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Altermagnetism in 6H perovskites

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Pith's one-line read This paper identifies a family of layered oxides, 6H perovskites with formula A$_3$BB'$_2$O$_9$, as altermagnets: their collinear antiferromagnetic order splits spin-up and spin-down bands without spin-orbit coupling, and two of the four…

desk verdict Clear symmetry case for two 6H altermagnets; 'giant piezomagnetism' and the Ba3SrIr2O9 entry need qualification. read the letter →

arxiv 2507.23232 v1 pith:CNRMAKKM submitted 2025-07-31 cond-mat.mtrl-sci cond-mat.str-el

S.V. Streltsov , S.-W. Cheong This is my paper
classification cond-mat.mtrl-scicond-mat.str-el
keywords altermagnetism6Hperovskitesnon-relativisticspinsplittingmagneto-opticalKerreffectpiezomagnetismBa3CoIr2O9Ba3NiRu2O9densityfunctionaltheory
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

Altermagnets are magnetic materials in which a collinear antiferromagnetic arrangement splits spin-up and spin-down electron bands even without spin-orbit coupling. This paper argues that 6H perovskites, a family of layered oxides with formula A$_3$BB'$_2$O$_9$, form a natural altermagnet platform: centrosymmetric in structure, yet lacking inversion between the magnetic B sites while possessing C$_2$ axes that connect the two magnetic planes. Symmetry analysis of fifteen known antiferromagnetic 6H perovskites yields four altermagnets, and density-functional calculations on representatives confirm non-relativistic spin splitting, magneto-optical activity in the M-type members, and a large piezomagnetic response in Ba$_3$NiRu$_2$O$_9$. Because these compounds can be grown as cleavable single crystals, the claim opens a practical route to imaging altermagnetic spin textures with surface-sensitive probes.

What carries the argument

The load-bearing object is the set of C$_2$ rotation axes perpendicular to the $c$-axis in the 6H perovskite space groups P63/mmc, Cmcm, and C2/c. These axes lie at 1/4 and 3/4 of the $c$-axis, and also at 0 and 1/2 in the hexagonal space group, and they map a magnetic B site in the plane at $z=0$ onto a B site in the plane at $z=c/2$. Because there is no inversion center connecting those sites, the combined operation of the C$_2$ rotation with time reversal changes the spin and the momentum of an electron in a way that makes the spin degeneracy at general $k$ points fail, while special mirror-plane points remain degenerate. The companion mechanism is the exchange pattern: ferromagnetic B-B' superexchange through nearly 180$^\circ$ B-O-B' bonds involving empty $e_g$ orbitals, together with antiferromagnetic intradimer B'-B' exchange, selects the required antiferromagnetic stacking of the two ferromagnetic B' planes. This combination of local orbital physics and crystal symmetry is what turns an otherwise conventional antiferromagnet into an altermagnet.

What would settle it

A decisive test is spin-resolved photoemission or tunneling spectroscopy on a cleaved surface below the Neel temperature: if Ba$_3$CoIr$_2$O$_9$ or Ba$_3$NiRu$_2$O$_9$ does not show non-relativistic spin-split bands along the predicted $k$-path (L-$\Gamma$-V for the cobalt compound, M-A for the nickel-ruthenium compound) that disappear above the ordering temperature, the altermagnetic assignment fails. For the piezomagnetic claim, an experiment applying about 1% tensile strain along $y$ to hole-doped Ba$_3$NiRu$_2$O$_9$ should find a magnetization along $y$ with the magnitude and doping dependence shown in the paper; absence of that response would falsify the quantitative prediction.

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Extended reading notes

Core claim

The central claim is that a centrosymmetric crystal structure combined with collinear antiferromagnetism can still produce altermagnetism. In the 6H perovskite structure there is no inversion center connecting the magnetic B sites, but there are C$_2$ rotation axes perpendicular to the $c$-axis that exchange the two $ab$ planes of B ions. When those two planes order antiferromagnetically with spins mostly along $c$, the C$_2$ operation flips spin and momentum in a way that forces spin-up and spin-down bands to split in momentum space even without spin-orbit coupling. Applying magnetic point group classification to the fifteen known antiferromagnetic 6H perovskites, the paper finds four altermagnets: Ba$_3$CoIr$_2$O$_9$ and Ba$_3$SrIr$_2$O$_9$ are M-type, with broken time-reversal symmetry, weak ferromagnetism, and magneto-optical response, while Ba$_3$NiRu$_2$O$_9$ and Ba$_3$TbRu$_2$O$_9$ are S-type, with pure spin splitting and piezomagnetism. DFT+U calculations for Ba$_3$CoIr$_2$O$_9$ show non-relativistic spin splitting along L-$\Gamma$-V; adding spin-orbit coupling opens a 40 meV gap with a net moment of 1.4 $\mu_B$ per formula unit and a single nonzero off-diagonal optical conductivity component. For Ba$_3$NiRu$_2$O$_9$, collinear calculations show spin splitting along M-A, and calculations with 1% tensile strain plus hole doping give a magnetization along $y$ whose piezomagnetic coefficient is an order of magnitude larger than earlier estimates.

