Field-tunable skyrmion phases in monolayer MXene for spintronic applications

Junais Habeeb Mokkath*
College of Integrative Studies, Abdullah Al Salem University (AASU), Block 3, Khaldiya, Kuwait. E-mail: junais.mokkath@aasu.edu.kw

First published on 22nd August 2025

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Introduction

Magnetic skyrmions^1–21 are topologically protected chiral spin textures that have enabled novel approaches for manipulating magnetization at the nanoscale.^22–34 Skyrmions can exist either as isolated entities—formed under weak magnetic fields and prone to instabilities—or as periodic lattices stabilized under stronger fields or reduced temperatures. While isolated skyrmions are ideal for data storage, lattice skyrmions offer enhanced robustness for spintronic applications. Their signature swirling spin configuration can be manipulated by external magnetic fields and has been observed in a variety of materials, including B20 non-centrosymmetric crystals³⁵ and thin films with perpendicular magnetic anisotropy.^36,37 The origin of skyrmions lies in the Dzyaloshinskii–Moriya interaction (DMI),^38–42 a relativistic effect arising from spin–orbit coupling in non-centrosymmetric systems, which fosters their non-collinear and energetically favorable spin configurations. Skyrmions are typically classified as Bloch-type or Néel-type, depending on the nature of their spin rotations in bulk or thin-film geometries, respectively.

The Mermin–Wagner theorem⁴³ prohibits long-range magnetic order in isotropic two-dimensional (2D) Heisenberg systems at finite temperature. However, the inclusion of magnetic anisotropy can suppress thermal fluctuations sufficiently to allow for stable magnetic order in 2D materials.^44–51 Among such materials, MXenes^52–56—2D transition metal carbides, nitrides, and carbonitrides—have emerged as promising candidates for electronic, optical, and electrochemical applications. Despite this, their magnetic properties remain relatively unexplored.^57–60 In particular, tailoring MXenes to support skyrmionic states offers a novel route toward next-generation magnetic technologies.^61,62 Crucially, the geometry and arrangement of skyrmions are strongly dependent on the applied magnetic field—a parameter yet to be thoroughly investigated in the context of MXenes.

In this work, we employ an atomistic spin dynamics approach based on the Landau–Lifshitz–Gilbert (LLG) equations^63,64 to study skyrmion formation and evolution in a monolayer Co₂C MXene flake. Recent findings indicate that monolayer Co₂C can stabilize skyrmions at room temperature,^61,65 attributed to its inherently strong DMI. This feature motivated our choice of material. Our simulations reveal a magnetic phase transition driven by the external magnetic field. At 0 T, spin spirals dominate. As the field increases from 0.5 T to 2.5 T, the system transitions from spiral states to a skyrmion lattice. Further field increase (3.0 T to 5.0 T) results in a more defined skyrmion crystal with field-dependent geometry and density. Beyond 6.0 T, skyrmions progressively vanish, and by 8.0 T, a saturated ferromagnetic state is achieved. This study provides a comprehensive understanding of field-controlled skyrmion evolution in Co₂C MXene and highlights its potential in 2D spintronic devices.

Computational aspects

Atomistic spin dynamics simulations were performed using the VAMPIRE 6.0 package.⁶⁶ A monolayer Co₂C MXene flake, measuring 20 nm × 20 nm × 0.40 nm and comprising 7956 atoms, was modeled with periodic boundary conditions applied along the x and y directions, as illustrated in Fig. 1. Previous studies have reported that Co₂C MXene exhibits a significant Dzyaloshinskii–Moriya interaction (DMI), the strength of which scales with the electric dipole moment of the MXene structure.⁶¹

The magnetic behavior of the system was governed by a spin Hamiltonian incorporating four key interactions: Heisenberg exchange, uniaxial anisotropy, Zeeman energy, and the DMI:

where _i and _j are unit vectors representing spin orientations at lattice sites i and j; J_ij is the nearest-neighbor exchange interaction; K_u denotes the uniaxial anisotropy constant, with ê_i indicating the easy axis; μ_S is the atomic magnetic moment; _app is the applied external magnetic field; and _ij is the DMI vector, defined as _ij = D(ẑ × _ij), where ẑ is the unit vector along the out-of-plane direction and _ij is the relative position vector between atoms i and j.

