Pith. sign in

REVIEW 3 major objections 4 minor 76 references

Dimensional crossover and local strain induced deflection of the spin spiral state in multiferroic NiI2

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Thickness and local bending continuously tune the spin-spiral wavevector in few-layer NiI2 films, giving two practical knobs for engineering helical magnetism and the ferroelectric polarization it carries.

desk verdict Solid new SP-STM thickness series and wrinkle-deflection observations in NiI2; the crossover mechanism is plausible but partly calibrated to the same data it explains, so treat the theory as corroboration, not proof. read the letter →

arxiv 2608.11944 v1 pith:RQ5UCXNO submitted 2026-08-12 cond-mat.mes-hall cond-mat.mtrl-sci

Tianxing Jiang , Lianchuang Li , Haiyan Zhu , Hongyu Wang , Junchao Tian , Wenzhao Wang , Weiyi Pan , Haitao Wang
show 5 more authors
Changlin Zheng Hongjun Xiang Changsong Xu Donglai Feng Tong Zhang
This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.mtrl-sci PACS 75.70.Ak75.25.-m68.37.Ef
keywords spinspiralNiI2dimensionalcrossoverspin-polarizedSTMmultiferroicinterlayerexchangecurvature-inducedmagnetismvanderWaalsmagnets
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

Using spin-polarized scanning tunneling microscopy on films grown one to seven atomic layers thick, this paper tracks how the helical magnetic order of NiI2 changes as the material approaches the two-dimensional limit. It finds that as thickness grows, the spin-spiral wavelength increases from about 1.7 nm toward 2.1 nm and the wavevector rotates continuously from near [110] toward [1-10], a behavior it attributes to antiferromagnetic interlayer exchange rebalancing the frustrated intralayer interactions. The same measurements show that wrinkles and bent regions deflect the spiral locally by 40–68 degrees, an effect reproduced in spin-lattice dynamics simulations and traced to curvature-induced changes in the exchange parameters. If the interpretation is right, film thickness and local strain become two independent tuning methods for non-collinear helical magnetism and the electric polarization that accompanies it in van der Waals multiferroics.

What carries the argument

The central object is a single-q spin-spiral energy functional for AA-stacked NiI2 built from the frustrated intralayer Heisenberg exchanges J1–J3, a Kitaev term K (a bond-directional anisotropic exchange), and a biquadratic term B, with interlayer Heisenberg exchanges included as effective intralayer modifications for an N-layer film. The load-bearing identity is the mapping of antiferromagnetic interlayer coupling onto effective ferromagnetic intralayer couplings, $J_{1,\mathrm{eff}} = J_1 - \frac{2(N-1)}{N} J_2^\perp$ and $J_{2,\mathrm{eff}} = J_2 - \frac{N-1}{N} J_{3,2}^\perp$, which lets the thickness trajectory be read off a monolayer phase diagram whose incommensurate ICα region supports continuous rotation of the wavevector between crystallographic directions. For curvature, the machinery is a machine-learned spin-lattice dynamics potential that relaxes a bent monolayer and predicts how bending redistributes J3/J1 and K spatially, producing the deflected spiral in simulation.

What would settle it

A layer-resolved spin-polarized STM experiment that images the spin phase on both sides of a step in a 3- or 4-ML NiI2 terrace would settle the interlayer assumption: finding a phase clearly different from 180 degrees, or finding that the measured 3-ML wavevector falls far from the model's q(N) curve when computed interlayer couplings are used without rescaling, would falsify the thickness-driven dimensional-crossover claim.

Watch

Extended reading notes

Core claim

The central claim is that the spin-spiral state in few-layer NiI2 is continuously tunable by thickness and by local curvature. In AA-stacked films from 1 to 7 monolayers, the authors observe a monotonic increase of the spiral wavelength and a rotation of its wavevector (the direction and period of the magnetic modulation) from about 7 degrees to about 26 degrees relative to [110], converging toward but not reaching the bulk value, and they attribute this to progressively stronger antiferromagnetic interlayer exchange acting through effective intralayer couplings that scale as (N−1)/N. At wrinkles with curvature around 10–20 inverse micrometers, the spiral direction locally reorients by 40–68 degrees and then recovers; spin-lattice dynamics simulations show that bending changes the J3/J1 ratio and the Kitaev interaction in the curved region. The paper concludes that thickness changes the global balance of competing interactions while local strain perturbs them at the nanoscale, each providing a distinct knob to reorient the helical magnetic order and to manipulate the associated ferroelectric polarization.

Load-bearing premise

The thickness model assumes adjacent NiI2 layers are strictly antiparallel, treats every layer as identical with averaged (N−1)/N effective couplings, and rescales the computed interlayer exchange ratios once to match the same experimental data being explained; if the interlayer angle, stacking registry, or the scale factor changes with thickness, the predicted q(N) trajectory would not be a clean test of dimensional crossover.

