisenberg exchange coupling, Dzyaloshinskii–Moriya interaction and magnetocrystalline anisotropy can be tuned by strain. Moreover, using micromagnetic simulations, we obtain the distinct phase diagram of chiral spin texture under different strains. Especially, we demonstrate that abundant chiral magnetic structures including ferromagnetic skyrmion, skyrmionium, bimeron, and antiferromagnetic spin spiral can be induced in the MnSeTe monolayer. We also discuss the effect of temperature on these magnetic structures. The findings highlight the Janus MnSeTe monolayer as a good candidate for spintronic nanodevices.

DOI:10.1088/0256-307X/40/1/017501 © 2023 Chinese Physics Society Article Text Topological chiral spin textures, such as magnetic skyrmions,[1,2] chiral domain walls (DWs),[3,4] and bimerons,[5,6] have attracted a great deal of attention due to their potential applications for next-generation spintronic devices. Their topological spin arrangements and consequent emergent electromagnetic fields give rise to unique features in transport and magnetoelectric properties, such as the topological Hall effect,[7] current-driven skyrmions,[8-10] and multiferroic behavior.[11] Over the past decade, many efforts have been devoted to studying magnetic skyrmion materials.[12] The most widely investigated magnetic skyrmion materials include the non-centrosymmetric bulk magnets, e.g., FeGe,[13] MnSi,[14] and the ferromagnet/heavy metal heterostructures such as Ir/Co/Pt[15] and Pt/Co/MgO.[16] Microscopically, the asymmetric Dzyaloshinskii–Moriya interaction (DMI), which arises from the spin-orbit coupling (SOC) and broken inversion symmetry, plays an essential role in formation of chiral magnetic nanostructures for above-mentioned magnetic materials.[17] Recently, two-dimensional (2D) van der Waals (vdW) magnets, such as CrI$_{3}$,[18] CrGeTe$_{3}$,[19] and Fe$_{3}$GeTe$_{2}$[20] monolayers, have been successfully realized in experiment, which provided an alternative avenue for exploring new exotic spin phenomena with potential applications in the development of novel spintronic devices. Inspired by the discovery of 2D vdW magnets, several theoretical and experimental studies have demonstrated the emergence of chiral spin structures in these materials and their heterostructures.[21-23] In particular, the previous research has shown that 2D magnetic Janus materials with intrinsic broken inversion symmetry have been predicted to be promising candidates for exhibiting skyrmionic state, e.g., Cr(I,$X$)$_{3}$ ($X$ = Br, Cl),[24] CrGe(Se,Te)$_{3}$,[25] Mn$X$Te($X$ = S, Se),[26] and Cr$X$Te ($X$ = S, Se) monolayers.[27] Moreover, it is also shown that the strain engineering can be an effective method for tuning the magnetic properties.[28-30] For instance, in the Janus thin films,[27,31] the increasing ferromagnetism is driven by rapidly quenched direct exchange interaction under biaxial strain, which substantially enhances the Curie temperature. As we all know, the competition of DMI, magnetocrystalline anisotropy (MCA) and Heisenberg exchange coupling in a proper way determines the novel spin textures. Especially, the ratio of DMI/Heisenberg exchange coupling (|$D/J$|) can provide information about the existence of magnetic skyrmions or not, which are within the typical range of 0.1–0.2 known to generate skyrmionic phases.[4] Interestingly, for the Cr(I,Br)$_{3}$ monolayer, the non-negligible second-nearest neighbors of DMI lead to the |$D_{2}/J_{2}$| ratios comparable to |$D_{1}/J_{1}$| counterparts, which can intrinsically host metastable domain wall skyrmionic phases. Therefore, it is interesting to investigate what chiral magnetic structures can be realized when the beyond-nearest exchange is considered. Here, we systematically study the DMI, MCA and Heisenberg exchange coupling of MnSeTe monolayer under biaxial strain by first-principles calculations. The magnetic constants of DMI and Heisenberg exchange coupling up to the third-nearest neighbors are explicitly considered in the calculations. Furthermore, the micromagnetic simulations demonstrate that the strain can induce abundant chiral magnetic structures in the MnSeTe monolayer. Calculation Methods. All of our firs-principles calculations are performed within the framework of density functional theory implemented in Vienna ab initio simulation package (VASP).[32-34] The exchange correlation energy is calculated within the generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof form.[35,36] Considering the strongly correlated 3$d$ electrons of Mn, the GGA + $U$ ($U_{\rm eff} = 2$ eV)[26,37] method is adopted. The cutoff energy is set to 520 eV. The centered $k$-point grids of $24 \times 24\times 1$ is good enough to sample the first Brillouin zones of the MnSeTe. A vacuum space of at least 15 Å in the direction normal to the layer is used to avoid interactions between the periodic images. All parameters are carefully tested and make the total energy of MnSeTe converge within $10^{-7}$ eV. The atom positions are fully relaxed until the Hellmann–Feynman force is less than $10^{-3}$ eV/Å. The magnetic anisotropic energy (MAE), defined as $E_{(001)}-E_{(100)}$, was determined by the comparison between the total energy of self-consistently calculated results with spin direction along $z$ (001) and $x$ (100) axes. We have used the chirality-dependent total energy difference approach to obtain the DMI strength, whose details are illustrated in the Supplemental Material.[38] Using the magnetic interaction parameters determined by the first-principles calculations, we perform Monte Carlo (MC) simulations with the Metropolis algorithm to explore the magnetic states. The investigated systems are gradually cooled down from 1000 K to the required low temperature. For each temperature, $10^{5}$ MC steps are employed to thermalize the system. In all MC simulations, a large supercell consisting of $180 \times 180\times 1$ unit cells and periodic boundary conditions are used in order to avoid the non-universal boundary effects. Results and Discussion—Geometric Structure. Figure 1(a) depicts the top and side views of 2D Janus MnSeTe monolayer that we investigate here. One can see that the Mn atoms with point group $C_{3v}$ form a hexagonal network sandwiched by two atomic planes of different chalcogenide atoms. The top atomic layer is formed by the lighter elements Se, while the heavier ions Te are on the bottom layers. The relaxed structural parameters of MnSe$_{2}$, MnSeTe and MnTe$_{2}$ monolayers including lattice constants $a$; bond lengths of Mn–Se ($d_{\rm Mn-Se}$) and Mn–Te ($d_{\rm Mn-Te}$), and bonding angles of Mn–Se(Te)–Mn $\theta_{1}(\theta_{2})$, are listed in Table 1, which yields reasonable comparison with previous work.[26] Due to the smaller atomic radii of Se compared with that of Te, the optimized lattice constant of MnSeTe is smaller than MnTe$_{2}$, while larger than MnSe$_{2}$. For the same reason, the $d_{\rm Mn-Te}$ and $\theta_{2}$ are always larger than $d_{\rm Mn-Se}$ and $\theta_{1}$, respectively. Similar relationship of structural parameters is also found in the Cr$X$Te ($X$ = S, Se) monolayers.[27]

