🔬 mesoscale physics

Sub-20 nm spin textures with arbitrary topological charge stabilized by higher-order interactions

This study predicts the emergence of thermally stable, sub-20 nm high-topological-charge spin textures in Janus monolayers, demonstrating that higher-order spin interactions are essential for stabilizing these complex structures and preventing their collapse into a ferromagnetic state.

Original authors: Zhen Sun, Shiwei Zhu, Moritz A. Goerzen, Megha Arya, Changsheng Song, Dongzhe Li

Published 2026-09-29

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CC0 1.0

Original authors: Zhen Sun, Shiwei Zhu, Moritz A. Goerzen, Megha Arya, Changsheng Song, Dongzhe Li

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of modern electronics, scientists are constantly searching for ways to store and move information more efficiently. One promising avenue involves tiny, stable swirls of magnetism called skyrmions. Imagine a magnetic field not as a uniform sheet, but as a collection of tiny, self-contained whirlpools. Each whirlpool carries a specific "charge," a number that defines its structure and how it behaves. For years, researchers have focused on creating these whirlpools with a charge of one or minus one. However, a new frontier has opened up: creating whirlpools with much larger charges. These high-charge structures could hold more data in a smaller space, but they have been notoriously difficult to create and keep stable, often requiring massive sizes or complex, artificial setups that are hard to build.

A team of researchers has now predicted a way to generate and stabilize these complex magnetic shapes at a scale small enough for practical use. By simulating a specific type of two-dimensional magnetic material, they discovered that a previously overlooked force within the material's atoms is the key to holding these intricate structures together. Their work suggests that it is possible to create magnetic whirlpools with charges ranging from two to five, all while keeping them smaller than twenty nanometers. This finding challenges the long-held belief that such complex shapes require specific, often unstable conditions, and instead points to a natural interaction within certain materials that can support them.

The researchers focused their study on a material known as a Janus monolayer, specifically a single layer of manganese, phosphorus, chlorine, and iodine atoms arranged in a triangular pattern. While this material has been studied before, the team took a closer look at the forces acting between its atoms. Standard models usually account for how neighboring atoms align their magnetic spins, but this team included a more complex force known as a higher-order interaction. This force involves groups of four atoms acting together, rather than just pairs. Using powerful computer simulations based on the laws of quantum mechanics, they mapped out how these atoms would behave under different conditions.

The results were striking. When the team included these four-atom interactions in their model, the material spontaneously formed stable magnetic whirlpools with high charges. They observed shapes that looked like rings of skyrmions, where multiple swirls formed a circle, and "bags" where smaller swirls were trapped inside a larger outer ring. These structures were remarkably small, with diameters ranging from about one to eight nanometers, well below the twenty-nanometer threshold needed for next-generation devices. Crucially, these shapes remained stable even with a significantly reduced external magnetic field influence, a feature that is essential for energy-efficient electronics.

One of the most significant findings was how these structures stayed intact. The researchers found that the four-atom interaction acted as a stabilizing glue. Without it, the complex shapes would quickly collapse into a simple, uniform magnetic state. This interaction worked by creating a kind of internal tension that favored the swirling, non-uniform patterns over the flat, uniform ones. It was so effective that it could stabilize these shapes even when the material's common chiral force, known as the Dzyaloshinskii-Moriya interaction (DMI), was reduced to just 5% of its natural strength. This means that materials with symmetrical structures, which were previously thought unsuitable for such tasks, could potentially host these complex textures, provided this specific higher-order interaction is strong enough.

The team also investigated how these high-charge structures would eventually disappear, a process known as collapse. They discovered that the way these shapes fall apart depends on whether their charge is an even or odd number. For shapes with an even charge, the collapse happens through a specific mechanism where a small part of the swirl reverses its direction locally. For shapes with an odd charge, a different process occurs: a small, isolated swirl breaks off from the main structure and carries away one unit of charge before the rest of the shape collapses. This discovery of two distinct decay pathways, which had not been reported before, provides a deeper understanding of the stability limits of these magnetic textures.

