🔬 mesoscale physics

Extrinsic anomalous Hall effect in altermagnets

This paper reveals that extrinsic anomalous Hall conductivity can be comparable to or even essential for the intrinsic contribution in altermagnets, a phenomenon driven by the nonanalytic dependence of the intrinsic effect on spin-orbit coupling and the lifting of spin degeneracy along nodal planes.

Original authors: A. Osin, A. Levchenko, M. Khodas

Published 2026-08-10

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CC BY 4.0

Original authors: A. Osin, A. Levchenko, M. Khodas

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 world where tiny magnets inside a material don't just point in one direction like a compass needle, or cancel each other out perfectly like a silent, invisible army. For decades, scientists thought these were the only two options: the loud, magnetic "ferromagnets" that stick to your fridge, and the quiet, balanced "antiferromagnets" that hide their magnetism. But recently, a new, quirky character has entered the stage: the altermagnet. Think of it as a material where the magnetic moments are balanced (zero net magnetism) but arranged in a complex, alternating pattern that breaks the usual rules of symmetry. This isn't just a curiosity; it's a potential goldmine for faster, more efficient electronics because these materials can conduct electricity in a very specific, twisty way without needing a strong external magnetic field.

To understand how electricity moves through these materials, we need to look at two invisible forces. First, there's spin-orbit coupling, which you can think of as a subtle "handshake" between an electron's spin (its internal magnetic direction) and its movement through the crystal lattice. Second, there's the Anomalous Hall Effect (AHE), a phenomenon where electrons, instead of flowing straight, get pushed sideways, creating a voltage across the material. Usually, scientists believed this sideways push came from two sources: an "intrinsic" push caused by the material's perfect, clean geometry (like a river flowing around a smooth rock), and an "extrinsic" push caused by electrons bumping into impurities or defects (like a river getting diverted by a sudden pile of rocks). For a long time, it was assumed that in these new altermagnets, the "clean" intrinsic push was the only thing that mattered, and the messy "extrinsic" push from impurities was negligible.

This paper, written by A. Osin, A. Levchenko, and M. Khodas, challenges that assumption with a surprising discovery. The authors set out to calculate exactly how much the "extrinsic" push (caused by disorder) contributes to the sideways voltage in altermagnets. They found that the answer depends entirely on the specific "symmetry group" of the material. In roughly half of the known altermagnetic candidates (which they call Class A), the extrinsic contribution is not just a tiny footnote; it is comparable in size to the intrinsic contribution. In fact, in these materials, the messy disorder actually helps drive the effect, making it just as important as the perfect crystal structure. However, in the other half of the candidates (Class B), the extrinsic contribution remains negligible, just as scientists previously thought.

The key to this difference lies in a subtle quantum trick. In Class A materials, the material's symmetry allows a specific type of interaction (similar to the Dzyaloshinskii-Moriya interaction) to create a small, induced magnetization that grows linearly with the spin-orbit coupling. This creates a situation where the "clean" and "dirty" contributions to the electricity flow are locked in a tight dance, both being essential. In Class B materials, this specific interaction is forbidden by symmetry, so the induced magnetization is much weaker, and the extrinsic contribution fades away. The authors show that this behavior is linked to how the material's energy levels split along specific "nodal planes" when the weak spin-orbit coupling is turned on.

The study uses theoretical models and mathematical simulations (specifically the Kubo-Středa formalism) to reach these conclusions. They didn't just guess; they built detailed mathematical representations of two specific types of altermagnets (one representing Class A, like FeSb₂, and one for Class B, like rutile structures) and calculated the electron flow under different conditions. They found that even when the disorder is incredibly weak—so weak that it's almost invisible—the extrinsic effect in Class A materials remains strong. This suggests that for about half of the altermagnets scientists are currently studying, you cannot ignore the "messy" side of the story. If researchers want to predict how these materials will behave in real-world devices, they must account for both the perfect crystal geometry and the inevitable imperfections, because in these specific cases, the imperfections are just as powerful as the perfection.

Technical Summary: Extrinsic Anomalous Hall Effect in Altermagnets

Problem Statement
Altermagnets represent a new class of magnetically compensated materials characterized by specific symmetries that lift spin degeneracy along nodal planes in momentum space, even in the non-relativistic limit (zero spin-orbit coupling, SOC). While the intrinsic anomalous Hall conductivity (AHC) in these systems has been extensively studied as a geometric property of Bloch wavefunctions, the role of disorder-induced extrinsic contributions remains less understood. Traditional classifications often treat AHC in gapless magnets as either purely intrinsic (topological) or dominated by skew scattering in the clean limit. This paper investigates the extrinsic AHC in altermagnets, specifically addressing whether disorder contributions are negligible or comparable to intrinsic ones, and how this depends on the specific symmetry class of the material and the strength of the exchange splitting.