Table 1. The optimized lattice constants $a$, bond lengths of Mn–Se $d_{\rm Mn-Se}$ and Mn–Te $d_{\rm Mn-Te}$, and bonding angles of atom planes Mn–Se–Mn, $\theta_{1}$, Mn–Te–Mn, $\theta_{2}$ of Mn$X_{2}$ ($X$ = Se, Te) and MnSeTe monolayers.
Pattern	$a$ (Å)	$d_{\rm Mn-Se}$ (Å)	$d_{\rm Mn-Te}$ (Å)	$\theta_{1}$ (deg)	$\theta_{2}$ (deg)
MnSeTe	3.681	2.74	2.52	93.85	84.23
MnSe$_{2}$	3.592	2.51		91.12
MnTe$_{2}$	3.834		2.74		88.92

Spin Model and Magnetic Parameters. Applying strain is an effective way to tune the magnetic properties of magnetic thin films.[39] We have calculated the magnetic parameters of MnSeTe as functions of strain in the range from $-6$% to 6%. The value of strain is defined as $\varepsilon = [(a - a_{0})/a_{0}]\times 100\%$, where $a$ and $a_{0}$ are the lattice constants of strained and unstrained MnSeTe monolayers. Note that the negative and positive strains represent the compressive and tensile strains, respectively. As one can see in Fig. S1, the main contribution to magnetism of MnSeTe comes from the Mn atoms, meanwhile, Se and Te are both effectively spin-polarized due to the hybridizations between Mn-$d$ and Se(Te)-$p$ orbitals. Interestingly, the biaxial strain can monotonically tune the value of spin polarization (see part I in the Supplemental Material[38]).

Fig. 1. (a) The top (upper) and side (lower) views of the crystal structure and the schematic diagram of exchange coupling (b) for the MnSeTe monolayer. The solid lines, $\theta_1$ and $\theta_2$ in (a) show the primitive cell, the bonding angles of atom planes Mn–Se–Mn and Mn–Te–Mn, respectively. The red, blue and black lines in (b) represent the nearest-$[J_{1}(d_{1})]$, next-nearest-$[J_{2}(d_{2})]$ and third-nearest-$[J_{3}(d_{3})]$ neighbors for spins, respectively. For clarity only the Mn atoms are shown in (b).