The evolution of spin configurations—including spin spirals and skyrmions—was simulated by solving the Landau–Lifshitz–Gilbert (LLG) equation:

where ħ is the reduced Planck constant, is the magnetization unit vector, α is the Gilbert damping constant, and denotes the effective magnetic field arising from local interactions. The equations were numerically integrated using the Heun method.⁶⁷ The magnetic parameters for cobalt were adopted from ref. 66, specifically: atomic magnetic moment μ_B = 1.72, exchange energy J_ij = 1.60 × 10⁻²² J, and anisotropy energy K_u = 5.0 × 10⁻²³ J m⁻². For other key simulation parameters, see Table 1. Long-range dipolar interactions are not included in the present model to reduce computational cost and isolate the role of exchange, DMI, and anisotropy. While this is a common approximation for ultra-thin systems, we note that including dipolar fields could shift phase boundaries and skyrmion stability windows—a refinement we aim to address in future work.

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The LLG equation with thermal noise is:

where the effective field H^eff_i includes a stochastic thermal field H^th_i, defined by:

This ensures that thermal fluctuations follow the fluctuation–dissipation theorem.

Results and discussion

In this work, we elucidate the effect of varying magnetic fields (0 T to 8 T) on spin states, with the spin states identified by parameters like skyrmion geometry, skyrmion number, and mean magnetization. Although the skyrmion-ferromagnet transition is a well-established phenomenon, our study contributes new insights by capturing its time-dependent evolution in a finite monolayer Co₂C system, quantifying the underlying energy changes, and confirming the topological character of intermediate textures through direct computation of the skyrmion number. Typically, magnetic materials exhibit collinear spin orientation due to the Heisenberg exchange interaction. However, strong Dzyaloshinskii–Moriya interaction (DMI) can lead to non-collinear spin structures⁶⁸ like skyrmions. In this work, we transform spin states into momentum space using Fourier transform to analyze long-range order:

S_k indicates the vector of the spin in the momentum space and s_j in the real space, while k and j indicate the position of the spin in momentum space and real space respectively. In the case of 2D square lattice, the coordinate in momentum space is . Initially, we delineate the spin configuration of magnetic skyrmions in a thin multi-ferroic Co₂C MXene flake at zero applied magnetic field (B_app = 0 T), see Fig. 2. The left panel shows a top-down view of the spin spiral structure in the X–Z plane, revealing periodic modulations indicative of a skyrmion lattice. The right panel presents a magnified cross-sectional view of the boxed region from the left panel, emphasizing the detailed spin arrangement. Spins are depicted by arrows and colored according to their z-component, with red corresponding to +1, blue to −1, and intermediate values represented by a continuous color gradient. Analyzing Fig. 2, it is evident that B_app = 0 T results in spin spiral states. This configuration is identified as the spiral state due to its spin pattern. The interplay of DMI and exchange interaction engenders these spin spiral states in the absence of B_app. Notably, spin spirals lack translational symmetry. However, a skyrmion remarkably emerges at the center of the monolayer.

Fig. 3 shows the evolution of magnetic spin textures from 0.5 T to 2.5 T. Each subfigure shows both a top-down (X–Z) and cross-sectional view of the spin configuration. The arrow colors represent the z-component of the spin, with red for +1 and blue for −1. (a) At B_app = 0.5 T, a deformed spin spiral or skyrmion-like lattice is observed, characterized by quasi-periodic modulations due to competing magnetic interactions. (b) At 1.0 T, the spin texture becomes more aligned with the applied field, forming elongated magnetic domains and suppressing the spiral components. (c) At 1.5 T, the system undergoes a transition toward a partially field-polarized state, where red-dominated regions indicate an increasing proportion of spins aligned along the +z direction, although topological features persist. (d) At 2.0 T, the spin configuration approaches a near-ferromagnetic state, with only faint remnants of non-uniform textures. (e) At 2.5 T, nearly all spins are aligned with the applied field, confirming the collapse of skyrmionic structures and the dominance of Zeeman energy. These results demonstrate a clear field-driven phase transition from a nontrivial skyrmion/spin spiral phase to a topologically trivial, uniformly magnetized state. The tunability of the magnetic texture in Co₂C MXene under applied field conditions underlines its potential for use in spintronic and magneto-electric devices.

