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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

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read In Fe3GaTe2, swapping gallium for germanium shrinks the K-point pockets, silencing the Fermi-sea torque's fourth harmonic, barely moving the Fermi-surface torque, and exposing hidden sublattice torques up to ~30 times the allowed size.

desk verdict Careful SOT comparison of two 2D magnets, but the main quantitative claim rests on a rigid-rotation approximation that freezes the chemical potential, which shifts by ~6 meV in FGaT and is never benchmarked for that material. read the letter →

arxiv 2608.05788 v1 pith:SJI7ZYJA submitted 2026-08-06 cond-mat.mtrl-sci

Gusthavo M. S. Brizolla , Stepan S. Tsirkin , Yaroslav Zhumagulov , Jaroslav Fabian This is my paper · ORCID
classification cond-mat.mtrl-sci
keywords spin-orbittorquetorkancemonolayerFe3GeTe2Fe3Gatime-reversal-evenhiddentorquesKubo-BastinlinearresponseWannierinterpolation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish which microscopic electronic states carry the spin-orbit torque in two nearly identical monolayer ferromagnets, Fe3GeTe2 and Fe3GaTe2. Because gallium has one fewer valence electron than germanium, the gallium compound is effectively a hole-doped twin of the germanium one, and the paper shows that this single change reshapes the torque landscape: the Fermi-surface (time-reversal-odd) torques stay similar, while the Fermi-sea (time-reversal-even) torque loses a pronounced fourth-harmonic angular component that is strong in Fe3GeTe2, an effect traced to the shrinkage of the $K$-point Fermi pockets. The paper also establishes that large hidden torques act on the two iron sublattices, reaching up to about thirty times the symmetry-allowed component in Fe3GaTe2, although they cancel exactly in the pristine monolayer. The finding matters because it identifies band filling, not crystal symmetry, as the control parameter for torque engineering in these van der Waals magnets, and it suggests that breaking the mirror symmetry between sublattices could activate otherwise hidden torques.

What carries the argument

The argument runs on the torkance tensor, the torque per unit applied electric field, computed by Kubo-Bastin linear response and split by time-reversal parity into a Fermi-sea (TR-even) channel $t^{even}_{ab}$ and a Fermi-surface (TR-odd) channel $t^{odd}_{ab}$. The electronic structure is represented by symmetry-adapted, fully spin-orbit-coupled Wannier functions built from relativistic density-functional theory; they carry the Hamiltonian, the velocity operator (including its Berry-connection contribution, which enforces symmetry-protected zeros of the response), and the spin operator used to build the exchange-torque operator. To sample the whole magnetization sphere the paper uses a rigid-exchange-rotation scheme: for each direction $\hat{m}$ only the TR-odd (exchange) part of the Hamiltonian is rotated by an SU(2) spinor rotation $U_R$, with the TR-even part fixed at the reference $\hat{m}\parallel\hat{z}$, so that a single reference calculation serves every magnetization direction. The angular maps are decomposed in vector spherical harmonics, the tangent families $\Psi_{\ell m}=\nabla_s Y_{\ell m}$ and $\Phi_{\ell m}=\hat{m}\times\nabla_s Y_{\ell m}$, and the fourth-harmonic content of the TR-even channel lives in the $m=4$ pair of harmonics; sublattice projectors resolve the same response onto individual atomic sites, which is what exposes the hidden torques.

What would settle it

Recompute the torkance at full self-consistent noncollinear DFT, without the rigid rotation, for both monolayers at the key in-plane directions, $\hat{m}\parallel\hat{x}$ where Fe3GeTe2's TR-even peak and fourth harmonic sit, and $\hat{m}\parallel\hat{y}$ where the benchmark already deviates by about 17%, and check whether the FGT/FGaT contrast in the TR-even peak or the fourth-harmonic coefficient survives. The experimental counterpart is to electrostatically hole-dope monolayer Fe3GeTe2 and measure the angular torque map: if the fourth-harmonic component does not shrink as holes are added, the K-pocket suppression mechanism is wrong.

