# A New Skyrmion Topological Transition Driven by Higher-Order Exchange Interactions in Janus MnSeTe

Megha Arya, Moritz A Goerzen, Lionel Calmels, Rémi Arras, Soumyajyoti

Haldar, Stefan Heinze, Dongzhe Li

► To cite this version:

Megha Arya, Moritz A Goerzen, Lionel Calmels, Rémi Arras, Soumyajyoti Haldar, et al.. A New Skyrmion Topological Transition Driven by Higher-Order Exchange Interactions in Janus MnSeTe. Nano Letters, 2025, 10.1021/acs.nanolett.5c04452. [https://doi.org/10.1021/acs.nanolett.5c04452](https://doi.org/10.1021/acs.nanolett.5c04452). [hal-05366108](https://HAL.inist.fr/hal-05366108v1).

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# A new skyrmion topological transition driven by higher-order exchange interactions in Janus MnSeTe

Megha Arya,† Moritz A. Goerzen,† Lionel Calmels,† Rémi Arras,†
Soumyajyoti Haldar,‡ Stefan Heinze,‡,¶ and Dongzhe Li*,†

†CEMES, Université de Toulouse, CNRS, 29 rue Jeanne Marvig, F-31055 Toulouse, France
‡Institute of Theoretical Physics and Astrophysics, University of Kiel, Leibnizstrasse 15, 24098

Kiel, Germany

¶Kiel Nano, Surface, and Interface Science (KiNSIS), University of Kiel, 24118 Kiel, Germany

E-mail: dongzhe.li@cemes.fr

**ABSTRACT:** Two-dimensional (2D) van der Waals magnets offer a promising platform for pushing skyrmion technology to the single-layer limit with high tunability. While Dzyaloshinskii–Moriya interaction (DMI) is often recognized as central to skyrmion formation, their stability, collapse, and topological transition in 2D materials remain largely unexplored. In particular, the effect of higher-order exchange interactions (HOI) on these phenomena is unknown. Here, using first-principles calculations and atomistic spin simulations, we report a new topological transition generated by HOI, which we term ‘ferric transition’, in single-layer MnSeTe. Surprisingly, skyrmion stability and collapse remain largely unaffected by HOI due to the dominant role of DMI near the saddle point, whereas the Bloch point is strongly modified, giving rise to this novel transition. Moreover, we predict that Janus MnSeTe exhibits remarkably high skyrmion energy barriers due to its strong DMI, among the highest reported for intrinsic 2D magnets.
**KEYWORDS:** Magnetic skyrmions, Dzyaloshinskii–Moriya interaction, Higher-order exchange interactions, ferric topological transition, single-layer MnSeTe

TABLE OF CONTENT:

<figure><img src="image_2.png" /></figure>

Magnetic skyrmions are vortex-like localized spin textures with non-trivial topology that have raised high hopes for next-generation spintronics devices. 1–5 During the last decade, most of the efforts were devoted to studying skyrmions in bulk crystals, 6–8 ultrathin films with interfaces, 9–12 and multilayers. 13–15 More recently, the study of skyrmions has progressed to two-dimensional (2D) van der Waals (vdW) magnets. 16–19 Stabilizing skyrmions in 2D vdW magnets offers several potential advantages. These include avoiding pinning by defects due to high-quality vdW interfaces, the possibility of intrinsic Dzyaloshinskii-Moriya interaction (DMI) within a single layer, and easy control of magnetism via external stimuli such as strain, 20 magnetic proximity, 21,22 electric field, 23,24 Moiré engineering, 25 or light. 26 An all-electrical scheme based on tunnel junctions was recently proposed to detect skyrmions in 2D magnets. 27

