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Anatomy of Spin--Orbit Torques in Monolayer Fe $_3$ GeTe $_2$ and Fe $_3$ GaTe $_2$ : Insights from atomistic and momentum-space decompositions

This first-principles study reveals that despite sharing identical crystal structures, monolayer Fe $_3$ GeTe $_2$ and Fe $_3$ GaTe $_2$ exhibit markedly different spin-orbit torques due to hole doping-induced changes in Fermi surface topology and spin polarization, which are elucidated through symmetry-adapted momentum-space decompositions and a phenomenological framework for self-torques.

Original authors: Gusthavo M. S. Brizolla, Stepan S. Tsirkin, Yaroslav Zhumagulov, Jaroslav Fabian

Published 2026-08-07

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CC BY 4.0

Original authors: Gusthavo M. S. Brizolla, Stepan S. Tsirkin, Yaroslav Zhumagulov, Jaroslav Fabian

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 you are trying to build a super-fast, super-small computer that doesn't just store data, but thinks with it. This is the dream of "spintronics," a field where scientists try to use the tiny magnetic spin of electrons—like a microscopic compass needle—to carry information instead of just their electric charge. The holy grail of this field is finding a way to flip these magnetic needles using electricity, which would let us switch computer memory on and off without needing bulky magnets or heat.

Usually, to flip a magnetic needle, you need to push it with a magnetic field or a stream of other spinning electrons. But there's a clever shortcut called "spin-orbit torque." Think of it like a dance floor: when electrons (the dancers) move through a material with heavy atoms (the bouncers), the heavy atoms' strong grip on the dancers' spins causes them to wobble and push against the magnetic needles, flipping them over. The big question for scientists is: how do we find the perfect dance floor? We need materials that are thin enough to fit on a chip, magnetic enough to hold data, and efficient enough to flip that data with a tiny push of electricity. Two promising candidates have emerged: a material called Fe3GeTe2 and its cousin, Fe3GaTe2. They look almost identical, but do they dance the same way?

This paper dives deep into the microscopic dance moves of these two materials, which are single layers of atoms (monolayers) that act like magnets. The researchers used powerful computer simulations to map out exactly how electricity pushes on the magnetism in these layers. They discovered that even though these two materials are twins in terms of their crystal structure and the atoms they are made of, they behave very differently when you try to flip their magnetic switches.

The main finding is that swapping just one type of atom in the recipe changes the entire dance. Fe3GeTe2 (let's call it "Ge-Twin") and Fe3GaTe2 ("Ga-Twin") are so similar that they share the same shape and the same main magnetic atoms (Iron). The only difference is that Ge-Twin has Germanium, while Ga-Twin has Gallium. Gallium has one fewer electron than Germanium, which acts like a tiny "hole" or missing piece in the electron crowd. The researchers found that this missing electron shifts the energy levels of the electrons near the surface where the action happens.

Here is where it gets interesting: despite being so similar, the "Ga-Twin" is much less efficient at the specific type of magnetic flipping that usually helps in these materials. The researchers found that the "Ge-Twin" has a strong, rhythmic push (a fourth-harmonic torque) that helps flip the magnet, but the "Ga-Twin" almost completely loses this rhythm. It's as if you replaced a drum in a marching band with a slightly different drum; the band looks the same, but the beat is gone.

However, the paper also uncovered a hidden secret. Inside these single layers, there are two different sets of iron atoms, one on the top half and one on the bottom half. The researchers found that these two sets of atoms actually push in opposite directions, creating a "hidden torque" that cancels itself out in the perfect, flat layer. But here's the kicker: in the "Ga-Twin," this hidden push is massive—about 30 times stronger than the visible push we can see. The paper suggests that if we could break the perfect symmetry of the layer (perhaps by adding a defect or stacking it differently), we could unlock this giant hidden force to flip magnets much more effectively.

The study confirms that the difference between the two materials comes down to how the electrons are filled up near the edge of the energy band, specifically around the "K" points in the material's structure. The "Ga-Twin" has smaller pockets of electrons there, which dampens the rhythmic pushing effect. While the "Ga-Twin" is weaker at the standard flipping moves, the discovery of these giant hidden forces offers a new roadmap. It suggests that by engineering these materials—perhaps by tweaking their defects or stacking them in specific ways—we might be able to harness these hidden forces to create the next generation of ultra-efficient, magnetic computer memory. The paper doesn't claim to have built this device yet, but it provides the microscopic blueprint for how to make it happen.

Technical Summary: Anatomy of Spin–Orbit Torques in Monolayer Fe $_3$ GeTe $_2$ and Fe $_3$ GaTe $_2$

Problem Statement
Electrical control of magnetization via spin–orbit torques (SOT) is a central objective in spintronics. While van der Waals (vdW) magnets like Fe $_3$ GeTe $_2$ (FGT) and Fe $_3$ GaTe $_2$ (FGaT) offer unique opportunities due to their low crystal symmetry and gate-tunable carrier density, the microscopic origins of their current-induced torques remain incompletely understood. Specifically, the dependence of torque harmonics on band filling, the role of specific electronic states near the Fermi level ( $\varepsilon_F$ ), and the existence of "hidden" torques arising from sublattice degrees of freedom in non-centrosymmetric monolayers require systematic investigation. Although FGT and FGaT share the same $D_{3h}$ crystal structure and dominant Fe 3d character, they differ by one valence electron (Ga vs. Ge), effectively making FGaT a hole-doped analogue of FGT. The paper addresses how this stoichiometric modification alters the SOT landscape without the need for electrostatic gating.