It is imperative now to analyze the variation of mean magnetization with B_app over time, as shown in Fig. 4. It is well understood that the properties of a magnetic material can be characterized by its mean magnetization, which is determined by the total magnetic moment per unit volume, providing insights into the magnetic configuration within the system. Within the atomistic framework, the average magnetization is determined as

where M_S signifies the saturation magnetization, N represents the total number of spins (or atoms), and S_i denotes the spin vector of each atom. The spin vector indicates a normalized, dimensionless spin moment and is defined by S_i = μ_i/μ_s, wherein μ_i is the magnetic moment of each atom, and μ_s constitutes the saturation value of this moment for each atom. Fig. 4 illustrates the time evolution of the mean magnetization length in a Co₂C MXene flake under applied magnetic fields ranging from 0.5 T to 2.5 T. The plot shows how the system's magnetic ordering responds dynamically over a simulation duration of 3 ns. At low field strength (0.5 T, black curve), the magnetization remains weak and fluctuates considerably, reflecting the persistence of non-uniform spin textures such as spirals and skyrmions. As the field increases to 1.0 T and 1.5 T (red and green curves), a more rapid magnetization buildup is observed, though saturation is not fully achieved—indicating partial alignment and residual topological features. For higher fields (2.0 T and 2.5 T, blue and magenta curves), the system exhibits a sharp rise in magnetization followed by stable saturation at significantly higher values. This behavior indicates a transition toward a homogeneous, field-polarized phase, where Zeeman energy dominates over competing magnetic interactions. These results demonstrate a clear field-dependent enhancement of spin alignment, confirming that increasing the external magnetic field facilitates a faster and more complete transition to a uniformly magnetized state.

Spin texture evolution in a Co₂C MXene flake under strong applied magnetic fields ranging from 3.0 T to 5.0 T is shown in Fig. 5. The left panels in (a)–(c) show the top-down (X–Z) projection of the spin configuration, while the right panels provide cross-sectional views along the Z-axis. Spin directions are represented by colored arrows, with red and blue indicating z-components of +1 and −1, respectively. (a) At B_app = 3.0 T, a well-defined lattice of magnetic skyrmions is observed, characterized by symmetric blue cores embedded within a predominantly red matrix. This ordered configuration indicates the stabilization of isolated skyrmions within a ferromagnetic background due to the balance of exchange, Dzyaloshinskii–Moriya, and Zeeman interactions. (b) At B_app = 4.0 T, the skyrmions persist but exhibit a slight decrease in core size and increased spacing, suggesting enhanced Zeeman energy further polarizing the surrounding matrix. The system remains in a skyrmion crystal phase, though the degree of non-collinearity is diminished. (c) at B_app = 5.0 T, the spin texture becomes more homogeneous, and the skyrmion cores appear more compressed and sparse. The dominance of red coloration across the spin map indicates stronger alignment of most spins along the +z direction, with skyrmionic features being increasingly suppressed. Overall, these results highlight a high-field regime where isolated skyrmions remain metastable but increasingly constrained by the applied field. The persistence of a skyrmion lattice up to 5.0 T underscores the robustness of topological magnetic textures in Co₂C MXene and their potential applicability in high-field spintronic devices.

