🔬 materials science

Intra-unit-cell resolved intertwining of multi- $Q$ charge and spin textures in an itinerant skyrmion magnet

This study utilizes atomically resolved imaging and numerical modeling to demonstrate that in the itinerant skyrmion magnet GdRu $_2$ Ge $_2$ , the alignment of Gd 4 $f$ spins directly dictates the multi- $Q$ charge texture of Ru 4 $d$ orbitals, revealing a microscopic intertwining of spin and charge degrees of freedom across multiple magnetic phases.

Original authors: Christopher J. Butler, Katsuki Nihongi, Haruto Yoshimochi, Nguyen Duy Khanh, Rina Takagi, Tetsuo Hanaguri, Shinichiro Seki

Published 2026-08-07

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CC BY 4.0

Original authors: Christopher J. Butler, Katsuki Nihongi, Haruto Yoshimochi, Nguyen Duy Khanh, Rina Takagi, Tetsuo Hanaguri, Shinichiro Seki

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 the inside of a magnet not as a solid, unchanging block, but as a bustling city where tiny, invisible magnets called "spins" live on a grid. Usually, these spins like to line up in neat, straight rows, all pointing the same way, like soldiers marching in formation. But sometimes, they get creative and start dancing in complex, swirling patterns called "skyrmions." These are like tiny, stable tornadoes of magnetism that can't be easily unraveled. Scientists have long wondered how these dances happen in materials that look perfectly symmetrical, where the usual rules for making them twist shouldn't apply. The big mystery is: what is the invisible conductor telling the spins how to move? Is it just the spins talking to each other, or is there a hidden crowd of wandering electrons acting as the middleman, passing notes and instructions back and forth to create these intricate patterns? Understanding this is like figuring out the secret recipe for a complex dish; if we can see how the ingredients mix, we might one day build better, faster, and more efficient computers that use magnetism instead of electricity.

In this study, researchers took a close-up look at a special crystal called GdRu2Ge2 to solve this mystery. They used a super-powerful microscope called a Scanning Tunneling Microscope (STM), which acts like a giant, sensitive finger that can feel the surface of a material atom by atom. Instead of just looking at the magnetic spins, they looked at the "local density of states" (LDOS), which is essentially a map of where the wandering electrons like to hang out. They found that as the magnetic spins changed their dance steps into five different complex patterns (including two types of skyrmion crystals), the electron clouds changed right along with them. It's as if the spins and the electrons were wearing matching outfits, twisting and turning in perfect sync.

The team discovered that the electrons didn't just react to the spins; they seemed to be intimately linked to the specific way the spins were aligned with their neighbors. By creating a simple computer model, they showed that the electron patterns could be predicted just by knowing how the neighboring spins were pointing. This suggests that the "itinerant" (wandering) electrons are the key glue holding these magnetic structures together. Interestingly, they ruled out a few other ideas, such as the electrons simply getting pushed around by the magnetic field or the spins interacting in a way that only happens at the very center of the atoms. Instead, the data points to a relationship where the electrons living on the "roads" (bonds) between the atoms are sensitive to the angle of the spins on either side.

The researchers observed these changes at extremely low temperatures, around 1.5 Kelvin, and applied magnetic fields ranging from 0 to 6 Tesla to switch the material between its different phases. They saw that in some phases, the electron patterns looked like a "basketweave," with stripes crossing over each other, while in others, they looked like a checkerboard. The paper suggests that these patterns are a direct result of the spins' arrangement, mediated by the Ru 4d orbitals (a specific type of electron cloud around the Ruthenium atoms). While the model they built successfully recreated the observed patterns, the authors note that there are still some small differences between their simulation and the real measurements, indicating that the full story is still being written. However, the core finding is clear: in this material, the dance of the spins and the flow of the electrons are so tightly intertwined that you can't have one without the other, offering a new, microscopic view of how complex magnetism is born.

