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Spontaneous in-plane anomalous Hall response observed in a ferromagnetic oxide

This study demonstrates that (111)-oriented ultrathin SrRuO $_3$ films exhibit a spontaneous, controllable in-plane anomalous Hall effect at zero magnetic field, driven by in-plane spin magnetization and modulated by the film's trigonal distortion.

Original authors: Shinichi Nishihaya, Yuta Matsuki, Haruto Kaminakamura, Yoshiya Murakami, Hiroaki Ishizuka, Masaki Uchida

Published 2026-08-04

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CC BY 4.0

Original authors: Shinichi Nishihaya, Yuta Matsuki, Haruto Kaminakamura, Yoshiya Murakami, Hiroaki Ishizuka, Masaki Uchida

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 world of electricity as a busy highway where tiny cars called electrons zoom along. Usually, when you push these cars with a magnetic field, they just speed up or slow down in a straight line. But sometimes, something magical happens: the magnetic field acts like a giant, invisible hand that swerves the cars sideways, creating a voltage across the road. This is called the Hall effect. For decades, scientists knew that if you pushed the cars with a magnetic field pointing straight up or down (perpendicular to the road), they would swerve sideways. This is the "out-of-plane" effect, and it's a well-known trick in the physics playbook.

However, a new puzzle has emerged. What happens if you push the cars with a magnetic field lying flat on the road, pointing sideways? In most materials, this flat push shouldn't make the cars swerve sideways at all; it should just push them forward. But recently, scientists found that in some special materials, a flat push does cause a sideways swerve. This is called the "in-plane" Hall effect. It's like pushing a car from the side and having it suddenly drift left or right on its own. This is exciting because it suggests a hidden connection between the direction the cars are spinning (their spin) and the invisible magnetic fields they carry (their orbit), offering a new way to control electricity without needing giant magnets.

Now, let's look at what this specific paper discovered. The researchers, led by Shinichi Nishihaya and Masaki Uchida, decided to test this idea on a very specific type of material: a thin film of a ferromagnetic oxide called Strontium Ruthenate ( $SrRuO_3$ ). Think of this material as a tiny, ultra-thin sheet of a hard, magnetic metal, grown on a crystal substrate. They cut this sheet so that the atoms were arranged in a specific pattern, like a triangular grid, which is crucial for the magic to happen.

The team found something surprising. When they applied a magnetic field flat against the sheet, the material generated a spontaneous sideways voltage even when the magnetic field was turned all the way down to zero. Usually, if you turn off the magnet, the swerving stops. But here, the material kept swerving on its own, as if it had a memory of the direction it was pushed. This "spontaneous" effect is a big deal because it means the material has an internal magnetic personality that doesn't need an external push to show off.

The researchers played with the direction of the magnetic field, spinning it around the flat sheet like a compass needle. They discovered that the sideways voltage changed in a very specific, three-sided pattern (like a triangle) as they rotated the field. This matched the triangular shape of the atoms in the crystal. More importantly, they proved that this effect wasn't caused by the magnetic field accidentally tilting the spins up and down. Instead, it was a direct, off-diagonal handshake between the spins lying flat and an invisible "orbital" magnetism pointing straight up and down.

To understand why the voltage behaved the way it did, the team looked at how the signal changed when they tilted the magnetic field up and down. They saw that the signal didn't just go up and down smoothly; it had bumps and dips that couldn't be explained by simple rules. They suggest that these wiggles are caused by "higher-order" effects—complex interactions allowed by the crystal's slight distortion. It's like a dance where the steps aren't just simple turns, but include fancy spins and dips that only happen because the dance floor (the crystal) is slightly uneven.

In short, this paper shows that in these special thin films, you can control a spontaneous sideways electrical current just by changing the direction of the magnetism lying flat on the surface. The researchers measured this effect at very low temperatures (2 Kelvin) and found it persisted up to 130 Kelvin. They ruled out the idea that the effect was just a simple tilt of the spins, confirming it is a genuine, intrinsic property of the material's orbital magnetism. While they used simulations to help explain the complex bumps in their data, the core discovery—that this spontaneous, controllable in-plane Hall effect exists in a hard ferromagnet—is a measured fact. This opens up a new playground for engineers who want to build devices that can switch electrical currents using the direction of magnetism, rather than just its strength.

Technical Summary: Spontaneous in-plane anomalous Hall response observed in a ferromagnetic oxide

Problem and Context
The manipulation of spin and orbital magnetization via magnetic fields is central to magnetotransport phenomena. While the conventional Hall effect arises from out-of-plane magnetic fields, recent observations have identified anomalous Hall effects (AHE) induced by magnetic fields applied within the Hall deflection plane (in-plane AHE). Unlike the diagonal Zeeman coupling, this in-plane AHE represents an off-diagonal response where a Hall vector component is generated perpendicular to the in-plane magnetic field.

