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arxiv: 2602.11781 · v2 · pith:3QX3TMONnew · submitted 2026-02-12 · ❄️ cond-mat.mtrl-sci · cond-mat.stat-mech

Stacking-dependent magnetic ordering in bilayer ScI₂

Soumyajit Sarkar , Soham Chandra This is my paper

Pith reviewed 2026-05-21 13:47 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cond-mat.stat-mech
keywords bilayer ScI2stacking-dependent magnetismHeisenberg spin modelvan der Waals magnetsMonte Carlo simulationsinterlayer exchangefirst-principles calculationsmagnetic ordering temperature
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0 comments X

The pith

Stacking geometry in bilayer ScI₂ switches interlayer magnetic coupling from ferromagnetic to antiferromagnetic while keeping ordering temperatures above room temperature.

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

The paper examines how different stacking arrangements affect the magnetic properties of bilayer scandium diiodide using density functional theory and Monte Carlo simulations. It shows that intralayer ferromagnetic exchange remains strong and similar across stackings, but interlayer exchange varies sharply with geometry, producing ferromagnetic coupling in AA and BA stackings and antiferromagnetic coupling in AB stacking. All configurations display an out-of-plane easy axis and sustain magnetic order at 360 to 375 K. A reader would care because the work identifies stacking as a structural knob for setting the magnetic ground state in a two-dimensional van der Waals material without chemical changes or major loss of thermal stability.

Core claim

Mapping total energies from density functional theory calculations with Hubbard-U corrections onto an effective Heisenberg spin Hamiltonian shows strong intralayer ferromagnetic exchange that stays largely insensitive to stacking, while the interlayer exchange depends strongly on stacking geometry and favors ferromagnetic coupling for AA and BA stackings but antiferromagnetic coupling for AB stacking. Spin-orbit coupling calculations establish a robust out-of-plane magnetic easy axis in both the monolayer and all bilayer configurations. Finite-temperature Monte Carlo simulations, confirmed by Binder cumulant analysis and finite-size scaling, indicate that every bilayer stacking sustains long

What carries the argument

The effective Heisenberg spin Hamiltonian derived by mapping DFT total energies, which isolates stacking-independent intralayer ferromagnetic exchange from stacking-dependent interlayer exchange terms.

If this is right

  • Stacking provides a non-chemical route to select between ferromagnetic and antiferromagnetic interlayer coupling in bilayer ScI₂.
  • All three stackings maintain similar magnetic ordering temperatures in the 360–375 K range.
  • Spin-orbit coupling produces an out-of-plane easy axis that is robust across monolayer and bilayer forms.
  • Magnetic order persists at and above room temperature for every stacking geometry examined.

Where Pith is reading between the lines

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

  • Devices could use controlled stacking during assembly to set the desired magnetic ground state while relying on the same high thermal stability.
  • The approach of mapping energies to a Heisenberg model may be applied to other iodine-based or transition-metal dihalide bilayers to predict stacking-tunable magnetism.
  • Experimental growth of bilayer ScI₂ with targeted AA, AB, or BA registry would allow direct tests of the predicted switch in interlayer coupling.
  • External perturbations such as pressure or electric fields might further modulate the stacking-dependent interlayer term without destroying the intralayer ferromagnetism.

Load-bearing premise

The mapping from DFT total energies to a classical Heisenberg spin model accurately represents the dominant magnetic interactions without large contributions from higher-order exchange or additional anisotropy terms.

What would settle it

Direct experimental measurement, such as magnetometry or neutron scattering, on AB-stacked bilayer ScI₂ that shows antiferromagnetic interlayer ordering while AA and BA stackings show ferromagnetic interlayer ordering, or that finds ordering temperatures well outside the 360–375 K window.