A major challenge in skyrmion-based device applications is that the activation energy for switching a magnetic bit is usually bounded by energy barriers, which ensure the thermal stability of the stored information. It is known that skyrmions are strictly topologically protected in the continuum, meaning a skyrmion cannot be continuously unwound into a uniform ferromagnetic (FM) state. Instead it is often assumed that the skyrmion annihilates by shrinking to a singular point so
that its energy reaches the topological Belavin-Polyakov limit,²⁸ which consequently gives rise to a finite energy barrier.²⁹,³⁰ However, this model breaks down for magnetic textures spanning only a few lattice constants³¹ and thus cannot capture more complex annihilation mechanisms that have been reported numerically and experimentally³² for skyrmions at the nanoscale. Therefore, the unbiased investigation of skyrmion collapse and topological transition mechanisms is crucial for enhancing the stability of skyrmions with technologically relevant sizes.²⁰,³²⁻³⁴

According to the extended Heisenberg model, the existence of skyrmions is ascribed to a delicate balance between various magnetic interactions, including Heisenberg exchange, DMI, magnetocrystalline anisotropy, Kitaev, and Zeeman interaction. Among these interactions, the DMI, which favors a canting of the spins of adjacent magnetic atoms, is often recognized as the key ingredient in assigning a unique rotational sense to skyrmions. DMI originates from spin-orbit coupling (SOC) and broken inversion symmetry. However, most 2D magnets are centrosymmetric, and several strategies have been proposed to achieve DMI in these materials by breaking inversion symmetry.³⁵⁻³⁷

While widely used, the extended Heisenberg model is a localized spin model, limiting its applicability to 3d transition metals due to their itinerant nature of magnetism. The alternative Hubbard model describes electrons on a lattice by their hopping between sites and mutual Coulomb repulsion. Note that the conventional Heisenberg exchange can be obtained in second-order perturbation theory from the $S = 1/2$ Hubbard model. If one includes higher-order terms in the perturbative expansion, new interactions termed higher-order exchange interactions (HOI) arise. In particular, the 4-spin 2-site (biquadratic) and the 4-spin 4-site interactions occur in fourth-order perturbative expansion for the $S = 1/2$ Hubbard model.³⁸,³⁹ It has recently been demonstrated that a 4-spin 3-site interaction also arises in $S \ge 1$ systems,⁴⁰ appropriate for describing crystals in which the magnetic moments are of the order of 2 to 3 $\mu_B$. It has been demonstrated that these HOI terms can play a crucial role for the magnetic properties of ultrathin films and 2D vdW magnets.⁹,³⁶,⁴¹⁻⁴⁹

Janus monolayer transition-metal dichalcogenides, with their inversion asymmetry and strong SOC, have been proposed as a promising platform for skyrmions. Over the last five years, Janus
monolayers have been extensively studied by different groups.³⁵,⁵⁰⁻⁵³ Two magnetic Janus monolayers have recently been synthesized, thus opening further opportunities for experimental investigation of topological magnetism in these materials.⁵⁴,⁵⁵ It is worth noting that previous studies on magnetic Janus only focused on realizing topological spin textures using the standard extended Heisenberg model. However, key aspects such as quantification of skyrmion stability, collapse, and topological transition – all crucial for device applications – have so far not been reported. In particular, the role of HOI in skyrmion collapse and topological transition are also unknown in 2D vdW magnets.

In this Letter, we report a new skyrmion topological transition induced by HOI in single-layer MnSeTe using first-principles density functional theory (DFT)-based calculations and atomistic spin simulations. This new transition mechanism is termed ferric, which is fundamentally different from the conventional radial²⁰,⁵⁶,⁵⁷ and chimera³²,⁵⁸ transitions reported in the literature. We explain its origin by using energy-decomposed density maps with HOI. Additionally, due to the strong DMI originating from the combined effects of broken inversion symmetry and SOC, Néel-type skyrmions are formed at zero magnetic field. Remarkably, we find an energy barrier exceeding 330 meV, which is among the highest reported for intrinsic 2D vdW magnets.

**First-principles atomistic spin model.** Fig. 1a shows the atomic structure of monolayer MnSeTe. The Mn atoms with point group $C_{3v}$ form a hexagonal lattice, sandwiched by two atomic planes of Se and Te atoms. We find that the lattice constant is 3.68 Å, which is in good agreement with previously reported values in Ref. ³⁵ (see Sec. I of Supporting Information (SI) for details).

