Interfacial contributions to spin-orbit torque and magnetoresistance in ferromagnet/heavy-metal bilayers

Alexey A. Kovalev; K. D. Belashchenko; M. van Schilfgaarde

REVIEW 3 major objections 5 minor 46 references

Interfacial contributions to spin-orbit torque and magnetoresistance in ferromagnet/heavy-metal bilayers

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that in ferromagnet/heavy-metal bilayers, both damping-like spin-orbit torque and the transverse magnetoconductance usually attributed to spin-Hall magnetoresistance carry comparably large interfacial contributions that…

desk verdict Serious first-principles SOT decomposition with a solid interfacial-torque result, but the magnetoresistance benchmark is dimensionally inconsistent as printed and the order-of-magnitude claim needs a re-derivation. read the letter →

arxiv 1908.02680 v2 pith:JRKK7LNC submitted 2019-08-07 cond-mat.mes-hall cond-mat.mtrl-sci

K. D. Belashchenko , Alexey A. Kovalev , M. van Schilfgaarde This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.mtrl-sci

keywords spin-orbittorquespin-Hallmagnetoresistanceinterfacialtransportnon-equilibriumGreen'sfunctionferromagnet/heavy-metalbilayerdamping-likevectorsphericalharmonicsmagnetoconductance

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 asks where spin-orbit torque and magnetoresistance actually arise in ferromagnet/heavy-metal bilayers such as Co/Pt and Co/Au. Using first-principles transport calculations with disorder, it finds that the damping-like torque decomposes into a bulk spin-Hall part that grows with heavy-metal thickness and a comparable interfacial part that does not. It further finds that the transverse magnetoconductance in Co/Pt is about one conductance quantum, more than an order of magnitude larger than the spin-Hall magnetoresistance model predicts. The conclusion is that both spin-orbit torque and magnetoresistance carry large interfacial contributions unrelated to the bulk spin-Hall effect, so measurements should not be interpreted solely through the spin-Hall mechanism.

What carries the argument

The argument rests on three tools: a first-principles non-equilibrium Green's function calculation with Anderson disorder that treats the whole bilayer quantum-mechanically; an expansion of the torquance tensor, the tensor mapping electric field to torque, in vector spherical harmonics, an orthonormal basis that cleanly separates damping-like from field-like torque terms; and a thickness-dependence analysis that fits the damping-like torque to $\tau_0+\tau_{\rm SH}[1-\mathrm{sech}(d_N/l_{\rm sf})]$ and compares the magnetoconductance with the spin-Hall SMR formula. The vector-harmonic expansion is what lets the paper identify which torque terms are damping-like and which are field-like, and the thickness fit is what separates the bulk spin-Hall piece from the interfacial piece.

What would settle it

Compute $\Delta_{yz}g$ for Co/Pt bilayers with the spin-orbit coupling in the Pt bulk turned off while leaving the interface unchanged. The paper's interfacial mechanism predicts $\Delta_{yz}g$ remains of order $e^2/h$; if it collapses to the spin-Hall prediction, the central claim is wrong.

Watch

Extended reading notes

Core claim

The central claim is that in Co/Pt and Co/Au bilayers, the damping-like spin-orbit torque is the sum of a bulk spin-Hall contribution that grows with heavy-metal thickness and a thickness-independent interfacial contribution of comparable size. In Co/Pt, the fitted interfacial constant is about 110 ns/m versus about 97 ns/m for the bulk spin-Hall part, and in Co/Au the interfacial part is also substantial. The paper additionally finds that the transverse magnetoconductance $\Delta_{yz}g$ in Co/Pt is of order $e^2/h$, which exceeds the spin-Hall magnetoresistance formula $\Delta_{yz}g^{\rm SH}=\theta_{\rm SH}^2\bar\rho\tanh^2(d_N/2l_{\rm sf})\tanh(d_N/l_{\rm sf})$ by more than an order of magnitude. The paper concludes that this magnetoconductance, like the damping-like torque, likely has an interfacial origin rather than a spin-Hall origin.

Load-bearing premise

The fit splits the torque into a bulk spin-Hall term with thickness dependence $1-\mathrm{sech}(d_N/l_{\rm sf})$ and a thickness-independent interfacial term; if the interfacial contribution itself changes with thickness, the split is not a physical decomposition.

