🔬 materials science

Relativistic spin-momentum locking in ferromagnets

Using density functional theory calculations, this study demonstrates that relativistic spin-momentum locking, previously established in non-collinear magnets, also generates strong momentum-dependent spin responses in a broad class of ferromagnets with rotational symmetry, offering a promising platform for observing emergent phenomena and spintronic applications.

Original authors: Xujia Gong, Amar Fakhredine, Carmine Autieri

Published 2026-07-15

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Xujia Gong, Amar Fakhredine, Carmine Autieri

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 microscopic world inside a magnet not as a static, frozen block of iron, but as a bustling, high-speed highway where tiny particles called electrons are zooming around. In this world, a particle's "spin" is like its internal compass needle, pointing either up or down, while its "momentum" is simply the direction it's speeding in. For decades, scientists thought these two things—where a particle is going and which way its compass points—were mostly independent, like a car's speed and the color of its paint. However, a fascinating new discovery called "spin-momentum locking" has changed that story. It turns out that in certain special materials, the direction you drive forces your compass to point a specific way, as if the road itself dictates the direction of your internal needle.

This phenomenon was first spotted in a weird new class of magnets called "altermagnets," which are like a hybrid between a regular magnet and a non-magnet. But here is the big question: Does this magical locking happen in the "regular" magnets we use every day, like the ones in your hard drive or a fridge magnet? These everyday magnets, called ferromagnets, are usually thought to be too simple and too "loud" with their magnetic noise to show off these subtle quantum tricks. Understanding if this locking exists in common magnets is a big deal because it could unlock new ways to build super-fast, energy-efficient computers that use the spin of electrons instead of just their charge.

In this study, the researchers decided to play detective with a powerful computer simulation tool called Density Functional Theory. They didn't just look at the famous altermagnets; they went hunting in the world of standard ferromagnets to see if the spin-momentum locking was hiding there, too. They focused on a specific group of magnets where the magnetic atoms are arranged in a pattern that has rotational symmetry (like a spinning top) but lacks a simple mirror or translation symmetry. They picked a cast of characters including a material called SrRuO3, some chromium compounds (CrTe and CrAs), a half-Heusler alloy named MnPtSb, and the classic face-centered cubic Nickel (Ni).

The team's simulations revealed a surprising twist. Even in these "regular" magnets, the spin-momentum locking is not only present but can be quite strong. In the material SrRuO3, the atoms are slightly tilted, creating a tiny "canting" effect where the spins aren't perfectly straight. Here, the locking behaves in a specific, predictable way. But the real magic happens in materials like CrTe and CrAs. In these crystals, the laws of physics strictly forbid the spins from tilting in real life—they must stay perfectly straight. Yet, when the researchers looked at the "map" of the electrons' energy (the k-space), they found that the spin-momentum locking was still there, roaring with significant strength, reaching up to 20% or even 25% of the main magnetic signal. It's as if the spins are perfectly straight on the ground, but in the energy map, they are dancing in a complex, locked pattern.

The study also looked at MnPtSb, a material that lacks a center of symmetry. Here, the locking takes on a different shape, resembling a "p-wave" pattern, which is a more complex, swirling motion compared to the simpler patterns seen in the other materials. Finally, they compared all these findings to the classic Nickel magnet. They found that Nickel is a bit more chaotic; its locking pattern is a messy mix of two different types, making it more complex than the neat, organized patterns found in the other ferromagnets they studied.

The bottom line of this research is that the "even-wave" spin-momentum locking, previously thought to be the exclusive party of exotic altermagnets, is actually a guest at the ferromagnet party too. The main difference is just the volume: in altermagnets, the locking is the main event, while in these ferromagnets, it's a sub-dominant component that is still loud enough to be heard. This discovery suggests that the common magnets we already know and use might be far more versatile and topologically interesting than we realized, offering a promising new playground for future technologies that rely on controlling electron spin.

Technical Summary: Relativistic Spin-Momentum Locking in Ferromagnets

Problem Statement
Relativistic spin-momentum locking has been rigorously established in time-reversal-breaking materials with zero net magnetization, specifically altermagnets and non-collinear magnets, where opposite spins are connected by rotational symmetries but not by translation or inversion. While the non-relativistic spin-momentum locking (even waves) in altermagnets is well-documented, the existence and nature of relativistic spin-momentum locking in conventional ferromagnets remain less explored. Specifically, it is unclear whether ferromagnetic materials with magnetic sites connected by rotational symmetry exhibit similar locking mechanisms, particularly in their subdominant spin components, and how these effects compare to prototypical ferromagnets like fcc Ni.

