Current-driven domain wall dynamics in ferrimagnets: micromagnetic approach and collective coordinates model

Eduardo Mart\'inez; \'Oscar Alejos; V\'ictor Raposo

arxiv: 1907.06431 · v1 · pith:JAIDA6MAnew · submitted 2019-07-15 · ⚛️ physics.app-ph · cond-mat.mes-hall

Current-driven domain wall dynamics in ferrimagnets: micromagnetic approach and collective coordinates model

Eduardo Mart\'inez , V\'ictor Raposo , \'Oscar Alejos This is my paper

Pith reviewed 2026-05-24 21:18 UTC · model grok-4.3

classification ⚛️ physics.app-ph cond-mat.mes-hall

keywords ferrimagnetsdomain wall dynamicscurrent-driven motionmicromagnetic simulationscollective coordinatesangular momentum compensationspintronics

0 comments

The pith

Two-subsystem micromagnetic and collective models show angular momentum compensation explains linear domain wall velocity increase with current in ferrimagnets at specific temperatures or compositions.

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

The paper develops micromagnetic simulations of ferrimagnetic alloys treated as two magnetic subsystems linked by exchange coupling, each subsystem having its own gyromagnetic ratio and temperature dependence. These simulations are supported by a collective coordinates model that likewise keeps the subsystems distinct rather than collapsing them into effective parameters. Both approaches identify angular momentum compensation as the mechanism behind the observed linear rise in domain wall speed as current increases, occurring at a particular temperature or alloy composition. The two-subsystem framework further allows extraction of predictions for experimental configurations that cannot be reached with effective single-parameter descriptions.

Core claim

Both the micromagnetic simulations and the collective coordinates model confirm that angular momentum compensation accounts for the linear increase with current of domain wall velocities in ferrimagnetic alloys at a certain temperature or composition. The representation of the alloy as two coupled subsystems, each carrying independently assignable gyromagnetic ratios, temperature dependences and torques, makes it possible to infer results relevant to future experimental setups that remain inaccessible when effective parameters are used instead.

What carries the argument

Representation of the ferrimagnetic alloy as two magnetic subsystems coupled by an additional exchange interaction, each subsystem having independently assignable gyromagnetic ratios, temperature dependences and torques.

If this is right

Domain wall velocity rises linearly with current exactly when angular momentum is compensated.
The velocity-versus-current relation can be tuned by temperature or composition through the compensation point.
Spin-orbit torques or anisotropic exchange can be assigned separately to each subsystem, producing distinct dynamical outcomes.
Predictions for experimental device geometries become available that effective single-parameter models cannot supply.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same two-subsystem treatment could be applied directly to antiferromagnetically coupled multilayers without first deriving effective parameters.
Device design for high-speed domain wall motion could exploit the compensation point to achieve current-linear response rather than saturating behavior.
If the two-subsystem picture holds, earlier effective models may have systematically underestimated the role of compensation in limiting or enhancing mobility.

Load-bearing premise

The ferrimagnetic alloy can be accurately represented as two distinct magnetic subsystems coupled only by an additional exchange interaction, each with independently assignable gyromagnetic ratios, temperature dependences, and torques.

What would settle it

A measurement showing that domain wall velocity does not increase linearly with current precisely at the angular momentum compensation temperature or composition would falsify the claim that compensation produces the linear dependence.

Figures

Figures reproduced from arXiv: 1907.06431 by Eduardo Mart\'inez, \'Oscar Alejos, V\'ictor Raposo.

