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Magneto-cubic and magneto-linear dependence observed in an in-plane anomalous Hall magnet

This study elucidates the multipolar dependence of off-diagonal coupling in the in-plane anomalous Hall effect of trigonal EuCd2Sb2 thin films, revealing a magneto-cubic dependence in paramagnetic and antiferromagnetic phases alongside a dominant magneto-linear dependence in the forced ferromagnetic phase.

Original authors: Ayano Nakamura, Shinichi Nishihaya, Mitsuru Akaki, Motoi Kimata, Kenta Sudo, Yuki Deguchi, Hsiang Lee, Tadashi Yoneda, Masaki Kondo, Hiroaki Ishizuka, Masashi Tokunaga, Masaki Uchida

Published 2026-08-04

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CC BY 4.0

Original authors: Ayano Nakamura, Shinichi Nishihaya, Mitsuru Akaki, Motoi Kimata, Kenta Sudo, Yuki Deguchi, Hsiang Lee, Tadashi Yoneda, Masaki Kondo, Hiroaki Ishizuka, Masashi Tokunaga, 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 electricity flowing through a wire like a stream of tiny, invisible cars zooming down a highway. Usually, if you push these cars with a magnetic field, they get nudged to the side, creating a voltage difference. This is the classic "Hall effect," a bit like wind pushing a sailboat off its course. But in certain special materials, the cars don't just get pushed; they seem to have a mind of their own, swerving sideways even without a direct push. This is the "anomalous Hall effect," and scientists think it happens because the electrons are dancing to a weird, invisible rhythm called "Berry curvature"—a kind of geometric twist in the fabric of the material itself.

Now, picture this dance happening on a flat stage. Usually, if you shine a magnetic light from above (out-of-plane), the dancers move in a predictable way. But what happens if you shine that light from the side (in-plane)? In most materials, the dancers just ignore it or move in a simple, straight line. However, in a specific type of crystal with a triangular shape (called a trigonal system), the rules change. The dancers might start spinning in complex patterns, creating a voltage that depends on the cube of the magnetic push rather than just the push itself. Understanding these strange, sideways dances is crucial because it could help us build faster, smarter electronics that use the hidden geometry of atoms instead of just their charge.

This is exactly the mystery a team of researchers set out to solve in a new study. They focused on a material called EuCd2Sb2, a thin film crystal that acts like a tiny, magnetic playground for electrons. By carefully sliding a magnetic field across the flat surface of this crystal and measuring how the electrons responded, they discovered that the electrons' behavior changes dramatically depending on how strong the magnetic push is and how hot the material is.

Here is what they found: When the magnetic field is very weak (near zero), the electrons don't just move in a straight line. Instead, their sideways voltage grows with the cube of the magnetic field strength. It's as if the electrons are waiting for a triple-whammy of force before they decide to dance. This "magneto-cubic" behavior was surprisingly stubborn; it happened even when the material was cold and its internal magnetic spins were locked in an orderly, anti-parallel arrangement (an antiferromagnetic state). The researchers measured this clearly, finding that the signal followed a perfect cubic curve, not a simple straight line.

However, the story gets even more interesting when they turned up the heat or the magnetic field. Above a certain temperature (7.5 Kelvin), the material becomes a paramagnet (where spins are jumbled), and the cubic behavior remains, but it fades away incredibly fast as the temperature rises—dropping off roughly with the cube of the inverse temperature. It's like the dancers are so sensitive to the heat that their complex triple-step routine falls apart almost immediately.

But the real twist happens when the magnetic field gets strong enough to force the material into a "ferromagnetic" state, where all the spins line up in the same direction. Once the field passes a specific threshold of 1.8 Tesla, the rules flip completely. The cubic dance stops, and the electrons suddenly start moving in a simple, straight line that grows directly with the magnetic field strength. This "magneto-linear" behavior doesn't just happen for a moment; it persists all the way up to very high fields, reaching 24 Tesla in their experiments.

The researchers explain this switch by looking at the crystal's internal structure. In the weak-field state, the electrons are navigating a landscape with tiny, protected gaps (Dirac points) that only break open in a complex, cubic way when pushed. But once the magnetic field is strong enough to force the spins to align, it splits these gaps into distinct pairs (Weyl points) that respond much more simply and directly to the magnetic push.

In short, this paper maps out a fascinating journey where electrons in a magnetic crystal switch from performing a complex, triple-step dance to a simple, straight-line march, depending entirely on the strength of the magnetic field and the temperature. It confirms that the "off-diagonal" coupling—the way a side-pull creates a sideways voltage—isn't just a simple linear trick, but a rich, multi-layered phenomenon that changes its very nature as the material's internal order shifts. This discovery helps scientists understand the hidden geometric rules that govern how electricity moves in magnetic materials, paving the way for future devices that might one day harness these quirky quantum dances.

Technical Summary: Magneto-cubic and magneto-linear dependence observed in an in-plane anomalous Hall magnet

Problem and Context
The anomalous Hall effect (AHE) is a fundamental transport phenomenon in magnetic materials, traditionally understood as a manifestation of the Berry curvature of Bloch electrons arising from spin-orbit coupling. While conventional AHE is observed in systems with out-of-plane magnetization, recent interest has focused on the in-plane AHE, where the Hall voltage emerges within the plane defined by the magnetization and current. Unlike the conventional case, in-plane AHE does not necessarily require net spin magnetization perpendicular to the Hall deflection plane but is instead associated with pure orbital magnetization under specific symmetry conditions.