Watch

Extended reading notes

Core claim

The central claim is that in two monolayers with the same point group ($D_{3h}$) and the same dominant Fe $3d$ bands, the substitution Ge$\to$Ga, which is equivalent to removing one valence electron per formula unit, changes the torque through band filling rather than symmetry. The density of states at $\varepsilon_F$ drops by roughly a factor of three and its spin polarization reverses sign, and individual band features shift by 30-130 meV relative to the Fermi level even though the two work functions differ by only about 28 meV. These shifts shrink the minority-spin pockets at $K$ and $K'$. As a result the time-reversal-odd (Fermi-surface) torkances of the two compounds are similar, whereas the time-reversal-even (Fermi-sea) torkance differs markedly: Fe3GeTe2 shows a strong fourth-harmonic angular component, concentrated at in-plane magnetization along $\hat{x}$, which is strongly reduced in Fe3GaTe2 because the positive and negative momentum-space contributions compensate more completely. A second discovery is site-resolved hidden torque: the two FeI sublattices, related by the horizontal mirror $\sigma_h$, carry components in directions the global symmetry forbids, and in Fe3GaTe2 these hidden components can exceed the symmetry-allowed torque by nearly a factor of 30; they cancel exactly in the ideal monolayer, but breaking the sublattice symmetry would activate them.

Load-bearing premise

Every angular torque map rests on the rigid-exchange-rotation approximation: the magnetization direction enters the Hamiltonian only by rigidly rotating its exchange (time-reversal-odd) part, with the time-reversal-even part frozen at its out-of-plane reference value; the paper's own benchmark shows this can deviate by about 17% for one TR-even component at in-plane magnetization, so the quantitative peak values of the maps inherit that uncertainty.

Editorial extensions

If this is right

  • The Fermi-surface (TR-odd) torque is largely transferable between Fe3GeTe2 and Fe3GaTe2, while the Fermi-sea (TR-even) torque changes sharply; the even channel is therefore the sensitive indicator of band filling in this family.
  • The band-feature shifts between the compounds (30-130 meV) lie within the reach of electrostatic or ionic gating, so the fourth-harmonic torque should be tunable in a single material by shifting the chemical potential toward or away from the $K$-point pockets.
  • Since the two FeI sublattices carry opposite hidden torques that cancel only while the horizontal mirror $\sigma_h$ is intact, any local or global breaking of that mirror, by a defect, adsorbate, substrate, or van der Waals partner, should activate torques that in Fe3GaTe2 reach up to about 30 times the symmetry-allowed size.
  • The symmetry phenomenology predicts that in the ideal monolayer the current-induced spin accumulation is strictly in-plane and linear in the magnetization, forbidding the conventional dampinglike torque; removing the mirror unlocks out-of-plane spin accumulation and dampinglike torques, a route the paper proposes for deterministic switching of perpendicular ferromagnets.
  • Because the fourth-harmonic content controls the field-free switching trajectory, as established for monolayer Fe3GeTe2 by the earlier work this paper builds on, the suppression of that harmonic in Fe3GaTe2 implies different current directions and thresholds for field-free switching in the two compounds.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural test the paper leaves implicit: electrostatically hole-doping monolayer Fe3GeTe2 should reproduce Fe3GaTe2's torque signature, namely smaller $K$-point pockets, a reduced Fermi-level density of states, and a weakened fourth-harmonic TR-even torque, without any chemical substitution.
  • The rigid-rotation benchmark error (up to about 17% in one TR-even component) is not propagated into the quantitative claims; a fully self-consistent check at the in-plane peak directions ($\hat{m}\parallel\hat{x}$ and $\hat{m}\parallel\hat{y}$) would pin down whether the factor-of-two TR-even contrast between the compounds is quantitatively robust.
  • Because the momentum-resolved maps show the TR-even integrand as alternating positive and negative densities concentrated near the $K$ pockets and at band anticrossings, strain or moire potentials that move those pocket energies should act as a continuous, non-chemical dial on the fourth-harmonic torque.
  • One internal inconsistency to note when reading: the abstract and main text state that the Fermi-level density-of-states polarization reverses from majority in Fe3GeTe2 to minority in Fe3GaTe2, but the tabulated values in Appendix F give the opposite (minority-like $P\approx -0.41$ in Fe3GeTe2, majority-like $P\approx +0.28$ in Fe3GaTe2); the torque conclusions do not depend on this direction, but
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper presents first-principles Kubo–Bastin linear-response calculations of the spin–orbit torkance in monolayer Fe3GeTe2 and Fe3GaTe2, using symmetry-adapted Wannier interpolation of fully relativistic DFT. The authors decompose the torkance by time-reversal parity (Fermi-sea/Fermi-surface), by atomic sublattice, and by momentum, and they compare the two compounds as a controlled hole-doping pair. Their central qualitative claims are that the time-reversal-odd (Fermi-surface) torkances are similar in the two materials, that the time-reversal-even (Fermi-sea) torkance has a pronounced fourth-harmonic component in FGT that is strongly suppressed in FGaT, and that site-resolved hidden torques on the FeI sublattices can exceed symmetry-allowed components by nearly a factor of 30 in FGaT. The full angular dependence is computed with a rigid-exchange-rotation scheme in which only the time-reversal-odd part of the Hamiltonian is rotated and all evaluations use a single reference chemical potential.

