Abstract
Topological spin structures, such as magnetic skyrmions, hold great promises for data storage applications, thanks to their inherent stability. In most cases, skyrmions are stabilized by magnetic fields in noncentrosymmetric systems displaying the chiral DzyaloshinskiiMoriya exchange interaction, while spontaneous skyrmion lattices have been reported in centrosymmetric itinerant magnets with longrange interactions. Here, a spontaneous antibiskyrmion lattice with unique topology and chirality is predicted in the monolayer of a semiconducting and centrosymmetric metal halide, NiI_{2}. Our firstprinciples and Monte Carlo simulations reveal that the anisotropies of the shortrange symmetric exchange, when combined with magnetic frustration, can lead to an emergent chiral interaction that is responsible for the predicted topological spin structures. The proposed mechanism finds a prototypical manifestation in twodimensional magnets, thus broadening the class of materials that can host spontaneous skyrmionic states.
Introduction
Magnetic skyrmions are localized topological spin structures characterized by spins wrapping a unit sphere, and carrying an integer topological charge Q^{1,2}. Their topological properties ensure the inherent stability that makes them technologically appealing for future memory devices^{3}. Noncoplanarity in the direction of the spin magnetic moments at different lattice sites is a necessary—albeit not sufficient—ingredient to obtain a net scalar spin chirality s_{i} ⋅ (s_{j} × s_{k}), in turn related to the topological invariants that characterize most of the appealing properties of skyrmions^{1,2}.
While the interplay of competing magnetic interactions may often lead to noncoplanarity, localized skyrmionlike magnetic textures with fixed chirality are generally believed to arise from the Dzyaloshinskii–Moriya (DM) interaction, driven by spin–orbit coupling (SOC) in systems lacking spaceinversion symmetry. Such shortrange antisymmetric exchange interaction, in fact, acts as a chiral interaction and fixes one specific rotational sense of spins, thus imposing a welldefined chirality to noncollinear and noncoplanar spin textures^{4,5,6,7,8,9,10}.
Conversely, in geometrically frustrated centrosymmetric lattices (such as triangular or Kagome), possible skyrmionlattice states triggered by competing exchange interaction manifest with various topologies^{11,12,13,14,15}, as there is no mechanism determining a priori their topology and chirality^{16,17,18,19,20}. These states, in fact, generally arise from nonchiral interactions, such as easyaxis magnetic anisotropy, longrange dipole–dipole and/or Ruderman–Kittel–Kasuya–Yosida (RKKY) interactions, and thermal or quantum fluctuations^{11,12,13,14,15,21}. Furthermore, skyrmionic spin structures are usually stabilized by external fields in both noncentrosymmetric and centrosymmetric materials. A spontaneous skyrmionlattice state has been proposed so far only in itinerant magnets displaying amplitude variations of the magnetization^{4}, where its microscopic origin was attributed to longrange effective fourspin and higherspin interactions that arise from conduction electrons^{6,22,23,24,25}.
In this work, we show that the spontaneous formation of thermodynamicallystable skyrmionic lattice with a unique, welldefined topology and chirality of the spin texture can also be driven by the anisotropic part of the shortrange symmetric exchange, in absence of DM and Zeeman interactions, but assisted only by the exchange frustration. In particular, by performing density functional theory (DFT) and MonteCarlo (MC) simulations, we report a spontaneous highQ antiskyrmion lattice and a fieldinduced topological transition to a standard skyrmion lattice in a potential semiconducting 2Dmagnet, the centrosymmetric NiI_{2} monolayer.