We have systematically considered the DMI and Heisenberg exchange coupling up to the third-nearest neighbors for spins, which are explicitly shown in Fig. 1(b). To investigate the magnetic properties of MnSeTe under biaxial strain, we adopt the following model Hamiltonian: \begin{align} H=H^{\scriptscriptstyle{\rm MCA}}{+H}^{\scriptscriptstyle{\rm EX}}+H^{\scriptscriptstyle{\rm DMI}} \tag {1} \end{align} with \begin{align} H^{\scriptscriptstyle{\rm MCA}}=\,&-K\sum\limits_i {{(\boldsymbol{S}_{i}^{z})}^{2},} \tag {2}\\ H^{\scriptscriptstyle{\rm EX}}=\,&-J_{1}\sum\limits_{\langle i, j \rangle } {\boldsymbol{S}_{i}\cdot \boldsymbol{S}_{j}} -J_{2}\sum\limits_{\langle \langle i, j\rangle\rangle}\boldsymbol{S}_{i}\cdot \boldsymbol{S}_{j}\notag\\ &-J_{3}\sum\limits_{\langle \langle\langle i, j \rangle \rangle\rangle} {\boldsymbol{S}_{i}{\boldsymbol{\cdot S}}_{j}} \tag {3}\\ H^{\scriptscriptstyle{\rm DMI}}=\,&-\sum\limits_{\langle i, j \rangle } {\boldsymbol{D}_{ij}\cdot (\boldsymbol{S}_{i}\times \boldsymbol{S}_{j})} -\sum\limits_{\langle \langle i, j\rangle\rangle } \boldsymbol{D}_{ij}\cdot (\boldsymbol{S}_{i}\times \boldsymbol{S}_{j})\notag\\ &-\sum\limits_{\langle \langle\langle i, j \rangle\rangle\rangle} {\boldsymbol{D}_{ij}\cdot (\boldsymbol{S}_{i}\times \boldsymbol{S}_{j}), } \tag {4} \end{align} where $\boldsymbol{S}_{i}$ ($\boldsymbol{S}_{j}$) is the unit spin vector at the $i$th ($j$th) Mn atom. The $H^{\scriptscriptstyle{\rm MCA}}$, $H^{\scriptscriptstyle{\rm EX}}$, and $H^{\scriptscriptstyle{\rm DMI}}$ represent the uniaxial MCA, Heisenberg exchange coupling, and DMI, respectively, with K, $J_{i}$, and $D_{ij}$ being the coupling constants. Moreover, the character of $\langle i,j\rangle$, $\langle \langle i,j \rangle\rangle$, and $\langle \langle \langle i,j \rangle\rangle\rangle$ show the summation of the nearest-, next-nearest-, and third-nearest neighbors for the spins, respectively. The dipole-dipole interaction is neglected in the calculations as it has negligible effect on the results (see Fig. S10 in the Supplemental Material[38]). We have also checked that the pseudodipolar interaction between the nearest neighboring spins can be neglected. We firstly discuss the variation of MCA of MnSeTe under biaxial strain, as shown in Fig. 2(a). Interestingly, the MCA of MnSeTe has oscillating behavior with the applied strain. One can see that pristine MnSeTe has perpendicular magnetic anisotropy (PMA) of 0.97 meV/unit cell. As the tensile strain is applied, the PMA is significantly enhanced nonmonotonically, and reaches the largest value of 2.68 meV at the 5% strain. However, the MCA changes from PMA to in-plane magnetic anisotropy (IMA) when the compressive strain ranges from 0 to $-3$%. If we keep increasing the compressive strain, for $\varepsilon > -3$%, the PMA can be obtained again.

Fig. 2. (a) The calculated MAE as a function of biaxial strain in MnSeTe. (b) Element-resolved MAE of MnTeSe monolayer as a function of the strain. The black, red and blue lines in (b) show the Mn, Se and Te atoms, respectively.

Fig. 3. The orbit-resolved MCA contributions from Te 5$p$ orbitals hybridization of MnSeTe under different strains.