Editorial extensions

If this is right

  • At 7 ML the spiral wavelength has nearly saturated while still differing from bulk NiI2, implying the AA stacking of the MBE-grown films defines a distinct thin-film limit whose crossover trajectory depends on stacking order.
  • Because the 2q charge modulation persists at every thickness, the spin-driven electric polarization is expected to persist in few-layer NiI2, so thickness tunes the magnetic and ferroelectric order parameters together.
  • Curvature-induced deflection is a local reorientation of a single-q spiral with no topological charge, distinguishing it from previously reported spin-spiral domain walls and offering a strain-based route to pattern multiferroic domains.
  • Substrate-induced charge transfer is not the controlling factor: NiI2 films on a strongly hole-doping NbSe2 substrate still show nearly the same wavevector as films on HOPG.

Reading between the lines

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

  • A testable extension would be to bend single-layer NiI2 by controlled amounts and measure the deflection angle as a function of curvature, checking whether it follows a monotonic relation set by the local change in J3/J1 and K.
  • The (N−1)/N effective-coupling argument is generic: other frustrated van der Waals helimagnets with antiferromagnetic interlayer stacking should show a similar thickness-driven rotation of the wavevector, and comparing materials such as CoI2 would separate the universal mechanism from NiI2-specific parameters.
  • Because the deflection is localized at the wrinkle, an array of wrinkles could act as a rewritable template: spin-polarized STM combined with local bias or field might write and erase the local spiral orientation, making the polarization pattern mechanically programmable.
  • The paper's wavevector extraction assumes a single-q spiral; if a thickness or strain regime with multi-q states or competing domains is found, the FFT-based analysis alone could misread the texture, so phase-resolved real-space imaging of deflected regions would sharpen the claim.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 manuscript reports a combined SP-STM and multi-scale modeling study of few-layer NiI2 films (1-7 monolayers) grown by MBE. The authors observe that as film thickness increases, the spin-spiral wavelength increases and the wavevector rotates from near [110] toward [1-10], which they attribute to a dimensional crossover driven by enhanced interlayer exchange interactions. They also observe that local wrinkles and bent regions deflect the spin-spiral wavevector, and they use SpinGNN++ spin-lattice dynamics simulations to argue that this deflection arises from curvature-induced modification of intralayer J3/J1 and Kitaev interactions. The experimental work includes FFT-based wavevector extraction from independent terraces, a NbSe2 substrate control, and stacking characterization by STM and STEM. The modeling includes an analytical single-q spin Hamiltonian, DFT extraction of interlayer parameters, and a machine-learned spin-lattice potential for the curvature simulations.

Significance. If the central mechanistic claims hold, the paper establishes two experimentally accessible tuning knobs—film thickness and local curvature—for non-collinear helical magnetism and associated electric polarization in a van der Waals multiferroic. The experimental dataset is a valuable systematic map of q(N) for 1-7 ML NiI2, and the NbSe2 control strengthens the claim that substrate charge transfer does not dominate the thickness evolution. The SpinGNN++ simulations provide an independent route to connect local structural deformation with exchange modification, and the reported test-set MAE of 0.036 meV/atom and reproduced energy rankings support the reliability of that potential. However, the mechanistic attribution of the thickness crossover is weakened by admitted fitting of model parameters to the same experimental data used for comparison, so the paper's central claim requires an uncalibrated or out-of-sample validation to be fully convincing.

major comments (3)
  1. [Supplement Part II-1 and Fig. 4d/e] The comparison in Fig. 4d/e is not an independent prediction of the dimensional-crossover mechanism. Supplement Part II-1 states that J3 was enhanced from 2.25 meV to 2.63 meV to match the monolayer wavevector and that the interlayer interaction magnitudes were scaled "to align with experimental data" after fixing their DFT ratios. Because the same experimental q(N) points are used to set at least one free scale, the calculated thickness trajectory can reproduce the trend by construction. The authors should provide an uncalibrated prediction using the raw DFT interlayer magnitudes (or a band reflecting the Ueff dependence shown in Table S1), or an out-of-sample test such as predicting q for a thickness or stacking variant not used in the fitting.
  2. [Supplement Part II-5 and Fig. 2f] The effective intralayer mapping J1,eff = J1 - 2(N-1)/N J2_perp assumes strictly antiparallel interlayer alignment (S_R0 = -S_R0,perp) for every adjacent layer and represents the film as N identical layers with (N-1)/N averaged couplings. The SP-STM step-edge data in Fig. 2f show a half-period offset at one 3ML/4ML step, which is suggestive but does not establish a 180-degree interlayer phase for all interior interfaces of all thicknesses, especially as q rotates with N. If the interlayer phase deviates from pi or if layer-resolved variations (surface versus interior layers, local stacking fluctuations) are present, the fitted scale factor can absorb these effects and the q(N) trajectory in Fig. 4c-e would no longer be a valid test of the interlayer-exchange mechanism. The authors should state what evidence constrains the interlayer phase and quantify the sensitivity of the predicted trajectory to phi and to layer-dependent parameters.
  3. [Section 3.3 and Supplement Part III-3] The attribution of the curvature-induced deflection to local reduction of J3/J1 and Kitaev interaction K rests on exchange tensors predicted by the SpinGNN++/Spin-Allegro++ potential on a curved lattice. While the potential's overall accuracy is documented for flat monolayer configurations, no direct DFT verification is provided for the bent supercell itself, where bond lengths and angles deviate substantially from the training distribution. A direct DFT calculation of exchange parameters for a few representative locally curved configurations (or at least a comparison of J3/J1 and K predicted by the ML potential against DFT for those configurations) would materially strengthen the claim that curvature-induced exchange modification is the microscopic cause of the observed deflection.
minor comments (4)
  1. [Fig. 4b caption] The caption describes the phase diagram as being in "J3-K space," while the text in Section 3.2 states that the diagram is in "(J1, K) space" with J1 set to -1 meV; please make the axes consistent.
  2. [Section 3.3 and Supplement Part I-5] The main text refers to "Fig. S6 (online)" for the bond-length and bond-angle changes in the curved lattice, but in the Supplement Fig. S6 is the NbSe2 substrate control data; the figure numbering for the curved-lattice structural analysis should be corrected.
  3. [Discussion and Conclusion] The phrase "turning knobs" in the final paragraph should read "tuning knobs" to match the abstract.
  4. [Supplement Part III] The main text consistently uses "SpinGNN++," while the supplement introduces "Spin-Allegro++" as the implementation; please clarify the relationship between these two names in the methods or supplement.