Results
Skyrmionic lattice in NiI_{2}
NiI_{2} is a centrosymmetric magnetic semiconductor long known for its exotic helimagnetism^{26,27,28,29}. It belongs to the family of transitionmetalbased van der Waals materials recently object of intense research activity due to their intriguing low dimensional magnetic properties^{30,31,32,33,34,35,36,37,38,39}. A single layer of NiI_{2} is characterized by a triangular net of magnetic cations at distance a_{0} and competing ferro (FM)magnetic and antiferro(AFM)magnetic interactions, resulting in strong magnetic frustration (Fig. S2 in Supplementary). Our DFT and MC calculations reveal that NiI_{2} monolayer displays a spontaneous transition below T_{c} ≃ 30 K to a tripleq state (with q_{1} = (δ, δ), q_{2} = (δ, − 2δ), q_{3} = (−2δ, δ) and δ ≃ 0.125, as detailed in Fig. 1). This state consists in a triangular lattice of antibiskyrmions (A2Sk) characterized by a topological charge ∣Q∣ = 2 with associated vorticity m = − 2, as shown in Fig. 1; see refs. ^{18,40,41} for further insights about skyrmionic spin structures. The magnetic unit cell (m.u.c.) of such A2Sk lattice, with lateral size L_{m.u.c} ≃ 8a_{0}, comprises three nanoscale antibiskyrmions, each with an approximate diameter of about 1.6 nm (Fig. S6) and surrounded by six vortices with vanishing net magnetization; the central spins (s_{i}) of the A2Sk have opposite s_{z} component with respect to the vortices center (in the specific case reported in Fig. 1a, spins in the antibiskyrmion core and in the vortices centers point upward and downward, respectively). When a perpendicular magnetic field (B_{z}) is applied, a conventional Blochtype skyrmion lattice with m = 1 (shown in Fig. 1b) is induced for a finite range of applied fields, before a ferromagnetic state is finally stabilized for large fields. Upon the fieldinduced transition, all spins surrounding one every two downward vortices align to the field, while the inplane spin pattern remains substantially unchanged. A sharp topological phase transition thus occurs as the charge ∣Q∣ changes from 2 to 1 at the critical ratio B_{z}/J^{1iso} > 0.4 (J^{1iso} being the nearestneighbor isotropic exchange interaction defined in the next section), while it changes to 0 for B_{z}/J^{1iso} ≃ 2.2, as displayed in Fig. 1d–f. The magnetization M also exhibits evidences of these phase transitions, signaled by abrupt changes in correspondence of the critical fields; the magnetization saturation, corresponding to all spin aligned with the magnetic field, takes place for B_{z}/J^{1iso} > 3.9, as shown in Fig. 1e.
A similar sequences of topological phase transitions and the spontaneous onset of the A2Sk lattice has been previously predicted in frustrated itinerant magnets described by a Kondolattice model on a triangular lattice^{24}. Nevertheless, the effective interaction between localized spins mediated by conduction electrons, that has been identified as the driving mechanism for the stabilization of the topological spin textures in such itinerant magnets^{25,42}, cannot be invoked to explain the A2Sk lattice in semiconducting NiI_{2}. Similarly, the DM interaction has to be excluded, being forbidden by the inversion symmetry of the lattice. As discussed below, here the A2Sk state, and related fieldinduced state, directly arise from the anisotropic properties of the shortrange symmetric exchange. As such, the underlying mechanism is not restricted to itinerant magnets and metallic systems, but rather has a more general validity, as it can apply also to centrosymmetric magnetic semiconductors.
Underlying microscopic mechanisms
Magnetic interactions between localized spins s_{i} can be generally modeled by the classical spin Hamiltonian
where A_{i} and J_{ij} denote the onsite or singleion anisotropy (SIA) and the exchange coupling interaction tensors, respectively^{43}. The latter is generally decomposed into three contributions^{44,45,46}: the isotropic coupling term \({J}_{{\mathrm{ij}}}^{{\mathrm{iso}}}=\frac{1}{3}{\rm{Tr}}{{\bf{J}}}_{{\mathrm{ij}}}\), defining the scalar Heisenberg model \({H}^{{\mathrm{iso}}}=\frac{1}{2}{\sum }_{{\mathrm{i}}\ne j}{J}_{ij}{{\bf{s}}}_{{\mathrm{i}}}\cdot {{\bf{s}}}_{{\mathrm{j}}}\); the antisymmetric term \({{\bf{J}}}_{{\mathrm{ij}}}^{A}=\frac{1}{2}({{\bf{J}}}_{{\mathrm{ij}}}{{\bf{J}}}_{{\mathrm{ij}}}^{{\rm{T}}})\), which corresponds to the DM interaction and vanishes in the presence of an inversion center on the spin–spin bond (as realized in the systems under investigation); the anisotropic symmetric term \({{\bf{J}}}_{{\mathrm{ij}}}^{{\mathrm{S}}}\,=\,\frac{1}{2}({{\bf{J}}}_{{\mathrm{ij}}}+{{\bf{J}}}_{{\mathrm{ij}}}^{{\rm{T}}}){J}_{{\mathrm{ij}}}^{{\mathrm{iso}}}{\bf{I}}\), also referred to as twosite anisotropy (J^{twosite aniso}), which is of particular interest here. The operator \({\rm{Tr}}\) and the superscript T label the trace and the transpose of the full J_{ij} matrix; I is the unit matrix. As the DM interaction, the twosite anisotropy arises from the spin–orbit coupling; while the former favors the canting of spin pairs, the latter tends to orient the spins along given orientations in space. When the principal (anisotropy) axes between different spin pairs are not parallel, an additional frustration in the relative orientation of adjacent spins may lead to noncoplanar magnetic configurations, in analogy with the mechanism devised by Moriya for canted magnetism arising from different singleion anisotropy axes^{47}.