Significance. If the quantitative claims hold, the paper offers a valuable microscopic picture of current-induced torques in two-dimensional ferromagnets and identifies a concrete materials-design route through sublattice engineering of hidden torques. The work has notable strengths: the Kubo–Bastin formalism and Wannier interpolation are standard but carefully implemented; the treatment of the Wannier velocity, including the Berry-connection term, is explicit and is supported by symmetry checks (Appendix E); convergence with respect to the k mesh and broadening is documented; and the calculation is essentially parameter-free apart from the phenomenological broadening Γ. The momentum-resolved decomposition in Fig. 5 is a useful diagnostic. However, the central FGaT-specific claims rest on the rigid-exchange-rotation approximation at a fixed chemical potential, and the only quantitative benchmark of that approximation is performed for FGT, where the chemical-potential shift under magnetization rotation is negligible. The 6 meV chemical-potential shift reported for FGaT in Sec. III A is therefore a load-bearing unresolved issue for the fourth-harmonic and hidden-torque conclusions.

major comments (3)
  1. [Sec. II D, Sec. III A, Table I, Fig. 5] The rigid-exchange-rotation scheme evaluates all magnetization directions from one reference Wannier model and therefore at a fixed chemical potential. Section III A states that rotating the magnetization into the plane changes the chemical potential by <0.1 meV in FGT but by about 6 meV in FGaT. The full-sphere maps, the harmonic coefficients in Table I (notably the TR-even m=4 coefficient ψ^c_5,4), and the momentum-space analysis of Fig. 5 are computed without re-adjusting εF to preserve charge neutrality for in-plane magnetization. Since the suppression of the fourth harmonic in FGaT is attributed to the small K-point Fermi pockets, an uncontrolled 6 meV misplacement of εF can alter the pocket sizes and the compensation pattern, potentially changing the harmonic coefficients or even their signs. The benchmark in Appendix E is performed only for FGT, where the shift is negligible, so it does not validate the FGaT results. Please provide a FGaT benchmark that either includes the self-consistent εF shift or quantifies the sensitivity of Table I and Fig. 5 to a 6 meV shift in εF.
  2. [Appendix E, Sec. III B 2, Table I] Even apart from the chemical-potential issue, the rigid-exchange-rotation approximation itself shows up to about 17% deviation in the TR-even component t^even_zx near m parallel to ±y (Fig. 9). The central claim of a strongly reduced fourth-harmonic TR-even torque in FGaT concerns precisely in-plane and near-in-plane magnetization angles, yet no error propagation from this benchmark is provided for the harmonic coefficients or for the site-resolved hidden torques. The authors should state how the 17% deviation of an individual tensor component translates into uncertainty in the coefficients of Table I and in the factor-of-30 hidden-torque enhancement, ideally by evaluating the harmonic decomposition on both the DFT and rigid-rotation data shown in Fig. 9.