Circularity Check

1 steps flagged · score 5.0 of 10

Thickness-dependence 'prediction' is calibrated: interlayer exchange magnitudes are scaled to align with experiment, and J3 is adjusted to monolayer q, so Fig. 4d/e is not an independent test.

  1. fitted input called prediction [Supplementary Part II-1 (Magnetic parameters for solving the spin Hamiltonian); Figs. 4d-e comparison]
    "The 𝐽𝐽3 was enhanced from 2.25 meV to 2.63 meV to account for the deviation of spin-spiral propagation in monolayer NiI2 ... Due to discrepancies in DFT calculation parameters between the intralayer and interlayer studies, we fixed the ratios of the interlayer interactions and scaled their magnitudes to align with experimental data. ... In Fig. 4d and e we show the calculated q vectors as function of N based on the above model. A good agreement is obtained between calculated orientation of q and the experimental data."

    The agreement in Fig. 4d/e is presented as evidence for the dimensional-crossover mechanism, but two model inputs are calibrated to the very data being compared: J3 is raised by 0.38 meV specifically to reproduce the monolayer propagation direction, and the interlayer exchange magnitudes are scaled by a fixed factor 'to align with experimental data' after fixing the DFT ratios. With a free scale factor absorbed into the effective interlayer couplings, the computed q(N) curve is constrained by the experimental q(N) points rather than independently produced. The DFT-computed ratios and the analytic (N-1)/N mapping provide partial independent content, so the curve is not fully predetermined, but the Fig.

full rationale

The main experimental dataset—monolayer-to-7ML spin-spiral wavelength and orientation—is solid and independently measured by SP-STM, and the curvature-induced deflection part of the paper is largely self-contained: the machine-learned SpinGNN++ potential is trained on DFT-computed spin-lattice energies, not on the wrinkle-deflection data, and the simulated deflection is an independent output. The circularity concern is confined to the thickness model. Supplement II-1 explicitly discloses two calibration steps: J3 is adjusted to match the monolayer spin-spiral, and the DFT-computed interlayer interaction magnitudes are scaled 'to align with experimental data' while only their ratios are kept from DFT. Since the same experimental q(N) data are then used to demonstrate 'good agreement' in Fig. 4d/e, the comparison is not an independent prediction of the dimensional-crossover mechanism; it contains fitted parameters that partially force the result. The self-citations to the prior NiI2 spin-model papers are not uniqueness theorems and do not themselves carry circular weight; the load-bearing issue is the parameter calibration. Overall this is partial circularity rather than complete reduction, because the interlayer ratios and the (N-1)/N layer-counting mapping still supply non-trivial structure to the thickness trajectory.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claims rest on the exchange parameters from prior work with one ad hoc enhancement, a DFT-based interlayer scale that is fitted, and the transferability of a machine-learned potential to curved geometries. No new physical entities are introduced.

free parameters (3)
  • J3 enhancement for monolayer = 2.63 meV (from 2.25 meV)
    Supplement Part II-1: J3 was enhanced from 2.25 to 2.63 meV to account for the deviation of spin-spiral propagation in monolayer NiI2, i.e., to match the observed monolayer wavevector.
  • Interlayer interaction scaling factor = not explicitly given; magnitudes scaled to align with experimental data
    Supplement Part II-1: due to DFT parameter discrepancies, the ratios of interlayer interactions were fixed and their magnitudes scaled to align with experimental data. This is a free scale fitted to the same data the model is later compared with.
  • Ueff choice = 4 eV
    Ueff = 4 eV was chosen because it provides good agreement with experimental data (Supplement Part II-1). Other U values were tested and ratios are similar, so its influence is modest, but it is a hand-picked parameter.
assumptions (5)
  • domain assumption NiI2 has no Dzyaloshinskii-Moriya interaction, so the Hamiltonian contains only Heisenberg, Kitaev, and biquadratic exchanges.
    Stated in Supplement Part II-2, 'there is no DMI in NiI2'. Underlies the analytical energy expression and the single-q minimization.
  • domain assumption The interlayer spin alignment is strictly antiparallel (S_R0 = -S_R0,perp) for all adjacent layers.
    Used in Part II-5 to map interlayer couplings to effective intralayer couplings. Supported by the half-period phase offset across step edges observed by SP-STM, but treated as exact in the model.
  • domain assumption The magnetic ground state is a single-q spin spiral.
    The analytical derivation in Supplement Part II-2 assumes a single-q state with two normal modes, consistent with the stripe-like SP-STM images.
  • domain assumption The J1-J3-K-B classical spin Hamiltonian from prior work [37] describes monolayer NiI2.
    Intralayer parameters are taken from Ref. [37]; the paper assumes this model transfers to few-layer AA-stacked films with interlayer couplings added.
  • domain assumption The machine-learned SpinGNN++ potential, trained on DFT data for flat monolayer NiI2, remains accurate for curved lattices with curvature about 10 μm^-1.
    The curvature simulation in Part III-3 relies on this transferability; the potential reproduces a [110] spiral for the flat monolayer, while the measured q is slightly canted, so the transfer is approximate.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Dimensional crossover and local strain induced deflection of the spin spiral state in multiferroic NiI2." pith.science (2026). https://pith.science/paper/RQ5UCXNO

