Alloying Driven Antiferromagnetic Skyrmions on NiPS3 Monolayer: A First‐Principles Calculation

Abstract Topological magnetic states are promising information carriers for ultrahigh‐density and high‐efficiency magnetic storage. Recent advances in two‐dimensional (2D) magnets provide powerful platforms for stabilizing various nanometer‐size topological spin textures within a wide range of magnetic field and temperature. However, non‐centrosymmetric 2D magnets with broken inversion symmetry are scarce in nature, making direct observations of the chiral spin structure difficult, especially for antiferromagnetic (AFM) skyrmions. In this work, it is theoretically predicted that intrinsic AFM skyrmions can be easily triggered in XY‐type honeycomb magnet NiPS3 monolayer by alloying of Cr atoms, due to the presence of a sizable Dzyaloshinskii–Moriya interaction. More interestingly, the diameter of the AFM skyrmions in Ni3/4Cr1/4PS3 decreases from 12 to 4.4 nm as the external magnetic field increases and the skyrmion phases remain stable up to an external magnetic field of 4 T. These results highlight an effective strategy to generate and modulate the topological spin texture in 2D magnets by alloying with magnetic element.

The discovery of intrinsic long-range magnetic ordering in two-dimensional (2D) materials, such as CrI 3 , Cr 2 Ge 2 Te 6 , Fe 3 GeTe 2 , and MPS 3 (M = Ni, Mn, Fe, Co), [11][12][13][14] provides new platforms for studying topologically nontrivial spin phenomena.Recently, many experimental and theoretical works have shown that dimensional reduction is capable of generating various types of small-size magnetic skyrmions.Experimentally, Bloch-type skyrmions have been observed in exfoliated van der Waals (vdW) materials Cr 2 Ge 2 Te 6 , whose diameter decreases from 120 nm at 11.7 mT to 77 nm at 195.8 mT. [15]Room-temperature clockwise and counterclockwise Bloch-type magnetic skyrmions were also observed in layered Cr 1+x Te 2 . [16]In 2D Fe 3 GeTe 2 sheet, Bloch-type skyrmions would transform into skyrmion bubbles with increasing magnetic field and the average width of skyrmions is ≈120 nm. [17]Néel-type skyrmions with a size of ≈150 nm at 94 K and ≈80 nm at 198 K have also been observed in Fe 3 GeTe 2 -based heterostructures. [18,19]Theoretically, various strategies have been proposed to further break the inversion symmetry of 2D magnets and stabilize the chiral spin textures.For instance, a fieldcontrolled Néel-type skyrmion-ferromagnet transition cycle has been predicted in a CrTe 2 /WTe 2 heterostructure, in which the diameter of skyrmions is ≈20 nm. [20]The transformation of bimeron-skyrmions was realized via perpendicular strain and electric field in CrISe/In 2 Se 3 heterostructure, the diameter of skyrmions is ≈14 nm. [21]Néel-type skyrmions with diameter of ≈10 nm were obtained in Janus CrInX 3 (X = Te, Se), and the skyrmion phases can be sustained up to ≈180 K. [22] In addition, skyrmion states and bimerons have also been detected in Janus MnXY (X, Y = S, Se, Te) and Cr(I, X) 3 (X = Br, Cl) monolayers. [23,24]However, most of the aforementioned systems host FM skyrmions, which are sensitive to stray magnetic fields.In contrast to FM skyrmions, [25] AFM skyrmions are a type of topological object consisting of similar but opposite spin texture on each sublattice, which have the advantage of the absence of stray magnetic fields and ultrafast dynamics. [26,27]30] Therefore, they offer greater potential for designing future nonvolatile logic computing devices with ultra-low energy consumption and high-density in AFM spintronics.
In principle, magnetic skyrmions are mainly stabilized by the competition between the antisymmetric Dzyaloshinskii-Moriya interaction (DMI) and the symmetric Heisenberg exchange (J). [31]A negative J value denotes that AFM coupling between magnetic ions.DMI originates from spin-orbit coupling (SOC) and broken inversion symmetry.DMI and J favor canted and collinear alignments between neighboring spins, respectively.Experimentally, large DMI has been observed in ultrathin films epitaxially grown on heavy metal substances. [32,33]oreover, large in-plane magnetic anisotropy also favors the formation of topological defects. [34,35]Considering the symmetrybreaking principle, alloying could be as a potential approach to induce nontrivial topological spin textures.First, incorporation of a second magnetic metal element can destroy the inherent inversion symmetry.Second, the additional spin lattice of alloying element can readily tune the exchange interaction and magnetic anisotropy.In a pioneering study, room-temperature Néel-type skyrmions were recently realized in 50% Co-doped Fe 5 GeTe 2 . [36]herefore, it is imperative to clarify the effect of alloying magnetic elements on the formation of AFM skyrmions in the emerging fields of 2D magnets.
In this work, we have theoretically proposed an alloying strategy to trigger AFM skyrmions in an AFM NiPS 3 monolayer.At the 2D limit, NiPS 3 was found to be an antiferromagnet with XY-type magnetic ordering.The alloying element Cr was selected from a large number of transition metals by comparing the bond length, magnetic ground state, exchange interaction, magnetic anisotropy energy (MAE) and DMI.Using first-principles calculations and Monte-Carlo simulations, we found that a sizable DMI can be induced in alloyed Ni 1-x Cr x PS 3 (x = 1/4, 1/2, and 3/4) monolayers.Consequently, Ni 1-x Cr x PS 3 monolayers exhibit chiral magnetic states of AFM skyrmions in the case of x = 1/4.The skyrmions are <12 nm in diameter and can be sustained up to an external magnetic field of 4 T.In addition to the DMI, the competing magnetic constants of Heisenberg exchange coupling and magnetic anisotropy were also discussed to clarify the underlying mechanism of the observed complex spin texture.Our results suggest that alloying is an effective strategy to modulate the topological spin textures of 2D antiferromagnets, highlighting promising applications in skyrmion-based spintronics.