Methodology
The authors employ a systematic first-principles approach combining density-functional theory (DFT) with linear-response theory:

Electronic Structure: Calculations were performed using Quantum ESPRESSO with fully relativistic optimized norm-conserving Vanderbilt (ONCV) pseudopotentials. Spin–orbit coupling (SOC) was treated self-consistently within a noncollinear framework. The Perdew–Wang LDA functional was used for response calculations to ensure accurate magnetic moments, consistent with previous studies on monolayer FGT.
Wannier Interpolation: To enable dense sampling of the Brillouin zone, symmetry-adapted Wannier functions (SAWFs) were constructed using WannierBerri. Crucially, SOC was included from the outset in the Bloch states, ensuring the interpolated model inherits ab initio spin–orbital entanglement rather than adding SOC as a perturbative term. The basis set included 48 spinor functions (projected onto Fe 3d, X p, and Te p orbitals) spanning the relevant $p$ – $d$ manifold.
Torkance Formalism: The spin–orbit torque (SOT) was computed using the Kubo–Bastin linear-response formalism. The torkance tensor ( $t_{ab}$ ), defined as torque per unit electric field, was decomposed into time-reversal (TR) even (Fermi-sea) and TR-odd (Fermi-surface) components.
Magnetization Rotation: To map the full angular dependence of the torque on the magnetization sphere, a "rigid-exchange-rotation" scheme was employed. The ab initio Hamiltonian was computed once for magnetization along $\hat{z}$ , then split into TR-even and TR-odd parts. The TR-odd exchange field was rotated via SU(2) spinor transformations to simulate arbitrary magnetization directions ( $\hat{m}$ ), while the TR-even part (orbital and SOC terms) remained fixed. This approach was benchmarked against fully self-consistent noncollinear DFT, showing agreement within $\sim 5\%$ for most orientations.
Decomposition: The resulting torques were analyzed via vector spherical harmonic expansion, atom-resolved decomposition (sublattices Fe $_I$ , Fe $_{II}$ , Te, X), and momentum-space density mapping.

Key Results

Electronic Structure Differences: Despite similar work functions ( $\sim 28$ meV difference), the hole doping in FGaT significantly alters the band filling near the $K$ and $K'$ points. The density of states at $\varepsilon_F$ in FGaT is reduced by a factor of three compared to FGT, and the spin polarization at the Fermi level reverses from majority (FGT) to minority (FGaT). The valley-Zeeman splitting remains robust at $\sim 40$ meV in both materials, dictated by the Te environment.
Angular Dependence of Torques:
- TR-Odd (Fermi-Surface) Torques: These dominate the total response in both materials. The angular dependence is remarkably similar between FGT and FGaT, with peak magnitudes in FGaT only $\sim 15\%$ smaller. This suggests that the TR-odd channel is largely transferable despite the distinct Fermi surface configurations.
- TR-Even (Fermi-Sea) Torques: These exhibit marked differences. FGT displays a pronounced fourth-harmonic contribution for in-plane magnetization, whereas this component is strongly suppressed in FGaT. The authors attribute this suppression to the reduction of $K$ -point Fermi pockets in FGaT, which leads to more complete momentum-space compensation of high-harmonic contributions.
Atom-Resolved and Hidden Torques:
- The analysis reveals that the two Fe $_I$ atoms (related by the horizontal mirror $\sigma_h$ ) contribute differently to the torque when the magnetization tilts from the out-of-plane direction.
- Hidden Torques: While the global $D_{3h}$ symmetry forbids a uniform current-induced spin accumulation, the sublattice symmetry allows for "hidden" spin-orbit fields. In FGaT, these hidden contributions on the Fe $_I$ sublattices can exceed the symmetry-allowed components by nearly a factor of 30. Although they cancel exactly in the ideal monolayer, the paper identifies them as a potential resource for torque engineering if sublattice symmetry is broken.
- The TR-odd torque density is confined to sharp contours near the $K$ points, while the TR-even density is spread over the zone, enhanced at band anticrossings.

Significance and Claims
The paper claims to provide a microscopic understanding of current-induced torques in 2D ferromagnets by isolating the effects of band filling from crystal symmetry. By comparing FGT and FGaT, the authors demonstrate that:

Stoichiometry as a Tuning Knob: Qualitative changes in torque harmonics (specifically the suppression of the fourth harmonic in the TR-even channel) can be achieved intrinsically through stoichiometric modification (Ge $\to$ Ga) rather than external electrostatic gating.
Hidden Degrees of Freedom: The study highlights that inversion-asymmetric monolayers host hidden torques arising from site-specific degrees of freedom (sublattices) that are not captured by conventional pictures of current-induced spin accumulation. These hidden torques, while canceling in the net response of the ideal layer, suggest that defect engineering or symmetry-breaking in vdW heterostructures could activate large, otherwise inaccessible torque components.
Methodological Rigor: The work establishes a robust computational pipeline for calculating SOTs in low-symmetry systems, emphasizing the necessity of self-consistent SOC treatment and symmetry-adapted Wannier functions to correctly capture the interplay between exchange fields and spin-orbit coupling.

The authors conclude that their results offer guidance for the "defect and van der Waals engineering of layered magnetic materials," suggesting that manipulating sublattice symmetry could be a viable route to enhance or tailor spin–orbit torques in 2D magnets.

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