To interpret the origin of the oscillating MCA, we consider the atom-resolved MAE of MnSeTe monolayer. Figure 2(b) shows the element-resolved MAE of the MnTeSe monolayer as a function of the strain. One can clearly see that the MAE is most contributed by the Te atom, and its variation trend follows that of the system in Fig. 2(a). To further elucidate mechanisms for the change of MAE of the Te atom, we perform a comparative analysis of the orbital-resolved MAE of the Te 5$p$ orbitals under $-6$%, $-4$%, $-3$%, 0%, 1%, 3%, 5%, and 6% strains, as shown in Fig. 3. For the pristine MnSeTe [Fig. 3(a)], the hybridization between $p_{z}$ and $p_{y}$ states of Te atoms (the blue bar in Fig. 3) gives rise to the dominant PMA. When the tensile strain is applied [Figs. 3(b)–3(e)], the PMA contributed by the ($p_{z}$, $p_{y}$) hybridization keeps the dominant role and further increases monotonically with the increase of tensile strain. However, the contribution from the hybridization of ($p_{x}$, $p_{y}$) (the red bar in Fig. 3) oscillates between the PMA and IMA as indicated by its sign variation. Such behaviors result in the nonmonotonic increment of PMA of the MnSeTe monolayer under the tensile strain. Based on the same analysis, we find that the contribution from the hybridization of ($p_{x}$, $p_{y}$) to MCA also oscillates as a function of the compressive strain [Figs. 3(f)–3(h)]. However, the present contribution from the hybridization of ($p_{z}$, $p_{y}$) is strongly suppressed by the compressive strain. With $-3$% strain [Fig. 3(f)], the hybridization of ($p_{x}$, $p_{y}$) constitutes IMA and the contribution of PMA arising from the hybridization of ($p_{z}$, $p_{y}$) is reduced, which finally gives rise to the relatively large IMA of the strained system. When $\varepsilon < -3$%, the contribution from the hybridization of ($p_{x}$, $p_{y}$) changes from IMA to PMA, which makes the system become PMA again. Therefore, it is the competition of the hybridization of ($p_{z}$, $p_{y}$) and ($p_{x}$, $p_{y}$) in the Te atom that results in the MCA oscillation. To investigate the impact of orbital contributions to the MAE, we have calculated the difference ($\Delta M_{\rm orbital}$) of the local orbital moments with the direction of magnetization along in-plane ($M_{||}$) or out-of-plane ($M_{\bot}$) axis of MnSeTe plane. From Fig. S2, we can see that the $\Delta M_{\rm orbital}$ values of the Mn, Se and Te elements are smaller than 0.01$\mu_{\scriptscriptstyle{\rm B}}$ per atom. Moreover, the variations of $\Delta M_{\rm orbital}$ for all the elements disagree with the change of MAE under biaxial strain. Therefore, there is no direct relation between orbital magnetic moments and MAE in the MnSeTe monolayer.[40] Next, we explore the Heisenberg exchange interaction under biaxial strain. To determine the exchange parameters between Mn atoms, we expanded the unit cell and mapped the DFT total energies of four different magnetic configurations into Eq. (3) (see Fig. S3 in the Supplemental Material[38]). The determined Heisenberg exchange constants $J_{1}$, $J_{2}$, and $J_{3}$ for the nearest-, next-nearest-, and third-nearest neighbors, respectively, are shown in Fig. 4(a). Here, we employ the sign convention that positive (negative) value of $J_{i}$ ($i=1$, 2, and 3) means ferromagnetic (antiferromagnetic) exchange coupling. For the pristine MnSeTe monolayer, the exchange constants are all ferromagnetic (FM), which is consistent with our previous work.[26] When we apply a tensile strain, the exchange parameter $J_{1}$ of the MnSeTe increases monotonically. However, the $J_{1}$ decreases monotonically by compressive strain and takes a transition from positive value to negative value for $\varepsilon < -3{\%}$. We can find that $J_{2}$ always prefers the FM ordering even though the biaxial strain is applied. The changing magnitude of $J_{2}$ is relatively small. Interestingly, when we gradually decrease the tensile strain and increase the compressive strain, the $J_{3}$ has a monotonic variation and takes a transition from negative value to positive value for $\varepsilon < 2{\%}$.

Fig. 4. (a) The nearest-, next-nearest-, and third-nearest-neighbor Heisenberg exchange coupling $J_{i}$ ($i = 1$, 2, and 3) and (b) distances of Mn–Mn, Mn–Te, and Mn–Se as functions of strain in Janus MnSeTe monolayers. Schematic diagrams of (c) Mn–Mn hopping, (d) Mn–Se/Te–Mn hopping, (e) Mn–Se/Te–Se/Te–Mn hopping, and (f) Mn–Se/Te–Mn–Se/Te–Mn hopping in Janus MnSeTe monolayer.