Load-bearing premise

The classification assumes the magnetic point group assigned to each compound is the true one, specifically that the two B-plane sublattices are antiferromagnetically stacked with spins along or near the $c$-axis; for Ba$_3$SrIr$_2$O$_9$ this rests on a theoretical ground state rather than a fully determined experimental magnetic structure, so an error in that assignment would remove the predicted spin splitting.

Editorial extensions

If this is right

  • Spin-resolved ARPES or STM on cleaved 6H perovskite surfaces should see spin-up and spin-down band splitting appear below the Neel temperature along the specified $k$-paths, providing a direct test of the altermagnetic state.
  • For Ba$_3$CoIr$_2$O$_9$, the M-type symmetry implies a measurable magneto-optical Kerr response: the paper lists eight nonzero components of the $q_{ijk}$ tensor and finds a single nonzero off-diagonal optical conductivity $\sigma_{xz}(\omega)$, so optical experiments can fingerprint the altermagnetic ordering.
  • For Ba$_3$NiRu$_2$O$_9$, stress along $x$ or $y$ with modest hole doping should generate transverse magnetization along $y$; the computed piezomagnetic coefficient is an order of magnitude larger than earlier estimates, making it a candidate for strain-controlled magnetization.
  • The same symmetry logic extends to other mixed perovskites with 4H, 9R, and different layer sequences, which may contain additional altermagnets.
  • When non-magnetic ions occupy the dimer sites, the in-plane order becomes antiferromagnetic and the compounds are not formal altermagnets; the paper calls these quasialtermagnets with a ferromagnetic point group and weaker magneto-optical or piezomagnetic effects.

Reading between the lines

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

  • A follow-up calculation could relax atomic positions under strain, since the paper fixed the crystal structure when computing the piezomagnetic response; relaxation might shift the size of the induced magnetization, and that shift is not yet quantified.
  • Because altermagnetism here depends on the B' dimers being magnetically active and isolated, varying the 3d/4d/5d occupancy in the B' site is a natural chemical lever for tuning the spin splitting and the piezomagnetic coefficient, which the paper only sketches.
  • The quasialtermagnet idea for 120-degree and spin-liquid systems suggests a continuum in which materials without a simple doubled magnetic cell may still show field-dependent magneto-optical and piezomagnetic effects; Ba$_3$CoSb$_2$O$_9$ and similar compounds could be tested for this weaker response.
  • If surface cleavage preserves the magnetic space group, spin-resolved STM on Ba$_3$NiRu$_2$O$_9$ could spatially resolve the spin-split dimer states; the paper proposes ARPES and STM experiments but does not estimate surface reconstruction effects.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper argues that several 6H perovskites of the form A3BB'2O9 are altermagnets, based on magnetic point group symmetry analysis and density functional theory (DFT+U and DFT+U+SOC) calculations. For Ba3CoIr2O9 (M-type) and Ba3NiRu2O9 (S-type), the authors present band structures with non-relativistic spin splitting, Fermi-surface analyses, and, for Ba3CoIr2O9, a computed off-diagonal optical conductivity indicating magneto-optical activity. For Ba3NiRu2O9, they report a large hole-doping-induced magnetization under 1% tensile strain, which they label 'giant piezomagnetism.' The symmetry classification is extended in Table I to Ba3SrIr2O9 and Ba3TbRu2O9 on the basis of magnetic point group assignments, with the former relying on a theoretical magnetic structure from previous work.

Significance. If the central claims hold, the paper identifies a new family of altermagnets in layered hexagonal perovskites, which are attractive because single crystals can be grown and cleaved for surface-sensitive probes such as spin-resolved STM and ARPES. The direct DFT+U calculations for Ba3CoIr2O9 and Ba3NiRu2O9 are a strength, as they provide concrete band-structure evidence consistent with the symmetry analysis, and the stated Hubbard parameters and numerical settings make the calculations reproducible. However, the significance is tempered by two overreaching claims: the abstract's 'several 6H perovskites are indeed altermagnets' is only directly verified for two compounds, and the 'giant piezomagnetism' is a doped, strain-assisted response rather than an intrinsic piezomagnetic tensor of the stoichiometric material. These issues affect the breadth and headline impact of the paper but do not undermine the core altermagnetism evidence for the two directly computed compounds.