Spin configuration in a Co₂C MXene flake under strong applied magnetic fields from 6.0 T to 8.0 T is shown in Fig. 6. The left panels in (a)–(c) show the top-down (X–Z) view of the spin textures, while the right panels provide the corresponding cross-sectional projections. Spin directions are represented using colored arrows, with red and blue denoting z-components of +1 and −1, respectively. (a) at B_app = 6.0 T, a small number of isolated skyrmions remain embedded in an otherwise uniformly magnetized background. The red-dominated field indicates strong spin alignment with the applied magnetic field, while the surviving skyrmions are localized and sparse, suggesting their metastability in a high-field environment. (b) at B_app = 7.0 T, the skyrmionic structures are nearly extinguished, with only a few remnant cores detectable. The magnetization is nearly uniform across the sample, indicating the system has largely transitioned into a ferromagnetic state. (c) at B_app = 8.0 T, the spin configuration is entirely homogeneous, with all spins fully aligned along the +z direction. The absence of any nontrivial textures confirms the complete collapse of skyrmions and the full dominance of Zeeman energy. These results reveal the final stage of the field-driven topological transition in Co₂C MXene, from a skyrmion-hosting state to a fully polarized ferromagnetic phase. The progressive suppression and ultimate annihilation of skyrmionic textures with increasing field emphasize the critical field thresholds for topological stability, providing insight into the control and manipulation of magnetic states in two-dimensional multiferroic systems.

While ref. 61 demonstrates the existence of room-temperature skyrmions in Co₂C MXene, our study builds upon and significantly extends these findings by exploring the field-tunable dynamical evolution of spin textures in a finite-sized monolayer system. Using atomistic spin dynamics simulations, we construct a complete spin-phase diagram as a function of external magnetic field strength, revealing a sequence of transitions from spin-spiral to skyrmion lattice to saturated ferromagnetic states. Furthermore, we track the time-resolved magnetization dynamics and decompose the energy contributions. The topological nature of the emergent skyrmion phases is quantitatively verified by calculating the skyrmion number from the spin texture. These dynamic and topological insights, which are absent in ref. 61, provide a deeper understanding of controllable skyrmion behavior in Co₂C MXene for spintronic applications.

Time evolution of the mean magnetization length in a Co₂C MXene flake under applied magnetic fields ranging from 3.0 T to 8.0 T is shown in Fig. 7. The plot illustrates the dynamic alignment of spins along the z-axis over a 3 ns simulation period. At lower field strengths (3.0 T and 4.0 T, black and red curves), the magnetization increases gradually, reaching moderate saturation values. This behavior corresponds to the coexistence of ferromagnetic alignment with residual non-collinear textures, such as skyrmions or distorted spin domains. As the field strength increases to 5.0 T and 6.0 T (green and blue curves), a sharper and more rapid rise in magnetization is observed, indicating enhanced spin alignment driven by stronger Zeeman coupling. These curves plateau at higher mean magnetization values, consistent with the increasing suppression of topological features. For the highest fields (7.0 T and 8.0 T, magenta and purple curves), the system rapidly approaches full saturation, with the magnetization nearing unity. This indicates that the spins are nearly completely aligned along the applied field direction, signifying a transition into a fully polarized ferromagnetic phase.

Next, we analyze the time evolution of the mean magnetization length at zero external magnetic field under finite temperature (100 K) conditions, see Fig. 8. The mean magnetization length exhibits low-temperature spin dynamics in a quasi-two-dimensional system. Initially, magnetization rises rapidly within the first 0.2 ns due to fast alignment of magnetic moments after thermal equilibration. From 0.2 ns to 1.2 ns, it stabilizes around 0.014, indicating transient spin ordering. Between 1.2 and 2.0 ns, magnetization increases to 0.017, possibly due to spin reorientation or domain merging. Subsequently, it declines gradually, reflecting thermal demagnetization without an external field. This aligns with expected behavior for a low-dimensional ferromagnet below its Curie temperature, where thermal fluctuations lead to spin decoherence. The decay is considered intrinsic, driven by spin–spin interactions and damping processes. Note that our model does not claim full thermal robustness at room temperature. A comprehensive set of finite-temperature simulations will be pursued in future work to evaluate stability under realistic operating conditions.

