Robust altermagnetism and compensated ferrimagnetism in MnPX$_3$-based (X = S or Se) heterostructures

Guifang Xu; HaiFeng Wang; Jijun Zhao; Nan Xia; Weiwei Gao; Xuefei Wang; Yanlong Liu; Yi Wang; Yunsong Liu

REVIEW 3 major objections 5 minor 47 references

Robust altermagnetism and compensated ferrimagnetism in MnPX$_3$-based (X = S or Se) heterostructures

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The stacking order of MnPX3-based van der Waals heterostructures determines whether they are altermagnets or compensated ferrimagnets, and a ferroelectric substrate makes the spin splitting switchable by an electric field.

desk verdict New material-specific altermagnet and ferrimagnet predictions with a flawed Luttinger-theorem argument for full compensation; the spin-splitting results are probably right, but the 'compensated' label needs computed moments. read the letter →

arxiv 2412.17232 v3 pith:7Z4U7OA2 submitted 2024-12-23 cond-mat.mtrl-sci physics.comp-ph

Yunsong Liu , Yanlong Liu , Xuefei Wang , Nan Xia , Guifang Xu , Yi Wang , Haifeng Wang , Weiwei Gao

show 1 more author

Jijun Zhao

This is my paper · ORCID

classification cond-mat.mtrl-sciphysics.comp-ph

keywords altermagnetismcompensatedferrimagnetismvanderWaalsheterostructuresMnPX3non-relativisticspinsplittingferroelectricswitchingtwo-dimensionalmagnets

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 shows that a single antiferromagnetic monolayer, MnPX3 (X = S or Se), can be turned into a magnet with momentum-dependent spin splitting simply by stacking it on a lattice-matched van der Waals substrate. With AA stacking the result is altermagnetic splitting—up- and down-spin bands split by up to tens of meV with the alternating four-lobed pattern characteristic of altermagnets—while with AB stacking the result is a ferrimagnetic-type splitting that the paper argues is exactly compensated, i.e., has no net magnetization, because the systems are semiconductors. Replacing the substrate with ferroelectric CuInP2X6 makes the size of the splitting depend on polarization, so an out-of-plane electric field can nonvolatilely switch the spin-related electronic structure. If these predictions hold, the work supplies a way to make tunable two-dimensional compensated magnets from materials that can already be exfoliated and stacked in experiments.

What carries the argument

The mechanism is the selective breaking of the combined inversion-and-time-reversal ($PT$) symmetry that protects spin degeneracy in the free-standing monolayer. In an AA-stacked heterostructure, inversion is lost but the spin-group operation $[C_2\,\Vert\,M_{\mathbf{a}}]$ survives, relating the up- and down-spin Mn sites; that surviving operation is what organises the splitting into the $d$-wave altermagnetic form. In an AB-stacked heterostructure, no such symmetry relates the two spin sublattices, and the splitting is ferrimagnetic-like; the paper then leans on the Luttinger theorem, which ties integer electron count per cell to quantisation of the total moment, to argue that the net magnetisation vanishes in these semiconductors. The ferroelectric CuInP2X6 substrate supplies the switching handle: its two polarisation states are separated by a calculated 20.1 meV/atom barrier, and reversing the polarisation changes the interlayer potential enough to alter the spin splitting.

What would settle it

Compute the total magnetisation per primitive cell of the ground-state AB-stacked MnPS3/MgPS3 heterostructure with the same first-principles method used in the paper; a nonzero integer moment (e.g., $2\,\mu_{\mathrm{B}}$ per cell) would contradict the fully compensated conclusion, while a zero moment would support it. Experimentally, a SQUID magnetometry or anomalous-Hall measurement on such a stack could settle the same question.

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Extended reading notes

Core claim

The paper's central discovery is that substrate-induced breaking of the combined inversion-and-time-reversal symmetry lifts the spin degeneracy of the Néel-ordered MnPX3 monolayer, and that the stacking sequence selects which type of compensated spin splitting appears. In the AA stack, the two opposite-spin Mn sites remain related by a spin-group rotation-mirror operation, so the splitting has the alternating d-wave pattern of an altermagnet, reaching about 46 meV in a deep valence band. In the AB stack no such operation relates the spin sites, producing a ferrimagnetic-type spin splitting with a sixfold-symmetric distribution in the Brillouin zone; because the heterostructures are semiconducting, the paper invokes the Luttinger theorem to conclude their total magnetization is exactly zero, and labels them Luttinger-compensated ferrimagnets. With CuInP2X6 as the substrate, reversing the ferroelectric polarization changes the band structure and the magnitude of the splitting substantially, with a calculated switching barrier of 20.1 meV/atom, so the spin splitting can be switched nonvolatilely by an external electric field.