For the MnSeTe monolayer, since the bonding angles of Mn–Se/Te–Mn are close to 90$^{\circ}$, the super-exchange coupling between two nearest neighboring Mn cations through intervening Se/Te anion is FM [Fig. 4(d)], according to the Goodenough–Kanamori–Anderson rules.[41-43] On the contrary, the direct exchange coupling originated from direct hopping between two nearest-neighboring Mn cations is antiferromagnetic (AFM) [Fig. 4(c)]. The competition between indirect FM and AFM couplings determines the final magnetic configuration of Mn atoms. When biaxial strain is applied, as shown in Fig. 4(b), the Mn–Te/Se distances hardly change, whereas the Mn–Mn distance obviously increases (decreases) by tensile (compressive) strain. Therefore, the tensile strain enhances ferromagnetism, while the AFM coupling is increased by compressive strain, which results in the monotonic variation of $J_{1}$. According to the distance between magnetic ions [Fig. 4(e)], the strength of $J_{2}$ is generally much weaker than those of direct interaction and super-exchange interaction. However, the $J_{3}$ can be enhanced by compressive strain due to the two times indirect exchange coupling through the Mn–Se/Te–Mn–Se/Te–Mn hopping path [Fig. 4(f)]. From the above discussions, we can know that the Heisenberg exchange interactions in the 2D magnets are rather complicated as the FM and AFM coupling may coexist in one typical system. Moreover, the strain can further lead to the competition of exchange interaction from different neighbors through the change of the distance between Mn pairs. Now we present the DMI of strained MnSeTe, which essentially affects the formation of topological magnetic states. We have used the chirality-dependent total energy difference approach to obtain the nearest-, next-nearest-, and third-nearest-neighbor DMI parameters $d_{i}$ ($i = 1$, 2, and 3) (the computational details are given in the Supplemental Material[38]). We define that the DMI $d_{i }> 0$ ($d_{i } < 0$) favors clockwise (anticlockwise) spin configurations. Based on Eq. (4) and the total energies of the spin spirals, we can determine the $d_{i}$ ($i = 1$, 2, and 3) of the free-standing MnSeTe as a function of biaxial strain, as shown in Fig. 5(a). We can find that the magnitude and chirality of $d_{1}$ and $d_{2}$ can be significantly tuned by biaxial strain, whereas $d_{3}$ shows the negligible variation. To elucidate the origin of the exceptional DMI in the MnSeTe monolayer, we plot atom-resolved localization of the associated SOC energy difference $\Delta E_{\rm soc}$ for $d_{i}$ ($i = 1$, 2, and 3). In Figs. 5(b) and 5(c), one can see that the dominant contribution to DMI stems from the competition of adjacent heavy Te and Se atoms, which accounts for variation of $d_{1}$ and $d_{2}$ by biaxial strain. Similar behavior has been identified for the FM/HM heterostructures,[44] where $E_{\rm soc}$ is dominated by the heavy 5$d$ transition metal at the interfacial layer. This is the so-called Fert–Levy mechanism of DMI.[4,45] However, for $d_{3}$ [Fig. 5(d)], the $\Delta E_{\rm soc}$ energy contributions from all atoms are small, which results in the negligible $d_{3}$ value.

Fig. 5. (a) The nearest-, next-nearest-, and third-nearest-neighbor DMI $d_{i}$ ($i = 1$, 2, and 3) as functions of strain in Janus MnSeTe monolayers. Atom-resolved localization of the DMI (b) $d_{1}$, (c) $d_{2}$, and (d) $d_{3}$ associated SOC energy $\Delta E_{\rm soc}$ for the Janus MnSeTe monolayers.

From the calculated DMI, MCA and Heisenberg exchange coupling constants discussed above, we can find that the Janus MnSeTe has rather complicated behaviors of magnetic properties with the variation of strain. More importantly, with the strain-enabled control of both the magnitude and sign of the magnetic parameters, we can expect that the competition of the magnetic interactions can make the MnSeTe monolayer become an ideal platform for the realization of rich chiral magnetic states in a single 2D thin film as discussed in the following. Spin Textures from Monte Carlo Simulations. Once all the parameters in the spin Hamiltonian are determined by first-principle calculation, we can perform MC simulations to explore the spin textures of the MnSeTe monolayer by biaxial strain. Figure 6 shows the relaxed magnetic structures at different strains. Strikingly, we can see that the rich chiral magnetic structures can be induced by application of the strain.

Fig. 6. Relaxed chiral magnetic textures for the 2D Janus MnSeTe monolayer under different biaxial strains. The color map indicates the out-of-plane spin component of Mn atoms.

