Introduction

Topological magnetic states are non-trivial magnetic structures. Based on their specific swirling configurations of spins, which can be characterized by topological charge number, they are classified as skyrmions, biskyrmions, merons, vortices, etc1,2,3,4,5. These topological spin configurations have received the most attention in recent years due to their intriguing electromagnetic properties, such as extraordinarily low current-driven threshold and highly efficient motion dynamics6. These effective interactions between electric fields (or currents) and the specific spin states enable materials to exhibit fascinating phenomena, such as the topological Hall effects7, spin transfer torque6,8, and spin-orbit torque9. These novel magneto-electric properties combined with their particle-like size and exceptional topological protected stability make the materials promising potential for applications in spintronics10 and orbitronics11, particularly in high-density magnetic storage devices12.

Another effective way of electric control involves the use of magneto-electric multiferroics. The search for these materials is driven by the prospect of controlling charges by applied magnetic fields and spins by applied voltages, thereby constructing new forms of multifunctional devices13,14,15. However, the performance of currently found multiferroic materials is still far from satisfying practical applications. The primary limiting factors are twofold: (i) the weak magnetoelectric coupling, and (ii) the operation temperature is substantially below room temperature. It is because ferroelectricity and magnetism tend to be mutually exclusive with each other when coexist, which is dictated by the nature of their respective d-electron occupations16. Whereas in an ideal multiferroic material, the dipolar order should be directly coupled to long-range magnetic configuration. One intriguing mechanism perfectly satisfies the requirement, in which the dipole moment is induced by having some specific patterns of non-collinear spin state. As a result, ionic displacement at the ligand site has been activated, and therefore, ferroelectric polarization is induced (the so-called inverse Dzyaloshinsky-Moriya mechanism)17. Typical types of spin texture that exhibit this dipole order include the well-studied Néel-type domain wall18, and more notably, the magnetic skyrmions, which have recently received widespread attention. Several experimental results have already shown the existence of multiferroic properties in magnetic skyrmions. For example, in GaV4S819, the ferroelectric polarization undergoes interesting changes along with the magnetic phase transition from a cycloidal order to Néel-type skyrmions. The polarization results deduced on Cu2OSeO3 indicate that each magnetic skyrmion locally carries an electric dipole or quadrupole moment20,21. Recently, an interesting theoretical study reported spin-chirality-drive multiferroicity in vanadium-halid compounds22. However, questions such as the relationship between the magnetic topological feature and the electric polarization, the precise manifestation of the polarization density, the potential role of this phenomenon as the origin of polar skyrmions, and the design strategies for generating multiferroic skyrmion with robust magneto-electric coupling at high temperatures, remain unanswered. To effectively address these issues, comprehensive and systematic theoretical studies based on the first-principles methods are a necessity.

Compared to bulk materials, low-dimensional materials, due to their broken symmetry, become an ideal platform for the emergence of numerous excellent properties and even new physical phenomena. The same for the study of magnetic skyrmions. Both experimental and theoretical studies have shown that a variety of magnetic topological states exist in low-dimension systems, such as Néel-type skyrmions in Fe/Ir(111)23, mixed Bloch-Néel skyrmions in Co/Pd multilayers24, bimerons in a thin plate of Co8Zn9Mn325, etc. Janus structure is an important member of 2D functional materials. Until now, most synthesized Janus 2D materials exhibit Janus characteristics due to anions asymmetrically occupying opposite surfaces, such as MoSSe26,27 and WSSe28, which can be prepared by modified chemical vapor deposition. Recently, a mechanically force-assisted synthesis of GeSe has been achieved29, and more cation-characterized Janus 2D materials are expected to be synthesized using unconventional synthesis methods such as pulsed laser deposition (PLD)30,31, and high-pressure and high-temperature apparatuses32. At the same time, there have been extensive theoretical studies on such materials26,27,33,34,35,36,37,38,39,40. Among them, monolayer CrInSe3 and CrInTe3 are exceptional examples39. These compounds are derived from van der Waals material In2Se3 that has been proven to exhibit room-temperature ferroelectricity with reversible spontaneous electric polarization41,42,43. With Cr partially substituted at the In site, CrInX3 are predicted to be stable and exhibit spontaneous magnetic skyrmions. However, the physical mechanism behind the magnetic exchange coupling remains to be fully understood. A thorough investigation, particularly into their magnetic ground state and excited state, is still required. More importantly, the intriguing possibility of multiferroicity, especially the presence of multiferroic skyrmions, remains unexplored.

In this paper, the magnetic properties of monolayer CrInSe3 and CrInTe3 are thoroughly investigated, starting with obtaining a rank-2 exchange matrix. The isotropic Heisenberg exchange J is analyzed by combining orbital-decomposed coupling results and electronic properties in specific local CrX6 crystal field. The antisymmetric Dzyaloshinskii-Moriya vector Dij is obtained and its mechanism is explained by intriguing variation of spin-orbit coupling strength on different X layers. Macroscopic magnetic studies on the two systems are conducted by investigating magnetic equilibrium states under various external temperatures and magnetic field perturbations. These magnetic structures are then used as input for studying the spin wave spectrum and consequently, a complete picture of magnetic properties from the ground state to the excited state is achieved. In addition, spontaneous electric polarization values are obtained by using both the modern theory of polarization and the phenomenological Landau treatment. By analyzing polarization densities calculated from various magnetic structures, we further conclude a direct correlation between electric polarization and the combined effects of topological magnetic state, non-magnetic ligand atoms, external magnetic field, and temperature. Based on which, the possibility of having a multiferroic skyrmion is also discussed.