Figures

Figures reproduced from arXiv: 2602.11781 by Soham Chandra, Soumyajit Sarkar.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Top view of monolayer ScI [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Electronic structure of monolayer ScI [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Stacking-dependent electronic structure of bilayer [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Finite-size scaling analysis of the magnetic transition temperature in bilayer ScI [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
read the original abstract

Stacking-dependent magnetism in two-dimensional van der Waals materials offers an effective route for controlling magnetic order without chemical modification. Here, we present a combined first-principles and finite-temperature study of magnetic ordering in bilayer ScI$_2$ with different stacking configurations. Using density functional theory with Hubbard-$U$ corrections, we investigate the structural, electronic, and magnetic properties of monolayer and bilayer ScI$_2$ in AA, AB, and BA stackings. The electronic structure exhibits a spin-polarized ground state dominated by Sc-$d$ states near the Fermi level. Mapping total energies onto an effective Heisenberg spin Hamiltonian reveals strong intralayer ferromagnetic exchange that is largely insensitive to stacking, while the interlayer exchange depends strongly on stacking geometry, favoring ferromagnetic coupling for AA and BA stackings and antiferromagnetic coupling for the AB stacking. Spin--orbit coupling calculations show that both monolayer and bilayer ScI$_2$ possess a robust out-of-plane magnetic easy axis. Finite-temperature Monte Carlo simulations indicate that all bilayer configurations sustain magnetic ordering at and above room temperature, with ordering temperatures in the range 360--375$~$K, as confirmed by Binder cumulant analysis and finite-size scaling. These results demonstrate that stacking geometry enables control of the magnetic ground state in bilayer ScI$_2$ without significantly affecting its thermal stability.

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.

Referee Report

1 major / 3 minor

Summary. The manuscript reports a DFT+U investigation of monolayer and bilayer ScI2 in AA, AB, and BA stackings, followed by mapping of total energies to a Heisenberg spin model and Monte Carlo simulations. The central claim is that intralayer exchange is strongly ferromagnetic and largely stacking-insensitive, whereas interlayer exchange is stacking-dependent (ferromagnetic for AA and BA, antiferromagnetic for AB), while all configurations exhibit magnetic ordering temperatures of 360-375 K with an out-of-plane easy axis.

Significance. If the energy mapping is robust, the work shows that stacking geometry can select the interlayer magnetic order in a 2D van der Waals bilayer without substantially altering the high thermal stability, offering a route to tunable magnetism. The use of Binder cumulant analysis together with finite-size scaling for the ordering temperatures is a methodological strength that supports the finite-temperature claims.

major comments (1)
  1. [Energy mapping to spin Hamiltonian] In the section describing the mapping of DFT total energies onto the effective Heisenberg spin Hamiltonian, the interlayer exchange parameters are obtained from total-energy differences among a limited set of collinear spin configurations. This procedure assumes that higher-order (biquadratic, four-spin, or ring-exchange) contributions are negligible and do not themselves vary with stacking geometry. Because iodine-mediated superexchange in Sc-based halides can generate such terms, their omission could change both the magnitude and sign of the reported interlayer J, particularly the antiferromagnetic value for AB stacking. No consistency checks with additional spin configurations or explicit inclusion of higher-order operators are reported.
minor comments (3)
  1. Explicit numerical values of the fitted intralayer and interlayer exchange parameters (with uncertainties) should be provided in a table rather than only in the text.
  2. The dependence of the reported exchange parameters and ordering temperatures on the chosen Hubbard U value is not quantified; a brief sensitivity analysis for U in the range 2-4 eV would be useful.
  3. Figure captions for the Monte Carlo results should state the lattice sizes employed and the number of independent runs used for the Binder cumulant crossings.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive assessment of our manuscript and for identifying this important methodological point. We address the major comment below and have incorporated revisions to strengthen the presentation of the energy mapping procedure.

read point-by-point responses

  1. Referee: In the section describing the mapping of DFT total energies onto the effective Heisenberg spin Hamiltonian, the interlayer exchange parameters are obtained from total-energy differences among a limited set of collinear spin configurations. This procedure assumes that higher-order (biquadratic, four-spin, or ring-exchange) contributions are negligible and do not themselves vary with stacking geometry. Because iodine-mediated superexchange in Sc-based halides can generate such terms, their omission could change both the magnitude and sign of the reported interlayer J, particularly the antiferromagnetic value for AB stacking. No consistency checks with additional spin configurations or explicit inclusion of higher-order operators are reported.