major comments (3)
  1. [Table I and Abstract] The abstract's claim that 'several 6H perovskites are indeed altermagnets' is over-broad relative to the evidence. Table I classifies Ba3SrIr2O9 as an M-type altermagnet using the footnote 'the theoretical magnetic ground state obtained in previous DFT+U+SOC calculations were used for analysis; moments are collinear in the ac plane with large ferromagnetic component.' This compound has no experimental TN and no measured magnetic structure. If the actual ground state differs (e.g., 120-degree in-plane order, or moments along c), the magnetic point group 2'/m' and the predicted rho_xy/rho_yz response would not apply. The two compounds computed in this work remain valid, but the 'several' wording implicitly weights all four entries. Please either provide additional verification of the assumed Ba3SrIr2O9 magnetic state or explicitly restrict the altermagnetism claim to the two compounds with direct computational evidence.
  2. [S-type altermagnet: Ba3NiRu2O9 and Methods] The 'giant piezomagnetic effect' reported in the abstract and conclusions is not the intrinsic piezomagnetic response of stoichiometric Ba3NiRu2O9. In the text, direct DFT+U+SOC calculations for the unstrained or 1%-strained undoped compound yield a vanishingly small net magnetization of about 10^-3 µB/2 f.u. The large signal in Fig. 6(a) appears only after hole doping (e.g., δ = 0.1 in Ba3−δNiRu2O9), and Fig. 6(b) is computed at a fixed doping of 0.1 holes/u.c. while varying strain. Moreover, the Methods state that 'atomic positions were not relaxed' when strain was applied. The piezomagnetic tensor Λijk is defined for the undoped linear response, so what is calculated is a doping-enhanced strain-induced magnetization, not a conventional piezomagnetic coefficient. Please qualify the 'giant piezomagnetism' claim accordingly, report the undoped piezomagnetic tensor explicitly, and discuss the sensitivity of the magnitude to atomic relaxation.
  3. [Symmetry analysis and robustness of spin orientation] The altermagnetic classification of Ba3NiRu2O9 and Ba3TbRu2O9 in Table I assumes the experimental moments are strictly along the c-axis. The paper does not discuss how robust the S-type classification is to small canting or to a spin reorientation (e.g., under strain or magnetic field). Since the piezomagnetic response is directly tied to the 6'/m'mm' magnetic point group, a quantitative statement about the allowable canting angles or a test with a canted configuration would strengthen the central claim. Without such analysis, the experimental relevance of the S-type classification for these two compounds is less certain than for the directly computed cases.
minor comments (5)
  1. [Fig. 2] The DFT+U+SOC band structure in Fig. 2 plots both spin projections, but because SOC mixes spin states, the spin-resolved character is not evident in the main figure; the spin-weighted version in Fig. S2 should be included in the main text or described in the caption.
  2. [Page 1] The line 'PACS numbers:' is followed by no entries; either provide the PACS codes or remove the placeholder.
  3. [Symmetry analysis, Eq. for spin degeneracy] The condition ε(k↑) = ε(k↓) is written without specifying the spin quantization axis; please define it (e.g., the z-axis) to avoid ambiguity.
  4. [S-type altermagnet, piezomagnetic tensor] The tensor components Λ112 = Λ211 = -2Λ and Λ222 = Λ are given without a sign convention or a definition of Λ; please specify the sign and the coordinate frame used in the DFT calculations.
  5. [Discussion] The statement 'there are two antiferromagnetic sub-lattices (Ni and Ru)' is imprecise; the unit cell contains two Ni and four Ru ions, so clarify that each sublattice consists of symmetry-equivalent ions.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the altermagnetic classification rests on a self-contained symmetry argument and independent DFT calculations; the one self-citation is not load-bearing.

full rationale

The paper's central claim that 6H perovskites are altermagnets is supported by a direct symmetry analysis ('There is no inversion center (I) connecting the magnetic B sites, but C2 rotation axes perpendicular to the c-axis ... This makes 6H perovskites altermagnetic if the two ab planes of magnetic B ions ... are antiferromagnetically ordered and the spins are (mostly) aligned along the c-axis.') and by DFT+U calculations for Ba3CoIr2O9 and Ba3NiRu2O9 that use experimental magnetic structures. These calculations are ab initio consequences of the assumed magnetic order, not fits to the target quantity. The classification 'according to [30]' cites a framework co-authored by S.-W. Cheong, but the same symmetry argument is given in the text and the DFT band structures independently confirm spin splitting in two representative compounds. The Table I entry for Ba3SrIr2O9 relies on a theoretical magnetic structure from prior DFT+U+SOC work, which is an acknowledged assumption and a potential correctness risk, but it is not a circular reduction: the classification is applied to an externally supplied magnetic structure rather than derived from the conclusion. The piezomagnetic and optical responses are computed within the same DFT framework after adopting the magnetic structure, so they are consequences of the model rather than circularly defined predictions. No equation or parameter is constructed from the quantity it is used to predict.

