🔬 materials science

Emergent Skyrmion Hall Effect in $d$ -wave Altermagnets at Finite Temperature

This paper demonstrates that while $d$ -wave altermagnetic skyrmions exhibit Hall-free motion in the absence of thermal fluctuations, finite temperatures induce a unique, anisotropic Skyrmion Hall effect driven by symmetry-dependent magnon scattering, enabling electrically controlled and thermally tunable transport.

Original authors: Tingting Liu, Bingyu Sun, Fengyue Zhu, Zhihui Zhang, Peiyu Zhang, Yang Liu, Minghui Qin

Published 2026-08-28

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CC BY 4.0

Original authors: Tingting Liu, Bingyu Sun, Fengyue Zhu, Zhihui Zhang, Peiyu Zhang, Yang Liu, Minghui Qin

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

In the quest to build faster, more efficient computers, scientists have long looked to the tiny magnetic swirls found inside certain materials. These swirls, called skyrmions, act like stable, particle-like knots of magnetism that can be pushed around to carry information. For decades, researchers have focused on two main types of magnetic materials: ferromagnets, where all the tiny atomic magnets point in the same direction, and antiferromagnets, where neighboring magnets point in opposite directions, canceling each other out. A newer, third category has recently emerged, known as altermagnets. These materials possess the best of both worlds: like antiferromagnets, their internal magnetic fields cancel out so they do not interfere with nearby electronics, but like ferromagnets, they exhibit a unique, directional response to electricity that makes them excellent candidates for next-generation data storage. The challenge has been figuring out how to move these skyrmions efficiently without using electric currents, which generate heat and waste energy.

A team of researchers has now discovered a way to guide these magnetic swirls using only electric fields, and they have uncovered a surprising twist that depends on temperature. By applying a voltage across a specific type of altermagnet, they created a gradient, or a slope, in the material's magnetic properties. This slope acts like a gentle hill, causing the skyrmions to roll toward areas of lower magnetic energy. In a perfectly cold environment, these magnetic swirls move in a perfectly straight line, following the slope without veering off to the side. This is a significant finding because, in most other magnetic materials, moving objects tend to drift sideways, a phenomenon known as the Hall effect, which makes precise control difficult. The researchers found that in these altermagnets, the opposing magnetic forces within the material cancel out this sideways drift, allowing for straight, efficient motion.

However, the story changes when the material is warmed up, even just slightly. In the real world, materials are rarely at absolute zero, and heat causes the atoms to vibrate, creating waves of magnetic energy called magnons. The researchers found that these heat-induced waves interact with the moving skyrmions in a complex way. As the skyrmion moves, it scatters these waves, and this scattering creates a new force that pushes the skyrmion sideways. Crucially, the direction of this sideways push depends entirely on which way the skyrmion is traveling. If the skyrmion is pushed along one specific axis of the crystal, the heat waves nudge it to the right. If it is pushed along a perpendicular axis, the same heat waves nudge it to the left. This happens because the internal structure of the altermagnet treats magnetic waves traveling in different directions differently, effectively swapping the influence of the waves depending on the path taken.

The team confirmed these findings using detailed computer simulations that modeled the behavior of individual atoms and the magnetic fields between them. They observed that as the temperature rose from absolute zero to just 0.1 Kelvin, the skyrmions began to drift sideways, but the direction of that drift flipped when they changed the direction of the driving force. There was one exception to this rule: when the skyrmion was pushed along a specific diagonal line through the crystal, the sideways forces from the heat waves canceled each other out perfectly. In this specific direction, the skyrmion continued to move in a straight line, even at higher temperatures. This behavior reveals a hidden symmetry in the material that protects the straight-line motion under certain conditions.

These results suggest that altermagnets offer a unique platform for controlling magnetic information carriers. The ability to steer skyrmions with electric fields avoids the energy waste of electric currents, while the temperature-dependent sideways drift offers a new way to tune the movement of these particles. The researchers propose that this directional sensitivity to heat could serve as a fingerprint for identifying altermagnetic materials and could be used to design devices where the path of information is controlled by both voltage and temperature. By understanding how these magnetic swirls interact with the thermal vibrations of the material, scientists are opening a path toward low-power, highly controllable spintronic devices that operate efficiently even in the presence of heat.