We describe the magnetic state of the MnSeTe monolayer by a set of classical magnetic moments $M_i$ localized on the atomic sites $i$ of a hexagonal lattice and with a dynamics governed by the following Hamiltonian of the beyond-extended Heisenberg model:

$$H = -\sum_{ij} J_{ij}(\mathbf{m}_i \cdot \mathbf{m}_j) - \sum_{ij} \mathbf{D}_{ij} \cdot (\mathbf{m}_i \times \mathbf{m}_j) - K_u \sum_i (m_i^z)^2 - \mu \sum_i (m_i^z \cdot \mathbf{B}) + H_{\text{HOI}} \quad (1)$$
where the z-axis is perpendicular to the MnSeTe layer and

$$ H_{\text{HOI}} = -B_1 \sum_{<ij>} (\mathbf{m}_i \cdot \mathbf{m}_j)^2 \\ -2Y_1 \sum_{<ijk>} (\mathbf{m}_i \cdot \mathbf{m}_j)(\mathbf{m}_j \cdot \mathbf{m}_k) \\ -K_1 \sum_{<ijkl>} [(\mathbf{m}_i \cdot \mathbf{m}_j)(\mathbf{m}_k \cdot \mathbf{m}_l) + (\mathbf{m}_i \cdot \mathbf{m}_l)(\mathbf{m}_j \cdot \mathbf{m}_k) - (\mathbf{m}_i \cdot \mathbf{m}_k)(\mathbf{m}_j \cdot \mathbf{m}_l)] \quad (2) $$

Here $\mathbf{m}_i = \mathbf{M}_i/M_i$ and $\mathbf{m}_j$ are normalized magnetic moments at positions $\mathbf{R}_i$ and $\mathbf{R}_j$, respectively. The first four magnetic interaction terms in Eq. 1 correspond to Heisenberg exchange, DMI, magnetocrystalline anisotropy energy (MAE), and Zeeman energy under an external magnetic field $\mathbf{B}$ respectively, and they are characterized by the constants $J_{ij}$, $\mathbf{D}_{ij}$, $K_u$, and $\mathbf{B}$ in the related terms. The fifth magnetic interaction term, detailed in Eq. 2 denotes HOI energies where $B_1$, $Y_1$, and $K_1$ are biquadratic, 4-spin 3-site, and 4-spin 4-site constants, respectively. In our material model, we systematically considered Heisenberg exchange parameters up to the tenth nearest-neighbor (NN), DMI up to the seventh NN (see Fig. 1b for the first three DMI vectors), and HOI only for NN spins. The notations $<ij>$, $<ijk>$, etc., in Eq. 2 indicate that the summation is restricted to NN spins (see Fig. 1c for NN hopping paths considered). The Kitaev interaction was neglected, as its contribution is an order of magnitude smaller than that of the other interaction terms.⁵⁰ On the other hand, the Kitaev interaction is pronounced in honeycomb lattices.⁵⁹