The study also looked at the energy required to create these shapes and the energy needed to destroy them. The simulations showed that the four-atom interaction significantly raised the energy barrier needed to collapse the structures, making them much more robust against thermal fluctuations. While the size of the whirlpools remained largely unchanged by this interaction, their stability increased dramatically. This suggests that by tuning the strength of these higher-order interactions, scientists could potentially design materials that host stable, high-charge magnetic states at room temperature.

The researchers identified that the key to this stability lies in the specific arrangement of atoms in the Janus monolayer, which naturally produces a strong four-atom interaction. They noted that while their simulations focused on one specific material, the principles they uncovered could apply to a broader family of similar materials. This opens the door to searching for new materials that naturally possess these strong interactions, rather than trying to engineer them artificially. The ability to create and control these high-charge states in such small, stable packages could revolutionize how information is stored and processed, offering a path toward devices that are faster, denser, and more energy-efficient than anything currently available.

In the broader context of magnetic research, this work shifts the focus from relying on external fields or complex layering to utilizing the intrinsic quantum forces within a material. By demonstrating that higher-order interactions can stabilize complex topological textures, the study provides a new blueprint for material design. It suggests that the future of spintronics may not lie in building more complex structures from the outside, but in understanding and harnessing the subtle, collective behaviors of atoms from the inside out. The predictions made in this study await experimental verification, but they offer a compelling vision of what might be possible when we look beyond the standard rules of magnetic interaction.

Technical Summary: Sub-20 nm Spin Textures with Arbitrary Topological Charge Stabilized by Higher-Order Interactions

Problem Statement
Magnetic topological solitons, characterized by an integer topological charge $Q$ , are central to modern magnetism and spintronics. While magnetic skyrmions ( $|Q|=1$ ) are well-studied, high- $Q$ textures ( $|Q| > 1$ ) such as skyrmion bundles, rings, and bags have typically been observed with lateral dimensions ranging from hundreds of nanometers to micrometers. For practical spintronic applications, there is a critical need for high- $Q$ states that simultaneously possess thermal stability near room temperature, sub-20 nm diameters, and efficient external control. Existing materials fail to meet all three criteria. Furthermore, while the coexistence of skyrmions and antiskyrmions has been reported in inversion-symmetric magnets, extending this to high- $Q$ textures with arbitrary charge remains elusive. The role of higher-order interactions (HOI) in stabilizing these specific states and their collapse mechanisms has remained largely unexplored.

Methodology
The authors employ an atomistic spin model parameterized by density functional theory (DFT) calculations to investigate the 2D van der Waals (vdW) Janus monolayer MnPCl. The magnetic Hamiltonian extends beyond the conventional bilinear Heisenberg exchange to include:

Standard Terms: Heisenberg exchange ( $J_{ij}$ ), Dzyaloshinskii-Moriya interaction (DMI, $D_{ij}$ ), magnetocrystalline anisotropy ( $K_u$ ), and Zeeman coupling.
Higher-Order Interactions (HOI): Biquadratic ( $B_1$ ), four-spin three-site ( $Y_1$ ), and four-spin four-site ( $K_1$ ) terms.

The study utilizes the SPINAKER code for atomistic spin simulations on $120 \times 120$ lattices with periodic boundary conditions. To analyze stability, the authors calculate Minimum Energy Paths (MEPs) using the geodesic nudged elastic band (GNEB) method. This allows for the determination of collapse energy barriers ( $\Delta E$ ) from metastable soliton states to the ferromagnetic (FM) state, as well as nucleation barriers ( $\Delta E_{fin \to ini}$ ).