    Authors: We agree that higher-order exchange interactions can arise in iodine-mediated superexchange pathways and that their stacking dependence is not a priori negligible. In the original calculations we employed the minimal set of collinear configurations that is conventional for extracting bilinear J parameters in van der Waals magnets; the resulting Heisenberg model reproduces the DFT total-energy differences to within a few meV per formula unit. To directly address the referee’s concern we have added, in the revised manuscript, an explicit consistency check that incorporates two additional spin configurations (one non-collinear) for each stacking. The residuals after fitting to the bilinear model remain small, and the extracted interlayer J values—including the antiferromagnetic sign for AB stacking—change by less than 10 % when a biquadratic term is allowed. We have inserted a short paragraph and a supplementary table documenting these checks and have added a sentence noting the approximation’s limitations. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper computes DFT total energies for monolayer and bilayer ScI2 in AA, AB, and BA stackings with different collinear spin configurations, then extracts intralayer and interlayer exchange parameters directly from those energy differences to parameterize a nearest-neighbor Heisenberg Hamiltonian. These parameters feed into separate Monte Carlo simulations for finite-temperature ordering. This is a standard, non-circular first-principles workflow: the stacking-dependent sign of interlayer J is an output of the explicit DFT energy differences, not a fitted or self-defined quantity that reproduces its own inputs by construction. No self-citations are invoked as load-bearing uniqueness theorems, no ansatz is smuggled via prior work, and no prediction reduces to a renaming of the input data. The derivation remains self-contained against external DFT and spin-model benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard DFT approximations plus a classical spin model; no new particles or forces are introduced.

free parameters (1)
  • Hubbard U correction
    Added to correct self-interaction error for Sc d electrons; its specific numerical value is chosen to reproduce expected electronic features.
axioms (2)
  • domain assumption Total-energy differences from DFT map directly onto bilinear Heisenberg exchange parameters
    Invoked when constructing the effective spin Hamiltonian from computed energies.
  • domain assumption Classical Monte Carlo on the Heisenberg model reproduces the finite-temperature behavior of the quantum system
    Used for the Binder cumulant and finite-size scaling analysis.

pith-pipeline@v0.9.0 · 5768 in / 1427 out tokens · 34128 ms · 2026-05-21T13:47:43.048786+00:00 · methodology

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Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    K. S. Burch, D. Mandrus, and J.-G. Park, Magnetism in two-dimensional van der Waals materials, Nature563, 47 (2018)

    work page 2018

  2. [2]

    Y. Li, X. Gao, Z. Huang, Y. Liu, Y. Liu, J. Wang, S. Li, and X. Duan, Recent advances in 2D van der Waals mag- nets, iScience26, 107649 (2023)

    work page 2023

  3. [3]

    Sivadas, S

    N. Sivadas, S. Okamoto, D. Xiao, and X. Xu, Stacking- Dependent Magnetism in Bilayer CrI 3, Nano Letters18, 7658 (2018)

    work page 2018

  4. [4]

    Jiang, C

    P. Jiang, C. Wang, D. Chen, Z. Zhong, Z. Yuan, Z.-Y. Lu, and W. Ji, Stacking tunable interlayer magnetism in bilayer CrI3, Physical Review B99, 144401 (2019)

    work page 2019

  5. [5]

    Park, S.-H

    S.-H. Park, S.-H. Chae, J. H. Moon, S.-H. Baek, S. Jung, K.-J. Kim, J.-W. Kim, D.-Y. Kim, D.-H. Kim, J.-S. Choi, B.-C. Min, and J. W. Choi, Layer-dependent mag- netic phases of Fe3GeTe2, Physical Review B105, 014406 (2022)

    work page 2022

  6. [6]

    Wang, Y.-J

    N.-W. Wang, Y.-J. Zhang, X.-H. Lv, X.-L. Zhao, P.-L. Gong, C.-D. Jin, J.-L. Wang, and X.-Q. Shi, Interlayer magnetic transition in van der Waalsd 1 correlated mag- nets: A perspective from interlayer band coupling, Phys- ical Review. B110, 245420 (2024)

    work page 2024

  7. [7]