Results and Discussion
Previous experiments have demonstrated that 2D metal phosphorous trichalcogenides MPS 3 are a new class of AFM semiconductors. [13,37]The monolayer structure of MPS 3 consists of octahedral MS 6 units with the layer group p 31m.Similar to the 1T phase of 2D MoS 2 , MPS 3 can be viewed as 1/3 of the M atom in MS 2 being replaced by P 2 dimers, corresponding to an equivalent stoichiometry of M 2/3 (P 2 ) 1/3 X 2 .As shown in Figure 1a, the (P 2 S 6 ) 4− anion is located at the center of the honeycomb lattice by M atoms.In experiments, the M cations in the synthetic MPS 3 monolayer materials are magnetic 3d transition metals, including Fe, Mn, Ni, and Co. Depending on the species of magnetic ions M in the 2D host lattice, the AFM ordering and critical Néel temperature (T N ) varies from Ising-type for FePS 3 (T N = 123 K), to Heisenberg-type for MnPS 3 (T N = 78 K), to XY-type for NiPS 3 (T N = 155 K) and XY-type for CoPS 3 (T N = 122 K). [38] It is also known that in-plane magnetic anisotropy and high magnetic transition temperature are crucial for the formation of topological spin textures. [34,35]Therefore, NiPS 3 monolayer was selected as a model compound for alloying to achieve the desired topological spin textures.
To investigate the influence of the alloying elements on the magnetic textures of NiPS 3 monolayer, we systematically calculated the structural parameters, electronic properties, magnetic interaction, and MAE of pristine NiPS 3 monolayer.As listed in Table S1 (Supporting Information), the optimized lattice parameters (a = 5.81 Å) and bandgap (E g = 1.72 eV) of NiPS 3 monolayer are in good agreement with the experimental results (a = 5.82 Å, E g = 1.6 eV). [39,40]To confirm the magnetic ground state of NiPS 3 , we compared the total energies of the FM state and three kinds of AFM states (Néel-type, Stripy-type, and Zigzag-type).Their spin configurations are displayed in Figure S1 (Supporting Information).We found that the Zigzag-type AFM is the lowest-energy state, whose energy is 82.7 meV/formula lower than that of the FM state.
We further adopted a Heisenberg spin Hamiltonian to describe the magnetic interactions in NiPS 3 monolayer: The first and second terms represent symmetric and antisymmetric parts of the exchange couplings, respectively.S i and S j are the spins of the i and j sites.J ij and D ij are the Heisenberg isotropic exchange coefficients and DMI strength between spins S i and S j , respectively.K is the single-ion anisotropy coefficient,