To determine the magnetic interaction constants in our system, DFT calculations were performed with the FLEUR code⁶⁰⁻⁶² based on the full-potential linearized augmented plane wave (FLAPW) formalism (see Sec. I of SI for methodology and computational details). Spin-spirals characterized by wave vector $\mathbf{q}$ are exact solutions of the extended Heisenberg model and often denoted as single-$\mathbf{q}$ states. There also exist multi-$\mathbf{q}$ states, which correspond to superpositions of these spin spirals. The converged energies of single-$\mathbf{q}$ states were utilized to determine pair-wise interaction constants ($J_{ij}$, $\mathbf{D}_{ij}$, and $K_u$), while the converged energy differences between multi-$\mathbf{q}$ and their corresponding single-$\mathbf{q}$ states were used to determine multi-spin interactions ($B_1$, $Y_1$, and
<table><caption>Table 1: Shell-resolved Heisenberg exchange constants ($J_i$), HOI-modified exchange constants ($J'_i$), DMI constants ($D_i$), MAE constant $K_u$, and NN HOI constants ($B_1$, $Y_1$, $K_1$) for MnSeTe. Positive (negative) $J_i$ indicates FM (AFM) coupling. Positive (negative) $D_i$ favors CW (CCW) spin spirals. $K_u > 0$ favors an out-of-plane magnetization.</caption><thead><tr><th colspan="10">Heisenberg exchange constants (meV/atom)</th></tr><tr><th>J<sub>1</sub> (J'<sub>1</sub>)</th><th>J<sub>2</sub> (J'<sub>2</sub>)</th><th>J<sub>3</sub> (J'<sub>3</sub>)</th><th>J<sub>4</sub></th><th>J<sub>5</sub></th><th>J<sub>6</sub></th><th>J<sub>7</sub></th><th>J<sub>8</sub></th><th>J<sub>9</sub></th><th>J<sub>10</sub></th></tr></thead><tbody><tr><td>3.91 (4.95)</td><td>3.43 (4.47)</td><td>2.22 (1.18)</td><td>-0.75</td><td>-0.84</td><td>0.09</td><td>0.53</td><td>0.16</td><td>0.06</td><td>-0.01</td></tr></tbody><thead><tr><th colspan="9">DMI constants and MAE (meV/atom)</th></tr><tr><th>D<sub>1</sub></th><th>D<sub>2</sub></th><th>D<sub>3</sub></th><th>D<sub>4</sub></th><th>D<sub>5</sub></th><th>D<sub>6</sub></th><th>D<sub>7</sub></th><th>K<sub>u</sub></th><th></th><th></th></tr></thead><tbody><tr><td>-1.04</td><td>0.19</td><td>-0.42</td><td>0.01</td><td>0.20</td><td>-0.01</td><td>-0.03</td><td>0.42</td><td></td><td></td></tr></tbody><thead><tr><th colspan="9">HOI constants (meV/atom)</th></tr><tr><th>B<sub>1</sub></th><th>Y<sub>1</sub></th><th>K<sub>1</sub></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>2.07</td><td>-1.04</td><td>-0.89</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

$K_1$).

Firstly, we focus on spin spirals **q** along the high-symmetry paths $\overline{\Gamma}\overline{M}$ and $\overline{\Gamma}\overline{K}\overline{M}$ of the Brillouin zone (BZ) (Fig. 2b and c). The high-symmetry points represent special magnetic states: $\overline{\Gamma}$ corresponds to the FM state, $\overline{K}$ to the Néel-state with an angle of 120° between adjacent spins, and $\overline{M}$ to the row-wise anti-ferromagnetic (RWAFM) state. A fit of the spin spiral energy dispersions, calculated with and without SOC, to the corresponding terms of the Heisenberg model, respectively, yields the DMI and the exchange energy constants displayed in Table 1. Without SOC, the energy dispersion shows a minimum at $\overline{\Gamma}$, which represents the FM state. Note that the chalcogen atoms exhibit small induced magnetic moments, causing deviations from an ideal magnetic monolayer structure. Consequently, the DFT energies at $\overline{M}$ for spin spirals along different directions are not equivalent. Furthermore, analytical functions derived from the monolayer model show slight deviations when fitted near $\overline{M}$ due to these deviations. When SOC is turned on, a significant DMI is observed as expected and in agreement with the literature,³⁵,⁵⁰ and it favors a counterclockwise (CCW) rotational sense, which corresponds to the negative value of energy $\Delta E_{SOC}(\mathbf{q})$. We further confirm that the DMI can be explained by the Fert-Lévy model⁶³ since it mainly originates from
interfacial nonmagnetic atoms (see Sec. II of SI). SOC is also responsible for the finite out-of-plane MAE as presented in Table 1. The MAE shifts the energy of spin spirals by $K_u/2$ with respect to the FM state since on average half of the spins are orientated in-plane and half of the spins out-of-plane. The magnetic ground state is found to be the FM state when we include all terms from DFT, i.e., $E_{ss}(\mathbf{q}) + \Delta E_{SOC}(\mathbf{q}) + K_u/2$.