In Table 1 we report such contributions to the exchange coupling for NiI_{2} monolayer as obtained by DFTcalculations. Further insights on the properties of the exchange coupling tensor are discussed in Section SI of the Supplementary. Moreover, in order to highlight chemical trends, we also report results for the isostructural NiCl_{2} and NiBr_{2} monolayer systems.
This allows, on the one hand, to analyze the effects of different SOC strengths (going from the weak contribution expected in Cl to the strongest one in I) and, on the other hand, to explore the range of the interactions (going from the localized 3p states in Cl to broader 5p in I).
As evident from Table 1, the FM nearestneighbor exchange interaction (J^{1iso}) becomes larger when moving from Cl to I. Interestingly, the resulting AFM thirdnearest neighbor interaction (J^{3iso}) is also strongly affected, as a consequence of the broader ligand p states mediating the superexchange^{48}: the J^{3iso}/J^{1iso} ratio is ≃ −0.33, −0.49, −0.83 for NiCl_{2}, NiBr_{2}, and NiI_{2}, respectively, revealing thus increasing magnetic frustration as a function of the ligand. A similar trend is observed for the exchange anisotropy, that also increases as the ligand SOC gets stronger along the series. Indeed, magnetism of NiCl_{2} results well described by the isotropic Heisenberg model, as opposed to NiBr_{2} and NiI_{2}. The strongest effect is predicted for NiI_{2}, where the twosite anisotropy associated to the nearestneighbor exchange interaction is one order of magnitude larger than in NiBr_{2}. In particular, the J_{yz}/J^{1iso} ratio, measuring the canting of the twosite anisotropy axes from the direction perpendicular to the monolayers, as argued below and in Section SI of the Supplementary, changes from 0.00 in NiCl_{2} to 0.02 and 0.20 in NiBr_{2} and NiI_{2}, respectively. This specific anisotropic contribution tends to vanish in the secondnearest and thirdnearest neighbor exchange interactions. Moreover, secondnearestneighbor and beyond thirdnearestneighbor interactions are at least one order of magnitude smaller than J^{1iso} and J^{3iso}. The SIA (whose values are reported in Supplementary Table SI) is also negligible with respect to these two main interactions and the twosite anisotropy; the largest value of SIA was found for NiI_{2}, that is predicted to display easyplane anisotropy with (A_{zz} − A_{xx}) ≃ +0.6 meV, the latter being not expected to play a relevant role in the stabilization of the tripleq state. In the following we will therefore focus on the crucial role played by the twosite anisotropy in driving the A2Sk lattice in NiI_{2}.
The exchange tensor discussed until now was expressed in the cartesian {x, y, z} basis, where x was chosen to be parallel to the NiNi bonding vector (see Fig. 2a and Fig. S1 in Supplementary). In order to better understand and visualize the role of the twosite anisotropy, it is useful to express the interaction within the local principalaxes basis {ν_{α}, ν_{β}, ν_{γ}} which diagonalize the exchange tensor, with eigenvalues (λ_{α}, λ_{β}, λ_{γ}), as detailed in Section SI of Supplementary. Principal values of the exchange tensor for the three compounds are reported in Table 1. It is also noteworthy that, within this local basis, the exchange interaction could be further decomposed into an isotropic parameter \(J^{\prime}\) and a Kitaev term K, as in refs. ^{49,50} and SupplementarySection SI, thus providing another estimate of the global anisotropy of the exchange interaction; the \( K/J^{\prime} \) ratio in fact evolves as ≃ 0.00, 0.05, 0.40 in NiCl_{2}, NiBr_{2}, and NiI_{2}, respectively.