S i
z is the z component of the spin at position i.The magnetic parameters J, D, and K are calculated by considering four specifically designed non-collinear spin configurations according to the well-established four-state method. [41,42]he calculated exchange coupling parameters for NiPS 3 monolayer are J 1 = 1.6 meV, J 2 = 0.36 meV, and J 3 = −7.45meV, consistent with the experimental values. [43]These results reveal FM J 1 and J 2 as well as a larger AFM J 3 , which are mainly determined by the Goodenough-Kanamori-Anderson (GKA) rules. [44]he magnetic Ni atoms are coordinated with six S atoms, forming an anti-triangular prism of NiS According to the threefold rotation operation symmetry, the third neighbor hopping is dominant in 2D NiPS 3 . [45]As a consequence, J 3 is negative and larger than the other exchange parameters, and the magnetic ground state of NiPS 3 is Zigzag-type AFM.To further illustrate the process of electron hopping, the schematic diagrams of Ni-S-Ni superexchange and Ni-S…S-Ni super-superexchange are displayed in Figure 3a,b.
Another important physical parameter is the MAE, which is defined as the difference between ground-state energies due to the rotation of magnetization direction.To determine the easy axis, MAE is calculated as follows: where E  and E 001 refer to the total energies of the states whose magnetization direction lies in the XY plane with the angle  and perpendicular to the XY plane, respectively.For NiPS 3 monolayer, the calculated MAE is −82.7 μeV per magnetic atom, meaning that the easy magnetization orientation is parallel to the XY plane.Figure 4a plots the angular dependence of MAE in the XY plane.One can see that MAE in the XY plane is nearly isotropic,  which coincides the experimental observation that NiPS 3 monolayer is a 2D XY-type antiferromagnet. [46]To further elucidate the origin of MAE, we decomposed the MAE of NiPS 3 into three coupling terms using the torque procedure, [47] namely the majority spin states (uu), minority spin states (dd), and cross spin states (ud+du), with respect to the Fermi level (Figure 4c).It can be seen that the ud+du channel makes the largest contribution to the in- Based on the aforementioned magnetic behavior of 2D NiPS 3 , the general principles for modulating the magnetic skyrmions in 2D AFM materials can be summarized as follows: i) breaking the inherent symmetry and inducing a large DMI; ii) moderately weakening the Heisenberg exchange interactions by increasing the distance between transition metal ions or that between transition metal and non-metal bridge ions; iii) achieving a larger in-plane MAE by weighting the contributions of SOC interactions between relevant coupling terms.Based on the above analysis, we first considered five elements of 3d magnetic transition metals (M = V, Cr, Mn, Fe, Co).For Ni 1/2 V 1/2 PS 3 and Ni 1/2 Cr 1/2 PS 3 , J 1 is reduced to 0.3 and 1.25 meV, J 2 is reduced to 0.079 and 0.25 meV, J 3 is reduced to 1.85 and −6.89 meV, respectively.Accordingly, sizable D/|J| is induced in alloy compounds of Ni 1/2 Cr 1/2 PS 3 (26.7%)and Ni 1/2 Co 1/2 PS 3 (59.9%).The calculated magnetic anisotropy still favors in-plane magnetization for V, Cr, Mn, and Co, but out-ofplane for Fe.Among them, the largest anisotropy energy barrier (77.3 μeV) between in-plane and out-of-plane direction is found in Ni 1/2 Cr 1/2 PS 3 , which is ≈7, 14, and 22 times higher than that of Ni 1/2 Mn 1/2 PS 3 , Ni 1/2 Co 1/2 PS 3 , and Ni 1/2 V 1/2 PS 3 , respectively.Therefore, Cr is chosen as the alloying element for the successive discussions about topological magnetic texture.
To further investigate Cr alloying-induced topological spin switching, we constructed a series of model structures of Ni 1-x Cr x PS 3 by varying Cr/Ni compositions, i.e., x = 1/4, 1/2, 3/4.For each concentration, we considered all possible six structurally ordered phases within a 2×2 supercell (Figure S3a-c, Supporting Information).These ordered phases have a definite structure and have been shown to be more stable than the disordered phases. [48,49]By comparing the relative energies (Figure S3d-f, Supporting Information), we finally identified the most stable structures for different Ni 1-x Cr x PS 3 monolayers (Figure 1b).The equilibrium lattice parameters, bond lengths and bond angles of Ni 1-x Cr x PS 3 are listed in Table S1 (Supporting Information).It can be seen from the Table S1 (Supporting Information) that the larger ionic radius of Cr compared to Ni leads to a gradual increase in the lattice constant of the structure and a disruption of the original crystal symmetry as the alloyed Cr content increases, These structures were then utilized to calculate the stability, electronic and magnetic properties.In order to characterize the energetics for alloying different Cr concentrations in the host NiPS 3 monolayer, we calculated the formation energies defined as  S3 (Supporting Information), all these systems prefer AFM as the magnetic ground states, which have lower energy than the FM state by 53.9 (x = 1/4), 54.3 (x = 1/2), 40.2 (x = 3/4) meV/formula, respectively.The projected electronic band structures revealed that all Ni 1-x Cr x PS 3 monolayers are semiconductors with moderate band gaps of 1.04 eV (x = 1/4), 0.94 eV (x = 1/2) and 0.98 eV (x = 3/4), respectively, as shown in Figure S6b-d  For these AFM semiconductors Ni 1-x Cr x PS 3 (x = 1/4, 1/2, 3/4) monolayers, the calculated magnetic parameters J, D, and MAE are listed in Table S4 (Supporting Information).With these parameters from first-principles calculations, MC simulations were carried out to investigate the possible topological spin textures in Ni 1-x Cr x PS 3 monolayers.As shown in Figure 5a, it is surprisingly to found spontaneous AFM skyrmions in Ni 3/4 Cr 1/4 PS 3 monolayers without an external magnetic field.This unique spin texture was not observed for the other two concentrations (as shown in Figure S7, Supporting Information).In particular, the diameter of AFM skyrmions in the Ni 3/4 Cr 1/4 PS 3 monolayer is 12 nm at zero field.Such a small size is urgently needed for both experimental and theoretical works.In addition, the topological charge (Q) is a crucial parameter describing topological properties, which is defined as, Q = 1 4 ∫ S ⋅ ( x S ×  y S)dxdy. [50]The calculated topological charge Q = 0, which is consistent with the typical topological charge of AFM skyrmions. [26]This result stems from the composition of the AFM skyrmions, which consist of two similar but opposite sublattices.
To unveil the underlying mechanism of intrinsic AFM skyrmions due to substituting 25% Ni with Cr, we first analyzed the alloying effect on the magnetic exchange coupling strength J.The calculated J 1 , J 2 and J 3 of Ni 3/4 Cr 1/4 PS 3 monolayers are listed in Table S4 (Supporting Information).For the exchange interaction between the first nearest neighbors, there are two new possible electron hopping paths introduced by alloying Cr, namely, the near-90°superexchange between the Cr-S-Ni and the direct spin exchange between Ni-Cr (the Ni-Cr distance is 3.21 Å).Due to the larger ionic radius of Cr, the distance of the electron hopping path becomes larger (the Ni-S and Cr-S distance are 2.44 and 2.54 Å, respectively).Based on PDOS in Figure 2c-e, a  S4, Supporting Information).Although the magnetic ground state of Ni 3/4 Cr 1/4 PS 3 monolayer is still AFM, the incorporation of the second magnetic element Cr provides some additional electron hopping pathways.As a result, J 1 and J 2 are reduced with regard to pristine NiPS 3 , while J 3 is enhanced.It is important to note that the current DMI is smaller than those of 2D CrInSe 3 , [22] Tl 2 NO 2 , [51] and MnSTe, [24] which can be explained by the Fert-Levy mechanism of DMI. [52]Compared to S in Ni 3/4 Cr 1/4 PS 3 compound, the heavier non-magnetic atoms (Se and Te), act as spin-orbit active sites and induce more significant spin-orbit scattering, ultimately leading to a larger DMI.Similar behavior has been discovered for the MnXY (X, Y = S, Se, Te) [24] and CrInX 3 (X = Se, Te). [22]The D 1 /|J 1 | ratio between DMI and the exchange parameter is 1.21% and 4.1% for Ni-Ni and Ni-Cr, respectively.Recently, it has also been shown that such amplitude of D/|J| ratio can induce topological spin textures in 2D AgCr 2 X 4 (X = S or Se). [53]In Ni 3/4 Cr 1/4 PS 3 monolayer, the weak anisotropy of XY-type magnetic ordering may be beneficial for the formation of skyrmions without large D/|J|.
To confirm this, we also calculate the angular dependence of MAE along the XY plane of Ni 1/2 Cr 1/2 PS 3 and Ni 1/4 Cr 3/4 PS 3 monolayers, which don't have the properties of AFM skyrmions.The results are shown in Figure S8 (Supporting Information) and one can observe a strong in plane MAE anisotropy.It is clear that the broken XY-type magnetic ordering is not conductive to the formation of skyrmions.Therefore, the emergence of AFM skyrmions in 2D Ni 3/4 Cr 1/4 PS 3 mainly relies on the enhanced DMI, suppressed J 1 and unchanged XY-type magnetic order during alloying.Based on the above factors, it is possible to extrapolate to other 2D magnetic semiconductor MPS 3 layers.In particular, CoPS 3 exhibits a significant potential to induce AFM skyrmions via alloying because it has a similar magnetic structure with NiPS 3 , which is also stabilized by the XY model.Moreover, Ni (1.24 Å) and Co (1.25 Å) also have a similar ionic radius.
Finally, we discuss the effects of magnetic field and temperature effect on the evolution of skyrmions.As can be seen in Figure 5b-f, the AFM skyrmions remain still visible under an external magnetic field up to 4 T.With increasing strength of the magnetic field, the diameter of the skyrmions generally decreases from 12 nm at 0 T to 4.4 nm at 4 T. The underlying reason for this phenomenon is that the external magnetic field promotes out-of-plane magnetization.The temperature effect on AFM skyrmions is depicted in Figure S9 (Supporting Information).The skyrmions remain clearly visible at 2 K.However, as the temperature increases to 5 K, the boundaries of the skyrmions become blurred, and they eventually disappear as the temperature further increases.Up to now, the skyrmion diameters reported experimentally are larger than 100 nm, which is an order of magnitude larger than the desired diameter (<10 nm) for memory applications. [54]Our finding of small-sized skyrmions is desirable for the synthesis of next-generation memory devices with higher storage density.