**Figure 2.** Figure 2: Transient response of a DW under the effect of a current pulse. Pulse amplitude [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Dependence on current density JHM of (a) DW terminal velocity vst, and (b) DW stationary angles ψ1,st and ψ2,st. Dots correspond to µM simulations, while continuous lines are the results drawn by the 1DM. 3.2. Stationary response of domain walls in ferrimagnets 3.2.1. Temperature dependence of CDDWD FIG.3 shows the dependencies of DW terminal velocities, vst = ˙qst, and stationary DW angles ψ1,st, and ψ2,s… view at source ↗

**Figure 4.** Figure 4: Dependence of the terminal velocity vst with temperature for different driving currents obtained from µM simulations (dots), and the 1DM (continuous lines). current direction: vst = γ0 ∆ α π 2 ¯hθSHJHM 2 |e| µ0tF iM g1g2 (g2Ms,1 + g1Ms,2) . (7) The general dependence of vst on temperature for different driving currents is plotted in FIG.4. Low currents (blue curve) are not sufficient to disorientate sublat… view at source ↗

**Figure 5.** Figure 5: Dependence on longitudinal applied field [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

**Figure 6.** Figure 6: Dependence on current density JHM with the transverse applied field By as a parameter of (a) DW terminal velocities vst at T = 240K, (b) DW stationary angles ψ1,st and ψ2,st at T = 240K, (c) DW terminal velocities vst at T = 260K, and (d) DW stationary angles ψ1,st and ψ2,st at T = 260K. Again, dots correspond to µM simulations, while continuous lines are the results drawn by the 1DM. 3.3. Realistic sample… view at source ↗

**Figure 7.** Figure 7: Emergence of a threshold current for strips with imperfections: (a) 5% anisotropy [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗

**Figure 8.** Figure 8: µM snapshots of the CDDWD of two adjacent DWs along curved paths. The current density in the straight paths, where it is uniform, is of JHM = 1.5 TA m2 , and the temperature chosen is T = TA. The panels represent: a) out of plane component mz,1 of the magnetization along the first sublattice, b) out of plane component mz,2 of the magnetization along the second sublattice, c) out of plane component of the n… view at source ↗

read the original abstract

Theoretical studies dealing with current-driven domain wall dynamics in ferrimagnetic alloys and, by extension, other antiferromagnetically coupled systems as some multilayers, are here presented. The analysis has been made by means of micromagnetic simulations that consider these systems as constituted by two subsystems coupled in terms of an additional exchange interlacing them. Both subsystems differ in their respective gyromagnetic ratios and temperature dependence. Other interactions, as for example anisotropic exchange or spin-orbit torques, can be accounted for differently within each subsystem according to the physical structure. Micromagnetic simulations are also endorsed by means of a collective coordinates model which, in contrast with some previous approaches to these antiferromagnetically coupled systems, based on effective parameters, also considers them as formed by two coupled subsystems with experimentally definite parameters. Both simulations and the collective model reinforce the angular moment compensation argument as accountable for the linear increase with current of domain wall velocities in these alloys at a certain temperature or composition. Importantly, the proposed approach by means of two coupled subsystems permits to infer relevant results in the development of future experimental setups that are unattainable by means of effective models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper builds a two-subsystem micromagnetic and collective-coordinate model for ferrimagnetic domain walls that assigns separate gyromagnetic ratios and temperature dependences to each sublattice instead of effective parameters.

read the letter

The main takeaway is that the authors treat the ferrimagnet as two exchange-coupled subsystems in both the micromagnetic simulations and the collective coordinates model, each with its own gyromagnetic ratio and temperature dependence. This lets them apply torques or anisotropies differently to each part and still recover the linear domain-wall velocity versus current at angular-momentum compensation. The collective model is written with experimentally assignable parameters for each subsystem rather than lumped effective ones, which is the clearest difference from earlier treatments. That framing is useful for anyone who needs to keep the two sublattices distinct when the material is not perfectly symmetric. The simulations and the reduced model are said to line up on the compensation point, which gives some internal consistency check. The approach also opens a route to include spin-orbit torques or anisotropic exchange on one subsystem only, which could matter for real alloys or multilayers. The soft spots are modest but real. The abstract states that the two-subsystem form permits results unattainable with effective models, yet no explicit comparison or new prediction is shown, so the practical gain remains unquantified. The central reinforcement of the compensation argument rests on the simulations and the collective model, but without the actual equations or parameter tables it is impossible to judge how strongly the evidence supports it or whether the extra variables simply reproduce known behavior. The assumption that the alloy splits cleanly into two subsystems with independent properties is presented as a modeling choice, but its accuracy for a given composition is not tested here. This work is for researchers already modeling current-driven walls in ferrimagnets or antiferromagnetically coupled stacks. A reader who needs a concrete way to keep sublattice parameters separate might extract a usable template, but the paper does not open a new class of devices or resolve an open experimental discrepancy. It is solid enough on its own terms to deserve a serious referee who can check the derivations and any direct comparisons to prior collective-coordinate forms.