  3. [Sec. III B 3, Fig. 4] The claim that site-resolved FeI hidden torques can exceed symmetry-allowed components by nearly a factor of 30 in FGaT is computed with the same fixed-εF rigid-rotation approximation and is not covered by any direct benchmark. Because this claim is presented as a promising engineering route, the authors should either validate it with the FGaT benchmark requested above or soften the conclusion and provide a sensitivity estimate with respect to εF and to the rigid-rotation error.
minor comments (4)
  1. [Fig. 3] Figure 3 appears twice in the manuscript with different color scales (the first set shows FGT TR-even up to 2.5×10^-2 and the second up to 4.2×10^-2, with different FGaT ranges as well). This looks like a duplicated figure from an earlier draft; please retain only the intended version with consistent scales and captions.
  2. [Sec. III B 3, after Eq. (11)] The sentence fragment "lead to hidden torques s" appears to contain a typo and should read "lead to hidden torques."
  3. [Sec. III B 2, around Eq. (10)] The phrase "We note that, due to D3h symmetry, magnetization-independent torque begins at ℓ = 2" is potentially confusing: in D3h a magnetization-independent torque is symmetry-forbidden, and the ℓ = 2 sector is the leading allowed magnetization-dependent contribution. Please rephrase to avoid implying a nonzero rigid-field torque.
  4. [Eq. (12) and surrounding text] The description of the parameters a, b, and c as "real parameters material-dependent coefficients" is grammatically incomplete and should be rephrased, for example as "real, material-dependent coefficients."

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: parameter-free linear response on a DFT Hamiltonian, with all load-bearing approximations benchmarked against external noncollinear DFT.

full rationale

The paper's derivation chain is DFT (Sec. II A) to symmetry-adapted Wannier interpolation (Sec. II B) to Kubo-Bastin torkance formulas (Sec. II C) to rigid-exchange-rotation angular sampling (Sec. II D). No torkance value, harmonic coefficient, or torque datum enters as an input; the only external parameters are standard DFT settings, the Wannier projection windows, and the broadening Gamma = 10 meV, whose dependence is explicitly mapped in Appendix C.1. The central claims, namely the TR-even fourth-harmonic reduction in FGaT and the large hidden FeI torques, are read off from the computed full-sphere maps and their momentum-resolved integrands, not fitted to targets. The rigid-rotation approximation is validated against independent self-consistent noncollinear DFT in Appendix E, with deviations up to about 17 percent in one component; the absence of an FGaT-specific benchmark is a validation-coverage limitation, not a circular reduction. Self-citations to WannierBerri [50] and IrRep [53] are code and tool references with author overlap through S. S. Tsirkin, but the manuscript spells out the operator definitions, Kubo formulas, and symmetry checks itself, so those citations do not carry the physical argument. No step was found that reduces to its own input by equations, fitted parameters, or a self-citation chain.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central computation rests on standard linear-response theory, the LDA approximation, and the paper-specific rigid-exchange-rotation assumption. There are no new physical entities or fitted parameters beyond the phenomenological broadening Gamma.