In Fig. 2 we show the principal axes for the most anisotropic system, NiI_{2}: ν_{α} and ν_{β} vectors lie in the NiI–NiI spin–ligand plaquette while ν_{γ} is perpendicular to it (cfr Fig. 2b). Therefore, the six ν_{α} and ν_{γ} vectors, being not parallel to any lattice vector, introduce a noncoplanar component in the interaction of the spins (Fig. 2c). This noncoplanarity is mathematically determined by the offdiagonal terms of the twosite anisotropy (see Section SI of Supplementary), arising from the SOC of the heavy I ligand, combined with the noncoplanarity of the spin–ligand plaquettes on each triangular spinspin plaquette displayed in Fig. 2c. Such noncoplanarity is the first main ingredient introduced by the twosite anisotropy and necessary to define the net scalar chirality of the spin texture. Its second main effect is also to fix the helicity, driving the inplane orientation of the spins. The topology and chirality of the spin texture is thus well defined. In fact, when looking at the inplane projection of the ν_{α} eigenvector reported in Fig. 2d, one can observe a direct relation to the antibiskyrmion spinpattern represented in Fig. 2e, f: spins on the nearestneighbor Ni atoms surrounding the central spin orient in plane according to the inplane components of the noncoplanar ν_{α} vector both in direction and sense (Fig. 2e, f); the accommodation of the antibiskyrmions in the spin lattice, along with the conservation of a vanishing net magnetization in the system, determine then the direction and orientation of the secondneighbor spins, and, as a consequence, the pattern of the associated magnetic defects (i.e., the vortices) surrounding the A2Skcore. In particular, the rotational sense of the spins, i.e., the chirality, is determined by the sign of the offdiagonal terms of the twosite anisotropy. We verified this dependence by artificially changing the sign of J_{yz} term in the Ni_{0}–Ni_{1} exchange tensor (that by symmetry also affects related terms of the other Ni–Ni coupling, as in Section SISupplementary) in the MC simulation; such operation would correspond to apply a reflection with respect to the xy plane. The resulting topological lattice still displays vorticity m = −2, i.e., the A2Sk lattice, but with opposite chirality: indeed, the inplane reflection leads to a change by π of the helicity, not affecting the zcomponent of the magnetization and keeping the orientation of the spin at the core of each A2Sk (see Fig. S4 in Supplementary). Moreover, to further verify the unique relation between the twosite anisotropy and the topology of the spin structure, we also artificially swapped the exchange interaction of the Ni_{0}–Ni_{2} and Ni_{0}–Ni_{3} pairs obtaining now a lattice with opposite vorticity, m = 2, i.e., a biskyrmion (2Sk) lattice. Such spin state can be obtained from the A2Sk lattice by inverting the sign of the xcomponent of each spin, namely (s_{x}, s_{y}, s_{z}) → (−s_{x}, s_{y}, s_{z}), thus leading to an opposite sign of the scalar spin chirality and a consequent change of the spinstructure topology (see Fig. S5 in Supplementary). In this case, an applied magnetic field would stabilize an antiskyrmion lattice (m = −1), as shown in Fig. S5e, f.