@misc{pith2026260811944,
  author       = {Pith},
  title        = {Pith review of: Dimensional crossover and local strain induced deflection of the spin spiral state in multiferroic NiI2},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RQ5UCXNO}},
  note         = {Machine review of arXiv:2608.11944}
}
read the original abstract

Low-dimensional multiferroics hold great promise for integrated magnetoelectric devices. Spin spiral state has recently been shown to induce ferroelectricity in single-layer van der Waals (vdW) material NiI2. However, how this state evolves and can be tuned towards the two-dimensional limit remain unclear. Here, we combine spin-polarized scanning tunneling microscopy, layer-by-layer film growth, and multi-scale theoretical modeling to investigate the spin spirals in NiI2 thin films. As the film thickness increases from 1 to 7 monolayers, we observed a continuous increase of spin-spiral wavelength and a rotation of wavevector from near [110] to [1-10] direction, which evidences a dimensional crossover primarily driven by enhanced interlayer exchange energy. Moreover, we find that the film wrinkles can cause deflection of the spin spiral wavevector, which is caused by local curvature induced modification of exchange interactions. Our findings establish thickness and local strain as two tuning methods for engineering non-collinear helical magnetism and accompanied electric polarization in vdW multiferroics.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

76 extracted references · 75 canonical work pages

  1. [110]

    Moreover, we find that the film wrinkles can cause deflection of the spin spiral wavevector, which is caused by local curvature induced modification of exchange interactions

    to [11�0] direction, which evidences a dimensional crossover primarily driven by enhanced interlayer exchange energy . Moreover, we find that the film wrinkles can cause deflection of the spin spiral wavevector, which is caused by local curvature induced modification of exchange interactions. Our findings establish thickness and local strain as two tuning...

    work page

  2. [1]

    Introduction The emergence of layered van der Waals (vdW) magnets has opened a new frontier for exploring exotic magnetic states and designing i ntegrated devices towards the two -dimensional (2D) limit [1-4]. Among them, the noncollinear magnets which break inversion symmetry, such as spin spirals, are of particular interest as they can induce ferroelect...

    work page

  3. [2]

    Sample Preparation: Few-layer NiI2 film were grown by molecular beam epitaxy (MBE) on highly oriented pyrolytic graphite (HOPG) and graphitized SiC(0001) substrates

    Materials and methods 2.1. Sample Preparation: Few-layer NiI2 film were grown by molecular beam epitaxy (MBE) on highly oriented pyrolytic graphite (HOPG) and graphitized SiC(0001) substrates. The HOPG was cleaved and outgassed at 400℃ under high vacuum. Graphitized SiC(0001) was prepared by annealing SiC at 900℃ under Si flux, followed by graphitization ...

    work page 2000

  4. [3]

    mirror domain

    Results 3.1 Surface structure characterization of NiI2 film. As shown in Fig. 1a and b, NiI2 is a layered vdW material with a rhombohedral (𝑅𝑅3�𝑚𝑚) structure, where Ni atoms form a triangular lattice with an in-plane constant of 0.39 nm. The Ni 3d orbitals split into half-occupied 𝑒𝑒𝑔𝑔 and fully occupied 𝑡𝑡2𝑔𝑔 states, producing a local moment of ~2μB per ...

    work page

  5. [4]

    The thickness dependence of q reveals how interlayer interactions reshape the spin spiral state

    Discussion and Conclusion: We have now identified two distinct mechanisms that can modify the spin-spiral wavevectors in few-layer NiI2 films. The thickness dependence of q reveals how interlayer interactions reshape the spin spiral state. For 1ML NiI2, the competing intralayer interactions of J1, J3, and K determine the wavevector q. Upon increasing the ...

    work page

  6. [5]

    Ferroelectricity in spiral magnets

    Mostovoy M. Ferroelectricity in spiral magnets. Phys Rev Lett 2006;96:067601

    work page 2006

  7. [6]

    Local reorientation of spin spirals at structural wrinkles uncovers a microscopic magneto-elastic coupling

    The continuous evolution of the spin spiral wavevector demonstrates layer- dependent dimensional crossover driven by interlayer interactions. Local reorientation of spin spirals at structural wrinkles uncovers a microscopic magneto-elastic coupling. Our findings establish film thickness and local strain as two accessible turning knobs for designing comple...