Technical Summary: Intra-unit-cell resolved intertwining of multi-Q charge and spin textures in an itinerant skyrmion magnet

Problem Statement
The mechanisms stabilizing non-collinear magnetic order, particularly skyrmion crystals, in centrosymmetric crystals remain unclear. While the Dzyaloshinskii-Moriya interaction (DMI) stabilizes skyrmions in chiral systems, its absence in centrosymmetric materials like $GdRu_2Ge_2$ suggests alternative stabilization mechanisms, likely involving spin-spin interactions mediated by itinerant electrons (e.g., RKKY interaction). Although finite-Q magnetic order is expected to induce electronic modulations, a detailed, microscopic relationship between the charge (electronic) and spin degrees of freedom in these non-collinear itinerant magnets has not been established. Specifically, it is unknown how the alignment of localized $Gd$ 4f spins translates into specific textures within the itinerant $Ru$ 4d orbital bands that mediate these interactions.

Methodology
The study utilizes Scanning Tunneling Microscopy (STM) to probe the local density of states (LDOS) at the cleaved Ge-terminated surface of single-crystal $GdRu_2Ge_2$ . This surface provides direct access to the $Ru$-Ge conduction network where magnetic interactions are mediated.

Experimental Conditions: Measurements were performed at $T = 1.5$ K across five successive magnetic phases (I–V), including two skyrmion crystal phases, by varying the magnetic field ( $\mu_0H$ ).
Data Acquisition: The authors acquired topographic images and normalized tunneling conductance maps, $L(r, V)$ , which serve as a proxy for the surface LDOS at $E = E_F + eV$ .
Signal Processing: To resolve the complex periodicities of multi-Q textures, the authors employed a Non-Uniform Discrete Fourier Transform (NUDFT). This allowed for the precise extraction of amplitude and phase information for a set of 42 specific wavevectors, including Bragg peaks, magnetic modulation vectors ( $Q_A, Q_B$ ), their harmonics, and satellite peaks.
Modeling: A numerical modeling scheme was developed to link the observed LDOS textures to the underlying spin structures. Ruling out Tunneling Magnetoresistance (TMR) and Tunneling Anisotropic Magnetoresistance (TAMR) effects, the authors adopted a Non-Collinear Magnetoresistance (NCMR) framework. They modeled the LDOS as a function of nearest-neighbor Gd spin alignments (spin-spin dot products) convolved with bond-centered kernel functions representing the $Ru$ 4d orbitals.

Key Results

Observation of Multi-Q Electronic Textures: In all five magnetic phases, distinct periodic LDOS patterns were observed. Fast Fourier Transforms (FFTs) revealed peaks corresponding to the magnetic modulation wavevectors $Q_A, Q_B$ and their second harmonics, confirming that the multi-Q magnetic order entails a corresponding multi-Q electronic texture.
Intra-unit-cell Resolution: The high-resolution STM images revealed that the electronic textures possess specific motifs within the magnetic unit cell. Notably, Phase II (an elliptical skyrmion crystal) exhibits a "basketweave" pattern composed of bond-order-like stripe components modulated in anti-phase. This pattern breaks the nominal $C_{4v}$ lattice symmetry at the bond level.
Spin-Charge Correlation: The modeling results demonstrate a tight correlation between the alignment of nearest-neighbor Gd spins and the local LDOS of the Ru 4d orbitals. Specifically, the LDOS intensity at bond-centered sublattices (where Ru ions reside) is governed by the spin-spin dot products of the adjacent Gd ions.
Revised Ground State: The observation of unexpected wavevectors in Phase I (the ground state) motivated a revision of the previously proposed spin screw structure. The data suggests the ground state includes an additional modulation perpendicular to the spin screw, aligning more closely with recent findings in the sister compound $GdRu_2Si_2$ .
Model Validation: The simple numerical model, based on spin-spin dot products and anisotropic bond kernels, successfully reproduced the qualitative features and Fourier spectra of the observed LDOS patterns across all five phases, including the complex basketweave texture and checkerboard-like patterns.

Significance and Claims
The paper claims to provide a microscopic view of the atomic-scale intertwining of charge and spin degrees of freedom in non-collinear itinerant magnets. By resolving the LDOS motifs at the bond-centered sublattices of the Gd square net, the authors establish a concrete relationship: the local electronic structure is directly dictated by the relative alignment of neighboring localized spins. This finding supports the paradigm that itinerant electrons (specifically Ru 4d-derived bands) mediate the magnetic interactions in centrosymmetric skyrmion magnets. The work moves beyond phenomenological descriptions to offer a mechanism where the electronic supermodulations are a direct consequence of the non-collinear spin texture, thereby clarifying the feedback loop between electronic reconstruction and magnetic stabilization.

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