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Magnetic skyrmion lattice disclinations in pentagon- and heptagon-shaped FeGe crystals

This study reports the stabilization and characterization of five-fold and seven-fold magnetic skyrmion lattice disclinations in pentagon- and heptagon-shaped FeGe nanocrystals, combining experimental imaging with micromagnetic simulations and analytical models to explore these previously unexplored angular defects.

Original authors: Thibaud Denneulin, Nikolai S. Kiselev, Vladyslav M. Kuchkin, Rafal E. Dunin-Borkowski

Published 2026-02-17

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CC BY 4.0

Original authors: Thibaud Denneulin, Nikolai S. Kiselev, Vladyslav M. Kuchkin, Rafal E. Dunin-Borkowski

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 crowded dance floor where everyone is trying to hold hands in perfect circles. In the world of tiny magnetic particles called skyrmions, this is exactly what happens. Usually, these particles are picky dancers; they love to arrange themselves in perfect honeycombs (hexagons) because that's the most efficient way to pack them together.

But what happens if you force them to dance in a room shaped like a pentagon (5 sides) or a heptagon (7 sides)? You can't fit a perfect honeycomb into a 5-sided or 7-sided box without breaking the rules. This creates a "traffic jam" or a disclination—a specific type of defect where the perfect order is twisted.

Here is a simple breakdown of what the scientists in this paper discovered:

1. The Setup: Building a Magnetic "Playground"

The researchers took a chunk of a special magnetic material called FeGe (Iron Germanium). Using a super-precise laser-like tool called a Focused Ion Beam (think of it as a microscopic sculptor), they carved out tiny islands of this material shaped like pentagons and heptagons.

They then cooled these shapes down and applied a magnetic field. This woke up the skyrmions, causing them to swirl and organize themselves inside these geometric cages.

2. The Problem: The "Missing Slice" and the "Extra Slice"

To understand the magic, imagine a pizza:

The Perfect Hexagon: A normal skyrmion lattice is like a pizza cut into perfect 6-slice wedges.
The Pentagon (5-fold Disclination): Imagine you take a pizza, cut out one slice, and tape the remaining five slices together. You now have a pentagon shape, but the crust is stretched, and the center is pinched tight. This is a 5-fold disclination. The scientists found that in their pentagon-shaped FeGe, the skyrmions naturally formed this "missing slice" pattern. The center skyrmion was squeezed into a pentagon shape, and the ones around it were stretched out like rubber bands.
The Heptagon (7-fold Disclination): Now, imagine you have a pizza and you add an extra slice, forcing the circle to bulge. This creates a 7-fold disclination. Here, the center skyrmion gets pushed out and becomes slightly larger, while the ones around it get squished together.

3. The Discovery: Catching the Defects in Action

Usually, these defects are messy and hard to see because the skyrmions are constantly moving and rearranging themselves. However, by trapping them in these specific 5-sided and 7-sided shapes, the scientists "stabilized" the defects. They forced the skyrmions to stay in these twisted patterns.

They used a special microscope (Lorentz TEM) that acts like a camera for magnetic fields. It allowed them to take "photos" of the skyrmions and even measure how much they were stretched or squeezed. They found that the real-world images matched perfectly with their computer simulations, proving that the skyrmions were behaving exactly like elastic rubber sheets being bent into weird shapes.

4. The "Wobble" Effect

Here is the most fun part. The scientists noticed that when they had 17 skyrmions in a pentagon (one more than the perfect 16), the lattice became unstable. It was like a wobbly table with one leg slightly too long.

The Tilt: They tilted the sample slightly (just a few degrees).
The Switch: This tiny tilt acted like a nudge, causing the "extra" skyrmion to jump from one corner of the pentagon to another.
The Wobble: When they tilted it to a middle angle, the skyrmions couldn't decide which corner to sit in. They started "wobbling" back and forth between the two spots so fast that in the camera, they looked like a blurry, smeared ghost.

Why Does This Matter?

Think of skyrmions as the future of computer memory. Instead of bits being just "on" or "off," we could use these tiny magnetic swirls to store data.

Defects are usually bad: In normal electronics, a crack or a defect ruins the circuit.
Defects are useful here: This paper shows that we can engineer these defects on purpose. By shaping the material, we can create specific "traffic jams" (disclinations) that might be used to trap data, move information around, or act as switches in future super-fast, low-energy computers.

In a nutshell: The scientists built tiny magnetic pentagons and heptagons to force magnetic swirls into awkward shapes. They discovered that these shapes create stable "defects" that stretch and squeeze the magnetic particles, and they even found a way to make these particles wobble back and forth like a pendulum. It's a bit like teaching a dance troupe to perform a perfect routine in a room that's the wrong shape, and then using that awkwardness to create something new.

1. Problem Statement

Magnetic skyrmions in chiral magnets (such as FeGe) typically self-organize into hexagonal lattices. While translational defects like dislocations and grain boundaries in these lattices are well-studied, angular defects known as disclinations (specifically 5-fold and 7-fold) remain largely unexplored in the context of skyrmion solid phases.

The Challenge: Disclinations are topological defects where rotational symmetry is violated (e.g., removing or adding a wedge of material). In chiral magnets, observing these in the hexatic or solid phase is difficult because skyrmions are highly mobile under large external magnetic fields, causing defects to dissociate or move rapidly.
The Gap: There is a lack of experimental evidence regarding the stabilization, structural characteristics, and mobility of isolated 5-fold and 7-fold disclinations in skyrmion lattices.