    work page

  8. [7]

    Magnetism in two -dimensional van der W aals materials

    Burch KS, Mandrus D, Park JG. Magnetism in two -dimensional van der W aals materials. Nature 2018;563:47-52

    work page 2018

Show all 76 references
  1. [8]

    Magnetic 2D materials and heterostructures

    Gibertini M, Koperski M, Morpurgo AF, et al. Magnetic 2D materials and heterostructures. Nat Nanotechnol 2019;14:408-19

    2019

  2. [9]

    Magnetism, symmetry and spin transport in van der Waals layered systems

    Kurebayashi H, Garcia JH, Khan S, et al. Magnetism, symmetry and spin transport in van der Waals layered systems. Nat Rev Phys 2022;4:150-66

    2022

  3. [10]

    Recent advances in two-dimensional ferromagnetic materials-based van der Waals heterostructures

    Zhang ZH, Sun R, Wang ZC. Recent advances in two-dimensional ferromagnetic materials-based van der Waals heterostructures. ACS Nano 2025;19:187-228

    2025

  4. [11]

    Lattice symmetries and regular magnetic orders in classical frustrated antiferromagnets

    Messio L , Lhuillier C, Misguich G . Lattice symmetries and regular magnetic orders in classical frustrated antiferromagnets. Phys Rev B 2011;83:184401

    2011

  5. [12]

    Classifying multiferroics: mechanisms and effects

    Khomskii D. Classifying multiferroics: mechanisms and effects. Physics 2009;2:20

    2009

  6. [13]

    General theory for the ferroelectric polarization induced by spin - spiral order

    Xiang HJ, Kan EJ, Zhang Y, et al. General theory for the ferroelectric polarization induced by spin - spiral order. Phys Rev Lett 2011;107:157202

    2011

  7. [14]

    Multiferroics of spin origin

    Tokura Y, Seki S, Nagaosa N. Multiferroics of spin origin. Rep Prog Phys 2014;77:076501

    2014

  8. [15]

    Multiferroics: different routes to magnetoelectric coupling

    Mostovoy M. Multiferroics: different routes to magnetoelectric coupling. npj Spintron 2024;2:18

    2024

  9. [16]

    Helical spin ordering—1 theory of helical spin configurations

    Nagamiya T. Helical spin ordering—1 theory of helical spin configurations. In: Seitz F, Turnbull D, Ehrenreich H, editors. Solid state physics. New York: Academic Press; 1968, p. 305-411

    1968

  10. [17]

    Dzyaloshinskii–Moriya interaction manipulation in multiferroic Janus monolayers

    Li XY, Wang Z W, Chen Z F, et al. Dzyaloshinskii–Moriya interaction manipulation in multiferroic Janus monolayers. npj Comput Mater 2025;11:123. 15

    2025

  11. [18]

    First -order phase transition with breaking of lattice rotation symmetry in continuous-spin model on triangular lattice

    Tamura R, Kawashima N. First -order phase transition with breaking of lattice rotation symmetry in continuous-spin model on triangular lattice. J Phys Soc Jpn 2011;80:074008

    2011

  12. [19]

    Multiple-q states and the skyrmion lattice of the triangular -lattice Heisenberg antiferromagnet under magnetic fields

    Okubo T, Chung S, Kawamura H. Multiple-q states and the skyrmion lattice of the triangular -lattice Heisenberg antiferromagnet under magnetic fields. Phys Rev Lett 2012;108:017206

    2012

  13. [20]

    Magnetic vortex induced by nonmagnetic impurity in frustrated magnets

    Lin SZ, Hayami S, Batista CD. Magnetic vortex induced by nonmagnetic impurity in frustrated magnets. Phys Rev Lett 2016;116:187202

    2016

  14. [21]

    Arc -shaped structure factor in the J 1−J2−J3 classical Heisenberg model on the triangular lattice

    Glittum C, Syljuåsen OF. Arc -shaped structure factor in the J 1−J2−J3 classical Heisenberg model on the triangular lattice. Phys Rev B 2021;104:184427

    2021

  15. [22]

    Effects of Kitaev interaction on magnetic order and anisotropy

    Li LC, Zhang B H, Chen Z F, et al. Effects of Kitaev interaction on magnetic order and anisotropy. Phys Rev B 2024;110:214435

    2024

  16. [23]

    Dzyaloshinskii-Moriya interaction and magnetic skyrmions induced by curvature

    Ga YL, Cui QR, Liang JH, et al. Dzyaloshinskii-Moriya interaction and magnetic skyrmions induced by curvature. Phys Rev B 2022;106:054426

    2022

  17. [24]

    Magnetism in curved geometries

    Streubel R, Fischer P, Kronast F, et al. Magnetism in curved geometries. J Phys D Appl Phys 2016;49:363001

    2016

  18. [25]

    Interplay between Kitaev interaction and single ion anisotropy in ferromagnetic CrI3 and CrGeTe3 monolayers

    Xu CS, Feng J S, Xiang HJ, et al. Interplay between Kitaev interaction and single ion anisotropy in ferromagnetic CrI3 and CrGeTe3 monolayers. npj Comput Mater 2018;4:57