The variation of the magnetic ground state with strain also shows complicated behavior similar to the magnetic parameters. For the pristine MnSeTe, we can obtain the chiral Néel skyrmionium phase when we consider the DMI and Heisenberg exchange coupling up to the third-nearest neighbors. Moreover, the diameter of skyrmionium phase is about 52 nm. Here, we should note that if only the nearest-neighbor interactions are considered, we can only find the skyrmion phase instead of the skyrmionium phase.[26] Thus, the appearance of the skyrmionium is related to the effect of the beyond nearest-neighbor interactions. Under 1% tensile strain, the strained MnSeTe has a ferromagnetic ground state due to the larger PMA and smaller DMI than the pristine MnSeTe. One can see that the Néel stripe-like ferromagnetic spirals can exist in the 2% strained MnSeTe, in which the width reaches 37 nm. Interestingly, the sign of $d_{1}$ changes from positive to negative when tensile strain reaches up to 3%. Nevertheless, the $d_{2}$ has the dominate contribution and still is positive. Thus, we can realize the Néel-type skyrmion phase with counterclockwise chirality in 3% strained MnSeTe. If we keep increasing tensile strain above 3%, the uniform ferromagnetic states appear due to the enhancement of ferromagnetic exchange coupling and PMA. When we apply the $-1$% compressive strain, we can realize the worm-like domains separated by chiral Néel DW. For $-2$% strained MnSeTe, the bimeron can be obtained as a result of transition from PMA to IMA. The size of bimeron is around 13 nm. As the compressive strain reaches up to $-3$%, the MnSeTe keeps a ferromagnetic state due to relatively large IMA and small Heisenberg exchange coupling. One can see that if the compressive strain is larger than $-3$%, we can realize the antiferromagnetic spirals. Additionally, we explore the temperature influence on the stabilization of spin textures under tensile and compressive strain as shown in Figs. S8 and S9. As the temperature increases, one can see that the images of ferromagnetic skyrmion, skyrmionium, bimeron, and antiferromagnetic spin spiral in MnSeTe all begin to be less well-defined above about 50 K and become more and more blurred up to 150 K. This blurring expresses the destabilization of the chiral spin textures by thermal fluctuations (see part V in the Supplemental Material[38]). According to results of MC simulations, we clarify the essential effect of DMI and Heisenberg exchange coupling up to the thirds-nearest neighbors for establishing the novel spin states. Moreover, the strain-enabled control method has shown the great potential for manipulating the chiral magnetic structures in the field of 2D magnets. In summary, using first-principles calculations and MC simulations, we have systematically investigated magnetic properties of Janus MnSeTe monolayers by biaxial strain. We find oscillatory MCA behavior induced by strain due to the competition of the hybridization of ($p_{z}$, $p_{y}$) and ($p_{x}$, $p_{y}$) of Te atoms. Moreover, we show that biaxial strain can effectively manipulate the Heisenberg exchange interactions and DMI. With the competition of MCA, Heisenberg exchange coupling and DMI, we can obtain abundant chiral topological magnetic structures including ferromagnetic skyrmion, skyrmionium, bimeron, and antiferromagnetic spirals. As temperature increases, the chiral spin states start fluctuating above 50 K before an evolution to a completely disordered structure at higher temperature. Altogether, we realize that the Janus MnSeTe monolayer can be a promising candidate for the realization of exotic chiral magnetic structures. Moreover, our results highlight the important role of the beyond nearest neighbor exchange coupling in the determination of magnetic states of 2D magnets. Acknowledgment. This work was supported by the National Key R&D Program of China (Grant No. 2022YFA1405102), the National Natural Science Foundation of China (Grant Nos. 11874059 and 12174405), the Key Research Program of Frontier Sciences, CAS (Grant No. ZDBS-LY-7021), the Ningbo Key Scientific and Technological Project (Grant No. 2021000215), the “Pioneer” and “Leading Goose” R&D Program of Zhejiang Province (Grant No. 2022C01053), the Zhejiang Provincial Natural Science Foundation (Grant No. LR19A040002), and Beijing National Laboratory for Condensed Matter Physics (Grant No. 2021000123). References Skyrmion Lattice in a Chiral MagnetNear room-temperature formation of a skyrmion crystal in thin-films of the helimagnet FeGeChiral spin torque at magnetic domain wallsSkyrmions on the trackDynamics of ferromagnetic bimerons driven by spin currents and magnetic fieldsMagnetic bimerons as skyrmion analogues in in-plane magnetsTopological properties and dynamics of magnetic skyrmionsElectric-field guiding of magnetic skyrmionsMotion tracking of 80-nm-size skyrmions upon directional current injectionsElectric-field control of skyrmions in multiferroic heterostructure via magnetoelectric couplingDzyaloshinsky–Moriya interaction (DMI)-induced magnetic skyrmion materialsDirect Imaging of a Zero-Field Target Skyrmion and Its Polarity Switch in a Chiral Magnetic NanodiskTopological Hall Effect in the

A

Phase of MnSiAdditive interfacial chiral interaction in multilayers for stabilization of small individual skyrmions at room temperatureRoom-temperature chiral magnetic skyrmions in ultrathin magnetic nanostructuresMagnetic skyrmions: advances in physics and potential applicationsLayer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limitDiscovery of intrinsic ferromagnetism in two-dimensional van der Waals crystalsGate-tunable room-temperature ferromagnetism in two-dimensional Fe3GeTe2Topological Magnetic-Spin Textures in Two-Dimensional van der Waals Cr₂ Ge₂ Te₆Observation of Magnetic Skyrmion Bubbles in a van der Waals Ferromagnet Fe₃ GeTe₂Néel-type skyrmion in WTe2/Fe3GeTe2 van der Waals heterostructureTopological spin texture in Janus monolayers of the chromium trihalides Cr(I,