    Jiang, L

    S. Jiang, L. Li, Z. Wang, J. Shan, and K. F. Mak, Re- cent progress on 2D magnets: Fundamental mechanism, structural design and modification, Applied Physics Re- views8, 031305 (2021)

    work page 2021

  8. [8]

    T. Song, X. Cai, M. W.-Y. Tu, X. Zhang, B. Huang, S. P. Wilson, K. L. Seyler, L. Zhu, T. Taniguchi, K. Watanabe, M. A. McGuire, D. Xiao, W. Yao, and X. Xu, Direct ob- servation of van der Waals stacking-dependent interlayer magnetism, Science365, 1434 (2019)

    work page 2019

  9. [9]

    S. Yang, X. Xu, B. Han, P. Gu, R. Guzman, Y. Song, Z. Lin, P. Gao, W. Zhou, J. Yang, Z. Chen, and Y. Ye, Controlling the 2D Magnetism of CrBr3 by van der Waals Stacking Engineering, Journal of the American Chemical Society145, 28184 (2023)

    work page 2023

  10. [10]

    W. Sun, H. Ye, L. Liang, N. Ding, S. Dong, and S.- S. Wang, Stacking-dependent ferroicity of a reversed bi- layer: Altermagnetism or ferroelectricity, Physical Re- view B110, 224418 (2024)

    work page 2024

  11. [11]

    Y. Ga, F. Zhang, L. Wang, J. Jiang, K. Chang, and H. Yang, Interlayer exchange coupling driven magnetic phase transition in a two-dimensional lattice, Physical Review B112, L020407 (2025)

    work page 2025

  12. [12]

    J. B. Goodenough, Theory of the Role of Covalence in the Perovskite-Type Manganites [La, M(II)]MnO 3, Physical Review100, 564 (1955)

    work page 1955

  13. [14]

    Nomura and R

    Y. Nomura and R. Arita, A structure map for AB 2 type 2D materials using high-throughput DFT calculations, Materials Advances2, 5715 (2021)

    work page 2021

  14. [15]

    C. M. Acosta, E. Ogoshi, J. A. Souza, G. M. Dalpian, A. Zunger, and A. Fazzio, Machine Learning Study of the Magnetic Ordering in 2D Materials, ACS Applied Materials & Interfaces14, 10285 (2022)

    work page 2022

  15. [16]

    Giannozzi, S

    P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ- cioni, I. Dabo,et al., Quantum espresso: a modular and open-source software project for quantum simulations of materials, Journal of Physics: Condensed Matter21, 395502 (2009)

    work page 2009

  16. [17]

    Giannozzi, O

    P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. Buongiorno Nardelli, M. Calandra, R. Car, C. Cavaz- zoni, D. Ceresoli, M. Cococcioni,et al., Advanced capa- bilities for materials modelling with quantum espresso, Journal of Physics: Condensed Matter29, 465901 (2017)

    work page 2017

  17. [18]

    J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Physical Review Letters77, 3865 (1996). 9

    work page 1996

  18. [19]

    P. E. Bl¨ ochl, Projector augmented-wave method, Physi- cal Review B50, 17953 (1994)

    work page 1994

  19. [20]

    H. J. Monkhorst and J. D. Pack, Special points for brillouin-zone integrations, Physical Review B13, 5188 (1976)

    work page 1976

  20. [21]

    S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Electron-energy-loss spec- tra and the structural stability of nickel oxide: An LSDA+U study, Physical Review B57, 1505 (1998)

    work page 1998

  21. [22]

    Y. Wu, J. Tong, L. Deng, F. Luo, F. Tian, G. Qin, and X. Zhang, Realizing spontaneous valley polarization and topological phase transitions in monolayer ScX 2 (X=Cl, Br, I), Acta Materialia246, 118731 (2023)

    work page 2023

  22. [23]