Referee Report

0 major / 2 minor

Summary. The manuscript develops a micromagnetic model treating ferrimagnetic alloys as two exchange-coupled subsystems with independent gyromagnetic ratios and temperature dependences, along with a collective-coordinates model that employs experimentally definite parameters for each subsystem rather than effective ones. Both approaches are used to argue that angular-momentum compensation accounts for the linear rise of domain-wall velocity with current density at a compensation temperature or composition, and that the two-subsystem framework permits predictions for experimental setups that effective models cannot provide.

Significance. If the central results hold, the work supplies a more physically grounded alternative to effective-parameter treatments of antiferromagnetically coupled systems. The explicit use of subsystem-specific gyromagnetic ratios, temperature dependences, and torques is a methodological strength that could aid the design of future spintronic experiments.

minor comments (2)

The abstract states that both simulations and the collective model 'reinforce' the compensation argument, yet the manuscript would benefit from an explicit statement (e.g., in §4 or §5) of the quantitative metric used to establish this reinforcement, such as the slope of v(J) or the temperature at which linearity is recovered.
Notation for the two subsystems (e.g., labels for M1, M2, γ1, γ2) should be introduced once in §2 and used consistently thereafter to avoid ambiguity when comparing to effective models.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The recognition that our two-subsystem treatment supplies a more physically grounded alternative to effective-parameter models is appreciated.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The abstract and available text present the two-subsystem micromagnetic model and collective coordinates approach as an explicit modeling choice using experimentally definite parameters, explicitly contrasted with prior effective-parameter models. No equations, fitted inputs renamed as predictions, self-citations, or ansatzes are quoted or described that would reduce any central claim to its own inputs by construction. The reinforcement of the angular-momentum compensation argument is positioned as an outcome of the simulations and model rather than a definitional tautology. The derivation chain is therefore self-contained against external benchmarks with no load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The central modeling choice of two exchange-coupled subsystems with independent gyromagnetic ratios is treated as a domain assumption whose validity cannot be assessed.

pith-pipeline@v0.9.0 · 5737 in / 1131 out tokens · 19780 ms · 2026-05-24T21:18:41.327352+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

two subsystems coupled... differing in their respective gyromagnetic ratios and temperature dependence... experimentally definite parameters
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

angular moment compensation argument as accountable for the linear increase with current of domain wall velocities

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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Thiaville, Y

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Stuart S. P. Parkin, Masamitsu Hayashi, and Luc Thomas. Magnetic do- main wall racetrack memory. Science, 320:190, 2008

work page 2008

[2] [2]

Electric control of multiple domain walls in pt/co/pt nanotracks with perpendicular magnetic anisotropy

Kab-Jin Kim, Jae-Chul Lee, Sang-Jun Yun, Gi-Hong Gim, Kang-Soo Lee, Sug-Bong Choe, and Kyung-Ho Shin. Electric control of multiple domain walls in pt/co/pt nanotracks with perpendicular magnetic anisotropy. Ap- plied Physics Express, 3:083001, 2010

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Thiaville, S

A. Thiaville, S. Rohart, E. Jue, V. Cros, and A. Fert. Dynamics of dzyaloshinskii domain walls in ultrathin magnetic ﬁlms. Europhysics Let- ters, 100:57002, 2012