Load-bearing premise

The paper's conclusion that the AB-stacked heterostructures are fully compensated rests on the premise that a semiconductor with an integer total moment per unit cell must have zero net magnetization; that premise is not generally valid by itself, and no computed total magnetic moments are reported for these structures.

Editorial extensions

If this is right

AA-stacked MnPX3/MPX3 heterostructures should behave as two-dimensional altermagnets, with d-wave spin splitting in the valence bands reaching about 46 meV.
AB-stacked semiconducting heterostructures should behave as compensated ferrimagnets, with sixfold-symmetric spin splitting and no net magnetization.
The splitting is largely strain-tolerant: biaxial strains between -1.5% and 1.5% shift the average splitting by only 0.2-1 meV.
MnPX3/CuInP2X6 heterostructures should allow nonvolatile electric-field switching of the spin splitting, with a calculated polarization-switching barrier of about 20.1 meV/atom.
The same recipe should extend to other intralayer antiferromagnets of the MPX3 family, such as NiPX3, FePX3, and CoPX3, when paired with suitable substrates.

Reading between the lines

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

The symmetry logic used here—remove inversion, let the stacking choose which spin-site symmetry survives—suggests a general construction rule for altermagnetic heterostructures from any 2D collinear antiferromagnet with a compatible substrate, which the authors only sketch for MnPX3.
Because ferromagnetic semiconductors exist with integer, nonzero total moments, the Luttinger-based compensation argument needs a direct test; computing the total moment of the AB-stacked cell would confirm or refute the 'fully compensated' label.
The ferroelectric switching scenario points toward a concrete device concept, a two-terminal stack whose polarization state encodes the spin-splitting state, readable through spin-resolved photoemission or magneto-optical measurements.

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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

Here's the short version before you read it. The paper's specific material predictions are new: AA-stacked MnPX3/MPX3 gives altermagnetic splitting, AB-stacked gives ferrimagnetic-type splitting, and CuInP2X6 substrates make it switchable. The DFT work is careful and the screening of stacking modes and magnetic orders is thorough. But the headline claim that the AB-stacked systems are 'fully compensated' is not supported. The Luttinger theorem argument after Fig. 4 is a non-sequitur: integer magnetization per cell can be 1, 2, ... μB, not just zero, and no computed total moments are given anywhere.

What it does well: It turns the general substrate-altermagnetism idea, already floated by Mazin et al. and others, into a concrete and experimentally plausible family. The spin-splitting maps in the BZ show the expected d-wave pattern, the strain dependence is sensible, and the ferroelectric switching calculation with NEB is a nice bonus. The methods are standard and reproducible: VASP, PBE+U with U=3 eV from literature, D3, scalar-relativistic collinear calculations.

The soft spots: (1) the Luttinger theorem misuse is load-bearing because 'compensated' is in the title and abstract. They need to report the actual total moments for the AB-stacked and CuInP2X6 systems. If the moment is nonzero, they are ordinary uncompensated ferrimagnetic semiconductors and the central claim collapses to something less interesting but still publishable. (2) The AA vs AB energy differences are around 0.1-1 meV/atom. That's within DFT error, so the predicted stacking preference is fragile; both stackings may be present in samples. (3) They don't discuss the strain energy cost of matching layers with 1.9% lattice mismatch. Minor.

The spin-splitting band structures themselves are probably fine, and I don't see a circularity problem—U is from prior work, not fit to the target splitting.

Bottom line: solid computational paper with one flawed central inference. It deserves a serious referee, but the referee should demand total moments and a corrected argument. I'd use this in my own work with caution, citing the AA altermagnetic results rather than the 'Luttinger-compensated' label.

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a general heterostructure design for nonrelativistic spin splitting in van der Waals antiferromagnets. Using PBE+U DFT with the Dudarev formalism, the authors study MnPX3 (X = S, Se) monolayers on MPX3 (M = Cd, Mg, Zn) and on ferroelectric CuInP2X6 substrates, considering AA and AB stackings and four collinear magnetic orders. They report that AA-stacked MnPX3/MPX3 heterostructures display altermagnetic spin splitting with d-wave symmetry, while AB-stacked semiconductors display ferrimagnetic-type spin splitting with C6-symmetric distribution in the Brillouin zone, which they label 'fully compensated' and attribute to protection by the Luttinger theorem. They also report that the spin splitting is weakly dependent on in-plane strain and that reversing the ferroelectric polarization of CuInP2X6 can nonvolatilely modulate the spin splitting, with an NEB barrier of 20.1 meV/atom. The central claim is that these heterostructures are tunable two-dimensional compensated magnets with nonrelativistic spin splitting.