Results and discussion

Structural model and ground state properties in the collinear limit

Monolayer CrInX3 is a Janus material consisting of five atomic layers and belongs to the Cs point group. As it is shown in Fig. 1b, indium atoms are situated on the upper side, with four nearest white ligand X atoms forming a tetrahedron structure. On the bottom side, a chromium layer is sandwiched between two layers of X atoms. Each chromium atom and its six nearest X neighbors form an octahedron. These two parts are connected by a shared X layer in the middle, which causes both the InX4 tetrahedron and the CrX6 octahedron to be slightly distorted from their ideal local structures. The dynamic and thermal stability of monolayer CrInX3 has been theoretically validated by the phonon spectra and ab initio molecular dynamic simulations39. Moreover, the stability of CrInX3 is further demonstrated by comparing its formation energies to those of the already synthesized parent compounds, monolayer In2X3, as shown in Supplementary Table 1.

Fig. 1: Structural model of CrInX3 (X=Se, Te).
figure 1

a Top view and first Brillouin zone with specific high symmetry points, b side view and a schematic picture of local CrX6 octahedron. Cr, In, X atoms are represented by blue, green, and white spheres, respectively.

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In order to determine the most stable magnetic structure, our studies were first performed in the collinear framework. Four different magnetic states are considered in our ab initio calculations. As is shown in Fig. 2, these include one ferromagnetic (FM) configuration, and three antiferromagnetic (AFM) configurations with parallel spin chains along a, b, and a+b direction, respectively. The results are shown in Supplementary Table 2. For both Se- and Te-based compounds, the structure with the lowest energy appears with the FM state. The three structures with different AFM patterns exhibit higher energy, all with exactly the same amount (~0.40eV), suggesting equivalence regardless of the three directions of the parallel spin chain. Compared to the Te-based compound with a lattice constant of 8.56 Å, the Se-based compound exhibits a shorter one at 7.92 Å. Moreover, the Se-based compound exhibits a bigger energy difference between FM and AFM states(~0.02eV). These characteristics suggest a stronger exchange coupling in the Se-based compound, as will be discussed in detail in Section “Inter-site magnetic exchange interactions and their mechanisms”.

Fig. 2: Four collinear magnetic structures are considered in this work.
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Cyan and red arrows indicate the orientations of spins on specific Cr atoms. Dashed lines in blue and purple represent two different nearest Cr-Cr distances due to parallel or antiparallel magnetic interactions.

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Moreover, when comparing the Cr-Cr distance along the directions with FM and AFM spin chains, it is found that the Cr-Cr pair with parallel spins (3.96 Å and 4.28 Å) is larger than those with antiparallel spins (3.95 Å and 4.26 Å). This provides evidence that the crystal structure is affected by the changes in magnetic configuration. The phenomenon is called magnetostriction, or spin-lattice coupling. By further comparing the Cr-Cr distance change in the two materials, we found that the value for the Te-based compound (0.02 Å) is doubled compared to the Se-based one (0.01 Å). This suggests a stronger magnetostriction and, consequently, a generally larger ionic displacement in the Te-based compound when the material’s magnetic configuration changes. In addition, as the magnitude of displacement is directly dictated by the relative orientations of the neighboring spin pairs, the lattice patterns in a material with complex spin textures, such as spin spiral or magnetic skyrmions, are expected to have various interesting features as well. These lattice patterns will then induce the positive and negative charges to be off-centered, which in turn induces ferroelectric polarization. Detailed discussions of ferroelectric polarization properties and direct magneto-electric coupling in these compounds will be addressed in Section “Complex ferroelectric polarization properties”.

The electronic band structure and density of states (DOS) for the two systems are shown in Fig. 3. The electronic bands for both systems exhibit similar features. Specifically, in the spin-up channel, the lowest unoccupied bands are strongly hybridized by all three atoms in the system. Among these, the contribution from the transition metal Cr atom is slightly larger than the others. The highest occupied bands are mainly contributed by the ligand Se and Te states. As a result, these features give rise to bandgaps of 0.75 eV and 0.18 eV for the two compounds. In the spin-down channel, the unoccupied states are primarily from the In and the same ligand. Similar to the other channel, the bands just below the Fermi level are largely occupied by these ligand states. The overall bandgap is dictated by the feature exhibited in the spin-up channel.

Fig. 3: Electronic band structures and density of states (DOS).
figure 3

Atomic projected DOS and the electronic band structure for up-spin and down-spin electrons of a CrInSe3 and b CrInTe3 in the FM state are presented. Total DOS is shown in gray, and the atomic contributions of Cr, In, Se, and Te are shown in blue, green, orange, and olive curves (circles) in DOS (band structure), respectively. The size of the circles in band structure is proportional to the contribution weight. The Fermi level is set to 0 eV.

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To determine the magnetocrystalline anisotropy energy (MAE), total energies with spins along [100], [010], and [001] are calculated. As it is revealed in Supplementary Fig. 1, the easy-magnetization axes of CrInSe3 and CrInTe3 are both out-of-plane. Moreover, CrInTe3 exhibits stronger anisotropy energy than CrInSe3, which reflects the strength of spin-orbit coupling of Te, and Se atoms.

Inter-site magnetic exchange interactions and their mechanisms

In the ideal ionic picture, the charge state of Cr in this material is 3+, yielding a 3d3 electronic configuration. Due to the octahedral crystal field environment, the d orbital splits into a twofold degenerate high-energy eg level and a three-fold degenerate low-energy t2g level. As a result of intra-atomic Hund’s interaction, the latter one is half-filled, resulting in a triplet S=3/2 state, which gives the moment of 3 Bohr magneton. The calculated moments for the two compounds, each with four different magnetic configurations, are shown in Supplementary Table 2. The first observation is that the moments range from 3.44 to 3.53 μB for CrInSe3, and from 3.64 to 3.71 μB for CrInTe3. These minor variations confirm the localized character of the Cr-d electrons, thereby validating the legitimacy of the Heisenberg model for accurately describing this system, as adopted in this paper. The second, and more significant observation, is that the calculated moments for the two compounds are noticeably larger—between 0.44 and 0.71 μB—than the values predicted from the above-described ideal picture. The moment deviation can often be attributed to local structural distortions or to electronic hybridization. As will be discussed in the following, this additional amount of electron plays a crucial role in determining electronic occupation, orbital structure, and consequently, the exchange routes. The three elements, in addition, serve as important factors in elucidating the magnetic exchange mechanism.