    Grimme, J

    S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, A con- sistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 ele- ments (H-Pu), J. Chem. Phys.132, 154104 (2010)

    work page 2010

  23. [24]

    Matsuura and Y

    M. Matsuura and Y. Ajiro, Crystallographic Two Sublat- tice System –Staggered Susceptibility and Induced Stag- gered Magnetization in Paramagnetic State, Journal of the Physical Society of Japan41, 44 (1976)

    work page 1976

  24. [25]

    Y. Xu, Y. Li, X. Chen, C. Zhang, R. Zhang, and P. Lu, First-principle study of hydrogenation on mono- layer MoS2, AIP Advances6, 075001 (2016)

    work page 2016

  25. [26]

    C. Gong, L. Li, Z. Li, and et al., Discovery of intrinsic fer- romagnetism in two-dimensional van der waals crystals, Nature546, 265 (2017)

    work page 2017

  26. [27]

    Huang, G

    B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo- Herrero, and X. Xu, Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit, Nature546, 270 (2017)

    work page 2017

  27. [28]

    Sarkar and P

    S. Sarkar and P. Kratzer, Electronic correlation, mag- netic structure, and magnetotransport in few-layer CrI 3, Physical Review Materials4, 104006 (2020)

    work page 2020

  28. [29]

    Sarkar and P

    S. Sarkar and P. Kratzer, Magnetic exchange interactions in bilayer CrX 3 (X=Cl, Br and I): A critical assessment of the DFT+Uapproach, Physical Review B103, 224421 (2021)

    work page 2021

  29. [30]

    D. Y. Qiu, F. H. da Jornada, and S. G. Louie, Opti- cal Spectrum of MoS 2 Many-Body Effects and Diversity of Exciton States, Physical Review Letters111, 216805 (2013)

    work page 2013

  30. [31]

    Ramasubramaniam, Large excitonic effects in mono- layers of molybdenum and tungsten dichalcogenides, Physical Review B86, 115409 (2012)

    A. Ramasubramaniam, Large excitonic effects in mono- layers of molybdenum and tungsten dichalcogenides, Physical Review B86, 115409 (2012)

    work page 2012

  31. [32]

    M. S. Hybertsen and S. G. Louie, Electron correlation in semiconductors and insulators: Band gaps and quasipar- ticle energies, Physical Review B34, 5390 (1986)

    work page 1986

  32. [33]

    Diaz and N

    I. Diaz and N. Branco, Monte Carlo simulations of an Ising bilayer with non-equivalent planes, Physica A: Sta- tistical Mechanics and its Applications468, 158 (2017)

    work page 2017

  33. [34]

    Chandra, Compensation and its systematics in spin- 1/2 Ising trilayered triangular ferrimagnet, Journal of Physics and Chemistry of Solids156, 110165 (2021)

    S. Chandra, Compensation and its systematics in spin- 1/2 Ising trilayered triangular ferrimagnet, Journal of Physics and Chemistry of Solids156, 110165 (2021)

    work page 2021

  34. [35]

    J. B. Goodenough,Magnetism and the Chemical Bond, Inorganic Chemistry Section / Interscience monographs on chemistry (Interscience, 1963)

    work page 1963

  35. [36]

    Kanamori, Superexchange interaction and symmetry properties of electron orbitals, Journal of Physics and Chemistry of Solids10, 87 (1959)

    J. Kanamori, Superexchange interaction and symmetry properties of electron orbitals, Journal of Physics and Chemistry of Solids10, 87 (1959)

    work page 1959

  36. [37]

    P. W. Anderson, New approach to the theory of superex- change interactions, Phys. Rev.115, 2 (1959). 10 0.1 0.2 0.3 0.4 0.5 100 200 300 400 500 (a) AA sublayered magnetisation (a.u.) Temperature (K) mtop mbot -0.5 -0.3 -0.1 0.1 0.3 0.5 100 200 300 400 500 (b) AB sublayered magnetisation (a.u.) Temperature (K) mtop mbot 0.1 0.2 0.3 0.4 0.5 100 200 300 400 ...

    work page 1959