Next, we focus on three prototypical multi-q states,⁴¹,⁶⁴ shown in Fig. 2a. Among them, the two *uudd* states are lower in energy, whereas the 3Q state is higher in energy compared to the corresponding 1Q states studied without SOC. Also, all three are far above the FM state (Fig. 2b). For the 3Q state, there are two possible choices of the corresponding single-q reference state: one at the $\overline{M}$ point along $\overline{\Gamma M}$ and another at the $\overline{M}$ point along $\overline{\Gamma KM}$. By comparing the magnetic moments of these RWAFM states at both $\overline{M}$ points with that of the 3Q state, we find that the one along $\overline{\Gamma M}$ provides a closer match. Therefore, we use the RWAFM state along $\overline{\Gamma M}$ for further analysis in the main text (note that we have also performed analysis and comparison using the RWAFM state along $\overline{\Gamma KM}$ and found that the general conclusions remain unchanged; see Sec. III of SI). The calculated HOI constants are displayed in Table 1. The biquadratic constant is comparable to the third NN exchange constant, while the 4-spin 3-site and 4-spin 4-site constants have smaller magnitudes and opposite signs relative to the biquadratic constant.

Note that the functional forms of the biquadratic and the 4-spin 3-site interactions for spin spirals resemble those of the first three exchange constants.⁴⁵ Hence, we cannot separate these exchange and HOI constants by fits and, to avoid double counting, we thus computed corrected exchange constants $J'_1$, $J'_2$, and $J'_3$, as displayed in Table 1. Nonetheless, the exchange interaction remains dominated by the first two NN constants.

**Atomistic spin simulations.** We performed atomistic spin simulations using the SPINAKER code to explore the properties of isolated skyrmions in Janus MnSeTe (see Sec. I of SI for methodology and computational details). Our simulations were performed using two distinct parameter sets from Table 1: one utilizing $J_i$ and $D_i$, referred to as "without HOI" simulations, and the other utilizing $J'_1-J'_3$, $J_4-J_{10}$, $D_i$, as well as $B_1$, $Y_1$, and $K_1$, referred to as "with HOI" simulations. On
the other hand, we have used MAE = $K_u/2$ in our calculations since this quantity, which is sensitive to doping or temperature, can easily be adjusted in experiments.<sup>65-67</sup> Moreover, as we have checked, the use of different MAE values such as $K_u/5$ or $K_u/10$ does not change the general conclusions of this work.

Using spin dynamics, we observe the emergence of Néel-type skyrmions at zero magnetic field, both with and without HOI. To quantify skyrmion size, we employ two definitions that are commonly used in the literature – one proposed by Bocdanov, *et al.*<sup>68</sup> ($R_{Boc}$) and another often used in experiments<sup>69</sup> ($R_{Exp}$) (see Sec. IV of SI). These two approaches yield noticeably different size estimates – for example, at $B = 0$ T without HOI, $R_{Boc} \approx 1.6R_{Exp}$. As the magnetic field increases, the skyrmion size shrinks rapidly according to both definitions, as shown in Fig. 3a. However, the inclusion of HOI has a negligible effect on the skyrmion size. Without HOI, $R_{Boc}$ decreases from about 17.1 nm at $B = 0$ T to about 5.0 nm at $B = 0.75$ T; with HOI, it changes slightly from 17.1 nm to 4.9 nm over the same field range. Similarly, $R_{Exp}$ decreases from about 10.7 nm to 2.8 nm without HOI, and from 10.7 nm to 2.6 nm with HOI, as $B$ increases from 0 to 0.75 T.