As shown in Fig. 1b, the octahedra exhibits distortion due to its Janus nature, characterized by two Cr-X bond lengths and three X-Cr-X bond angles. As the detailed data is listed in Supplementary Table 3, the differences in bond length are less than 0.22 Å, and the bond angles deviate from 90∘ by approximately 8∘. This relatively minor distortion indicates that one can assuredly investigate the exchange mechanism and electron occupation within a standard octahedral crystal field. Schematic picture of the first three inter-site exchange pairs and the isotropic exchange parameters as a function of distance are plotted in Fig. 4a. The first observation is that the J values decrease rapidly as the Cr-Cr distance increases. Specifically, only the first-nearest and third-nearest couplings make a noticeable contribution to the system. Take CrInSe3 as an example, the magnitude of the first nearest FM coupling is more than nineteen times bigger than the third nearest AFM coupling, with values of 7.39 and –0.37 meV, respectively. This results explains why systems show a strong inclination towards an FM ground state in the collinear limit. By incorporating orbital information into the on-site potential and the inter-site Green’s function in the LKAG formula, one can obtain the orbital-resolved exchange parameters. In an octahedral crystal structure, the total exchange parameter (black lines) is contributed by the coupling from three different orbital channels: t2g-t2g (red lines), eg-eg (blue lines), and t2g-eg (green lines). And overall exchange coupling for any specific Cr-Cr pair is determined by the competition among the three. Taking the first nearest Jtot as an example, it is decomposed to competition between a significant positive t2g-eg coupling (9.12 meV) and a substantially negative eg-eg coupling (-1.39 meV), while the other one, t2g-t2g, keeps almost zero (-0.35 meV).

Fig. 4: Heisenberg exchange parameters and corresponding exchange mechanism.
figure 4

a Orbital-decomposed exchange parameters plotted as a function of Cr-Cr distance, obtained for CrInSe3 (solid points and lines) and CrInTe3 (hollow points and only plotted the first-, second-, and third-nearest neighbor Cr-Cr pair), positive number indicates FM contribution, and negative means AFM. Inserts are Cr lattice of CrInX3, the first-, second-, and third-nearest neighbor Cr-Cr pair are represented by green, rose red, and orange arrows, respectively. b The rotated DOS for decomposed 3d orbital of CrInSe3 with the state of FM. c Schematic picture of orbital-dependent exchange mechanism.

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To understand this coupling pattern, it’s necessary to clarify the physical nature of the electronic occupation, as well as the orbital states in the systems. For these compounds, the electronic quantities must be projected from the global Cartesian coordinates into a more physically meaningful, local octahedral basis. Figure 4b and Supplementary Table 4 present the lm-decomposed DOS and the respective DOS integration up to the Fermi level. After the rotation, a three-fold t2g degeneracy and a twofold eg degeneracy can be clearly seen in the DOS. Due to the strong hybridization, the three states that belong to the lower energy t2g level contain around 0.80 electrons each, and interestingly, the two states of the higher energy eg level are occupied by more than 0.71 electrons each. As a result, it gives the Cr atom about 2.4 and 1.4 electrons for t2g and eg levels, respectively. This atypical eg occupation is the direct reason for the system’s remarkable t2g-eg and eg-eg couplings, as the exchange mechanism depicted in a schematic picture shown in Fig. 4c. The hopping of the form t2g-eg leads to an exchange coupling that is predominantly FM due to the local Hund’s rule. The hopping of the form eg-eg is energy favorable with FM alignment, but due to the specific exchange pathway, Cri(\({d}_{{x}^{2}-{y}^{2}}\))-X(px)-Crj(\({d}_{{x}^{2}-{y}^{2}}\)), formed by the hopping of electrons from an occupied \({d}_{{x}^{2}-{y}^{2}}\) orbital at site i to an unoccupied \({d}_{{x}^{2}-{y}^{2}}\) orbital at site j through a pdσ bonding state, it gives a sizable AFM contribution instead (detailed discussions in Supplementary Note 1). Additionally, the hopping of the form t2g-t2g is prohibited for FM alignment due to the already half-filled t2g states on the other site. Therefore, a weak AFM coupling is shown in Fig. 4a. This unconventional exchange mechanism due to the eg occupation was also confirmed in a previous study on monolayer CrI344.

By comparing the dominant exchange coupling (J1) in the two compounds, we find that the FM exchange coupling strength diminishes from 7.39 meV in the Se-based compound to 4.75 meV in the Te-based one. This is markedly influenced by the larger lattice constant in the Te-based compound. More fundamentally, it is directly dictated by the changes in orbital occupation, which in turn, alters the exchange pathways. In specific, as the number of electrons occupying eg states increases from 1.25 to 1.34 in CrInTe3 (as shown in Table 1), it exhibits stronger AFM eg-eg coupling and therefore, provides slightly more compensation for the dominant FM t2g-eg coupling.

Table 1 The integrated electrons number of d decomposed orbitals of CrInX3 (X=Se, Te) with FM magnetic structure
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In systems where spin-orbit coupling plays an unignorable effect, the off-diagonal elements in Eq. (2) become non-zero. In some cases, the elements in the matrix are not exactly symmetric, from which a non-zero vector can be derived. This three-dimensional (3D) vector, stemming from the aforementioned time-reversal symmetry breaking, is known as the Dzyaloshinskii-Moriya (DM) vector, and the corresponding interaction is referred to as DM interaction. In contrast to the Heisenberg exchange parameter which depicts the collinear magnetic coupling strength of the system, the DM vector is a key determinant of whether it is possible to show a non-linear magnetic structure. Therefore, how to understand them correctly is an essential question that must be addressed.