Next, we analyze the time evolution of the mean anisotropy energy and mean exchange energy under different applied magnetic fields. Fig. 9a) shows the time evolution of mean magnetocrystalline anisotropy energy under different magnetic fields. Initially, the anisotropy energy rapidly decreases within 0.5 ns as the system moves toward a more stable magnetic state. With increasing external magnetic field, the stabilized anisotropy energy becomes more negative, indicating better spin alignment and enhanced ferromagnetic order, stabilizing uniform magnetization at high fields. Lower fields (0–3 T) result in shallower energy minima, typical of non-collinear textures like spirals or skyrmions. This trend underscores the role of uniaxial anisotropy in transitions between magnetic phases. Fig. 9b) depicts the time evolution of mean exchange energy under different fields. Initially, exchange energy decreases rapidly, indicating quick spin alignment. Over time, the system reaches field-dependent quasi-equilibrium. Lower fields (0–3 T) lead to deeper exchange energy minima, aligning with non-collinear textures that maintain strong local exchange interaction. Higher fields (6–8 T) move the system to a saturated ferromagnetic state, with reduced overall exchange energy due to less local angular deviation. This inverse field-exchange energy relationship highlights varying magnetic ordering mechanisms across the phase diagram.

We assessed our simulation's robustness through a sensitivity analysis by altering the exchange interaction strength by ±25% of the baseline, see Fig. 10. Increasing the strength enhances spin coupling, leading to faster magnetic alignment and higher magnetization. Reducing it slows magnetization and lowers equilibrium. These trends align with theoretical predictions, confirming the dynamics aren’t due to exact parameter tuning. The stability of transitions and spin textures across this range supports the model's reliability, even without Co₂C-specific exchange data. This validates the physical relevance of our findings and justifies using bulk Co parameters in the absence of specific inputs.

In summary, the results from this study imply that exploring skyrmions in MXenes experimentally [via magnetic force microscopy (MFM: mapping the magnetic texture of the material) or Lorentz transmission electron microscopy (TEM): direct visualization of skyrmion structures in real-time] will further clarify their microscopic nature and boost the development of spintronic devices based on MXenes. We remark that the nanoscale dimensions and topological protection of skyrmions make them exceptionally suitable for both data storage and logic applications. Prospective studies will further explore skyrmion dynamics, focusing on factors such as mobility, stability in the presence of current, and response to external stimuli.

Conclusion and outlook

In this work, we have employed atomistic spin dynamics simulations based on the Landau–Lifshitz–Gilbert formalism to systematically investigate the formation, evolution, and annihilation of magnetic skyrmions in a monolayer Co₂C MXene flake under applied magnetic fields ranging from 0 T to 8 T. Our results reveal a field-driven topological transition in the magnetic texture of the system. At zero field, the system stabilizes into a spin-spiral configuration due to strong Dzyaloshinskii–Moriya interaction (DMI). With increasing magnetic field strength (0.5–2.5 T), these spin spirals evolve into skyrmion lattices. A more ordered and dense skyrmion crystal emerges between 3.0 T and 5.0 T. At higher fields (6.0–8.0 T), the skyrmions become progressively destabilized, and the system transitions into a uniformly magnetized ferromagnetic phase, with complete skyrmion annihilation occurring by 8.0 T. These findings were corroborated by temporal magnetization dynamics, which demonstrated faster and more complete saturation under stronger applied fields. While the present study does not evaluate the annihilation energy barrier, future work using minimum-energy path methods (e.g., NEB) could quantitatively assess their robustness against thermal or field-driven collapse. This study highlights the tunable nature of skyrmionic phases in Co₂C MXene and establishes it as a promising platform for two-dimensional spintronic devices. The robust DMI and controllable topological transitions make Co₂C particularly suitable for next-generation magnetic memory and logic applications. Future investigations could explore the effects of temperature, strain, and electric fields on skyrmion stability and mobility in Co₂C MXene, as well as the influence of flake size and edge termination. Additionally, integrating Co₂C with other 2D materials in heterostructures may offer new pathways for electric-field-controlled magnetism and topologically protected transport. Experimental validation and synthesis of skyrmion-hosting Co-based MXenes will be essential to bridge the gap between theoretical predictions and real-world device implementation.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data that support the findings of this study, including simulation input files, output data, and analysis scripts, are available from the corresponding author upon reasonable request.

Acknowledgements

The research reported in this publication was supported by Abdullah Al Salem University (AASU), Kuwait.

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