Competing magnetic frustrations
It is important to stress that magnetic frustration is a necessary prerequisite for the stabilization of the skyrmionic tripleq state, further determining its periodicity and, hence, the size of individual topological objects (as shown in Fig. S7), whose character is then dictated by the frustrated twosite anisotropy. Indeed, the latter competes with the isotropic exchange that, in the absence of magnetic frustration, would favor collinear spin configurations. This is, for instance, the case of the prototypical 2D ferromagnet CrI_{3}, which displays an easyaxis FM groundstate despite the strong exchange anisotropy^{49,51}. Indeed, Xu and coauthors^{49} reported a strong anisotropic symmetric exchange for monolayer CrI_{3}, akin to what we obtained for monolayer NiI_{2} and consistently with the similar metalhalide arrangement mediating the exchange interaction. Nevertheless, in CrI_{3} the magnetic frustration due to thirdneighbor interaction is rather weak^{52}, because of the honeycomb lattice adopted by the Cr cations, preventing the stabilization of a noncoplanar spin structure and favouring, instead, an easyaxis FM state. On the other hand, the presence of a strong twosite anisotropy is necessary to obtain the topological tripleq state, when magnetic frustration is already strong enough to stabilize a noncollinear spin structure. In fact, in monolayer NiBr_{2}, for which we found a much weaker contribution of the twosite anisotropy to the exchange interaction, the groundstate is a singleq helimagnetic state with no net scalar chirality (Fig. 3a). Its anisotropy still introduces frustration in the spins direction, but it is energetically not strong enough to stabilize the skyrmionic pattern; thus, a complex noncoplanar helimagnetic texture takes place, as detailed in Fig. 3a, c. Furthermore, neither the singleion nor the twosite anisotropies are strong enough to support a fielddriven topological transition to a skyrmion lattice; rather, a singleq conical cycloid state, shown in Fig. 3b, d, develops under applied field before a purely FM state is stabilized at B_{z}/J^{1iso} ≳ 0.9.
Further insight on the role of competing interactions is provided by comparing the total internal energy of NiI_{2} evaluated on various aforementioned spin structures including firstneighbor and thirdneighbor interaction parameters listed in Table 1 and the easyplane singleion anisotropy. As shown in Fig. 4a, the A2Sk lattice shows the lowest total energy among the possible topological tripleq states, namely the anti(bi)skyrmion and (bi)skyrmion lattices. As shown in Fig. 4b, the ∣Q∣ = 1 configurations (skyrmion, Sk, and antiskyrmion, ASk) are destabilized with respect to the ∣Q∣ = 2 ones (biskyrmion, 2Sk, and antibiskyrmion, A2Sk) mostly by the AFM third nearest neighbor interaction J^{3iso}, whose associated energy cost overcomes the energy gain due to FM nearestneighbor exchange J^{1iso}. On the other hand, the twosite anisotropy J^{twosite aniso} is responsible for the stabilization of the A2Sk lattice over the 2Sk one, since both would be energetically degenerate in the presence of purely isotropic (and competing) exchange interactions. Furthermore, both the A2Sk and the Sk lattices can be realized with different chiralities of the spin configurations, related by an inplane reflection, whose degeneracy is lifted by the twosite anisotropy, as shown in Fig. 4b. Due to their specific multichiral nature^{1,20,53}, no such chiraldegeneracy lifting is observed for the ASk and 2Sk lattices. Our results thus demonstrate that the twosite anisotropy may behave as an emergent chiral interaction for this class of centrosymmetric systems, determining a unique topology and chirality of the spin structure.
We also considered the total energies of possible singleq spin configurations, namely a coplanar cycloidal helix (with spins rotating in the xy plane, consistently with the SIA easyplane anisotropy) and a noncoplanar helix maximizing the twosite anisotropy energy gain, analogous to the complex helimagnetic state predicted for NiBr_{2}. For NiI_{2}, such spin configurations display an overall energy gain associated to isotropic exchanges with respect to the tripleq states, at the expense of an energy cost associated with the twosite anisotropy, as shown in Fig. 4b. Therefore, the competition bewteen isotropic and anisotropic symmetric exchanges determines which spin state is eventually stabilized. Interestingly, the noncoplanar helix is found to be almost degenerate with the A2Sk lattice, implying that the thermodynamic stability of the latter in NiI_{2} has to be ascribed to thermal fluctuations and entropic contributions to the free energy accounted for in our MC simulations. Finally, the chiral nature of the twosite anisotropy emerges also for singleq states, lifting the degeneracy between the two chiral partners of the noncoplanar helix.
Discussion
In this work, we have identified the twosite anisotropy, arising from the shortrange symmetric exchange interaction, as the driving mechanism for the stabilization of a spontaneous topological spinstructure. Specifically, we predict a spontaneous antibiskyrmion lattice below T_{c} ≃ 30 K along with a fieldinduced topological transition in NiI_{2} monolayer, representative of the class of 2D magnetic semiconductors.