The emergence of DM interaction can be understood well by analyzing the symmetry elements in CrInX3. As is illustrated in Supplementary Fig. 8a, b, DMI originates from the deviation of ligand atoms positioned relative to the midpoint of the nearest Cr-Cr pairs. Specifically, as the upward and downward atoms (Seup and Sedn) deviate from the positions of the pink atoms, the point group of the structure is reduced from D6h to C3v. This process involves the following four symmetry operation changes: 1. The disappearance of the mirror plane perpendicular to the six-fold rotation axis; 2. The replacement of the six-fold rotation by a three-fold rotation axis; 3. The original six mirror planes parallel to the c-direction are reduced to three; 4. The disappearance of all the twofold rotation axes that were originally aligned with the six mirrors.

In addition, according to Moriya’s symmetry rule45, the orientation of this interaction is determined by the cross product of two vectors as per the equation Dij ~ rij × a. Here, rij is a vector that connects the two magnetic atoms, and a represents the displacement of the ligand atom relative to the midpoint of the two magnetic atoms. Whereas in CrInX3, the physical mechanism of DM vectors is more complex than what this simplified theory suggests. The calculated vectors of the nearest neighbor Cr-Cr pair are shown in Fig. 5a. Six vectors around each Cr atom exhibit (a slightly deviated) clockwise pattern. Upon examining the crystal structures, we found that each nearest Cr-Cr pair is connected by two ligand atoms, positioned in two different layers (as depicted by white and orange spheres in Fig. 5a). Therefore, two Cr-Xup∣dn-Cr triangles are formed, and as a result, two vectors, which are proven to be in opposite directions, are generated. The overall DM vector between this specific Cr-Cr pair is determined by competition between the two. As DM interaction directly originates from the spin-orbit coupling effect of ligand atoms, a detailed analysis of MAE contribution is performed to further investigate this interaction mechanism. As shown in Fig. 5b, compared to all the other atoms, the ligand X at the bottom layer (Xdn) carries a majority of the spin-orbit effect into the system. As a result, the DM vector generated by the lower Cr-Xdn-Cr triangle is bigger than the other one and, therefore, plays a dominant role in determining the overall vector orientation. It is remarkable that the same atoms, merely due to their different positions, play entirely different roles in the MAE contribution. Subsequent investigation uncovered that the origin of this phenomenon lies in the different bonding environments of the X atoms in the two layers: the upper layer is connected to In atoms, while the lower layer is not. In addition, we measured the respective Xdn displacements relative to the midpoint of Cr-Cr pairs in CrInSe3, as shown in Supplementary Fig. 8c. The values from d1 to d6 are listed in Supplementary Table 5. These six displacements can be classified into three categories: d1 = d3, d2 = d4, and d5 = d6. As a result, this breaks the C3 rotational symmetry and reduces the three mirror planes to a single mirror plane that is diagonal to the system’s unit cell, thereby giving the system a Cs point group, as illustrated in Fig. 5a.

Fig. 5: Dzyaloshinskii-Moriya interaction and the corresponding exchange mechanism.
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a Calculated DM parameters of the nearest neighbor bonds in CrInSe3 compounds. White and orange spheres represent upper and lower layers of the ligand Se atoms, respectively. The direction and size of the DM vectors between specific Cr-Cr pairs are represented by colored arrows, with the length of arrow scaling with the DMI strength. b Calculated MAE contributions for each atom in CrInX3 compounds.

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On the other hand, it is relatively easy to understand the magnitude of the calculated DM vectors. In any of the two systems, the X atoms in each layer are equivalent, therefore, the size of the generated DM vector is the same. This specific displacement pattern provides CrInX3 with a corresponding DMI symmetry (Cs). As the magnitudes of any two mirror-reflected DMI vectors are nearly identical, larger displacements of d4 and d2 result in greater magnitudes of these DMI vectors — 0.77 meV (shown as purple and yellow arrows in Supplementary Fig. 8c), compared to an average of 0.57 meV for the other displacements in CrInSe3. When comparing the DM vectors in Se-based and Te-based compounds, it can be concluded that, with a heavier element in the system, the magnitude of the DM vector is bigger. As it is shown in Supplementary Table 6, ∣Dij∣ in CrInTe3 is approximately 2.4 times bigger than in CrInSe3, with values of 1.86 and 0.77 meV, respectively.

Magnetic topological states and their equilibrium and excited states

To obtain the compound’s dynamic properties on a relatively macroscopic scale under different external magnetic fields and temperatures, classic Monte Carlo simulations were carried out. Calculations were performed with 5 ensembles under the same conditions to illustrate the relative stability of different magnetic structures and to properly average the properties. Among them, the one with the lowest energy was considered as the ground state, leading to the phase diagrams shown in Fig. 6.

Fig. 6: Magnetic structures as a function of temperature and magnetic field.
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Calculated B-T phase diagrams of a CrInSe3 and b CrInTe3. Two insets are shown in each, representing typical zoomed-up magnetic configurations in phase diagrams (as indicated by yellow parallelograms). Moment orientations are indicated by a heat bar at the right side panel, where blue, white, and red represent the moment that is polarized along +z, in xy-plane, and -z, respectively. c, d Side views of two 0K, 0T magnetic configurations, along with their specific coordinate vectors. Zoomed-in images illustrate the single q spin spiral structure in CrInSe3 and the multi-q spin spiral structure in CrInTe3. Relevant length scales are inserted at the bottom right corners of each panel.