    2018

  19. [26]

    Strain gradient mediated magnetism and polarization in monolayer VSe2

    Shi WH, Guo YF, Zhang ZH, et al. Strain gradient mediated magnetism and polarization in monolayer VSe2. J Phys Chem C 2019;123:24988-93

    2019

  20. [27]

    Nonlocal chiral symmetry breaking in curvilinear magnetic shells

    Sheka DD, Pylypovskyi OV, Landeros P, et al. Nonlocal chiral symmetry breaking in curvilinear magnetic shells. Commun Phys 2020;3:128

    2020

  21. [28]

    New dimension in magnetism and superconductivity: 3D and curvilinear nanoarchitectures

    Makarov D, Volkov OM, Kákay A, et al. New dimension in magnetism and superconductivity: 3D and curvilinear nanoarchitectures. Adv Mater 2022;34:2101758

    2022

  22. [29]

    Evidence for a single -layer van der Waals multiferroic

    Song Q, Occhialini CA, Ergeçen E, et al. Evidence for a single -layer van der Waals multiferroic. Nature 2022;602:601-5

    2022

  23. [30]

    Strain gradient induced skyrmion in a van der Waals magnet by wrinkling

    Jin S Z, Liu Y J, Deng Z Y, et al. Strain gradient induced skyrmion in a van der Waals magnet by wrinkling. Adv Mater 2025;37:2501935

    2025

  24. [31]

    Shear‐mediated stabilization of spin spiral order in multiferroic NiI2

    Tseng Y, Occhialini CA, Song Q, et al. Shear‐mediated stabilization of spin spiral order in multiferroic NiI2. Adv Mater 2025;37:2417434

    2025

  25. [32]

    Magnetic and structural investigations on NiI2 and CoI2

    Kuindersma SR, Sanchez JP, Haas C. Magnetic and structural investigations on NiI2 and CoI2. Physica B+C 1981;111:231-48

    1981

  26. [33]

    Magnetoelectric responses induced by domain rearrangement and spin structural change in triangular-lattice helimagnets NiI2 and CoI2

    Kurumaji T, Seki S, Ishiwata S, et al. Magnetoelectric responses induced by domain rearrangement and spin structural change in triangular-lattice helimagnets NiI2 and CoI2. Phys Rev B 2013;87:014429

    2013

  27. [34]

    Possible persistence of multiferroic order down to bilayer limit of van der Waals material NiI2

    Ju H, Lee Y, Kim KT, et al. Possible persistence of multiferroic order down to bilayer limit of van der Waals material NiI2. Nano Lett 2021;21:5126-32

    2021

  28. [35]

    Orientation-selective spin-polarized edge states in monolayer NiI 2

    Wang Y, Zhao XL, Yao L, et al. Orientation-selective spin-polarized edge states in monolayer NiI 2. Nat Commun 2024;15:10916

    2024

  29. [36]

    Atomic‐scale visualization of multiferroicity in monolayer NiI2

    Amini M, Fumega AO, González‐Herrero H, et al. Atomic‐scale visualization of multiferroicity in monolayer NiI2. Adv Mater 2024;36:2311342

    2024

  30. [37]

    Giant chiral magnetoelectric oscillations in a van der Waals multiferroic

    Gao FY, Peng X Y, Cheng X L, et al. Giant chiral magnetoelectric oscillations in a van der Waals multiferroic. Nature 2024;632:273-9

    2024

  31. [38]

    Coexistence of ferroelectricity and antiferroelectricity in 2D van der Waals multiferroic

    Wu YL, Zeng ZZ, Lu HP, et al. Coexistence of ferroelectricity and antiferroelectricity in 2D van der Waals multiferroic. Nat Commun 2024;15:8616

    2024

  32. [39]

    Observation of electromagnons in a monolayer multiferroic

    Amini M, Antão TVC, Jing L W, et al. Observation of electromagnons in a monolayer multiferroic. arXiv:2510.08253; 2025

    2025 arXiv

  33. [40]

    Microscopic evidence of spin- driven multiferroicity and topological spin textures in monolayer NiI2

    Wang H T, Jiang T X, Pan W Y, et al. Microscopic evidence of spin- driven multiferroicity and topological spin textures in monolayer NiI2. Phys Rev Lett 2026;136:026402

    2026

  34. [41]

    Generalized gradient approximation made simple

    Perdew JP, Burke K, Ernzerhof M. Generalized gradient approximation made simple. Phys Rev Lett 1996;77:3865-8

    1996

  35. [43]

    Realistic spin model for multiferroic NiI 2

    Li X Y, Xu C S, Liu B Y, et al. Realistic spin model for multiferroic NiI 2. Phys Rev Lett 2023;131:036701

    2023

  36. [44]

    Electrical switching of a p-wave magnet

    Song Q, Stavrić S, Barone P, et al. Electrical switching of a p-wave magnet. Nature 2025;642:64-70

    2025

  37. [45]

    Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set

    Kresse G, Furthmüller J . Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys Rev B 1996;54:11169-86

    1996

  38. [46]

    Projector augmented-wave method

    Blöchl PE. Projector augmented-wave method. Phys Rev B 1994;50:17953-79

    1994

  39. [47]