Conclusion
To summarize, we theoretically propose a feasible alloying strategy to regulate the AFM skyrmions in 2D magnets based on symmetry considerations and analysis of exchange interactions.Starting from an AFM NiPS 3 monolayer, a series of stable Ni 1-x Cr x PS 3 monolayers with high thermodynamic stability are predicted by first-principles calculations.Monte-Carlo simula-tions reveal that Ni 3/4 Cr 1/4 PS 3 monolayer exhibits intrinsic AFM spin textures at zero field.When the external magnetic field is applied, the diameter of the skyrmions decreases from 12 to 4.4 nm and the skyrmion phase can be retained up to an external field of 4 T. The emergence of AFM skyrmions is attributed to the alloying Cr element, which induces considerable DMI, suppresses the exchange coupling strength, and maintains the weak easy-plane (XY plane) magnetic anisotropy.All these results demonstrate that alloying might be an effective way to induce topological spin textures in 2D magnets.Hence, further comprehensive experimental and theoretical investigations are desired to substantiate this pivotal argument.

Experimental Section
The spin-polarized density functional theory calculations were performed with the Vienna Ab Initio Simulation Package. [55]The projector augmented wave [56] method was used for ion-electron interactions and the Perdew-Burke-Ernzerhoffunctional within the generalized gradient approximation (GGA) [57] for exchange-correlation interactions.The criteria for energy and force convergences were 10 −6 eV and 10 −3 eV Å −1 , respectively.The cutoff energy of the plan-wave basis was 600 eV.To minimize the interaction between neighboring images, a vacuum region of 20 Å was applied along the z-direction.A Γ-centered Monkhorst-Pack k-point grid with a uniform spacing of 0.02 Å −1 was used for sampling the Brillouin zone.Considering the strong correlation effect of d electrons, GGA plus on-site repulsion U method with an effective Coulomb parameter U eff = 4 eV was adopted for the d orbitals of Ni and Cr. [58]To investigate the spin dynamics of Ni 1-x Cr x PS 3 monolayers, Monte Carlo simulations were performed using the Metropolis algorithm implemented in the Spirit package. [59]An 80 × 80 × 1 periodical supercell with 51200 spin sites was used to simulate the evolution of spin textures.For each temperature and magnetic field strength, at least 7 × 105 MC steps were simulated.