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For the ground state at 0 K and 0 T, as shown at the left bottom corner of the phase diagram, both compounds show spin stripe texture. As can be seen in the side view (Fig. 6c, d), they exhibit clockwise spin helical configurations along specific real-space directions. Particularly, in CrInSe3, the spin helical propagation is along the \([\sqrt{3}/2,1/2,0]\) direction (forming a 60∘ stripe-type domain) with a periodicity of 13.77 nm. With spin chains that are parallel to this direction having the exact same propagation pattern (as shown in the zoomed-in image in Fig. 6), this spin stripe state can be described by a single q vector. Whereas in CrInTe3, the ground-state spin-spiral stripe is complex and cannot be characterized properly by a single propagation vector, a multi-q description with different propagation directions or periods is needed. Thereby a multi-q with a 40∘ stripe-type domain and a significantly smaller periodicity of 2.22 nm is formed. Given that the D/J ratio changes from 0.10 to 0.39 in Se- and Te-based compounds, the difference observed here is expected.

The stripe-like magnetic state evolves under the influence of an external magnetic field and temperature. As the magnetic field increases, the stripe width decreases slowly. The stripe orientation in CrInSe3 may also change due to the need for width adjustment. When the field reaches a threshold, the stripe texture breaks and skyrmions are created. Due to the variation of the magnetic parameters in two compounds, the thresholds are different (0.4 T and 5.5 T for CrInSe3 and CrInTe3, respectively). As the magnetic field keeps increasing, skyrmions change their shape from elliptical to circular. Taking CrInSe3 as an example, the major axis of the ellipses at 0.4 and 0.5 T are 27.8 nm and 25.3 nm, respectively. As the field changes from 0.6 to 0.8 T, the diameter of the circular skyrmion shrinks from 12.02 nm to 8.24 nm. Finally, when the magnitude of the field exceeds 0.9 T, the topological protection of the swirling spin state is destroyed, resulting in an FM state. The overall process for CrInTe3 is similar, but with a more than seven times stronger critical magnetic field, and a much denser of stripes and skyrmions, as a result of the bigger D/J ratio. Interestingly, the evolution of the morphology of these magnetic structures is directly connected to the anisotropic DMI pattern. In specific, along the diagonal direction, which features two relatively larger DMI vectors, a spin stripe phase is formed. This anisotropy remains robust upon the formation of a skyrmion under an external magnetic field, resulting in an elliptical magnetic skyrmion with its semimajor axis aligned along the mirror plane, as shown in Fig. 6a and Supplementary Fig. 8d. This phenomenon is consistent with the recently theoretically predicted elliptical antiskyrmion in monolayer CrI346.

Different from external magnetic fields, limited temperature introduces thermal perturbations into the system, thereby diminishing the correlations between the spin pairs. As can be seen in Fig. 6, with increasing temperature, the spin fluctuation initially occurs at the edge of the periodic spin stripe and the topological skyrmions state. Subsequently, penetrating deeper into these phases and ultimately destroys both long- and short-range spin orders, leading to a paramagnetic phase. The critical temperatures for destroying the spin texture in CrInSe3 and CrInTe3 are approximately 200 K and 150 K. Primarily because the J1 in CrInSe3 being more than 1.5 times larger than in CrInTe3.

The above-mentioned macroscopic magnetic properties, observed under various external perturbations, are results that reflect the competitive nature of the systems from an energy perspective. More specifically, two aspects of the competitions can be classified as: first, a stability competition among different topological magnetic states; second, the significance of various terms in the spin Hamiltonian (Eq. (4)). As shown in Supplementary Fig. 11, the energy differences of various spin states at specific magnetic fields clearly illustrate the first competition. For instance, increasing the magnetic field from 0.5 T to 0.6 T helps resolve the competition among multiple configurations (60∘ stripe, 60∘ + 1 skyrmion, and 30∘ 1 skyrmion), stabilizing the 1 skyrmion state. This results in a phase transition from the 60∘ stripe state to the 1 skyrmion state. On the other hand, the latter competition is fundamentally more important, as it rises from terms used to characterize magnetic features, which in turn directly determine the equilibrium state under each set of external conditions. To be specific, each term in the spin Hamiltonian responds to an external field by varying its contribution to the total energy. As it is shown in Fig. 7, the total energy decreases as the magnetic field increases, with the primary positive contributions coming from EEx, EZeeman. EAni shows a similar but more moderate trend. On the contrary, the magnitude of the energy contribution from the DM interaction decreases as a function of the magnetic field. By investigating only the ground state configurations, we found that the emergence of skyrmions requires significant contributions from both EDMI and EZeeman. Specifically, when the magnetic field is weaker than 0.3 T, the energy contribution of the Zeeman term becomes negligible. Similarly, when the magnetic field is stronger than 0.9 T, the contribution of the DM term becomes unimportant. When the applied magnetic field lies between 0.3 and 0.9 T, the two terms exhibit contrasting variations and compete with each other. In fact, the relative magnitude of these two terms is the decisive factor in determining the skyrmion’s morphology. When the DM interaction has a substantial significant role, it results in an elliptical form, whereas when the Zeeman term is more dominant, the skyrmion forms a well-defined swirling configuration.

Fig. 7: Average topological charge, total energies, and decomposed energies of the spin Hamiltonian for CrInSe3 as functions of the external field.
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The shaded area indicates the magnetic field range where there exists skyrmion(s) in the ground state.