free parameters (1)
  • Gamma (broadening) = 10 meV
    Constant phenomenological broadening in the Kubo-Bastin Green functions (Sec. II C). Chosen for all angular sweeps; torkance amplitudes and sign structure depend on it (Fig. 6).
assumptions (5)
  • standard math Kubo-Bastin linear response with a constant broadening correctly describes the intrinsic spin-orbit torkance.
    Invoked in Sec. II C, equations (2)-(3), following Freimuth et al. This is the accepted framework for intrinsic SOT.
  • domain assumption LDA exchange-correlation functional provides accurate electronic structure and magnetic moments for monolayer FGT and FGaT.
    Sec. II A: LDA chosen because PBE overestimates Fe moments in bulk FGT; no monolayer-specific experimental benchmark for torkance exists.
  • ad hoc to paper The magnetization-direction dependence of the Hamiltonian is captured by rigidly rotating only the TR-odd (exchange) part, leaving the TR-even part fixed.
    Sec. II D and Appendix E. The benchmark against self-consistent DFT shows up to about 17 percent deviation in t_even_zx, so this is a load-bearing approximation.
  • domain assumption SOC-induced spin-flip (noncollinear) terms in the exchange Hamiltonian are negligible for the torque operator and can be projected out.
    Sec. B 3 and Appendix D quantify Rnc around 0.9 percent on the 24 by 24 by 1 mesh; the projection is applied only to the torque operator, not to the band structure.
  • standard math The interpolated Wannier spin operator and velocity, including the Berry-connection contribution, correctly represent the true operators in the low-energy subspace.
    Appendices A and B; the full velocity is shown to be required for symmetry-protected zeros (Appendix E).

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Cite this review

Pith. "Pith review of Anatomy of Spin--Orbit Torques in Monolayer Fe$_3$GeTe$_2$ and Fe$_3$GaTe$_2$: Insights from atomistic and momentum-space decompositions." pith.science (2026). https://pith.science/paper/SJI7ZYJA

@misc{pith2026260805788,
  author       = {Pith},
  title        = {Pith review of: Anatomy of Spin--Orbit Torques in Monolayer Fe$_3$GeTe$_2$ and Fe$_3$GaTe$_2$: Insights from atomistic and momentum-space decompositions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SJI7ZYJA}},
  note         = {Machine review of arXiv:2608.05788}
}
abstract

We present a systematic first-principles study of the spin-orbit torques in the ferromagnetic monolayers Fe$_3$GeTe$_2$ (FGT) and Fe$_3$GaTe$_2$ (FGaT). Despite sharing the same crystal structure (point group $D_{3h}$) and predominantly Fe~$3d$ spin-polarized bands, the two materials exhibit markedly different current-induced torques. We reveal these differences by computing the full angular dependence of the torkance---the torque per unit applied electric field---using linear-response theory with symmetry-adapted spin--orbit-coupled Wannier functions. FGaT may be viewed as a hole-doped analogue of FGT, since Ga contributes one valence electron fewer than Ge. Although the work functions differ by only about $28$~meV, the band filling near $K$ and $K'$ changes substantially: the density of states at $\varepsilon_F$ is reduced by a factor of three and its spin polarization reverses from majority in FGT to minority in FGaT. These electronic changes are reflected in the torques resolved by time-reversal parity, sublattice, and momentum. In particular, we identify pronounced hidden torques in FGaT and relate the suppression of its fourth-harmonic Fermi-sea component to the evolution of momentum-space pockets. Finally, we discuss the emergence of such self-torques, which are not captured by the conventional picture of current-induced spin accumulation, within a symmetry-based phenomenological framework. Our results provide microscopic insight into current-induced torques in two-dimensional ferromagnets and offer guidance for defect and van der Waals engineering of layered magnetic materials.

Figures

Figures reproduced from arXiv: 2608.05788 by the authors.