Assumptions & free parameters 7 free parameters · 3 assumptions · 0 invented entities

The central altermagnetism claim requires the assigned magnetic structures and the DFT+U approximation; the U/J values are parameters but are standard and not fitted to the target result. No new entities are introduced.

free parameters (7)
  • Hubbard U for Ni (UNi) = 8 eV
    Coulomb repulsion parameter in DFT+U for Ni 3d states; chosen as typical for transition metal oxides, not fitted to data.
  • Hubbard U for Co (UCo) = 7 eV
    Coulomb repulsion parameter in DFT+U for Co 3d states; robustness tested by increasing to 10 eV, still metallic in DFT+U.
  • Hubbard U for Ru (URu) = 3 eV
    Coulomb repulsion parameter in DFT+U for Ru 4d states.
  • Hubbard U for Ir (UIr) = 1.5 eV
    Coulomb repulsion parameter in DFT+U for Ir 5d states; robustness tested by increasing to 3 eV.
  • Hund's J for Ni and Co = 1 eV
    Exchange parameter in DFT+U for Ni and Co.
  • Hund's J for Ru = 0.7 eV
    Exchange parameter in DFT+U for Ru.
  • Hund's J for Ir = 0.5 eV
    Exchange parameter in DFT+U for Ir.
assumptions (3)
  • domain assumption The experimental crystal and magnetic structures of Ba3CoIr2O9, Ba3NiRu2O9, Ba3TbRu2O9 (and theoretical for Ba3SrIr2O9) are correct.
    Symmetry analysis and DFT calculations use these structures as input; a wrong magnetic structure would invalidate the classification.
  • domain assumption DFT+U with the chosen U and J values reliably describes the electronic structure and magnetic ground state of these 4d/5d oxides.
    The band splittings and piezomagnetic response are computed within this approximation; the paper tests robustness only for UCo and UIr, not for the other parameters.
  • domain assumption Goodenough-Kanamori rules for exchange interactions (ferromagnetic for empty/eg half-filled, antiferromagnetic for t2g dimers) determine the interplane magnetic order.
    Used in the Discussion to rationalize the AFM order between B' planes and the resulting altermagnetic splitting.

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Cite this review

Pith. "Pith review of Altermagnetism in 6H perovskites." pith.science (2026). https://pith.science/paper/CNRMAKKM

@misc{pith2026250723232,
  author       = {Pith},
  title        = {Pith review of: Altermagnetism in 6H perovskites},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CNRMAKKM}},
  note         = {Machine review of arXiv:2507.23232}
}
abstract

The combination of a centrosymmetric crystallographic structure with local structural alternations and collinear antiferromagnetism can lead to broken PT (Parity $\times$ Time-reversal) symmetry, resulting in altermagnets with non-relativistic spin-split bands. The 6H perovskites with composition A$_3$BB'$_2$O$_9$ exhibit unique layered structural alternations and typically adopt an antiferromagnetic ground state. Here, we report the discovery that several 6H perovskites are indeed altermagnets exhibiting non-relativistic spin-split bands. We also explore the possible presence of net magnetization due to spin-orbit coupling in these materials, as well as the manifestation of giant piezomagnetism. Since the single crystals of 6H perovskites can be readily grown and cleavable, our findings provide a new avenue to study the cleaved atomically-flat surfaces of altermagnets with advanced experimental techniques such as spin-resolved scanning tunneling microscopy (STM) or spin-resolved angle-resolved photoemission spectroscopy (ARPES) to explore their spin splitting nature.

Figures

Figures reproduced from arXiv: 2507.23232 by the authors.

Figure 1
Figure 1. FIG. 1: The crystal structure of 6H perovskites. Magnetic [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The band structure of Ba [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4: The real part of the optical conductivity for [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: FIG. 5: Band structure of Ba [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Illustration of the piezomagnetic effect. (a): depen [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Sketches illustrating origin of antiferromagnetic in [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Nanoscale Imaging of Strain-Controlled Altermagnetic Domains in {\alpha}-MnTe

    cond-mat.mtrl-sci 2026-07 conditional novelty 7.0 of 10

    In alpha-MnTe, compression makes magnetic domains grow by merging, and unloading leaves them fragmented in a different, metastable pattern, so the material remembers the strain history.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.