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Adiabatic spin wave spectrum for CrInX3 (X=Se, Te), corresponding to the collinear FM state under a high external magnetic field and the spin spiral ground state with zero external fields, are obtained by adopting the adiabatic approach. As it is shown in Fig. 8, some general features can be observed from the FM spectra. First, the spin waves of the two materials exhibit different maximum excitation values, which reflect the difference in overall magnetic interaction strength, more specifically, the dominant exchange parameter J1. Second, the spin waves of both materials exhibit a quadratic dependence around the Γ point, consistent with the FM ordering adopted in the calculation. Moreover, the magnon softening observed in CrInTe3 indicates a stronger instability, and therefore, a greater tendency toward a non-collinear state. This is verified by the ground state macroscopic spin textures obtained in the Monte Carlo simulations, which reveal a significantly larger spin-spiral propagation vector. Third, the energy gaps at the Γ point for the two systems are 0.66 and 9.70 meV, respectively, representing the threshold energy required to excite each material. It directly reflects the MAE of 0.17 meV for CrInSe3 and 2.43 meV for CrInTe3.

Fig. 8: Calculated adiabatic spin wave spectrum for CrInX3 (X=Se, Te).
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Solid and dashed lines represent results obtained from their FM and ground state stripe states, respectively.

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On the other hand, the magnon spectrum of the spin spiral ground state reveals several interesting features attributed to its noncollinear nature. To begin with, the previously lowest energy point in the spectrum has shifted away from the Γ points, which represent the FM state, to a point located within the first Brillouin zone, corresponding to the spin stripe texture. Specifically, as shown in Fig. 8, the dips exhibited by Se- and Te-based compounds along the Γ − M path are [0.036,0,0] and [0.188,0,0], respectively. They represent the propagation vectors in that particular direction, with the magnitude providing insights into the spin spiral modulation period. The bigger the period, the smaller the magnitude. Moreover, this magnitude varies along different high symmetry paths, as demonstrated by the vectors [0.188,0,0] along Γ − M and [0.096,0.096,0] along Γ − K for CrInTe3, respectively. This variation is due to the fact that the spin spiral propagation vector in a real material typically does not align with any high-symmetry directions within the first Brillouin zone. Therefore, a combination of multiple vectors is required for accurate representation. Following this shift, the threshold for exciting specific spin textures in CrInSe3 has been reduced from 0.66 meV to 0.11 meV, and for another compound, from 9.70 meV to 0.34 meV. And the maximum eigenvalue for these compounds has been reduced by 1.27 meV and 16.65 meV, respectively. Last but not least, the spin-stripe texture breaks time-reversal symmetry due to the fact that dip positions, which used to be equivalent, now exhibit non-degeneracy, with each showing different vector magnitudes as well as distinct gap values. As demonstrated in a more detailed spin wave spectrum shown in Supplementary Fig. 13.

Complex ferroelectric polarization properties

Similar to In2Se341,42,43, which exhibits excellent room-temperature ferroelectricity due to the displacement of its middle-layer Se atoms, the middle layer of X atoms in CrInX3 is also displaced from its high-symmetry position following the same pattern. The systems are polar due to uncompensated dipole moments originating from Cr-X, and X-In polar bonds. As shown in the inset of Fig. 9a, the spontaneous polarized state corresponds to a structure in which the X atom is positioned directly under the In atom. In contrast, the reference structure aligns with the one that the X atom is located at the midpoint between the Cr and In atoms. This displacement is along [–0.84 0.84 0.27] direction, associated with a polarization of 13.19 μC/cm2 for CrInSe3 and 7.38 μC/cm2 for CrInTe3, respectively, along the exact same direction. Further investigation shows that the electronic contribution denotes a polarization that is along the displacement direction, while the ionic contribution is along the opposite direction with a slightly weaker value. The total polarization results from the counterbalance between these two contributions, as the detailed data shown in Supplementary Table 7 and Supplementary Fig. 14. Nonetheless, the uncompensated polar bonds between Cr-X and In-X are more pronounced as compared to the case in In2Se3 implying larger polarization values in our systems.

Fig. 9: Electric polarization.
figure 9

a Calculated ferroelectric polarization by using the modern theory of polarization. The data is plotted as a function of X atoms displaced from their high-symmetry position. Purple and blue lines represent the results obtained for CrInSe3 and CrInTe3, respectively. Data denoted by solid (empty) points are the results of calculation (fitting). Insets are the side and top views of the crystal structure with different X displacements. Purple solid and dashed circles illustrate the position of the displaced X atom. Black arrow in the lower inset indicates the displacement direction. b Calculated spin-spiral induced polarization for five magnetic ensembles. Solid and dashed lines represent the results for CrInSe3 and CrInTe3, respectively. Different line colors correspond to the results obtained from various external magnetic fields, ranging from 0 to 6.6 T. c Polarization density and d the corresponding spin texture of the five ensembles, obtained from CrInSe3. Double-headed arrows colored in black and green denote the spin stripe width and skyrmion diameters.

Full size image

Except for the polarization induced by the ionic displacement and the corresponding electronic charge redistribution, cycloidal spiral spin state-induced electric polarization plays a surprisingly important role in such material. According to the KNB model, polarization occurs when the spin spiral propagation vector q and spin rotation axis e form a non-zero angle, and reach its maximum when the angle is equal to 90∘. The orientation of the propagation vector q is closely related to the chirality of the spin spiral’s evolution, as discussed in the computational details in Section “Ferroelectric polarization calculations”. In CrInX3, both the spin stripe and the magnetic skyrmions exhibit as Néel-type. As can be seen in Supplementary Fig. 15, both q and e are orientated within the xy-plane and, more importantly, are nearly perpendicular to each other. This configuration makes them ideal candidates for this form of polarization.

