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Skyrmions in 2D chiral magnets with noncollinear ground states stabilized by higher-order interactions

This study proposes and demonstrates, through first-principles calculations and simulations, that unconventional skyrmions can be stabilized in noncollinear ground states of Rh/Co and Pd/Co bilayers on Re(0001) via higher-order spin exchange interactions, offering new prospects for topological spintronics and hybrid systems.

Original authors: Mathews Benny, Moinak Ghosh, Moritz A. Goerzen, Bjarne Beyer, Hendrik Schrautzer, Stefan Heinze, Souvik Paul

Published 2026-02-04

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CC BY 4.0

Original authors: Mathews Benny, Moinak Ghosh, Moritz A. Goerzen, Bjarne Beyer, Hendrik Schrautzer, Stefan Heinze, Souvik Paul

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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

Imagine a vast, flat dance floor where thousands of tiny dancers (atoms) are holding hands and spinning. In most magnetic materials, these dancers are very orderly: they all face the same direction, like a synchronized marching band. This is called a "ferromagnetic" state.

However, in this new study, the researchers discovered a way to make these dancers form a much more complex, swirling pattern that doesn't just march in a straight line. They call this a "noncollinear" state, where the dancers face different directions in a specific, repeating pattern.

Here is the breakdown of their discovery using simple analogies:

1. The Unexpected Twist: Breaking the Rules

Usually, if you have a material made of Cobalt (Co)—which is famous for being a very strong, orderly magnet—you expect the dancers to stay in that straight-line formation. You wouldn't expect them to suddenly start dancing in a complex, swirling circle.

But the researchers found that in very thin layers of Cobalt sandwiched between other metals (like Rhodium or Palladium) on a specific surface, something strange happens. A hidden force, which they call "higher-order interactions," kicks in.

The Analogy: Think of the standard magnetic force as a rule that says, "Everyone must face North." The "higher-order interactions" are like a new, secret rule that says, "Actually, if you are standing next to two other specific people, you must face East, and the person next to you must face West."
The Result: This secret rule is so strong that it breaks the "face North" habit. Instead of a straight line, the dancers form a complex, non-straight pattern (the noncollinear ground state).

2. The New "Skyrmion" Dance Move

Once the dancers are in this complex, swirling pattern, the researchers found they could create a special, isolated move called a Skyrmion.

What is a Skyrmion? Imagine a whirlpool in a river. The water spins around a center point, but the water far away is calm. A skyrmion is a tiny, stable whirlpool of magnetic spins.
The Discovery: Usually, these whirlpools are found in the orderly "marching band" (ferromagnetic) materials. This paper shows that you can create these whirlpools inside the complex, swirling "noncollinear" dance floor.
The Surprise: It's like finding a stable whirlpool in the middle of a chaotic, swirling storm, rather than in a calm lake. The researchers call these "unconventional skyrmions."

3. Why Don't They Fall Apart? (The Energy Barrier)

You might wonder: "If the dancers are already in a complex pattern, why doesn't the whirlpool just collapse and disappear?"

The researchers used computer simulations to see how hard it is to destroy these whirlpools. They found that there is a massive "energy wall" protecting them.

The Analogy: Imagine the whirlpool is a marble sitting at the bottom of a deep, steep bowl. To get the marble out (to destroy the skyrmion), you have to push it all the way up the steep side of the bowl. It takes a lot of energy to do that.
The Finding: The "walls" around these new, unconventional whirlpools are just as high and steep as the ones around the old, standard whirlpools. This means they are very stable and won't just vanish on their own.

4. How Do We See This?

Since these patterns are happening at the atomic scale (smaller than a virus), you can't see them with a regular microscope. The researchers simulated what these patterns would look like using a special tool called SP-STM (Spin-Polarized Scanning Tunneling Microscopy).

The Visual: If you could take a picture of this atomic dance floor, the "noncollinear" background would look like a honeycomb pattern of bright and dark spots. The "skyrmions" would look like distinct, round blobs sitting on top of that honeycomb. The simulation shows that these patterns look very different from the standard magnetic patterns, making them easy to identify if an experiment is run.

Summary of the Claim

The paper claims that by using specific atomic layers (Rh/Co and Pd/Co on a Re surface), they can force a material that is usually a simple, straight-line magnet to become a complex, swirling magnet. Inside this complex swirl, they can create stable, isolated magnetic whirlpools (skyrmions) that are protected by high energy barriers.

They did not claim these are ready for use in computers or medical devices yet; they only claim that the physics allows these structures to exist and that they are stable enough that scientists should be able to find them in a lab using the right microscope.

Technical Summary: Skyrmions in 2D Chiral Magnets with Noncollinear Ground States Stabilized by Higher-Order Interactions

Problem Statement
Magnetic skyrmions are typically studied as topological spin textures on ferromagnetic (FM), antiferromagnetic, or ferrimagnetic backgrounds, stabilized by the competition between Heisenberg exchange, Dzyaloshinskii-Moriya interaction (DMI), and magnetocrystalline anisotropy (MAE). While skyrmions have been observed in bulk materials, transition-metal interfaces, and 2D van der Waals magnets, their existence on complex noncollinear magnetic ground states remains largely unexplored. Specifically, it is unclear whether metastable isolated skyrmions can emerge on noncollinear backgrounds stabilized by higher-order exchange interactions (HOI), particularly in systems containing prototypical ferromagnets like Cobalt (Co), where strong pairwise exchange usually dominates.