Methodology
The authors employ Density Functional Theory (DFT) calculations, including spin-orbit coupling (SOC), to investigate the electronic structure and spin textures of several ferromagnetic systems. The study focuses on materials where magnetic sites are connected by rotational symmetries (proper or improper, symmorphic or nonsymmorphic) but not by translation or inversion. The specific materials analyzed include:

Centrosymmetric orthorhombic SrRuO $_3$
Hexagonal CrTe and CrAs (NiAs structure)
Noncentrosymmetric MnPtSb (half-Heusler)
Face-centered cubic (fcc) Ni (as a comparative simple ferromagnet)

The analysis involves calculating Fermi surfaces at specific $k_z$ planes (e.g., $k_z = 0, \pm 0.25$ ) and examining spin-resolved band structures to identify nodal planes and the symmetry of spin-splitting. The authors map the spin-momentum locking patterns to specific wave symmetries (s-wave, d-wave, p-wave) and magnetic multipole moments (quadrupoles).

Key Contributions and Results

Existence of Relativistic Locking in Ferromagnets: The paper demonstrates that relativistic spin-momentum locking is a robust feature in ferromagnets with rotational symmetry connections, even when real-space spin canting is symmetry-forbidden.
Real-Space vs. k-Space Discrepancy: A critical finding is the disparity between real-space and momentum-space spin components.
- In SrRuO $_3$ , antisymmetric exchange (staggered Dzyaloshinskii-Moriya interaction) induces a small spin canting orthogonal to the easy axis (4% for $S_x$ and 0.3% for $S_y$ relative to $S_z$ ).
- In CrTe, CrAs, and MnPtSb, real-space spin canting is forbidden by symmetry. However, in k-space, the spectral weight of these subdominant components reaches approximately 20–25% of the dominant component.
Symmetry Classification of Spin-Momentum Locking:
- SrRuO $_3$ (Centrosymmetric): The dominant component exhibits s-wave locking. The subdominant components exhibit d-wave locking ( $d_{xz}$ , $d_{yz}$ , $d_{xy}$ ) depending on the magnetization direction and spin component.
- CrTe and CrAs (Centrosymmetric NiAs structure): Similar to SrRuO $_3$ , the dominant component is s-wave, while subdominant components show d-wave textures ( $d_{xy}$ , $d_{yz}$ , $d_{xz}$ ). The breaking of $C_6$ rotational symmetry by the magnetization vector drives these d-wave lockings.
- MnPtSb (Noncentrosymmetric): Due to the lack of inversion symmetry and strong SOC from heavy elements (Pt, Sb), the system hosts relativistic p-wave locking ( $p_x \pm p_y$ ) for subdominant components, coexisting with magnetic quadrupoles. The dominant $S_z$ component retains s-wave symmetry.
- fcc Ni (Simple Ferromagnet): The locking pattern is more complex than in the rotationally symmetric cases. It cannot be described by a single magnetic quadrupole; instead, it requires a combination of two quadrupole terms ( $Q_{(x-y)z}$ and $Q_{(x+y)z}$ ) with different weights, indicating a more intricate spin texture.
Model Hamiltonian: The authors propose that the same model Hamiltonians used for altermagnets can describe these ferromagnets, with the primary distinction being that the dominant component in ferromagnets is an s-wave, whereas in altermagnets it is an even-wave.

Significance and Claims
The paper claims to establish that relativistic spin-momentum locking is a fundamental property of ferromagnets with specific rotational symmetries, analogous to altermagnets but distinguished by the nature of the dominant spin component (s-wave vs. even-wave).

Platform for Emergent Phenomena: Because ferromagnets typically possess larger bandwidths than altermagnets, they are proposed as a promising platform for observing even-wave relativistic spin-momentum locking and associated emergent phenomena.
Symmetry and Response: The authors assert that this locking governs symmetry-allowed spin Hall currents and other momentum-dependent spin responses in k-space.
Multipolar Physics: The presence of these lockings implies the existence of octupoles and higher-order multipoles, offering a pathway to study multipolar anisotropy in the anomalous Hall effect and nonlinear responses (e.g., nonlinear photocurrents, circular photogalvanic effects).
Tunability: The spin-momentum locking patterns can be tuned by changing the magnetization direction, allowing for the manipulation of specific spin components (e.g., switching $S_z$ locking from $d_{yz}$ to $d_{xz}$ ).

The authors conclude that these relativistic effects should be included in the fundamental characterization of materials alongside electronic and topological properties.

Technical Summary: Relativistic Spin-Momentum Locking in Ferromagnets

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