Key Contributions and Results

Stabilization of Sub-20 nm High- $Q$ Textures: The study predicts the emergence of sub-20 nm high- $Q$ textures in MnPCl, including skyrmion rings ( $Q = -2$ to $-5$), antiskyrmion rings ( $Q = 2$ to $5$), and skyrmion bags ( $Q = 1$ to $4$). These textures are stabilized even in the absence of significant DMI, provided HOI are present.
Critical Role of the Four-Spin Three-Site Interaction ( $Y_1$ ): The authors identify the four-spin three-site interaction ( $Y_1$ $Y_{1}$ ) as the key ingredient for stabilizing high- $Q$ $Q$ textures. In MnPCl, $Y_1$ $Y_{1}$ is large and negative ($-3.13$ meV). This term introduces effective exchange frustration that favors non-collinear spin alignments.
- Energy Barriers: HOI substantially enhance the collapse energy barriers ( $\Delta E$ ) for both skyrmions and antiskyrmions (e.g., increasing the antiskyrmion barrier from 23 meV to 122 meV at 5 T) while leaving their physical size nearly unchanged.
- Mechanism: The enhancement arises because the negative $Y_1$ raises the energy of the saddle point (SP) configuration, where large-angle spin rotations occur locally at the soliton core. This creates a localized energy barrier rather than a global modification of the energy landscape.
Skyrmion Bags: Unlike rings, skyrmion bags (where an outer shell encloses inner skyrmions) nucleate only in the presence of HOI. Without HOI, these states collapse immediately. The four-spin three-site interaction prevents high- $Q$ textures from collapsing into the ferromagnetic state.
Parity-Dependent Collapse Mechanisms: The paper identifies previously unreported collapse mechanisms for high- $Q$ $Q$ textures that depend on the parity of $Q$ $Q$ :
- Rings: For even $Q$ , rings undergo a "chimera" collapse where the topological charge reverses locally within the ring wall. For odd $Q$ , a novel "ejection collapse" occurs, where a localized soliton detaches from the ring, carrying away one unit of topological charge.
- Bags: For odd $Q$ , two inner skyrmions merge and annihilate ("combining collapse"). For even $Q$ , an inner skyrmion couples to the outer boundary, triggering a local reversal at a corner kink ("kink collapse").
Material Scope: While MnPCl serves as the primary example, the phase diagrams suggest that other Mn- and Cr-based Janus monolayers (e.g., CrSeCl, MnPBr) may also host these states. The study notes that Cr-based Janus monolayers generally appear less promising for this specific stabilization mechanism compared to Mn-based ones.

Significance and Claims
The paper establishes higher-order interactions, specifically the four-spin three-site interaction, as an overlooked but essential mechanism for the nucleation and stabilization of high- $Q$ topological spin textures. The findings demonstrate that:

Arbitrary Charge: HOI enable the stabilization of spin textures with arbitrary topological charge $Q$ , overcoming the limitations of DMI which typically restricts stability to $|Q|=1$ .
Size-Stability Trade-off: The mechanism breaks the conventional size-stability limitation, offering sub-20 nm textures with high thermal stability (barriers up to ~228 meV for skyrmions and ~170 meV for bags with $Q \ge 1$ ).
DMI Independence: These states can be stabilized even in systems with inversion symmetry or in the absence of DMI, broadening the search for suitable materials beyond non-centrosymmetric compounds.
Experimental Feasibility: The predicted textures and their specific collapse mechanisms (ejection, chimera, kink, combining) are proposed to be testable via Spin-Polarized Scanning Tunneling Microscopy (SP-STM) or Magnetic Force Microscopy (MFM).

The authors conclude that these high- $Q$ states are promising candidates for high-density spintronic devices due to their compact size, large energy barriers, and the potential for tuning topological charge via external stimuli. They also note that while the large barriers suggest increased thermal lifetimes, a quantitative assessment of room-temperature stability requires further calculation of the pre-exponential factor ( $\Gamma_0$ ), which remains a direction for future work.

Technical Summary: Sub-20 nm Spin Textures with Arbitrary Topological Charge Stabilized by Higher-Order Interactions

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