{X)}_{3}

Emergence of skyrmionium in a two-dimensional

{CrGe (Se, Te)}_{3}

Janus monolayerVery large Dzyaloshinskii-Moriya interaction in two-dimensional Janus manganese dichalcogenides and its application to realize skyrmion statesStrain-tunable ferromagnetism and chiral spin textures in two-dimensional Janus chromium dichalcogenidesNon-Volatile 180° Magnetization Reversal by an Electric Field in Multiferroic HeterostructuresHeterogeneous integration of single-crystalline complex-oxide membranesInterfacial Dzyaloshinskii-Moriya interaction arising from rare-earth orbital magnetism in insulating magnetic oxidesHigh-temperature ferromagnetic semiconductors: Janus monolayer vanadium trihalidesEfficient iterative schemes for ab initio total-energy calculations using a plane-wave basis setEfficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis setAb initio molecular dynamics for liquid metalsFrom ultrasoft pseudopotentials to the projector augmented-wave methodGeneralized Gradient Approximation Made SimpleVoltage-controllable colossal magnetocrystalline anisotropy in single-layer transition metal dichalcogenidesTight-binding approach to the orbital magnetic moment and magnetocrystalline anisotropy of transition-metal monolayersTheory of the Role of Covalence in the Perovskite-Type Manganites

[La, M (II)] Mn O_{3}

Superexchange interaction and symmetry properties of electron orbitalsNew Approach to the Theory of Superexchange InteractionsAnatomy of Dzyaloshinskii-Moriya Interaction at

Co / Pt

InterfacesRole of Anisotropic Exchange Interactions in Determining the Properties of Spin-Glasses