Figure 1 .
Figure 1.a) Top and side views of the crystal structure of monolayer NiPS 3 .The dotted lines represent the unit cell.b) The most stable alloying configurations of Ni 1-x Cr x PS 3 (x = 1/4, 1/2, and 3/4).Only magnetic atoms are shown here, with gray and green atoms representing Ni and Cr, respectively.

Figure 2 .
Figure 2. Projected density of states (PDOS) of magnetic atoms of Ni 1-x Cr x PS 3 monolayers.PDOS of a) S and b) Ni of NiPS 3 .PDOS of c) S, d) Ni, and e) Cr of Ni 3/4 Cr 1/4 PS 3 .

6 .
Under the corresponding crystal field, five Ni-3d orbitals split into two-fold degenerate e 1 (d xz + d yz ), e 2 (d x2-y2 + d xy ) and a single a 1 (d z2 ) states.For the first nearest neighbors, there is no direct Ni-Ni exchange since the corresponding overlapping orbitals for Ni 2+ are filled.The near-90°Ni-S-Ni superexchange interaction (with Ni-S distance of 2.44 Å) contributes to the FM interaction of the first nearest Ni atoms.According to the projected density of states (PDOS) of NiPS 3 monolayers (Figure 2a,b), a near-90°superexchange interaction between spin-up occupied Ni-d yz/xz orbital and S-p z /p x/y orbitals and spin-down unoccupied Ni-d x2-y2 orbital leads to weak ferromagnetism.For the second nearest Ni atoms, there is neither direct exchange path.The superexchange between the M1 and M3 Ni atoms occurs via two S atoms from different layers (Figure S2, Supporting Information).Therefore, the J 2 value is positive and very small.For the third nearest exchange interaction, the super-superexchange path M1-S1…S5-M4 is activated because two S atoms are in the same layer.The Ni-S and S-S distances are 2.44 and 3.47 Å, respectively, and the Ni-S-Ni angle is 131°.Based on the PDOS of NiPS 3 , the super-superexchange interaction arises from the electrons hopping between spin-up occupied Ni-d yz/xz orbital and S-p z /p x/y orbitals, resulting in the antiferromagnetism.

Figure 3 .
Figure 3.The schematic diagrams of a) superexchange paths of Ni-d yz , S-p z /p y , and Ni-d x2-y2 , b) super-superexchange paths of Ni-d yz , S-p z /p y , S-p y /p z and Ni-d yz in NiPS 3 .