To quantify the thermal stability of skyrmions, we calculate the energy barriers that stabilize skyrmions against collapse, both with and without HOI (see Fig. 3b). This has been done using the geodesic nudged elastic band (GNEB) method.<sup>70</sup> Surprisingly, these skyrmions exhibit high stability with energy barriers exceeding 330 meV at zero magnetic field – even surpassing those reported for state-of-the-art ultrathin films.<sup>56</sup> The energy barriers exhibit a nonlinear dependence on the magnetic field and remain sufficiently high for $B$ values up to 0.7 T, making them technologically desirable for skyrmionic devices. In the insets of Fig. 3, we show the difference between the cases with and without HOI. Relatively small deviations are observed for both the radius and energy barriers. In view of previous work on the large enhancement of skyrmion stability due to HOI in ultrathin films,<sup>45</sup> it is quite unexpected that we find the energy barrier with HOI to be only slightly lower than that obtained neglecting HOI.

To get more insight into the effect of HOI, we plot in Fig. 4a the minimum energy path (MEP)
for the skyrmion collapse into the FM ground state calculated using the GNEB method at $B = 0$ T.
The positions of saddle point (SP) and Bloch point (BP) are marked along the MEP for both cases,
with and without HOI. The SP is the point of maximum energy w.r.t the initial skyrmion state. The
BP is identified by calculating the topological charge, given by $Q = \frac{1}{4\pi} \int_{\mathbb{R}^2} \mathbf{m} \cdot (\frac{\partial \mathbf{m}}{\partial x} \times \frac{\partial \mathbf{m}}{\partial y}) dx dy$,
which jumps from -1 to 0 at the BP. Interestingly, we find that the SP and the BP do not coincide
in either case. While this separation of SP and BP is unusual in continuum models, where the
zero-size skyrmion<sup>29,30</sup> resembles both at the same time, there is no law for a discrete spin model
stating that the BP has to be the point of highest energy on a reaction coordinate (i.e., SP). Thus,
our findings underline the complexity of skyrmion topological transitions, even though, to the best
of our knowledge, this separation has not explicitly been addressed in the literature so far.

From the analysis of spin structures and topological charge densities (see Sec. V of SI) at dif-
ferent points along MEP (see Fig. 4b), we find that the SPs are almost identical in both cases. How-
ever, surprisingly, the skyrmion topological transitions are very different. Without HOI, we observe
a radially symmetric transition, where the skyrmion shrinks symmetrically until the BP and finally
collapses into the FM state. Remarkably, with HOI, a new skyrmion topological transition, which
we denote as ferric transition, is observed. We call it ferric because of the quasi-ferrimagnetic (FI)
state that emerges as a metastable state in the transition induced by HOI. This new transition is
fundamentally distinct from both radial<sup>20,56,57</sup> and chimera<sup>32,58</sup> transitions reported in the litera-
ture. The skyrmion transitions into a quasi-FI state wherein opposing topological charge density
substructures are formed near the center, leading to partial topological cancellation before finally
collapsing into the FM state (see Sec. VI of SI for additional figures illustrating ferric transition
at finite magnetic fields). From site-resolved energy contributions along the MEP, presented in
Sec. VII of SI, it becomes clear that 4-spin 3-site and 4-spin 4-site interactions favor the local FI
modulation. Considering the Hamiltonian utilized in this work, the energy difference per site be-
tween FM and FI state on the hexagonal lattice is $E_{\text{FI}} - E_{\text{FM}} = 16(Y_1 + K_1) + E_{\text{RWAFM}}^{\text{exc}} - E_{\text{FM}}^{\text{exc}}$,
where the last two terms are the Heisenberg exchange energies of the RWAFM and FM state, re-
spectively, leading to the FI being favored by both HOI terms in MnSeTe (See Sec. VII of SI for
details). Since the dominating frustrated exchange interaction still enforces a FM ground state, the FI modulation at the BP appears due to $Y_1, K_1 < 0$ and consequently leads to a reduction in energy compared to the case without HOI, as shown in Fig. 4a. So, HOI significantly decreases the energies after the SP, leading to very different BP configurations. However, HOI have only a minor effect on the energy barrier, as it is dominated by DMI. As a result, the interplay between DMI and HOI results in almost identical SPs with and without HOI, but their BPs are completely different.