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G. Chen, J. Zhu, A. Quesada, J. Li, A. T. N’Diaye Y. Huo, T. P. Ma, Y. Chen, H.Y. Kwon, C. Won, Z. Q. Qiu, A. K. Schmid, and Y. Z. Wu. Physical Review Letters, 110:177204, 2013. 16

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Tetienne, T

J.-P. Tetienne, T. Hingant, L.J. Mart´ ınez, S. Rohart, A. Thiaville, L. Her- rera Diez, K. Garcia, J.-P. Adam, J.-V. Kim, J.-F. Roch, I.M. Miron, G. Gaudin, L. Vila, B. Ocker, D. Ravelosona, and V. Jacques. The nature of domain walls in ultrathin ferromagnets revealed by scanning nanomag- netometry. Nature Communications, 6:6733, 2015

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Mougin, M

A. Mougin, M. Cormier, J. P. Adam, P. J. Metaxas, and J. Ferr´ e. Do- main wall mobility, stability and walker breakdown in magnetic nanowires. Europhysics Letters, 5:57007, 2007

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Eduardo Martinez. Static properties and current-driven dynamics of domain walls in perpendicular magnetocrystalline anisotropy nanostrips with rectangular cross-section. Advances in Condensed Matter Physics , 2012:954196, 2012

work page 2012

[8] [8]

Eduardo Martinez, Satoru Emori, Noel Perez, Luis Torres, and Geoﬀrey S. D. Beach. Current-driven dynamics of dzyaloshinskii domain walls in the presence of in-plane ﬁelds: Full micromagnetic and one-dimensional analysis. Journal of Applied Physics , 115:213909, 2014

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Garg, S.-H

C. Garg, S.-H. Yang, T. Phung, A. Pushp, and S. S. P. Parkin. Dramatic inﬂuence of curvature of nanowire on chiral domain wall velocity. Science Advances, 3, 2017

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[10] [10]

Robin Bl¨ asing, Tianping Ma, See-Hun Yang, Chirag Garg, Fasil Kidane Dejene, Alpha T N’Diaye, Gong Chen, Kai Liu, and Stuart S. P. Parkin. Exchange coupling torque in ferrimagnetic co/gd bilayer maximized near angular momentum compensation temperature. Nature Communications, 9:4984, 2018

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Current-driven domain wall dynamics in ferromagnetic layers synthetically exchange-coupled by a spacer: A micromagnetic study

´Oscar Alejos, V´ ıctor Raposo, Luis Sanchez-Tejerina, Riccardo Tomasello, Giovanni Finocchio, and Eduardo Martinez. Current-driven domain wall dynamics in ferromagnetic layers synthetically exchange-coupled by a spacer: A micromagnetic study. Journal of Applied Physics, 123(1):013901, 2018

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G¨ unther, Piet Hessing, Alexandra Churikova, Christopher Klose, Michael Schneider, Dieter Engel, Colin Marcus, David Bono, Kai Bagschik, Stefan Eisebitt, and Geoﬀrey S

Lucas Caretta, Maxwell Mann, Felix B¨ uttner, Kohei Ueda, Bastian Pfau, Christian M. G¨ unther, Piet Hessing, Alexandra Churikova, Christopher Klose, Michael Schneider, Dieter Engel, Colin Marcus, David Bono, Kai Bagschik, Stefan Eisebitt, and Geoﬀrey S. D. Beach. Fast current-driven domain walls and small skyrmions in a compensated ferrimagnet. Nature Na...

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Spin-orbit torque-induced switching in ferrimagnetic alloys: Experiments and model- ing

Soong-Geun Je, Juan-Carlos Rojas-S´ anchez, Thai Ha Pham, Pierre Val- lobra, Gregory Malinowski, Daniel Lacour, Thibaud Fache, Marie-Claire Cyrille, Dae-Yun Kim, Sug-Bong Choe, Mohamed Belmeguenai, Michel Hehn, St´ ephane Mangin, Gilles Gaudin, and Olivier Boulle. Spin-orbit torque-induced switching in ferrimagnetic alloys: Experiments and model- ing. App...