To clarify the dependency of ferroelectric polarization on magnetic texture, magnetic structures for each compound are selected under several representative external magnetic fields, with five ensembles examined for each field. The calculated total and projected polarization results are shown in Fig. 9b and Supplementary Fig. 16. First and foremost, the magnitude of the total polarization, Ptot, is primarily attributed to its Pz component. This result is perfectly consistent with the one derived from the KNB model, as both q and e align in-plane, the induced polarization P, which is a product of the two, points out-of-plane. Secondly, there is a one-to-one correlation between polarization density and the corresponding spin texture, suggesting the strong interplay between their electric polarization and magnetic characteristics. Take CrInSe3 as an example, when the magnetic structure exhibits a perfect long-range spiral texture, as seen in Fig. 9d at B equal to 0 T, a strong average polarization value of 50 μC/m2 is observed (indicated by the cyan solid line in Fig. 9b). With increasing external fields, the long-range spiral state is destroyed, transforming into a short-range ordered skyrmion state. This change is directly reflected in the polarization value, as the average drops to 25 μC/m2 (shown by the green solid line). When the magnetic field is strong enough to disrupt the previously stabilized cycloidal state, an approximate collinear FM state occurs at 1.0 T, where spins are nearly aligned with the magnetic field, deviating by less than six degrees (illustrated in Supplementary Fig. 17). As shown in Fig. 9, the polarization disappears due to a phase transition from the long-range spin spiral state to the approximate collinear FM state. This is reflected in the polarization density, which changes from a purely out-of-plane ordered state (blue) to a randomly distributed state, where clockwise (blue) and anticlockwise (red) spin deviations from the easy axis. With the external magnetic field further increasing to 3.0 T, the noncollinear character disappears (q = 0, as there is no spin spiral propagation), and the system becomes strictly FM, thereby resulting in a non-polar state (white). As a result, the system exhibits an interesting series of phase transitions under an increasing magnetic field, starting with the ferroelectric phase, transitioning through the paraelectric phase, and ultimately reaching the non-polar phase. As the total polarization value is the sum of the polarization densities at each lattice site, a more detailed analysis of the magneto-electric coupling can be obtained by comparing the spin textures with their corresponding polarization densities.

For the spin spiral states, in addition to having the largest polarization values in general, variations in texture, such as stripe width and stripe angle, also affect their polarization outcomes. According to Eq. (5), changes in the magnitude of q and the relative angle between q and e can affect the polarization value. A narrower spin stripe, being represented by a larger propagation vector, consequently generates a larger polarization. For example, a spin stripe with a width of 15.46 nm corresponds to a polarization value of 35.67 μC/m2, and one that is 7.73 nm wide corresponds to a polarization of 55.42 μC/m2. Moreover, the relatively lower polarization of ensemble 2 is ascribed not only to a wider spin stripe (16.05 nm) but also to a non-90∘ angle between q and e, as depicted in Supplementary Fig. 15. When it comes to magnetic skyrmions, the polarization density exhibits a white area around the skyrmion boundary, as it is clearly seen in ensemble 4 at B=0 T. It happens because of the swirling nature of skyrmions: for any spin along a spin spiral chain across the magnetic skyrmion center, the resulting wave vectors always point toward the center. More fundamentally, as the origin of this type of polarization can be traced to the ionic displacement induced by the non-collinear spin structure (referred to as the inverse DM effect)47,48. Consequently, one can anticipate variations in the ionic displacement corresponding to this specific circular polarization density. Such realizations pave the way for the emergent formation of a ferroelectric skyrmion, which in itself represents a novel entity in the field of spintronics. With an increasing magnetic field, smaller skyrmions emerge alongside an enlarged FM area. In this case, the characteristic of polarization density is an increase in the white region, accompanied by a randomly interspersed distribution of red and blue areas that represent upward (red) and downward (blue) polarization, respectively. As a result, this leads to a reduction in polarization.

The results obtained for CrInTe3 are similar, while its polarization value, as indicated by the peak of 194.93 μC/m2 shown in Fig. 9b, not only more than four times bigger than the CrInSe3 but also significantly exceeds those of the most promising candidates previously identified with this mechanism for type-II multiferroic applications21. This phenomenon in CrInX3 not only demonstrates the direct magneto-electric nature but also, more importantly, unveils the potential of controllable magnetoelectric multiferroic skyrmions within a defined magnetic field range.

In summary, the magnetic features of the Janus material CrInX3 (X=Se, Te) were first extracted from first-principles simulations. Based on these, the exchange mechanisms of both Heisenberg and Dzayaloshinskii-Moriya interactions have been fully explained. Two intriguing aspects have been identified: firstly, the unusual eg electron occupation in Cr-d orbitals, which determines the exchange pathway and, consequently, the related FM and AFM exchange couplings. Secondly, the surprising variation in spin-orbit coupling strength on different X layers, which dictates the orientations of the 3D Dzayloshinskii-Moriya vectors and, therefore, the non-collinear inclination of the spin system. Additionally, detailed phase diagrams have been obtained, from which the importance of EDMI and EZeeman in the spin Hamiltonian have been emphasized, the mechanisms of magnetic skyrmion formation as well as the decisive factors affecting the swirling shape have been established. Together with magnetic excited states fully exemplified by investigating spin wave spectrum, a complete description of magnetic properties has been achieved. Moreover, further polarization studies have shown that these Janus structures not only exhibit well-defined in-plane polarization but also possess surprisingly high out-of-plane polarization induced by spin spirals. More importantly, by investigating the direct correlation between spin textures and polarization densities, we conclude that multiferroic skyrmions are expected to be observed in experiments. Together with their high magnetic transition temperatures, the CrInX3 compounds emerge as perfect candidates for overcoming the main limiting factors of the multiferroic materials and therefore, pave the way for the discovery of room-temperature, strong magnetoelectric materials for spintronic applications.