Furthermore, we investigated the energy decomposition of MEP at $B = 0$ T for both cases (see Fig. 5). We find that the energy barriers arise from the competition between various interactions, with DMI strongly favoring the skyrmion state, while exchange and MAE favor the FM state. Since HOI terms modify the exchange constants, a meaningful comparison would be between the exchange contribution without HOI and the combined exchange (which includes Heisenberg exchange, biquadratic, and 4-spin 3-site interactions) contribution with HOI.⁴⁵ Interestingly, these contributions are of comparable magnitude at the SP in both cases. Also, the DMI and MAE contributions are of similar magnitudes at the SP with and without HOI. The slight difference is an associated effect of the HOI caused by the changes in relative spin angles during the collapse process. The biquadratic and 4-spin 3-site interactions exhibit nearly compensating contributions. A striking observation is that the 4-spin 4-site interaction acts in a qualitatively different way compared to all other interactions, in agreement with previous work on the role of HOI for skyrmion stability in transition-metal films.⁴⁵ It remains nearly constant along most of the MEP but exhibits a dip just after the SP. This explains why the transition mechanism is altered, but the energy barriers remain largely unchanged with and without HOI. In addition, we observe that the 4-spin 3-site interaction favors the skyrmion state, whereas the biquadratic and the 4-spin 4-site interaction favor the FM state.

To conclude, we demonstrate that a fundamental skyrmion transition emerges in the presence of HOI. We term this topological transition as ferric transition and show that it possesses unique characteristics that make it distinct from the conventional radial and chimera transitions. We explain
the origin of this new skyrmion topological transition and further identify the relevant HOI terms
that cause it. Our predictions can be experimentally tested using, for instance, spin-polarized scan-
ning tunneling microscopy,³² which could resolve the flanks of the differential tunnel conductivity
(dI/dU) signal. These flanks carry information on skyrmion collapse, and significant fingerprints
of enhanced non-collinear magnetoresistance⁷¹ contributions to the flanks should be identifiable
in the case of ferric transition. Regarding the promising materials, HOI are expected to be most
pronounced in large S systems (e.g., S = 1, 3/2, 2), in low-dimensional systems, and in metals.⁴⁰
Additionally, due to the strong DMI in Janus MnSeTe, we find skyrmion energy barriers exceeding
330 meV at zero field. These values are among the highest reported for intrinsic 2D magnets. Our
work demonstrates that taking into account the contribution of HOI is likely to be essential for
skyrmion topological transitions.

**Acknowledgement:** This study has been supported through the ANR grant no. ANR-22-CE24-0019. This work is supported by France 2030 government investment plan managed by the French National Research Agency under grant reference PEPR SPIN – [SPINTHEORY] ANR-22-EXSP-0009. This study has been (partially) supported through the grant NanoX no. ANR-17-EURE-0009 in the framework of the "Programme des Investissements d'Avenir". S.Ha and S.He gratefully acknowledge financial support from the Deutsche Forschungs Gemeinschaft (DFG, German Research Foundation) through SPP2137 "Skyrmionics" (project no. 462602351) and project no. 418425860. This work was performed using the HPC resources provided by CALMIP (Grant 2022/2025-[P21008]).

**Supporting Information Available**

The Supporting Information is available free of charge at https://pubs.acs.org/. Method-
ology and computational details on first-principles calculations and atomistic spin simulations, the
interfacial DMI mechanism, the choice of single-q reference states for the estimation of HOI con-
stants, skyrmion size estimation, topological charge density, ferric transitions at finite magnetic
fields, and energy density maps with and without HOI.