Figure 4 .
Figure 4. Angular dependence of MAEs of a) NiPS 3 , b) Ni 3/4 Cr 1/4 PS 3 monolayer with magnetization direction lying on XY planes.Fermi level dependent decomposed and total MAEs of c) NiPS 3 , d) Ni 3/4 Cr 1/4 PS 3 monolayer.The uu, dd, and ud+du represent the spin coupling between the spin-up channels, spin-down channels, and spin-up with spin-down channels, respectively.The Fermi level is set to zero.
plane MAE.Combining the PDOS of Ni atoms in Figure 2a,b, one can conclude that the negative MAE of NiPS 3 mainly originates from the ud+du coupling channel between spin-up occupied Nid xz orbital and spin-down unoccupied Ni-d yz orbital, as well as spin-up occupied Ni-d x2-y2 and spin-down unoccupied Ni-d xy orbital.On the other hand, the inherent inversion symmetry of 2D NiPS 3 leads to the absence of DMI.
where E Ni 1−x Cr x PS 3 is the energy of Ni 1-x Cr x PS 3 monolayers per formula, and E Ni , E Cr , E P , and E S are the energy per atom of Ni, Cr, P, and S elements in their most stable solid states, respectively.By definition, a negative E f value indicates that formation of Ni 1-x Cr x PS 3 monolayer is exothermic.The calculated E f of Ni 1-x Cr x PS 3 monolayers are −0.54eV/atom (x = 1/4), −0.68 eV/atom (x = 1/2), and −0.83 eV/atom (x = 3/4), respectively.They are all more stable than the pristine NiPS 3 , whose E f is −0.41 eV/atom.In addition, ab initio molecular dynamics (AIMD) simulations at 300 K were performed for these Ni 1-x Cr x PS 3 monolayers.As shown in FigureS4(Supporting Information), there are no significant lattice deformation after 10 ps simulation, indicating satisfactory thermal stability.The preferred magnetic ground states of Ni 1-x Cr x PS 3 monolayers were confirmed by comparing the energies of FM state and various possible AFM configurations, whose spin densities are displayed in FigureS5(Supporting Information).As shown in Table (Supporting Information).As expected, the local magnetic moments in these Ni 1-x Cr x PS 3 monolayers are mainly contributed by the d orbitals of Ni and Cr atoms, i.e., 1.4 μ B on Ni and 3.8 μ B on Cr.

Figure 5 .
Figure 5. Top views of the real-space distribution of magnetic moments from snapshots of MC simulations under different out-of-plane magnetic fields for Ni 3/4 Cr 1/4 PS 3 monolayer.The color map represents the out-of-plane spin component of the magnetic atoms.The insets show the enlarged image of AFM skyrmions.

For Ni 3 / 4
Cr 1/4 PS 3 monolayer, the angular dependence of MAE along the XY plane was calculated and plotted in Figure 4b.Similar to NiPS 3 monolayer, Ni 3/4 Cr 1/4 PS 3 system also exhibits inplane XY type magnetic ordering.The energy barrier of magnetic easy axis along the XY plane and perpendicular to the XY plane is −70.1 μeV per magnetic atom, which is slightly lower than that of NiPS 3 monolayer (−82.7 μeV).To further analyze the relationship between MAE and Cr content, we decomposed the MAE of Ni 3/4 Cr 1/4 PS 3 monolayer into three coupling terms with respect to the Fermi level (Figure 4d).It can be seen that the large negative MAE originates mainly from the strong coupling through the ud+du channels.Combining PDOS of d orbitals of the magnetic atoms in Figure 2c-e, the coupling orbitals are still spin-up occupied Ni-d yz and spin-down unoccupied Ni-d xz .Moreover, the contribution of uu coupling channel between spin-up occupied Crd yz and spin-up unoccupied Cr-d xz is enhanced in Ni 3/4 Cr 1/4 PS 3 monolayer, giving rise to a positive MAE.This contribution lowers the total MAE to a smaller negative value.The inclusion of a second transition metal element not only modulates the amplitudes of exchange coupling J and MAE, but also destroys the inherent inversion symmetry of NiPS 3 .Therefore, a nonzero DMI value is generated by Cr alloying.In Ni 3/4 Cr 1/4 PS 3 monolayer, the induced DMI values (D 1 ) are 0.017 meV between the first nearest neighbor Ni-Ni and 0.054 meV between Ni-Cr, respectively.The D 2 values are 0.009, 0.007, and 0.001 meV for second nearest neighbors Ni-Ni, Ni-Cr, and Cr-Cr, respectively, and D 3 are 0.007 and 0.002 meV for the third nearest neighbors Ni-Ni and Ni-Cr, respectively.The DMI amplitudes of Ni 3/4 Cr 1/4 PS 3 monolayer decrease rapidly with increasing distance between magnetic ions and are mainly dominated by the Cr ions.