Editorial extensions

If this is right

For Co/Pt at typical disorder, the interfacial damping-like torque constant is about 110 ns/m, comparable to the bulk spin-Hall value near 97 ns/m, so neglecting the interface would overestimate the bulk spin-Hall torque.
The spin-Hall magnetoresistance formula underpredicts the calculated $\Delta_{yz}g\sim e^2/h$ by more than an order of magnitude, so measured SMR in metallic bilayers cannot be taken as direct evidence for the spin-Hall effect.
Since the angular functions of SMR, AMR, and interfacial anomalous magnetoresistance are linearly dependent, angular scans alone can determine only two independent parameters; separating the mechanisms requires thickness dependence or interface control.
If the interfacial contribution is included, the effective spin-Hall angle inferred for Co/Pt rises from about $\theta_{\rm SH}\approx 0.02$ to about $0.06$, matching typical experimental values.
The linear growth of SMR at small thicknesses, often cited in favor of spin-Hall theory, may instead indicate the thickness at which a continuous metal film forms, because the interface properties change over a few monolayers.

Reading between the lines

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

If this interfacial picture is right, spin-Hall angles extracted from torque experiments on nanometer-thick bilayers are not bulk material parameters; they depend on interface quality and disorder, so comparing samples requires controlling the interface as well as the metal.
A direct test would be to insert a single monolayer spacer at the Co/Pt interface or to vary interfacial roughness while keeping the Pt bulk unchanged; the interfacial torque and the large $\Delta_{yz}g$ should change substantially if they are truly interfacial.
A natural computational extension is to repeat the thickness series with spin-orbit coupling scaled separately in the bulk and at the interface, mapping where the torque and magnetoresistance are generated and turning the fitted separation into a microscopic assignment.
The same vector-spherical-harmonic expansion could be applied to other bilayer combinations, including systems with weak bulk spin-Hall effect, potentially revealing interfacial spin-orbit torque in materials previously classified as spin-Hall dominated.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Worth knowing: this paper has two real results and one botched comparison. The SOT decomposition is genuinely new and mostly solid; the magnetoresistance benchmark is not reproducible as printed.

What is new: thickness-resolved NEGF-LMTO calculations of damping-like torque in Co/Pt and Co/Au with explicit Anderson disorder, decomposed using a vector-spherical-harmonics expansion. The clean result is that tau0, the zero-thickness intercept, is comparable to tauSH, the spin-Hall coefficient, in both systems, and larger in Co/Pt. The cross-check idea is good: they fit theta_SH and l_sf to the SOT thickness data, then use those parameters to predict the spin-Hall magnetoconductance without fitting to the MR data. That is not circular.

Main problem: Eq. (4) is dimensionally inconsistent. Delta_yz g is defined as LG/w, which has units of siemens. The right-hand side is theta_SH^2 rho_bar times dimensionless tanh factors, which has units of resistivity. So the dashed curve in Fig. 2, “scaled by a factor of 10,” cannot be reproduced without some unstated conversion. The authors cite Refs. [12,41] but do not translate those results into their units. This matters because the claim that interfacial magnetoconductance exceeds the spin-Hall prediction by more than an order of magnitude is quantitative. The qualitative conclusion may survive—a rough estimate from their own parameters at d_N = 12 ML gives something near 0.1 e^2/h, below the computed conductance—but the paper as printed does not establish the stated excess. They need to re-derive the benchmark with correct units, show the unscaled curve, and give error bars on the SOT fit parameters.

Second soft spot: the tau0/tauSH split depends on the assumed functional form Eq. (1), which presumes a sharp interface and a thickness-independent interfacial torque. They acknowledge this themselves. The thickness-independent intercept is suggestive, but it is not a unique physical decomposition. I would treat it as a model-dependent estimate, not a direct measurement of a well-defined quantity.

Citation pattern is fine; they extend their own Ref. [26] and cite the relevant prior experimental and theoretical work. No invented entities, no data overclaim beyond the MR issue.

Who this is for: spintronics theorists and experimentalists working on SOT/SMR in Co/Pt and Co/Au. It deserves a serious referee. Send it to peer review with the expectation of heavy revision: Eq. (4) must be fixed and the quantitative MR claim re-derived. If that works out, the paper is a solid contribution. If not, the SOT decomposition alone is still worth publishing.

Referee Report

3 major / 5 minor

Summary. The manuscript reports first-principles non-equilibrium Green's function (NEGF) calculations of spin-orbit torque (SOT) and magnetoresistance in Co/Pt and Co/Au bilayers with explicit Anderson disorder. The damping-like SOT is fitted to τ0 + τSH[1 - sech(dN/lsf)], yielding a thickness-independent interfacial part τ0 comparable to the spin-Hall part τSH. The transverse magnetoconductance Δyzg in Co/Pt is reported to be of order e²/h, exceeding the spin-Hall magnetoresistance prediction of Eq. (4) by more than an order of magnitude, which the authors attribute to an interfacial contribution. The paper also analyzes the field-like torque and the planar-Hall-like term, and proposes that the spin-Hall mechanism cannot account for the observed magnetoconductance.