2. Methodology

The researchers employed a combination of advanced nanofabrication, electron microscopy, and computational modeling:

Sample Fabrication:
- Material: Bulk FeGe single crystals.
- Technique: Focused Ion Beam (FIB) milling was used to pattern the crystals into specific pentagon (5-sided) and heptagon (7-sided) shapes.
- Protection: Carbon layers were deposited via Ion Beam-Induced Deposition (IBID) to protect the edges. A damaged layer of ~20 nm was acknowledged and accounted for in simulations.
- Geometry: Samples were thinned to ~150 nm for Transmission Electron Microscopy (TEM).
Experimental Characterization:
- Lorentz TEM (LTEM): Used Fresnel imaging to visualize magnetic domain structures at 220 K.
- Off-axis Electron Holography: Used to reconstruct the magnetic phase ( $\phi_{mag}$ ) and in-plane magnetic induction fields ( $B_\perp$ ) with high resolution, allowing for quantitative analysis of the skyrmion cores and inter-skyrmion distances.
- Field Cycling: External magnetic fields were cycled (0–210 mT) to nucleate skyrmions and observe the formation of specific lattice configurations.
- Tilt Experiments: The sample was tilted relative to the electron beam to introduce an in-plane magnetic field component, testing the stability and mobility of defects.
Theoretical Modeling:
- Micromagnetic Simulations: Performed using the Excalibur and Mumax3 codes. Parameters for FeGe (exchange, DMI, saturation magnetization) were adjusted for 220 K. Simulations included FIB-damaged layers and curved edges to match experimental conditions.
- Linear Elastic Theory: Analytical models based on isotropic linear elasticity were used to predict displacement fields and orientational order parameters for disclinations.
- Minimum Energy Path (MEP): Calculated using the Geodesic Nudged Elastic Band (GNEB) method to study the energy barriers between different defect configurations.

3. Key Contributions

First Stabilization of Skyrmion Disclinations: The study successfully stabilized isolated 5-fold and 7-fold disclinations in the skyrmion solid phase of FeGe by geometrically confining the lattice within pentagon and heptagon shapes.
Quantitative Structural Analysis: Provided the first detailed experimental mapping of the strain fields around skyrmion disclinations, confirming predictions from linear elasticity theory.
Defect Mobility Mechanism: Discovered and explained a "wobbling" phenomenon where skyrmions switch between energetically equivalent configurations when the rotational symmetry of the outer ring is broken by dislocations.

4. Key Results

A. Stabilization and Geometry

Pentagon (5-fold): A 5-fold disclination state corresponds to a specific number of skyrmions ( $S$ $S$ ). For a sample with 2 rings ( $N=2$ $N = 2$ ), the ideal state is $S=16$ $S = 16$ (1 central + 5 inner + 10 outer).
- Observation: The central skyrmion adopts a pentagonal shape and is slightly smaller due to compressive strain.
- Strain: Surrounding skyrmions are elongated along the circular direction (tangential) due to the removal of a $2\pi/6$ wedge.
Heptagon (7-fold): A 7-fold disclination requires $S=22$ $S = 22$ for $N=2$ $N = 2$ .
- Observation: The central skyrmion is slightly larger due to tensile strain from the addition of a wedge.
- Strain: Surrounding skyrmions are compressed along the circular direction and elongated radially.

B. Structural Metrics

Inter-skyrmion Distance:
- In 5-fold disclinations, the distance increases along the circular direction and decreases radially.
- In 7-fold disclinations, the distance decreases along the circular direction and increases radially.
Order Parameters: The local hexagonality ( $|\Psi_6|$ $∣ Ψ_{6} ∣$ ) and orientation ( $\theta$ $θ$ ) maps derived from experiments matched linear elastic theory predictions.
- 5-fold: $\langle|\Psi_6|\rangle \approx 0.81$ , orientation change $\Delta\theta \approx 12^\circ$ .
- 7-fold: $\langle|\Psi_6|\rangle \approx 0.85$ , orientation change $\Delta\theta \approx 9^\circ$ .

C. Defect Mobility and "Wobbling"

When the skyrmion count deviates from the perfect disclination number (e.g., $S=17$ in a pentagon), a 5-7 dislocation pair forms.
The 7-coordinated skyrmion can occupy any of the five corners, creating five geometrically equivalent energy states.
Tilt-Induced Switching: Tilting the sample introduces an in-plane field component that breaks the degeneracy, stabilizing one specific configuration.
Wobbling Effect: At zero tilt (intermediate angle), the energy barrier between configurations is minimal. Thermal energy at 220 K allows the system to fluctuate between states, causing the skyrmions in the outer ring to appear diffuse or "smeared" in Fresnel images. This effect vanishes at low temperatures where the system gets trapped in a local minimum.

5. Significance

Fundamental Physics: This work bridges the gap between crystallographic defect theory (disclinations in solids) and topological spin textures. It validates that skyrmion lattices behave as elastic media governed by linear elasticity theory, even at the nanoscale.
Defect Engineering: It demonstrates that geometric confinement is a powerful tool for engineering specific topological defects in chiral magnets, offering a route to control skyrmion lattice topology without complex external field protocols.
Spintronic Applications: Understanding the stability and mobility of disclinations is crucial for future spintronic devices. Disclinations could act as pinning centers for skyrmion motion or serve as stable topological bits. The ability to control defect switching via small field tilts suggests potential mechanisms for low-energy data manipulation.
Methodological Advance: The combination of FIB patterning, holography, and micromagnetic simulation provides a robust framework for studying topological defects in 2D magnetic systems.