Methods

Density functional theory (DFT) based calculations are performed to study the structural, electronic, magnetic, and electric polarization of CrInX3 systems. The structural model was built with a 20 Å vacuum layer along z direction. A 2 × 2 × 1 supercell is employed to simulate structures with different magnetic states. The generalized gradient approximation (GGA)49 is used to take into account the exchange-correlation effect in our density functional theory calculations50,51. The strong correlation effect on Cr d-electrons is corrected by adopting the on-site Hubbard Ueff52,53 with a value of 3 eV, which is proven to be good for the Cr-based compounds54. All geometry optimization and standard electronic property calculations were carried out using the Vienna ab initio Simulations Package (VASP), a plane wave basis set and projector-augmented wave (PAW)55,56 code. The kinetic energy cutoff of the plane waves is set to 520 eV. A k-mesh integration is chosen as 9 × 9 × 1 and 15 × 15 × 1, with the convergence criteria of 1 × 10−6 eV atom−1 and 0.005 eV Å−1 for the two calculations, respectively.

Magnetic parameters calculations

Furthermore, utilizing the magnetic force theorem and the Lichtenstein-Katsnelson-Antropov-Gubanov (LKAG) formula57, a complete rank-2 exchange matrix is obtained, from which both the orbital-decomposed isotropic Heisenberg exchange J and the Dzyaloshinskii-Moriya vector Dij58 can be numerically derived, as implemented in the full-potential linear muffin-tin orbital (FP-LMTO) electronic structure code RSPt59. The validity of using the Heisenberg model in this system has been further exemplified, as detailed in Supplementary Note 2. The generalized Heisenberg model is written as

$$H=-\sum\limits_{i\ne j}{J}_{ij}^{\alpha \beta }{{\bf{e}}}_{i}^{\alpha }{{\bf{e}}}_{j}^{\beta },\,\,\,\,\,\,\,\{\alpha ,\beta =x,y,z,\}$$
(1)

where the unit vectors e are the direction of local spins, and the inter-site exchange matrix \({\hat{J}}_{ij}\) has the following form,

$${\hat{J}}_{ij}=\left(\begin{array}{rcl}{J}^{xx}&{J}^{xy}&{J}^{xz}\\ {J}^{yx}&{J}^{yy}&{J}^{yz}\\ {J}^{zx}&{J}^{zy}&{J}^{zz}\end{array}\right),$$
(2)

in which the diagonal components represent isotropic Heisenberg exchange and the DM vector can be written in the form of off-diagonal components, e.g., z component:

$${D}_{ij}^{z}=({J}_{ij}^{xy}-{J}_{ij}^{yx})/2.$$
(3)

Spin Hamiltonian model

By using the magnetic parameters extracted from the firs-principles calculations mentioned above, the following effective spin Hamiltonian is formulated,

$$\begin{array}{ll}{{\mathcal{H}}}_{{\rm{eff}}}=&-\sum\limits_{i,j}{J}_{ij}\,{{\bf{e}}}_{i}\cdot {{\bf{e}}}_{j}-\sum\limits_{i,j}{{\bf{D}}}_{ij}\cdot \left({{\bf{e}}}_{i}\times {{\bf{e}}}_{j}\right)\\ &-\sum\limits_{i}\,{K}_{i}{\left({\hat{{\bf{e}}}}_{i}\cdot {{\bf{e}}}_{i}^{K}\right)}^{2}-\sum\limits_{i}{{\bf{B}}}^{ext}\cdot {{\bf{e}}}_{i}\,\end{array}$$
(4)

where the third and fourth terms represent energies induced by single-ion anisotropy and external magnetic field, respectively.

Classical Monte Carlo simulations

In this way, the equilibrium magnetic state under any thermal and magnetic field perturbation can be obtained by performing classical Monte Carlo simulations, and the excited spectrum can be achieved by having Fourier transformation of real space parameters or by analyzing dynamic structure factors obtained in spin dynamics simulations. In the simulations, a lattice size of 80 × 80 × 1 is used with an additional consideration of periodicity in the xy-plane, which corresponds to an area of 31.2 nm × 31.2 nm.

Ferroelectric polarization calculations

Electric polarization originating from three sources ionic, electronic, and non-collinear spin texture is calculated separately. Among them, the first two parts are calculated by using the modern theory of polarization60,61,62,63, and the last one is obtained by adopting the approach proposed by Katsura et al.64, which is equivalent to the Mostovoy’s expression48,65

$${\bf{P}}=c\,({\bf{e}}\times {\bf{q}}),$$
(5)

where c = γ χe, with γ stands for the spin-orbit coupling strength and χe denotes the dielectric susceptibility. q is the wave vector of the spiral, and e ~ ei × ej is the spin rotation axis. In CrInX3, γ is complex due to the varying nature of spin-orbit coupling strengths introduced by ligand atom X, as illustrated in Fig. 5b. On the other hand, there is a lack of experimental data regarding dielectric susceptibility. Therefore, we have chosen c as a constant value of 1 in all of our polarization calculations. It is exemplified by the fact that e and q alone sufficiently reflect the significant differences in polarization between the two systems. Therefore, we believe that the approximation of c = 1 is adequate to capture the main features of the system.

The wave vector q is a quantity that represents information on the spin spiral, including both its propagation directions as well as the propagation modulus. The orientation of the wave vector is defined in close relation to the chirality of the spin spiral’s evolution65, as shown in the schematic picture in Supplementary Fig. 18a. Specifically, when the spin spiral evolves clockwise, the wave vector q is oriented in the same direction; when it evolves anticlockwise, q is oriented in the opposite direction, as illustrated in Supplementary Fig. 18c, d. In order to take into account the average effect, five layers of nearest neighbors are considered for each site i, comprising a total of 32 j-atoms.

Topological charge calculations

In addition, to quantitatively describe the topological properties of magnetic textures, topological charge Q is calculated by the following formula66:

$$Q=\frac{1}{4\pi }\int{\bf{m}}\cdot \left(\frac{\partial {\bf{m}}}{\partial x}\times \frac{\partial {\bf{m}}}{\partial y}\right){\rm{d}}x\,{\rm{d}}y.\,$$
(6)