Figure 1
Figure 1. Geometry and crystal structure. Top: response geometry. An in-plane field E ∥ xˆ drives a current jx; the magnetization direction mˆ is parametrized by spherical an￾gles (θ, ϕ). The axis σˆ ≡ zˆ × Eˆ and the conventional bilayer torques—fieldlike τFL ∥ mˆ × σˆ (time-reversal odd) and damp￾inglike τDL ∥ mˆ × (σˆ × mˆ ) (time-reversal even)—are drawn only to fix axes and nomenclature: in the D3h monolayer both rigid f… view at source ↗
Figure 1
Figure 1. Because SOC ties the electronic structure to the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Spin-projected band structures along Γ–K–M–K [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figures from the paper (9 more)
Figure 3
Figure 3. Figure 3: For magnetization along ±zˆ, the torque vanishes, given the presence of the σh mirror. Once the magneti￾zation tilts, σh is no longer a symmetry of the magnetic configuration, allowing the emergence of torque [32]. The TR-odd torkances exceed the TR-even ones in peak m…
Figure 3
Figure 3. Figure 3: Angular dependence of the spin-orbit torkance on the unit sphere of magnetization directions for monolayer Fe [PITH_FULL_IMAGE:figures/full_fig_p007_3.png]
Figure 4
Figure 4. Figure 4: Calculated atom-resolved torkance E ∥ xˆ, along the xz and yz meridians and the equator θ = π/2 (columns). For each compound the upper two rows are the channels the magnetic point group allows in the total, the lower two those it forbids. Curves give the sublattice gro…
Figure 5
Figure 5. Figure 5: Momentum-resolved torkance density for FGT (left) [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 1
Figure 1. Figure 1: Certainly, being effective requires strong electric [PITH_FULL_IMAGE:figures/full_fig_p010_1.png]
Figure 6
Figure 6. Figure 6: Broadening dependence of the spin–orbit torkance in [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: Band decomposition of monolayer Fe3GeTe2 [(a)–(c)] and Fe3GaTe2 [(d)–(f)] at mˆ = zˆ: spectra of the full Hamiltonian, its TR-even component, and its TR-odd component [Eq. (B3)]. The dashed line marks the Fermi level of thev full Hamiltonian, repeated on the Heven pane…
Figure 8
Figure 8. Figure 8: Symmetry-protected torkance components of mono [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 10
Figure 10. Figure 10: Calculated spin-resolved DOS of monolayer [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]

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Reference graph

Works this paper leans on

77 extracted references · 60 canonical work pages · cited by 1 Pith paper

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    Full-sphere overview We plot the calculated torkances for FGT and FGaT in Fig. 3. For magnetization along ±ˆz, the torque vanishes, given the presence of the σh mirror. Once the magneti- zation tilts, σh is no longer a symmetry of the magnetic configuration, allowing the emergence of torque [32]. The TR-odd torkances exceed the TR-even ones in peak magnit...

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    Expand- ing these in spherical harmonics Yℓm( ˆm) gives the two tangent familiesΨ ℓm = ∇sYℓm andΦ ℓm = ˆm× ∇sYℓm, with ∇s the surface gradient [32, 56]

    Symmetry expansion and closed forms We decompose the torkance over the magnetization sphere in the vector-spherical-harmonics (VSH) basis gen- erated by two scalar potentials ψ( ˆm) and χ( ˆm). Expand- ing these in spherical harmonics Yℓm( ˆm) gives the two tangent familiesΨ ℓm = ∇sYℓm andΦ ℓm = ˆm× ∇sYℓm, with ∇s the surface gradient [32, 56]. The torkan...

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    Atom- and momentum-resolved torkances Having analyzed the torques for the monolayers, let us now look at how the torques affect the magnetization of the individual atoms. This is particularly useful in order to see how the atoms work together to induce the magnetization dynamics. The atom-resolved torques are in general richer, especially if the atomic si...

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    Exchange torque operator The total spin-torque operator is ˆτ = (i/ℏ)[H, ˆS]. In the magnetization-frame SOT formulation [ 33] the rele- vant torque generates magnetization rotations: under a uniform SU(2) rotation about axis a only the TR-odd part ofHvaries, so that ∂H ∂θa θa=0 = i ℏ Hodd, ˆSa .(B8) This presupposes that Hodd transforms rigidly under SU(...

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    Time-reversal decomposition of the Hamiltonian With ˆΘ = (Iorb ⊗iσ y)K the time-reversal unitary of Appendix A 1, the time-reversed Bloch Hamiltonian is HTR(k) = ˆΘH ∗(−k) ˆΘ†, and the time-reversal even and odd parts of the Hamiltonian can be introduced as Heven/odd(k) = 1 2 H(k)±H TR(k) .(B3) Here, Heven collects the kinetic, crystal-field, and pure spi...

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