We found that metastable multiq skyrmionic states, that may occur in frustrated magnets, can stabilize as groundstate with welldefined topology and chirality in the presence of competing twosite anisotropies characterized by noncoplanar principal axes. Such kind of additional frustration in the relative orientation of spins acts as an emergent chiral interaction, fixing the topology and the chirality of the localized spin textures and of the resulting skyrmion lattice, whereas its size and periodicity are mostly determined by competing isotropic exchanges. Interestingly, our findings are not limited to centrosymmetric systems, as the twosite anisotropy can be found in noncentrosymmetric magnets as well; its competition with the Dzyaloshinskii–Moriya interaction could thus also reveal interesting phenomena in many other systems, including the recently proposed Janus Cr(I,Br)_{3} monolayer^{50}.
In conclusion, the proposed mechanism, which we have predicted here in a Nihalide monolayer, enlarges both the kind of magnetic interactions able to drive the stabilization of topological spin structures, and the class of materials able to host spontaneous skyrmionic lattices with definite chirality, including also magnetic semiconductors with shortrange anisotropic interactions.
Methods
Firstprinciples calculations
Magnetic parameters of the interacting spin Hamiltonian (1) were calculated by performing firstprinciples simulations within the density functional theory (DFT), using the projectoraugmented wave method as implemented in the VASP code^{54,55,,–56}. The following orbitals were considered as valence states: Ni 3p, 4s and 3d, Cl 3s and 3p, Br 4s and 4p, and I 5s and 5p. The Perdew–Burke–Erzenhof (PBE) functional^{57} within the generalized gradient approximation (GGA) was employed to describe the exchangecorrelation (xc) potential; the plane wave cutoff energy was set to 600 eV for NiCl_{2} and NiBr_{2}, and 500 eV for NiI_{2}, which is more then 130% larger than the highest default value among the involved elements. The U correction^{58} on the localized 3d orbitals of Ni atoms was also included. Exchange energies reported in this article, and used to run the MC simulations, were calculated by employing U = 1.8 eV and J = 0.8 eV within the Liechtenstein approach^{59}. We also adopted the Dudarev approach^{60} to test the results solidity against different effective U values; we choose U equal to 1, 2, and 3 eV and a fixed J equal to 0 eV. Results remain qualitatively the same: the J^{3iso}/J^{1iso}, and J_{yz}/J^{1iso} ratios (i.e., the quantities with relevant physical meaning, even more than the absolute values of the interactions) are almost unaffected, as shown in Table SII in Supplementary.
We calculated the magnetic parameters reported in this paper via the fourstate energy mapping method, which is explained in detail in the refs. ^{43,49,50}, performing noncollinear DFT calculations plus spin–orbit coupling (SOC) and constraints on magnetic moments direction. It is based on the use of large supercells, also allowing to exclude the coupling with unwanted distant neighbors. By means of this method we can obtain all the elements of the exchange tensor for a chosen magnetic pair, thus gaining direct access to the symmetric anisotropic exchange part (the twosite anisotropy) and the antisymmetric anisotropic part (the DM interaction) of the full exchange. In particular, we performed direct calculations on the magnetic Ni–Ni pair parallel to the x direction, here denoted Ni_{0}–Ni_{1} (Fig. 2a). The interaction between the five other nearestneighbor pairs can be evaluated via the threefold rotational symmetry, as commented in SupplementarySection SI. In all our systems, the tensor turned out to be symmetric or, equivalently, excluding any antisymmetric (DMlike) contribution.
We performed calculations of the SIA, firstneighbors and secondneighbors interaction using a 5 × 4 × 1 supercell; while a 6 × 3 × 1 supercell for the estimate of the third neighbors interaction. Such cells should exclude a significant influence from next neighbors. We built supercells from the periodic repetition of the NiX_{2} monolayer unit cell, with lattice parameters (a_{0}) and ionic positions optimized by performing standard collinear DFT calculations with a ferromagnetic spin ordering. The obtained lattice parameters are: 3.49 Å and 3.69 Å for NiCl_{2} and NiBr_{2}, respectively^{61}, which are in agreement (within 0.3% uncertainty) with the values known for the bulk compounds, and 3.96 Å for NiI_{2}, which is 1.5% larger than the 3.89 Å bulk value^{30}. We thus checked the stability of the NiI_{2} magnetic parameters by extracting them from a cell with lattice constant fixed to the experimental value, not obtaining significant changes (as reported in Table SIII in Supplementary). In all cases, the length of the outofplane axis, perpendicular to the monolayer plane, was fixed to 20.8 Å, which provides a distance of more than 17.5 Å with respect to the periodic repetition of the layer along this direction. The sampling of the Brillouin zone for the monolayer unit cell relied on a 18 × 18 × 1kpoints mesh; meshes for the supercells have been chosen according to the latter.