[1]	Mühlbauer S, Binz B, Jonietz F, Pfleiderer C, Rosch A, Neubauer A, Georgii R, and Böni P 2009 Science 323 915
[2]	Yu X Z, Kanazawa N, Onose Y, Kimoto K, Zhang W Z, Ishiwata S, Matsui Y, and Tokura Y 2011 Nat. Mater. 10 106
[3]	Ryu K S, Thomas L, Yang S H, and Parkin S 2013 Nat. Nanotechnol. 8 527
[4]	Fert A, Cros V, and Sampaio J 2013 Nat. Nanotechnol. 8 152
[5]	Shen L, Li X, Xia J, Qiu L, Zhang X, Tretiakov O A, Ezawa M, and Zhou Y 2020 Phys. Rev. B 102 104427
[6]	Göbel B, Mook A, Henk J, Mertig I, and Tretiakov O A 2019 Phys. Rev. B 99 060407(R)
[7]	Nagaosa N and Tokura Y 2013 Nat. Nanotech. 8 899
[8]	Upadhyaya P, Yu G, Amiri P K, and Wang K 2015 Phys. Rev. B 92 134411
[9]	Schott M, Bernand-Mantel A, Ranno L, Pizzini S, Vogel J, Béa H, Baraduc C, Auffret S, Gaudin G, and Givord D 2017 Nano Lett. 17 3006
[10]	Yu X, Morikawa D, Nakajima K, Shibata K, Kanazawa N, Arima T, Nagaosa N, and Tokura Y 2020 Sci. Adv. 6 eaaz9744
[11]	Ba Y, Zhuang S, Zhang Y, Wang Y, Gao Y, Zhou H, Chen M, Sun W, Liu Q, Chai G, Ma J, Zhang Y, Tian H, Du H, Jiang W, Nan C, Hu J, and Zhao Y 2021 Nat. Commun. 12 322
[12]	Wei W S, He Z D, Qu Z, and Du H F 2021 Rare Met. 40 3076
[13]	Zheng F S, Li H, Wang S S, Song D S, Jin C M, Wei W S, Kovács A, Zang J D, Tian M L, Zhang Y H, Du H F, and Dunin-Borkowski R E 2017 Phys. Rev. Lett. 119 197205
[14]	Neubauer A, Pfleiderer C, Binz B, Rosch A, Ritz R, Niklowitz P G, and Böni P 2009 Phys. Rev. Lett. 102 186602
[15]	Luchaire C M, Moutafis C, Reyren N, Sampaio J, Vaz C A F, Van Horne N, Bouzehouane K, Garcia K, Deranlot C, Warnicke P, Wohlhüter P, George J M, Weigand M, Raabe J, Cros V, and Fert A 2016 Nat. Nanotechnol. 11 444
[16]	Boulle O, Vogel J, Yang H, Pizzini S, de Chaves D S, Locatelli A, Mentes T O, Sala A, Buda-Prejbeanu L D, Klein O, Belmeguenai M, Stashkevich Y R A, Chérif S M, Aballe L, Foerster M, Chshiev M, Auffret S, Miron I M, and Gaudin G 2016 Nat. Nanotechnol. 11 449
[17]	Fert A, Reyren N, and Cros V 2017 Nat. Rev. Mater. 2 17031
[18]	Huang B V, Clark G, Navarro-Moratalla E, Klein D R, Cheng R, Seyler K L, Zhong D, Schmidgall E, McGuire M A, Cobden D H, Yao W, Xiao D, Jarillo-Herrero P, and Xu X 2017 Nature 546 270
[19]	Gong C, Li L, Li Z, Ji H, Stern A, Xia Y, Cao T, Bao W, Wang C, Wang Y, Qiu Z Q, Cava R J, Louie S G, Xia J, and Zhang X 2017 Nature 546 265
[20]	Deng Y, Yu Y, Song Y, Zhang J, Wang N Z, Sun Z, Yi Y, Wu Y Z, Wu S, Zhu J, Wang J, Chen X H, and Zhang Y 2018 Nature 563 94
[21]	Han M G, Garlow J A, Liu Y, Zhang H, Li J, DiMarzio D, Knight M W, Petrovic C, Jariwala D, and Zhu Y 2019 Nano Lett. 19 7859
[22]	Ding B, Li Z, Xu G, Li H, Hou Z, Liu E, Xi X, Xu F, Yao Y, and Wang W 2020 Nano Lett. 20 868
[23]	Wu Y, Zhang S, Zhang J, Wang W, Zhu Y L, Hu J, Wong K, Fang C, Wan C, Han X et al. 2020 Nat. Commun. 11 3860
[24]	Xu C, Feng J, Prokhorenko S, Nahas Y, Xiang H, and Bellaiche L 2020 Phys. Rev. B 101 060404(R)
[25]	Zhang Y, Xu C, Chen P, Nahas Y, Prokhorenko S, and Bellaiche L 2020 Phys. Rev. B 102 241107(R)
[26]	Liang J H, Wang W W, Du H F, Hallal A, Garcia K, Chshiev M, Fert A, and Yang H X 2020 Phys. Rev. B 101 184401
[27]	Cui Q R, Liang J H, Shao Z J, Cui P, and Yang H X 2020 Phys. Rev. B 102 094425
[28]	Yang S, Peng R, Jiang T, Liu Y, Feng L, Wang J, Chen L, Li X, and Nan C 2014 Adv. Mater. 26 7091
[29]	Kum H S, Lee H, Kim S, Lindemann S, Kong W, Qiao K, Chen P, Irwin J, Lee J H, Xie S, Subramanian S, Shim J, Bae S, Choi C, Ranno L, Seo S, Lee S, Bauer J, Li H, Lee K, Robinson J A, Ross C A, Schlom D G, Rzchowski M S, Eom C, and Kim J 2020 Nature 578 75
[30]	Caretta L, Rosenberg E, Büttner F, Fakhrul T, Gargiani P, Valvidares M, Chen Z, Reddy P, Muller D A, Ross C A, and Beach G S D 2020 Nat. Commun. 11 1090
[31]	Ren Y, Li Q, Wan W, Liu Y, and Ge Y F 2020 Phys. Rev. B 101 134421
[32]	Kresse G and Furthmüller J 1996 Phys. Rev. B 54 11169
[33]	Kresse G and Furthmüller J 1996 Comput. Mater. Sci. 6 15
[34]	Kresse G and Hafner J 1993 Phys. Rev. B 47 558
[35]	Kresse G and Joubert D 1999 Phys. Rev. B 59 1758
[36]	Perdew J P, Burke K, and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
[37]	Sui X, Hu T, Wang J, Gu B L, Duan W, and Miao M S 2017 Phys. Rev. B 96 041410(R)
[38]	See Supplemental Material for (1) the calculations of Heisenberg exchange parameters $J$ and (2) the calculations of DMI $d$.
[39]	Wang Z, Liang J, Cui Q, Ren W, and Yang H 2021 J. Magn. Magn. Mater. 535 168068
[40]	Bruno P 1989 Phys. Rev. B 39 865
[41]	Goodenough J B 1955 Phys. Rev. 100 564
[42]	Kanamori J 1959 J. Phys. Chem. Solids 10 87
[43]	Anderson P W 1959 Phys. Rev. 115 2
[44]	Yang H X, Thiaville A, Rohart S, Fert A, and Chshiev M 2015 Phys. Rev. Lett. 115 267210
[45]	Fert A and Levy P M 1980 Phys. Rev. Lett. 44 1538