Significance. If the central quantitative claim holds, the paper would provide first-principles evidence that interfacial transport processes dominate the magnetoresistance of Co/Pt bilayers, complementing similar conclusions for damping-like SOT. The methodological strengths include the use of NEGF with explicit disorder, the systematic expansion of SOT in vector spherical harmonics, and a genuine cross-observable check: θSH and lsf are fitted to SOT and then used to predict the magnetoresistance, not fitted to it. However, the dimensional inconsistency in Eq. (4) and the model dependence of the SOT decomposition currently prevent the central quantitative claim from being substantiated as written.

major comments (3)

[Magnetoresistance, Eq. (4)] Equation (4) as printed is dimensionally inconsistent: Δyzg is defined in Eq. (3) as a reduced conductance with units of siemens, while the right-hand side θ_SH² ρ̄ tanh²(d_N/2l_sf) tanh(d_N/l_sf) has units of resistivity (Ω·m). The dashed line in Fig. 2 is additionally 'scaled by a factor of 10' without showing the unscaled curve, so the claim that the spin-Hall mechanism is too weak by more than an order of magnitude is not reproducible from the manuscript. The authors should re-derive the spin-diffusion SMR expression from the cited references, state any unit convention or missing length factor explicitly, and plot the unscaled benchmark.
[Thickness dependence of SOT, Eq. (1)] The decomposition of the damping-like torque into τ0 + τSH[1 - sech(dN/lsf)] assumes a strictly thickness-independent interfacial contribution. The paper itself warns that this 'assumes a geometrical interface between homogeneous bulk regions and ignores thickness-dependent perturbations and finite-size effects.' Because the central claim of a comparable interfacial contribution rests on the fitted τ0, the authors should provide a sensitivity analysis, for example by allowing τ0 to have a weak thickness dependence or by fitting only data in a range where the interface is expected to be converged. Without such a test, the separation of τ0 and τSH in Table I is model-dependent.
[Magnetoresistance, comparison with computed Δyzg] Even after fixing the units in Eq. (4), the benchmark uses the same θSH and lsf values that were obtained from the model-dependent SOT decomposition of Eq. (1). The comparison in Fig. 2 therefore inherits the assumptions of that decomposition. The manuscript should acknowledge this coupling explicitly and, if possible, show how the benchmark would change if the SOT fit parameters are varied within their uncertainty.

minor comments (5)

[Abstract and Conclusions] The abstract and conclusions state that Δyzg is 'of the order of a conductance quantum per interfacial atom,' but the body text (Magnetoresistance section) states it is 'of the order of one conductance quantum' without a per-atom normalization. Please clarify which quantity is meant and define the normalization.
[Fig. 2] The dashed line is described as 'scaled by a factor of 10' but the unscaled prediction is not shown. Please include the unscaled curve (or an inset) so the reader can directly compare the computed points with the theoretical benchmark.
[Thickness dependence of SOT, first paragraph] There is a typo: 'τ0 andτSH' should have a space after 'and'.
[Magnetoresistance, last paragraph] The notation '∆gyz/g' in the paragraph on the growth at small thicknesses is inconsistent with the earlier definition '∆µνg/g'; please unify the notation.
[Eq. (2)] The quantity 'M/A' is used in the definition of θSH but is not explicitly defined in the main text; please define it as the total magnetic moment per unit area and specify its units.

Circularity Check

1 steps flagged · score 5.0 of 10

The interfacial SOT contribution is defined by the fitted form of Eq. (1) rather than independently derived, while the magnetoresistance benchmark is a genuine cross-check; circularity is partial.

self definitional [Thickness dependence of SOT, Eq. (1) and the paragraph after Fig. 1]
"The damping-like SOT coefficient C (1) 1,−1 is well described by the function C (1) 1,−1 = τ0 + τSH [1 − sech(dN/lsf)] (1) with the parameters listed in Table I, where τ0 represents the thickness-independent interfacial contribution to SOT and τSH the conventional spin-Hall-generated part. Importantly, we find that τ0, which can appear due to interface scattering [8, 9], is comparable with τSH in both systems."