Significance. If the calculated spin-splitting patterns are correct, the paper offers a materials-realizable route to two-dimensional altermagnets and compensated ferrimagnets, and a nonvolatile electric-field switch for spin splitting. The paper has clear strengths: it systematically screens stacking modes and magnetic orders, provides an explicit spin-group symmetry analysis for the altermagnetic case, tests strain sensitivity, and estimates the ferroelectric switching barrier. The design principle is falsifiable and experimentally testable. However, the 'fully compensated' label, which is central to the title and abstract, is not currently demonstrated because the Luttinger-theorem inference is not valid and no total magnetic moments are reported. The spin-splitting results themselves are not in question, but the compensated-ferrimagnet classification needs direct verification.

major comments (3)

[Section I (paragraph after Fig. 4); also Abstract and Conclusions] The statement 'the heterostructures considered in this work are semiconductors, which require the total magnetization per cell to be an integer according to the Luttinger theorem. Therefore, the full compensation of the macroscopic magnetization is guaranteed' is not logically valid. In a collinear, spin-conserving insulator the spin-resolved electron count per cell is an integer, but the difference between spin-up and spin-down counts can be any integer, not only zero. Ferromagnetic semiconductors are a direct counterexample. The paper never reports computed total magnetic moments or spin-resolved electron counts for the AB-stacked heterostructures, so the 'fully compensated' and 'Luttinger-compensated' claims are unsupported. Please provide the total magnetic moment per cell (and, if desired, spin-resolved occupation numbers) for all AB-stacked systems; if any moment is nonzero, the abstract, Table 1 labels, and title must be revised.
[Section II (paragraph after Fig. 6)] The same invalid inference is used for the MnPX3/CuInP2X6 heterostructures: the text states that 'their semiconducting nature guarantees the total magnetization is zero.' As in the AB-stacked MnPX3/MPX3 case, a semiconducting gap does not force the magnetization to vanish; an integer nonzero moment is allowed. No total moments are reported for these systems either. The Conclusions statement that 'since the systems studied in this work are semiconductors' something is guaranteed inherits the same problem. Please verify the zero-magnetization claim by direct calculation or remove the compensation claim from the abstract and conclusions.
[Tables 1 and 2] The total-energy differences between AA and AB stackings are 0.1-1.0 meV/atom, and differences between magnetic orders can be as small as about 0.3 meV/atom (for example, MnPSe3/CdPSe3 AB Stripy is -0.3 meV/atom relative to Néel). These values are at or below the typical accuracy of semilocal DFT with D3 corrections. The statements that AB stacking is 'generally more stable' and that the Néel order remains the lowest-energy magnetic state are therefore not strongly supported. Please provide explicit convergence checks with respect to k-mesh, smearing, and van der Waals functional, or soften the ground-state claims to reflect that the differences are within numerical uncertainty.

minor comments (5)

[Fig. 6 caption] The caption refers to 'MnPS3/MgPS3' and 'MnPSe3/MgPSe3' in a section about MnPX3/CuInP2X6 heterostructures; this appears to be a typo for MnPS3/CuInP2S6 and MnPSe3/CuInP2Se6.
[Computational Methods] The sentence 'the force and energy convergence criteria are set to be set to 0.01 eV/Å and 1 × 10−5 eV' contains a duplicated phrase and should be revised.
[Introduction] 'Kramer’s spin degeneracy' should be 'Kramers spin degeneracy'.
[Figs. 2 and 3] The high-symmetry point M2 is used but never defined; please define it in the text or in the figure caption.
[Table 2 header] The table heading 'The energy difference between AFM (Néel, Stripy, Zigzag) and Néel states is taken into account...' is unclear; please rephrase to specify the reference energy directly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spin-splitting results are direct DFT outputs and are not fitted to the target; the contested Luttinger step is a correctness concern, not a circular reduction.