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<figure><img src="image_3.png" /><figcaption>Figure 1: (a) Top (upper) and side (lower) views of the crystal structure of Janus MnSeTe. The black dashed lines indicate the 2D primitive cell. (b) Illustration of DMI vectors on the hexagonal lattice. For clarity, only the Mn atoms are shown. (c) Illustration of HOI constants on the hexagonal lattice. One of the six possible minimal hopping paths for the biquadratic interaction ($B_1$) is depicted by a black curve. One of the six possible minimal hopping triangular paths for the 4-spin 3-site interaction ($Y_1$) is depicted in yellow. Two of the twelve possible minimal hopping diamond-shaped paths for the 4-spin 4-site interaction ($K_1$) are depicted in gray.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 2: (a) Spin structures of multi-q states: two *uudd* states along $\overline{\Gamma}\overline{M}$ and $\overline{\Gamma}\overline{K}$, and the $3Q$ state at the $\overline{M}$ point along $\overline{\Gamma}\overline{M}$ of the 2D BZ. (b) Energy dispersion of flat spin spirals ($E_{ss}$) for the MnSeTe monolayer along the high-symmetry path $\overline{M}\overline{\Gamma}\overline{K}\overline{M}$ without SOC. The filled circles represent DFT total energies, while the solid lines are fits to the Heisenberg exchange interaction up to the tenth NN. The energies of the two *uudd* and the $3Q$ states are also denoted by red squares at the *q* values of the corresponding single-*q* states. (c) Energy contribution of flat spin spirals due to SOC ($\Delta E_{SOC}$), also referred to as the DMI contribution, is fitted up to the seventh NN. All energies are measured with respect to the FM state ($E_{FM}$) at the $\overline{\Gamma}$ point. To accurately map the position and curvature of the ground state, a dense sampling around the $\overline{\Gamma}$ point was done.</figcaption></figure>
<figure><img src="image_5.png" /></figure>

Figure 3: (a) Bogdanov (Boc) and experimental (Exp) skyrmion radii as functions of the applied magnetic field *B* with (red) and without (black) HOI in Janus MnSeTe. The inset shows the radii difference with (*R*₂) and without (*R*₁) HOI, with the solid blue line for Boc and the dashed line for the Exp model. Since the Exp model corresponds to a zero-order method, whereas the Boc model represents a first-order approach that incorporates information about the slope of the profile, the Boc variation exhibits smoother behavior than the Exp model and is therefore numerically more stable. The inset further shows that the radii difference with and without HOI is very small, reaching maxima of about 0.15 nm at *B* = 0.75 T for the Boc model, and about 0.2 nm at *B* = 0.4 T and 0.22 nm at *B* = 0.75 T for the Exp model. (b) Total energy barriers of isolated skyrmions versus *B* with and without HOI. The blue curve in the inset shows the difference between the two curves: black (*ΔE*₁) and red (*ΔE*₂).
<figure><img src="image_6.png" /><figcaption>Figure 4: (a) MEP obtained for the transition from the skyrmion (sky) to the FM state through the SP for MnSeTe at *B* = 0 T with (red) and without (black) including HOI. The energies are shown w.r.t the skyrmion state. A zoom on the MEP near the SPs and the BPs is provided in the lower panel. SPs are marked as vertical dashed lines. BPs, identified via the magenta *y*-axis representing *Q*, are marked by vertical solid lines. (b) Corresponding spin textures (left panels) and topological charge densities (right panels) are also shown near SP and BP. Note that the topological phase transition occurs between BP-1 and BP. Interestingly, the skyrmion topological transition is radial without HOI, whereas it is ferric with HOI.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 5: (a) Decomposition of the MEP obtained for the skyrmion to FM transition in MnSeTe with and without including HOI at $B = 0$ T. The energy contributions of the different interactions are shown w.r.t the skyrmion state energy $E_{sky}$ (see legend). On the right axis the topological charge along the MEP is shown (magenta curves). SP and BP are marked by the vertical dashed lines. (b) Energy decomposition at the SP as shown in (a) with respect to the skyrmion state, both with and without HOI. Note that combined exchange and exchange denote the same quantity without HOI.</figcaption></figure>