τ0 is not an independently computed or predicted interfacial torque; it is, by construction, the zero-thickness intercept of the chosen fitting function. Any thickness-independent residual after subtracting the assumed spin-diffusion term sech(dN/lsf) is assigned the label 'interfacial.' The paper's conclusion that damping-like SOT has an interfacial contribution comparable to the spin-Hall effect therefore restates the fit assumption rather than deriving it from first principles. The paper itself concedes that Eq. (1) 'assumes a geometrical interface between homogeneous bulk regions and ignores thickness-dependent perturbations and finite-size effects in the electronic structure of the bilayer.'

full rationale

The circularity is localized to the SOT decomposition. The magnetoresistance portion is not circular in the same sense: θ_SH, l_sf, and ρ̄ are fitted to the SOT thickness dependence and then used to form the Δyzg_SH benchmark for an independently computed magnetoconductance, which is a legitimate cross-observable test. However, Eq. (4) as printed is dimensionally inconsistent (left side is a reduced conductance while the right side retains resistivity units), and the dashed benchmark is only shown scaled by a factor of 10; these are correctness concerns rather than circularity. The self-citations to Ref. [26] (NEGF methodology and the previous statement about Co spin-orbit coupling) are not load-bearing in a circular way: the methodology is externally anchored, and the Co-SOC statement is re-verified for additional stacks in this paper. Overall, the central SOT claim reduces appreciably to the definition of τ0 in the fitting model, while the MR claim retains independent content, so the paper is partially but not wholly circular.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on model assumptions rather than invented entities. The SOT fit introduces three free parameters whose separation into interfacial and bulk terms is model-dependent. The SMR comparison relies on a phenomenological spin-Hall formula. No new particles, forces, or conserved quantities are introduced.

free parameters (4)

tau0 (thickness-independent SOT intercept) = 110.3 ns/m (Co/Pt, Vm=1.09 eV); 54.6 ns/m (Co/Au, Vm=1.09 eV)
Thickness-independent term in Eq. (1), interpreted as interfacial SOT. It is an extrapolated intercept, not a directly measured quantity.
tauSH (spin-Hall SOT coefficient) = 96.5 ns/m (Co/Pt, Vm=1.09 eV); 108.9 ns/m (Co/Au, Vm=1.09 eV)
Saturating spin-Hall term in Eq. (1), fitted to the same thickness series and used to estimate the spin-Hall angle via Eq. (2).
lsf (spin-diffusion length) = 1.94 nm (Co/Pt, Vm=1.09 eV); 3.61 nm (Co/Au, Vm=1.09 eV)
Spin-diffusion length in Eq. (1), fitted along with tau0 and tauSH.
Vm (Anderson disorder amplitude) = 0.77 or 1.09 eV
Chosen disorder strengths, used to test robustness. Not fitted to the target data, but a hand-chosen model parameter.

assumptions (4)

domain assumption Anderson disorder model captures dominant disorder scattering in these metallic bilayers.
The calculation replaces real microstructure with a uniform random potential Vm; the central thickness comparisons depend on this model.
domain assumption Fermi-surface contribution to SOT dominates at room temperature.
Invoked in the methods section; the Fermi-sea term is omitted based on Ref. 26.
ad hoc to paper The bulk spin-Hall SOT contribution follows tauSH [1 - sech(dN/lsf)].
Eq. (1) is assumed as the fitting form; the authors explicitly warn it ignores thickness-dependent perturbations. The tau0/tauSH split depends on this form.
domain assumption Spin-Hall SMR can be estimated from Eq. (4) with theta_SH and lsf taken from the SOT fit.
The comparison of magnetoconductance to the spin-Hall model uses a phenomenological formula with parameters transferred from SOT; the printed formula lacks a stated length prefactor.

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Cite this review

Pith. "Pith review of Interfacial contributions to spin-orbit torque and magnetoresistance in ferromagnet/heavy-metal bilayers." pith.science (2026). https://pith.science/paper/JRKK7LNC

@misc{pith2026190802680,
  author       = {Pith},
  title        = {Pith review of: Interfacial contributions to spin-orbit torque and magnetoresistance in ferromagnet/heavy-metal bilayers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JRKK7LNC}},
  note         = {Machine review of arXiv:1908.02680}
}

read the original abstract

The thickness dependence of spin-orbit torque and magnetoresistance in ferromagnet/heavy-metal bilayers is studied using the first-principles non-equilibrium Green's function formalism combined with the Anderson disorder model. A systematic expansion in orthogonal vector spherical harmonics is used for the angular dependence of the torque. The damping-like torque in Co/Pt and Co/Au bilayers can be described as a sum of the spin-Hall contribution, which increases with thickness in agreement with the spin-diffusion model, and a comparable interfacial contribution. The magnetoconductance in the plane perpendicular to the current in Co/Pt bilayers is of the order of a conductance quantum per interfacial atom, exceeding the prediction of the spin-Hall model by more than an order of magnitude. This suggests that the "spin-Hall magnetoresistance," similarly to the damping-like torque, has a large interfacial contribution unrelated to the spin-Hall effect.

Figures

Figures reproduced from arXiv: 1908.02680 by the authors.

**Figure 2.** FIG. 2. Dependence of the magnetoconductances ∆ [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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K. D. Belashchenko, A. A. Kovalev, and M. van Schilf- gaarde, Phys. Rev. Materials 3, 011401 (2019)

work page 2019

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