Methodology
The authors employ a microscopic approach based on the Kubo-Středa formalism to calculate the Hall conductivity tensor. The study focuses on the limit of large exchange splitting ( $2|N|$ ), where the system can be effectively reduced from a four-band model (accounting for sublattice and spin degrees of freedom) to an effective two-band model.

Key methodological steps include:

Model Construction: Two representative altermagnetic candidates are modeled: a Class A candidate (FeSb $_2$ , space group $2m2m1m$ ) and a Class B candidate (rutile structure, space group $24/1m2my1md$). Both are treated as $d$ -wave altermagnets with short-range, spin-independent Gaussian disorder.
Symmetry Classification: The materials are classified based on whether the SOC lifts the spin degeneracy at the $\Gamma$ -point (Class A, $\lambda_{k=0} \neq 0$ ) or preserves it (Class B, $\lambda_{k=0} = 0$ ).
Green's Function Approach: The disorder-averaged Green's functions are computed using the Born approximation. The authors explicitly separate the self-energy into scalar and vector parts, noting that the vector part arises from the $k$ -independent SOC component ( $\lambda_0$ ) and is crucial for the extrinsic effect.
Diagrammatic Expansion: The extrinsic AHC is calculated by evaluating the disorder-averaged correlation function. The authors include both non-crossing (bubble) diagrams and diffractive contributions (X and $\Psi$ diagrams) arising from scattering off pairs of impurities. They demonstrate that for short-range disorder, skew scattering is absent, and vertex corrections to the current operator vanish due to the odd parity of the current operator in momentum space.

Key Contributions and Results

Classification of Extrinsic AHC: The paper establishes a dichotomy between two classes of altermagnets:
- Class A: In materials where SOC lifts the $\Gamma$ -point degeneracy (e.g., FeSb $_2$ ), the extrinsic AHC is comparable in magnitude to the intrinsic AHC, even in the clean limit (vanishing disorder potential strength $U \to 0$ at fixed impurity concentration).
- Class B: In materials where the $\Gamma$ -point degeneracy remains (e.g., rutile structures), the extrinsic AHC is negligible.
Quantitative Ratios: For Class A altermagnets in the weak SOC limit, the authors derive the ratio of extrinsic to intrinsic conductivity as $\hat{\sigma}_{ex} \approx -\frac{1}{3} \hat{\sigma}_{in}$ . In the SOC-dominated limit, this ratio approaches $-\frac{2}{5}$ . These results indicate that the extrinsic contribution is not a minor correction but is of the same order as the intrinsic one.
Physical Origin: The substantial extrinsic contribution in Class A is linked to the non-analytic dependence of the intrinsic AHC on SOC. This non-analyticity arises from the Berry curvature being strongly peaked at the intersections of nodal planes (where spin degeneracy is lifted by weak SOC) with the Fermi surface. The extrinsic contribution shares this origin, stemming from the vector part of the self-energy induced by the constant SOC field ( $\lambda_0$ ) which causes spin-flip transitions between states.
Role of Magnetization: The paper correlates the magnitude of the extrinsic AHC with the induced magnetization ( $M$ ). In Class A, the magnetization is linear in SOC (arising from Dzyaloshinskii-Moriya type coupling), and the extrinsic AHC is large. In Class B, the magnetization appears only at higher orders in SOC and is weak, corresponding to the negligible extrinsic AHC.
Clean Limit Definition: The authors clarify that the "clean limit" of AHC is distinct from the AHC of a perfectly clean system. By defining the extrinsic contribution as the difference between the conductivity in the presence of weak disorder ( $U \to 0$ ) and the clean system ( $U=0$ ), they show that a finite extrinsic term persists even when skew scattering is suppressed.

Significance and Claims
The paper claims that the extrinsic AHC is an essential component of the total Hall response in roughly half of the altermagnetic candidates (Class A), particularly in the limit of large exchange splitting. This challenges the common assumption that AHC in such systems is primarily an intrinsic, disorder-independent geometric effect.

The authors argue that the non-analytic behavior of the AHC with respect to SOC is a unifying feature that governs both the intrinsic Berry curvature and the extrinsic disorder scattering. Consequently, for quantitative comparisons with experimental data in Class A altermagnets, the extrinsic contribution cannot be ignored. The findings suggest that the interplay between non-relativistic spin symmetries and weak relativistic SOC creates a unique transport regime where disorder effects are intrinsically tied to the topological properties of the band structure, distinguishing altermagnets from other systems like Weyl semimetals where extrinsic contributions are typically negligible unless the Fermi surface is large.

Technical Summary: Extrinsic Anomalous Hall Effect in Altermagnets

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