In trigonal systems, such as the antiferromagnet EuCd $_2$ Sb $_2$ , the emergence of in-plane AHE is constrained by magnetic point group symmetries. Theoretical considerations suggest that for systems belonging to point groups like $\bar{3}m$ (e.g., EuCd $_2$ Sb $_2$ and Fe $_3$ Sn $_2$ ), magneto-linear coupling ( $\propto B$ ) to the Hall resistivity $\rho_{yx}$ is prohibited by symmetry, making magneto-cubic coupling ( $\propto B^3$ ) the leading term. However, in magnetic phases where time-reversal symmetry is broken (such as the antiferromagnetic state), the leading order of coupling could theoretically shift. Previous observations confirmed a three-fold symmetric signal but left the detailed field dependence and the nature of the intrinsic off-diagonal coupling elusive.

Methodology
The study investigates the multipolar dependence of off-diagonal coupling in the in-plane AHE by systematically measuring EuCd $_2$ Sb $_2$ thin films. The samples are (001)-oriented single-crystalline thin films grown via molecular beam epitaxy on (111)A CdTe substrates. Transport measurements were conducted using a standard four-probe method on Hall bar devices.

Key experimental parameters included:

Temperature: Measurements spanned from 2 K up to temperatures well above the Néel temperature ( $T_N = 7.5$ K).
Magnetic Field: Fields were applied in-plane (within the (001) plane) and out-of-plane. Field sweeps were performed from zero up to 24 T.
Geometry: The magnetic field was rotated within the (001) plane, with the azimuthal angle $\phi$ measured from the [110] direction.
Analysis: The in-plane anomalous Hall resistivity ( $\rho_{yx, AHE}$ ) was extracted by isolating the three-fold symmetric component from the total signal. The field dependence around zero field was fitted to cubic and linear functions to extract response coefficients, specifically the octupolar tensor component ( $o_{zyyy}$ ) and the dipolar component ( $o_{zz}$ ).

Key Results

Magneto-cubic Dependence in PM and AFM Phases:
Around zero magnetic field, the in-plane anomalous Hall resistivity exhibits a clear magneto-cubic dependence ( $\rho_{yx, AHE} \propto B_y^3$ ). This behavior is observed not only in the paramagnetic (PM) phase but also persists in the antiferromagnetic (AFM) ground state. This finding indicates that the breaking of $C_3$ rotational symmetry by the AFM ordering has little effect on the leading order of the off-diagonal coupling in the in-plane AHE. The response is well-described by the octupolar tensor component $o_{zyyy}$ , formulated as $\rho_z = o_{zyyy}B_y^3$ .
Unconventional Temperature Decay:
The study analyzes the temperature dependence of the response coefficients. The octupolar component $o_{zyyy}$ (in-plane) decays significantly faster with increasing temperature than the dipolar component $o_{zz}$ (out-of-plane). Specifically, above the Néel temperature ( $T_N$ ), $o_{zyyy}$ follows a dependence roughly proportional to the inverse temperature to the third power ( $T^{-3}$ ), whereas $o_{zz}$ follows a linear inverse temperature dependence ( $T^{-1}$ ). This is consistent with a model where the in-plane AHE is induced by the coupling of the in-plane field to internal magnetization, which follows the Curie-Weiss law.
Magneto-linear Dependence in the Forced FM Phase:
In the forced ferromagnetic (FM) phase, achieved above the in-plane saturation field ( $B_{sat,in} = 1.8$ T), the behavior changes drastically. The in-plane AHE exhibits a dominant magneto-linear dependence ( $\propto B_y$ ) that persists up to very high fields (24 T). This linear increase continues even after the spin magnetization saturates, suggesting that the splitting of Weyl points along the $k_y$ and $k_z$ directions continues to evolve with the field, driving the linear response.

Significance and Claims
The authors claim that these findings clarify key aspects of the off-diagonal coupling in in-plane AHE, distinguishing it from conventional out-of-plane AHE and other magnetic Hall effects.

Symmetry and Coupling Order: The work demonstrates that in trigonal systems, the leading order of off-diagonal coupling is determined by the magnetic phase. While the AFM ground state retains the magneto-cubic dependence expected from the crystal symmetry, the forced FM phase allows for a magneto-linear term due to the specific alignment of spins and the resulting band structure evolution.
Orbital Magnetization Model: The observed behaviors—specifically the $B^3$ dependence in the AFM/PM phases and the $B^1$ dependence in the FM phase, along with the $T^{-3}$ temperature scaling—are consistent with the orbital magnetization model. This model posits that the in-plane AHE arises from the splitting of Weyl points and the induction of orbital magnetization by the in-plane magnetic field, mediated by relativistic cross terms (e.g., $k_y^n \sigma_z$ ) involving spin-orbit coupling.
Distinction from Altermagnets: The paper contrasts these results with "altermagnets" (anomalous Hall antiferromagnets without net magnetization). In altermagnets, the Hall response typically relies on small spin magnetization induced by spin-orbit coupling. In contrast, the in-plane AHE in EuCd $_2$ Sb $_2$ is driven by orbital magnetization induced by the field, allowing for a finite Hall response even when spin magnetization lies entirely within the Hall deflection plane.

The authors conclude that their demonstration of multipolar off-diagonal coupling provides a framework for understanding non-trivial Hall effects in generic systems, moving beyond conventional expectations of the Hall effect.

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