    Stacking-dependent interlayer coupling in trilayer MoS2 with broken inversion symmetry

    Yan JX, Xia J, Wang XL, et al. Stacking-dependent interlayer coupling in trilayer MoS2 with broken inversion symmetry. Nano Lett 2015;15:8155-61

    2015

  40. [48]

    Electron-energy-loss spectra and the structural stability of nickel oxide: an LSDA+U study

    Dudarev SL, Botton GA, Savrasov SY, et al. Electron-energy-loss spectra and the structural stability of nickel oxide: an LSDA+U study. Phys Rev B 1998;57:1505-9

    1998

  41. [49]

    Microscopic origin of multiferroic order in monolayer NiI 2

    Fumega AO, Lado JL. Microscopic origin of multiferroic order in monolayer NiI 2. 2D Mater 2022;9:025010

    2022

  42. [50]

    Microscopic origin of magnetism in monolayer 3d transition 16 metal dihalides

    Riedl K, Amoroso D, Backes S, et al. Microscopic origin of magnetism in monolayer 3d transition 16 metal dihalides. Phys Rev B 2022;106:035156

    2022

  43. [51]

    Sliding- and twist-tunable valley polarization in bilayer NiI2

    Li LZ, Li X, Lin LY, et al. Sliding- and twist-tunable valley polarization in bilayer NiI2. Phys Rev B 2024;110:205119

    2024

  44. [52]

    One-dimensional edge state of Bi thin film grown on Si(111)

    Kawakami N, Lin CL, Kawai M, et al. One-dimensional edge state of Bi thin film grown on Si(111). Appl Phys Lett 2015;107:031602

    2015

  45. [53]

    Moiré-induced magnetoelectricity in twisted bilayer NiI2

    Zhu HY, Yu HY, Zhu WQ, et al. Moiré-induced magnetoelectricity in twisted bilayer NiI2. Phys Rev Lett 2025;135:196701

    2025

  46. [54]

    Molecular beam epitaxy of highly crystalline monolayer molybdenum disulfide on hexagonal boron nitride

    Fu D Y, Zhao X X, Zhang YY, et al. Molecular beam epitaxy of highly crystalline monolayer molybdenum disulfide on hexagonal boron nitride. J Am Chem Soc 2017;139:9392-400

    2017

  47. [55]

    Electric field control of moiré skyrmion phases in twisted multiferroic NiI2 bilayers

    Antão TVC, Lado JL, Fumega AO. Electric field control of moiré skyrmion phases in twisted multiferroic NiI2 bilayers. Nano Lett 2024;24:15767-73

    2024

  48. [56]

    Stacking- engineered ferroelectricity and multiferroic order in van der Waals magnets

    Bennett D, Martínez-Carracedo G, He X, et al. Stacking- engineered ferroelectricity and multiferroic order in van der Waals magnets. Phys Rev Lett 2024;133:246703

    2024

  49. [57]

    Controlling the magnetic properties of the van der Waals multiferroic crystals Co1–xNixI2

    Lukovkina A, López -Paz SA, Besnard C, et al. Controlling the magnetic properties of the van der Waals multiferroic crystals Co1–xNixI2. Chem Mater 2024;36:6237-45

    2024

  50. [58]

    First principles study of magnetism and polarization in 1T -NiI2 bilayer

    Mnich J, Milivojević M, Gmitra M. First principles study of magnetism and polarization in 1T -NiI2 bilayer. AIP Conf Proc 2024;3251:020001

    2024

  51. [60]

    Competing multiferroic phases in monolayer and few -layer NiI2

    Liu NS, Wang C, Yan C L, et al. Competing multiferroic phases in monolayer and few -layer NiI2. Phys Rev B 2024;109:195422

    2024

  52. [62]

    In- plane strain tuning multiferroicity in monolayer van der Waals NiI2

    Ni XS, Yao DX, Cao K. In- plane strain tuning multiferroicity in monolayer van der Waals NiI2. J Magn Magn Mater 2025;613:172661. 1 Supplementary Materials for Dimensional crossover and local strain induced deflection of the spin spiral state in multiferroic NiI2 Tianxing Jian...

    2025

  53. [63]

    S 1a for example, the exposed graphene or HOPG substrate can be easily identified in topographic images, as they display much less point defects than NiI2 terraces

    At low-coverage regions, as shown Fig. S 1a for example, the exposed graphene or HOPG substrate can be easily identified in topographic images, as they display much less point defects than NiI2 terraces. Once the substrate is identified, the thicknesses of adjacent NiI2 terrac...

  54. [64]

    As shown in Fig

    The tunneling spectrum provides another independent criterion for thickness identification. As shown in Fig. S2a and b. The spectra display a large insulating gap of ~2 eV on all measured thicknesses. As thickness increases, the conduction band edge shifts to positive energy s...