Technical Summary: Emergent Skyrmion Hall Effect in d-wave Altermagnets at Finite Temperature

Problem Statement
Altermagnets represent a new class of magnetic materials combining compensated collinear magnetic order with unconventional, symmetry-dependent spin splitting. While they offer advantages for spintronics—such as ultrafast dynamics and robustness against external perturbations—controlling their topological spin textures (skyrmions) remains challenging. Conventional current-driven manipulation (e.g., spin-transfer torque) is often inefficient due to the insulating or semiconducting nature of candidate altermagnets. Furthermore, while voltage-controlled magnetic anisotropy (VCMA) offers a current-free driving mechanism, the dynamical behavior of altermagnetic (ATM) skyrmions under VCMA gradients, particularly in the presence of thermal fluctuations, is not fully understood. Specifically, it is unclear how the unique, anisotropic magnon spectra of altermagnets influence skyrmion transport at finite temperatures, and whether the "Hall-free" motion characteristic of compensated magnets at zero temperature persists when thermal magnons are present.

Methodology
The authors employ a dual approach combining collective-coordinate theory and atomistic spin simulations to investigate ATM skyrmion dynamics.

Theoretical Framework: They utilize a two-sublattice altermagnetic model with anisotropic intrasublattice exchange interactions and interfacial Dzyaloshinskii-Moriya interaction (DMI). The system is driven by a spatially varying magnetic anisotropy gradient ( $K(\xi)$ $K (ξ)$ ), simulating a VCMA device.
- At zero temperature, they derive equations of motion for the skyrmion's guiding center, incorporating the magnetic quadrupole induced by $C_4T$ symmetry.
- At finite temperatures, they introduce a thermal magnon bath. The interaction is modeled via magnon–skyrmion scattering, resolving the magnon spectrum into two handedness branches ( $\nu = \pm$ ). They calculate the transverse force arising from handedness-dependent skew scattering, which depends on the anisotropic dispersion and group velocities of the magnons.
Numerical Simulations: The theoretical predictions are verified using atomistic spin simulations based on the stochastic Landau–Lifshitz–Gilbert (sLLG) equation. These simulations incorporate thermal noise to model finite-temperature effects and track the skyrmion trajectory under various anisotropy gradients and temperatures.

Key Contributions and Results
The study establishes the transport characteristics of ATM skyrmions driven by VCMA gradients across different temperature regimes:

Zero-Temperature Dynamics: In the absence of thermal fluctuations, the strongly compensated gyrotropic response of the two magnetic sublattices leads to nearly rectilinear motion. The skyrmion moves toward regions of reduced magnetic anisotropy with negligible transverse drift (Hall angle). The longitudinal velocity is linearly proportional to the anisotropy gradient ( $\Delta K$ ) and inversely proportional to the Gilbert damping ( $\alpha$ ), while being tunable by the background anisotropy ( $K_0$ ).
Finite-Temperature Dynamics (Anisotropic Hall Effect): Thermal excitations introduce two distinct effects:
1. Longitudinal Drag: Magnon–skyrmion scattering increases dissipation, reducing the longitudinal velocity as temperature rises.
2. Transverse Drift: Asymmetric skew scattering of thermally excited magnons generates a transverse reaction force. Crucially, due to the anisotropic magnon band structure of d-wave altermagnets, the relative transport weights of the two magnon handednesses depend on the propagation direction.
  - Along the $x$ and $y$ axes: The dispersion and group velocities of the right- and left-handed magnon branches differ. This leads to unequal thermal weights and incomplete cancellation of transverse forces, resulting in a net transverse drift. Notably, the sign of this drift reverses when the driving direction switches from $x$ to $y$ , as the relative dominance of the magnon handednesses inverts.
  - Along the High-Symmetry ( $\Gamma \to M$ ) Direction: The two magnon branches remain degenerate. Consequently, their transverse contributions cancel exactly, preserving Hall-free motion even at finite temperatures.
Validation: Atomistic simulations confirm the theoretical predictions, showing linear velocity dependence on the gradient at $T=0$ and the emergence of direction-dependent transverse drifts at finite $T$ that vanish along the high-symmetry axis.

Significance
The paper identifies a "thermally emergent anisotropic skyrmion Hall effect" as a generic finite-temperature dynamical feature of ATM skyrmions. The direction-dependent magnitude and sign of this effect originate directly from the intrinsic symmetry-dependent magnon spectrum of the altermagnet. This finding suggests that the anisotropic Hall response can serve as a dynamical signature for identifying altermagnetic order. Furthermore, the work establishes a low-power route for electrically controlled and thermally tunable ATM skyrmion transport, leveraging VCMA gradients to manipulate skyrmions without the energy costs associated with Joule heating from electrical currents.

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