MC simulations
MC calculations were performed using a standard Metropolis algorithm on L × L triangular supercells with periodic boundary conditions. At each simulated temperature, we used 10^{5} MC steps for thermalization and 5 × 10^{5} MC steps for statistical averaging. Average total energy, magnetization and specific heat have been calculated. The lateral size of the simulation supercells has been chosen as L = nL_{m.u.c.}, where n is an integer and L_{m.u.c.} is the lateral size of the magnetic unit cell needed to accomodate the lowestenergy noncollinear helimagnetic spin configurations. Accordingly, we estimated L_{m.u.c.} as 1/q, where q is the length of the propagation vector q minimizing the exchange interaction in momentum space J(q). For the isotropic model and neglecting second nearestneighbor interactions, the propagation vector is given by \(q=2{\cos }^{1}[(1+\sqrt{12{J}^{1{\mathrm{iso}}}/{J}^{3{\mathrm{iso}}}})/4]\)^{14,21}, resulting in L_{m.u.c.} ≃ 8 and ≃ 10 for NiI_{2} and NiBr_{2}, respectively. Results are shown for calculations performed on supercells with lateral size L = 3L_{m.u.c.}, but we verified the accuracy of our choice by performing benchmark calculations on commensurate and incommensurate cells with L ranging from 8 to 64. Further insight on the magnetic configurations was obtained by evaluating the spin structure factor:
where r_{i} denotes the position of spin s_{i} and N = L^{2} is the total number of spins in the supercell used for MC simulations. The braket notation is used to denote the statistical average over the MC configurations. The spin structure factor provides direct information on the direction and size of the propagation vectors, which are found to agree with the analytical estimate provided above.
In order to assess the topological nature of the multipleq phase, we evaluate the topological charge (skyrmion number) of the lattice spin field of each supercell as 〈Q〉 = 〈∑_{i}Ω_{i}〉, where Ω_{i} is calculated for each triangular plaquette as^{62}:
The corresponding topological susceptibility has been evaluated as:
Data availability
Main results are reported in this article and related Supplementary Material. All other data that support the findings discussed in this study are available from the corresponding author upon reasonable request.
Code availability
Firstprinciples calculations were performed using the licensed VASP code^{56}. Monte Carlo calculations were performed using a code implementing the standard Metropolis algorithm, that is available from the authors upon reasonable request.
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Acknowledgements
This work was supported by the Nanoscience Foundries and Fine Analysis (NFFAMIUR Italy) project. P.B. and S.P. acknowledge financial support from the Italian Ministry for Research and Education through PRIN2017 projects “Tuning and understanding Quantum phases in 2D materials—Quantum 2D” (ITMIUR Grant No. 2017Z8TS5B) and “TWEET: Towards Ferroelectricity in two dimensions” (ITMIUR Grant No. 2017YCTB59), respectively. Calculations were performed on the highperformance computing (HPC) systems operated by CINECA, supported by the ISCRA C (IsC66I2DFM, IsC722DFmF) and ISCRA B (IsB17COMRED, grant HP10BSZ6LY) projects. We thank Krisztian Palotás, Bertrand Dupé, Mario Cuoco, Changsong Xu, Hongjun Xiang, Laurent Bellaiche, Hrishit Banerjee, Roser Valentí, Sang Wook Cheong, Sergey Artyukhin, and Stefan Blügel for helpful and illuminating discussions.
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D.A. performed firstprinciples calculations. P.B. and D.A. performed Monte Carlo simulations and analyzed the results. S.P. supervised the project. All the authors discussed the results and contributed to the manuscript writing.
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Amoroso, D., Barone, P. & Picozzi, S. Spontaneous skyrmionic lattice from anisotropic symmetric exchange in a Nihalide monolayer. Nat Commun 11, 5784 (2020). https://doi.org/10.1038/s4146702019535w
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DOI: https://doi.org/10.1038/s4146702019535w
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