  55. [65]

    + 𝜆𝜆𝒌𝒌,2(𝑎𝑎2 2 + 𝑏𝑏2

  56. [66]

    To find the ground spiral state, the energy density 𝐸𝐸/𝑁𝑁 should be minimized under the constrains

    + 𝜆𝜆𝒌𝒌,3(𝑎𝑎3 2 + 𝑏𝑏3 2) 𝑎𝑎1 2 + 𝑎𝑎2 2 + 𝑎𝑎3 2 = 1 4 , 𝑏𝑏1 2 + 𝑏𝑏2 2 + 𝑏𝑏3 2 = 1 4 , 𝑎𝑎1𝑏𝑏1 + 𝑎𝑎2𝑏𝑏2 + 𝑎𝑎3𝑏𝑏3 = 0. To find the ground spiral state, the energy density 𝐸𝐸/𝑁𝑁 should be minimized under the constrains. A simple insight suggests that 𝑎𝑎3 2 + 𝑏𝑏3 2 = 0, i.e., 𝑎𝑎3 = 𝑏...

  57. [67]

    Then 𝐸𝐸𝐵𝐵 can be simplified as 𝐸𝐸𝐵𝐵 = 1 2 𝐵𝐵 � 𝑐𝑐𝑐𝑐𝑐𝑐2[𝒌𝒌 ⋅ (𝑹𝑹 − 𝑹𝑹′)] 𝑹𝑹,𝑹𝑹′ = 𝑁𝑁 4 𝐵𝐵 �[𝑐𝑐𝑐𝑐𝑐𝑐(2𝒌𝒌 ⋅ 𝑹𝑹1) + 1] 𝑹𝑹1

    This constrain should be satisfied by all 𝑹𝑹, thus we have 𝑨𝑨𝒌𝒌 ⋅ 𝑨𝑨𝒌𝒌 ∗ = 1 2 and 𝑨𝑨𝒌𝒌 2 = 0. Then 𝐸𝐸𝐵𝐵 can be simplified as 𝐸𝐸𝐵𝐵 = 1 2 𝐵𝐵 � 𝑐𝑐𝑐𝑐𝑐𝑐2[𝒌𝒌 ⋅ (𝑹𝑹 − 𝑹𝑹′)] 𝑹𝑹,𝑹𝑹′ = 𝑁𝑁 4 𝐵𝐵 �[𝑐𝑐𝑐𝑐𝑐𝑐(2𝒌𝒌 ⋅ 𝑹𝑹1) + 1] 𝑹𝑹1 . where 𝑹𝑹1 indicates the vectors of the first nearest-neighbors...

  58. [68]

    Photoconductivity of the nickel dihalides and the nature of the energy gap

    Ronda CR, Arends GJ, Haas C. Photoconductivity of the nickel dihalides and the nature of the energy gap. Phys Rev B 1987;35:4038-43

    1987

  59. [69]

    Atomically thin 1T-FeCl2 grown by molecular-beam epitaxy

    Zhou XH, Brzostowski B, Durajski A, et al. Atomically thin 1T-FeCl2 grown by molecular-beam epitaxy. J Phys Chem C 2020;124:9416-23

    2020

  60. [70]

    Microscopic evidence of spin-driven multiferroicity and topological spin textures in monolayer NiI2

    Wang HT, Jiang TX, Pan WY, et al. Microscopic evidence of spin-driven multiferroicity and topological spin textures in monolayer NiI2. Phys Rev Lett 2026;136:026402

    2026

  61. [71]

    Realistic spin model for multiferroic NiI2

    Li XY, Xu CS, Liu BY, et al. Realistic spin model for multiferroic NiI2. Phys Rev Lett 2023;131:036701

    2023

  62. [72]

    Evidence for a single-layer van der Waals multiferroic

    Song Q, Occhialini CA, Ergeçen E, et al. Evidence for a single-layer van der Waals multiferroic. Nature 2022;602:601-5

    2022

  63. [73]

    Layer thickness crossover of type- II multiferroic magnetism in NiI 2

    Wu S, Chen X Y, Hong CY, et al. Layer thickness crossover of type- II multiferroic magnetism in NiI 2. arXiv:2307.10686, 2023

    2023 arXiv

  64. [74]

    Learning local equivariant representations for large- scale atomistic dynamics

    Musaelian A, Batzner S, Johansson A, et al. Learning local equivariant representations for large- scale atomistic dynamics. Nat Commun 2023;14:579

    2023

  65. [75]

    Physics-informed time-reversal equivariant neural network potential for magnetic materials

    Yu HY, Liu BY, Zhong Y, et al. Physics-informed time-reversal equivariant neural network potential for magnetic materials. Phys Rev B 2024;110:104427

    2024

  66. [76]

    Atomic -scale visualization of multiferroicity in monolayer NiI2

    Amini M, Fumega AO, González- Herrero H, et al. Atomic -scale visualization of multiferroicity in monolayer NiI2. Adv Mater 2024;36:2311342

    2024

  67. [77]

    Orientation-selective spin-polarized edge states in monolayer NiI2

    Wang Y, Zhao XL, Yao L, et al. Orientation-selective spin-polarized edge states in monolayer NiI2. Nat Commun 2024;15:10916

    2024

  68. [78]

    Spin-resolved imaging of atomic-scale helimagnetism in mono- and bilayer NiI2

    Miao MP, Liu NS, Zhang WH, et al. Spin-resolved imaging of atomic-scale helimagnetism in mono- and bilayer NiI2. Proc Natl Acad Sci USA 2025;122:e2422868122

    2025

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.