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Low magnetic damping of ferrimagnetic gdfeco alloys

Duck-Ho Kim, Takaya Okuno, Se Kwon Kim, Se-Hyeok Oh, Tomoe Nishimura, Yuushou Hirata, Yasuhiro Futakawa, Hiroki Yoshikawa, Arata Tsukamoto, Yaroslav Tserkovnyak, Yoichi Shiota, Takahiro Moriyama, Kab-Jin Kim, Kyung-Jin Lee, and Teruo Ono. Low magnetic damping of ferrimagnetic gdfeco alloys. Physical Review Letters, 122:127203, 2019

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P. P. J. Haazen, E. Mure, J. H. Franken, R. Lavrijsen, H. J. M. Swagten, and B. Koopmans. Nature Materials, 12:299, 2013

work page 2013

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Zhang and Z

S. Zhang and Z. Li. Physical Review Letters, 93:1, 2004

work page 2004

[17] [17]

Ma, Xiaopu Li, and S

Chung T. Ma, Xiaopu Li, and S. Joseph Poon. Micromagnetic simulation of ferrimagnetic tbfeco ﬁlms with exchange coupled nanophases. Journal of Magnetism and Magnetic Materials , 417:197–202, 2016

work page 2016

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Nature Nanotechnology, 8:527–533, 2013

Kwang-Su Ryu, Luc Thomas, See-Hun Yang, and Stuart Parkin. Nature Nanotechnology, 8:527–533, 2013

work page 2013

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Satoru Emori, Uwe Bauer, Sung-Min Ahn, Eduardo Mart´ ınez, and Geoﬀrey S. D. Beach. Nature Materials, 12:611–616, 2013

work page 2013

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Mart´ ınez, S

E. Mart´ ınez, S. Emori, and G. S. D. Beach. Applied Physics Letters , 103:072406, 2013

work page 2013

[21] [21]

Vogel, M

J. Vogel, M. Bonﬁm, N. Rougemaille, O. Boulle, M. Miron, S. Auﬀret, B. Rodmacq, G. Gaudin, J. C. Cezar, F. Sirotti, and S. Pizzini. Direct observation of massless domain wall dynamics in nanostripes with perpen- dicular magnetic anisotropy. Physical Review Letters, 108:247202, 2012

work page 2012

[22] [22]

Tunable in- ertia of chiral magnetic domain walls

Jacob Torrejon, Eduardo Mart´ ınez, and Masamitsu Hayashi. Tunable in- ertia of chiral magnetic domain walls. Nature Communications, 7:13533, 2016

work page 2016

[23] [23]

Calculation of the gilbert damping matrix at low scattering rates in gd

Jonas Seib and Manfred F¨ ahnle. Calculation of the gilbert damping matrix at low scattering rates in gd. Physical Review B, 82(6):064401, 2010

work page 2010

[24] [24]

Giant magnetoresistance in spin-valve multilayers.Journal of Magnetism and Magnetic Materials , 136:335–359, 1994

Bernard Di´ eny. Giant magnetoresistance in spin-valve multilayers.Journal of Magnetism and Magnetic Materials , 136:335–359, 1994

work page 1994

[25] [25]

Sun, and Stuart S

Xin Jiang, Li Gao, Jonathan Z. Sun, and Stuart S. P. Parkin. Temperature dependence of current-induced magnetization switching in spin valves with a ferrimagnetic cogd free layer. Physical Review Letters, 97:217202, 2006

work page 2006

[26] [26]

Thiaville, Y

A. Thiaville, Y. Nakatani, J. Miltat, and N. Vernier. Journal of Applied Physics, 95(11):7049, 2004. 18

work page 2004