Methodology
The authors employed a multi-scale approach combining first-principles calculations and atomistic spin simulations:

First-Principles Calculations: Density Functional Theory (DFT) was used to calculate magnetic interaction parameters for Rh/Co and Pd/Co atomic bilayers on the Re(0001) surface. This included the Heisenberg pairwise exchange ( $J_{ij}$ ), DMI vectors ( $D_{ij}$ ), MAE ( $K$ ), and higher-order interactions: biquadratic ( $B_1$ ), three-site four-spin ( $Y_1$ ), and four-site four-spin ( $K_1$ ) terms. These HOI terms were derived from fourth-order perturbation theory of the Hubbard model.
Atomistic Spin Simulations: An extended spin Hamiltonian incorporating all calculated interactions was used to relax various spin configurations (homogeneous spin spirals, skyrmion lattices, and FM states) at zero and finite magnetic fields. The Landau-Lifshitz dynamics were combined with velocity projection optimization (VPO) to find energy minima.
Stability Analysis: The geodesic nudged elastic band (GNEB) method was utilized to calculate the minimum energy path (MEP) between isolated skyrmions and the ground state, determining annihilation and creation energy barriers.
Experimental Simulation: Spin-polarized scanning tunneling microscopy (SP-STM) images were simulated to predict experimental signatures of the proposed spin structures.

Key Contributions and Results

Stabilization of Noncollinear Ground States: The study demonstrates that in Rh/Co/Re(0001) and Pd/Co/Re(0001) bilayers, the standard FM ground state is destabilized by higher-order exchange interactions. Instead, a noncollinear ground state (nc-GS) emerges.
- For Rh/Co/Re(0001), the nc-GS becomes the ground state at a critical magnetic field of ~1.7 T.
- For Pd/Co/Re(0001), the nc-GS is the ground state even at zero magnetic field.
- This nc-GS is characterized by a magnetic unit cell containing three spins, where two align close to the out-of-plane direction and the third is inclined toward the in-plane direction. It is stabilized by the negative sign and specific magnitude of the three-site and four-site four-spin interactions, which compensate for the energy cost of deviating from the FM state.
Emergence of Unconventional Skyrmions: On this noncollinear background, the authors predict the existence of:
- Noncollinear Skyrmion Lattices (nc-SkX): Metastable lattices where skyrmions form on the nc-GS.
- Noncollinear Isolated Skyrmions (nc-ISk): Metastable isolated skyrmions with diameters ranging from ~10 nm down to smaller sizes as the magnetic field increases.
Topological Properties: Despite the complex background, these structures retain a topological charge of $Q=1$ . The authors show that the nc-GS can be decomposed into three ferromagnetic sublattices with different magnetization axes. The skyrmions on these sublattices are topologically protected, governed by the second homotopical group $\pi_2(S^2) = \mathbb{Z}$ .
Stability and Energy Barriers: Transition-state theory calculations reveal that these unconventional skyrmions are protected by significant energy barriers. For Rh/Co/Re(0001) at 2.4 T, the annihilation barrier is approximately 335 meV. This is comparable to barriers observed in conventional skyrmions on FM backgrounds (e.g., Pd/Fe/Ir(111)), suggesting they are stable enough to be observed experimentally.
Collapse Mechanism: The annihilation of these skyrmions follows a radial collapse mechanism similar to conventional skyrmions, though the energy decomposition along the MEP shows distinct contributions from DMI, exchange, and higher-order terms.
Phase Diagram Analysis: By varying the HOI constants relative to the effective exchange interaction, the authors constructed a phase diagram showing that negative values of the three-site and four-site four-spin interactions favor the noncollinear phase over the FM phase. This mechanism is not limited to the specific films studied but suggests a broader class of materials where such states could exist.

Significance
The paper claims that these findings extend the family of topological quasiparticles beyond conventional magnetic backgrounds. The primary significance lies in demonstrating that:

Prototypical Ferromagnets Can Host Noncollinear States: Even in Co-based systems, typically dominated by strong pairwise exchange, higher-order interactions can stabilize complex noncollinear ground states.
New Platform for Skyrmions: Unconventional skyrmions can emerge spontaneously on these noncollinear backgrounds, stabilized by the interplay of DMI and frustrated exchange interactions.
Experimental Feasibility: The large energy barriers and distinct SP-STM signatures (such as hexagonal arrangements with varying intensity and specific spin textures) suggest that these states are observable using current experimental techniques like SP-STM.
Potential Applications: The authors suggest these unconventional skyrmions could open new possibilities for topological spin transport or magnet-superconductor hybrid systems, though they do not propose specific device architectures.

The work provides a theoretical framework for identifying and stabilizing topological textures in materials where higher-order interactions play a dominant role, challenging the standard view that skyrmions require a simple ferromagnetic background.

1. The Unexpected Twist: Breaking the Rules

2. The New "Skyrmion" Dance Move

3. Why Don't They Fall Apart? (The Energy Barrier)

4. How Do We See This?

Summary of the Claim

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