full rationale

The central spin-splitting results are produced by first-principles DFT calculations with a fixed exchange-correlation functional (PBE), vdW correction, and a literature Hubbard Ueff = 3 eV; no parameter is fitted to reproduce the reported altermagnetic or ferrimagnetic spin splitting, so the predicted band structures are not equivalent to their inputs by construction. The symmetry classification (altermagnetism vs. ferrimagnetic-type splitting) follows from standard spin-group analysis of the computed structures. The only self-citations in the reference list (e.g., [23]) are review/examples of 2D altermagnets and are not load-bearing for the present calculation. The skeptical concern that the 'Luttinger theorem' inference from semiconducting character to zero total magnetization is invalid is a scientific-validity objection, not circularity: the paper does not define 'compensated' in terms of the spin-splitting output, nor does it fit anything to force the conclusion. Therefore no circular step satisfying the quoted-reduction standard is identifiable.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No new physical entity is introduced. The main numerical input is the Hubbard U of 3 eV, and the main conceptual premise is that Luttinger theorem fixes zero magnetization for gapped AB-stacked heterostructures, which is questionable.

free parameters (1)

Hubbard Ueff for Mn 3d = 3 eV
Taken from the literature (Moore et al., PRM 2024) and applied with the Dudarev formalism. It affects the Mn magnetic moment, band gap, and spin-splitting magnitude, though it is not fitted to the target spin-splitting values.

assumptions (4)

domain assumption GGA-PBE with DFT-D3 gives reliable relative energies and band structures for these van der Waals heterostructures.
Used throughout the Methods and Results without validation against hybrid functionals or experiments.
domain assumption Collinear magnetic order and scalar-relativistic treatment capture the non-relativistic spin splitting.
Spin-orbit coupling is neglected, which is appropriate for the non-relativistic claim but still an approximation for S and Se.
domain assumption The Neel order remains the ground-state magnetic order in the heterostructures.
Supported by Table 1 energy differences, though some non-Neel orders are only a few tenths of meV/atom higher.
ad hoc to paper The Luttinger theorem applied to Kohn-Sham occupations forces zero net magnetization in gapped AB-stacked heterostructures.
Invoked after Fig. 4. The inference from an integer total magnetization to zero magnetization is not valid in general, so this premise is questionable.

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Cite this review

Pith. "Pith review of Robust altermagnetism and compensated ferrimagnetism in MnPX$_3$-based (X = S or Se) heterostructures." pith.science (2026). https://pith.science/paper/7Z4U7OA2

@misc{pith2026241217232,
  author       = {Pith},
  title        = {Pith review of: Robust altermagnetism and compensated ferrimagnetism in MnPX$_3$-based (X = S or Se) heterostructures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7Z4U7OA2}},
  note         = {Machine review of arXiv:2412.17232}
}

abstract

The recent research interests in the non-relativistic spin splitting of electronic band structures have led to the exploration of altermagnets and other compensated magnets. Here, we show that various types of non-relativistic spin splitting can be robustly induced by constructing Van der Waals heterostructures consisting of materials with intra-plane anti-ferromagnetic orders and suitable substrates. Using MnPX$_3$ (X = S or Se) as an example, which has a N\'eel magnetic order, we demonstrate that altermagnetic spin splitting can arise in the AA-stacking MnPX$_3$/MPX$_3$ (M = Cd, Mg, or Zn) heterostructures. For the AB-stacking heterostructures that are semiconducting, ferrimagnetic-type spin splitting emerges, and the fully compensated magnetization is protected by the Luttinger theorem. By combining with a Van der Waals ferroelectric substrate like CuInP$_2$S$_6$, MnPX$_3$-based heterostructures can show tunable spin splitting and spin-related properties that depend on the electronic band structures and ferroelectric polarization, which can be non-volatilely reversed by applying an out-of-plane electric field. Our study provides a route to induce tunable non-relativistic spin splitting in experimentally synthesizable two-dimensional magnets.

Figures

Figures reproduced from arXiv: 2412.17232 by the authors.

**Figure 4.** (a) Monolayer MnPS3 has a Néel magnetic order and satisfies the PT-symmetry; The differences in the spin-charge density (localized around MnPS3) between the free-standing monolayer MnPS3 and MnPS3/MgPS3 heterostructures with the (b) AA-stacking and (c) ABstacking sequences, respectively. The red and green isosurfaces indicate positively and negatively valued regions. And the isosurface value is ±1.9 × 10−5 𝑒/Bohr3 … view at source ↗

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Šmejkal, J

L. Šmejkal, J. Sinova, and T. Jungwirth, Physical Review X 12 (3), 031042 (2022)

work page 2022

[2] [2]

L.-D. Yuan, Z. Wang, J.-W. Luo, E. I. Rashba and A. Zunger, Physical Review B 102 (1), 014422 (2020)

work page 2020

[3] [3]

I. I. Mazin, K. Koepernik, M. D. Johannes, R. González -Hernández and L. Šmejkal, Proceedings of the National Academy of Sciences 118 (42), e2108924118 (2021)

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