Previous experimental studies on in-plane AHE have been limited to non-magnetic, antiferromagnetic, or soft-magnetic materials (e.g., Weyl semimetals Fe $_3$ Sn $_2$ and EuCd $_2$ Sb $_2$ ) where magnetic hysteresis is negligible. In these systems, the effect is understood as a consequence of band structure and quantum geometry modulations induced by the field, strictly following crystal symmetry. However, the existence of a spontaneous in-plane AHE in hard ferromagnets—specifically one that persists at zero magnetic field and is intrinsically coupled to in-plane spin magnetization—remains unexplored. A major challenge in this domain is distinguishing true off-diagonal coupling from artifacts caused by spin canting or reorientation, which can induce conventional out-of-plane AHE components.

Methodology
The authors investigated (111)-oriented ultrathin films of the ferromagnetic perovskite oxide SrRuO $_3$ (SRO), a material known to host Weyl point pairs near the Fermi level. The films (4.1 nm and 7.0 nm thick) were epitaxially grown on (111) SrTiO $_3$ substrates, resulting in a structure with threefold rotational symmetry about the out-of-plane [111] direction and trigonal distortion.

The study employed systematic magnetotransport measurements at low temperatures (down to 2 K) using a superconducting magnet and a sample rotator. The experimental protocol involved:

Field Scans: Sweeping magnetic fields in both out-of-plane and in-plane directions to measure Hall resistivity ( $\rho_{yx}$ ).
Azimuthal Angle ( $\phi$ ) Dependence: Rotating the in-plane magnetic field within the (111) plane to map the angular dependence of $\rho_{yx}$ at both high fields (9 T) and zero field (after cycling the field).
Polar Angle ( $\theta$ ) Dependence: Tilted the magnetic field from the out-of-plane [111] direction toward the in-plane direction to probe magnetic anisotropy and hysteresis behavior.
Data Processing: Raw Hall resistivity data were antisymmetrized to isolate the transverse component, defined as $\rho_{yx}(\phi) = (\rho_{yx,raw}(\phi) - \rho_{yx,raw}(\phi + 180^\circ))/2$ .

Key Results

Spontaneous In-Plane AHE: The (111) SRO ultrathin films exhibit a large spontaneous AHE at zero magnetic field when the field is applied in-plane. This response persists up to the Curie temperature ( $T_C \approx 130$ K) and displays a hysteresis loop characteristic of hard ferromagnets.
Coupling to In-Plane Magnetization: The films possess in-plane easy axes along the [11 $\bar{2}$ ] direction and its equivalents. The spontaneous AHE is directly coupled to this in-plane spin magnetization. Crucially, the sign of the AHE can be switched by controlling the direction of the in-plane magnetization, confirming that the effect is not an artifact of out-of-plane spin canting.
Symmetry and Angular Dependence:
- At high fields (9 T), the in-plane AHE exhibits a sinusoidal curve with three-fold symmetry as the field rotates, consistent with the trigonal distortion of the film.
- At zero field, the AHE displays a square-wave-like curve with three-fold symmetry, reflecting the magnetic anisotropy and the switching of the in-plane magnetization direction.
- Polar angle scans reveal a plateau structure in $\rho_{yx}$ near the in-plane direction, further confirming the in-plane easy axis.
Higher-Order Effects: The polar angle dependence at high fields shows non-monotonic behavior with local maxima and minima, particularly a sign change around the [11 $\bar{1}$ ] direction compared to [111]. This behavior cannot be explained by standard linear or simple cubic third-order terms ( $B^3$ ). The authors attribute this to higher-order terms allowed by the trigonal distortion of the film (e.g., terms involving products like $B_{[100]}B_{[010]}B_{[001]}$ ), which effectively generate an out-of-plane orbital magnetization component coupled to the in-plane field.

Key Contributions

Discovery of Spontaneous In-Plane AHE: The paper reports the first observation of a spontaneous in-plane AHE in a hard ferromagnet (SrRuO $_3$ ) that persists at zero field.
Orbital-Spin Coupling Mechanism: The study elucidates that this response is an off-diagonal coupling between in-plane spin magnetization and out-of-plane orbital ferromagnetism.
Symmetry Analysis: The work demonstrates how trigonal distortion in (111)-oriented films allows for specific higher-order terms in the Hall conductivity expansion, leading to complex angular dependencies and sign reversals not seen in ideal cubic systems.
Control of Hall Response: The authors show that the sign and magnitude of the in-plane Hall response can be controlled via the direction of the in-plane spin magnetization, offering a new degree of freedom for engineering Hall transport.

Significance
The paper claims that these findings highlight "versatile and controllable in-plane Hall responses with out-of-plane orbital ferromagnetism." By demonstrating that hard ferromagnets can host spontaneous off-diagonal Hall responses, the work broadens the horizons of Hall physics. It establishes that the Hall conduction in conventional ferromagnets can be essentially controlled by the in-plane magnetic field and its history, providing a platform for engineering Hall responses through symmetry and magnetic anisotropy without relying on the negligible hysteresis found in soft magnets or topological semimetals. The results suggest that the interplay between crystal symmetry, magnetic anisotropy, and higher-order orbital effects is critical for realizing and understanding spontaneous in-plane AHE.

Technical Summary: Spontaneous in-plane anomalous Hall response observed in a ferromagnetic oxide

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