Pith. sign in

REVIEW 3 major objections 3 minor 75 references

Spin-split flat bands at the band edge and two-dimensional hole gases towards quantum Hall effect in altermagnetic CoF$_2$

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read Altermagnetic CoF2's flat valence bands should form spin-dependent 2D hole gases with quantized Hall conductance.

desk verdict A plausible DFT-based proposal for spin-dependent QHE in altermagnetic CoF2, but the central prediction hinges on an unquantified exact flatness that needs much stronger evidence. read the letter →

arxiv 2411.16188 v3 pith:2WVXN447 submitted 2024-11-25 cond-mat.mtrl-sci cond-mat.mes-hall

Bo-Wen Yu , Bang-Gui Liu This is my paper · ORCID
classification cond-mat.mtrl-scicond-mat.mes-hall
keywords altermagnetismflatbandstwo-dimensionalholegasquantumHalleffectCoF2spinsplittingLandaulevelsrutilestructure
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 claims that rutile CoF2, a wide-gap antiferromagnetic semiconductor, has spin-split flat bands at the top of the valence band: spin-up holes are immobile along one in-plane direction and spin-down holes along the perpendicular direction. Because each spin is confined to its own plane, the material should act as two independent two-dimensional hole gases. The paper shows that an in-plane magnetic field quantizes these hole gases into spin-resolved Landau levels and predicts a spin-dependent quantum Hall effect whose Hall conductance can be tuned by rotating the field. If right, this gives a concrete bulk material where the altermagnetic spin splitting directly produces spin-polarized topological transport.

What carries the argument

The load-bearing object is the spin-resolved effective low-energy Hamiltonian (Eq. 1), $H_\uparrow = (\hat p_x)^2/(2M_0) + (\hat p_z)^2/(2M_z)$ and $H_\downarrow = (\hat p_y)^2/(2M_0) + (\hat p_z)^2/(2M_z)$, which encodes that each spin has exactly one flat direction (no kinetic term) and one dispersive direction. In an in-plane magnetic field, the minimal coupling (Eq. 2) turns each Hamiltonian into a one-dimensional harmonic oscillator with Landau-level frequencies $\omega_\uparrow = |qB_y|/\sqrt{M_z M_0}$ and $\omega_\downarrow = |qB_x|/\sqrt{M_z M_0}$; the effective masses $M_0 = 0.1603\,m_e$ and $M_z = 0.2461\,m_e$, fitted to the band structure, set the energy scale. The Chern numbers $C_\uparrow = -r_\uparrow \,\mathrm{sign}(\sin\theta)$ and $C_\downarrow = -r_\downarrow \,\mathrm{sign}(\cos\theta)$, computed from the Berry curvature of the occupied Landau levels, convert these oscillator levels into quantized Hall conductivities. The entire argument is the chain: flat band edge to two independent 2D hole gases, to spin-resolved Landau levels, to spin-dependent Chern numbers.

What would settle it

Measure the valence band dispersion along the nominally flat $\Gamma$–$M$ and $M'$–$\Gamma$ directions with high-resolution angle-resolved photoemission or a very fine $k$-grid calculation: if the bandwidth along that direction is comparable to or larger than $\hbar\omega \approx 2\,\mathrm{meV}$ at $B = 3.6\,\mathrm{T}$ (from $M_0 = 0.1603\,m_e$ and $M_z = 0.2461\,m_e$), the predicted Landau-level structure cannot survive. Alternatively, Hall measurements on hole-doped CoF2 films with in-plane field should show spin-resolved plateaus at the stated densities and angles; their absence would falsify the central claim.

Watch

Extended reading notes

Core claim

The central discovery, as the authors state it, is that the valence band edge of altermagnetic CoF2 is made of spin-split flat bands: in the ground state with moments along [001], the spin-up flat band runs along $\Gamma$–$M$ and the spin-down flat band along $M'$–$\Gamma$, and both persist when spin-orbit coupling is included. The flatness is nonrelativistic in origin and is lost when the moments are tilted away from [001]. Around the band edge the band structure is captured by two independent effective Hamiltonians, $H_\uparrow = \hat p_x^2/(2M_0)+\hat p_z^2/(2M_z)$ and $H_\downarrow = \hat p_y^2/(2M_0)+\hat p_z^2/(2M_z)$, with $M_0 = 0.1603\,m_e$ and $M_z = 0.2461\,m_e$, which describe two-dimensional hole gases in the $xz$ and $yz$ planes. In a magnetic field in the $xy$ plane these become oscillator problems with spin-dependent frequencies, and the Berry curvature gives Chern numbers $C_\uparrow = -r_\uparrow \,\mathrm{sign}(\sin\theta)$ and $C_\downarrow = -r_\downarrow \,\mathrm{sign}(\cos\theta)$. The paper consequently predicts spin-dependent quantized Hall conductivity and Hall resistance, along with anisotropic spin-selective longitudinal transport.

Load-bearing premise

The prediction of quantized Landau levels and Hall plateaus assumes the valence band is exactly flat along one direction for each spin (no $p_y$ term in $H_\uparrow$ and no $p_x$ term in $H_\downarrow$) and that these bulk flat bands behave as independent 2D hole gases at the assumed density $n = 8\times 10^{11}\,\mathrm{cm}^{-2}$; the paper does not quantify how much residual dispersion in the flat direction could be tolerated before the plateaus are destroyed.

Editorial extensions

If this is right

  • Hole doping CoF2 should produce two independent spin-polarized two-dimensional hole gases, one in the $xz$ plane (spin-up) and one in the $yz$ plane (spin-down), with no inter-spin coupling near the valence band edge.
  • With an in-plane magnetic field at angle $\theta$ from the $x$ axis, both spins form Landau levels whose spacings scale as $|\sin\theta|$ and $|\cos\theta|$; at $\theta = \pi/4$ the two spins share the same levels, while at $\theta = 0$ or $\pi/2$ only one spin channel is quantized.
  • The Hall conductivity of each spin is quantized in units of $e^2/h$ with Chern numbers $C_\uparrow = -r_\uparrow \,\mathrm{sign}(\sin\theta)$ and $C_\downarrow = -r_\downarrow \,\mathrm{sign}(\cos\theta)$, so the spin-resolved Hall response can be reversed by rotating the field through $\theta = \pi/2$.
  • Longitudinal transport is also spin-selective: an in-plane electric field along $x$ drives only the spin-up channel and along $y$ only the spin-down channel, giving a current polarization of $\cos(2\theta_E)$ that is tunable by the electric-field direction.

Reading between the lines

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

  • The mechanism is not specific to CoF2: any rutile altermagnet with $d_{xz}/d_{yz}$ valence-band-edge states and the same enforced flat direction should produce the same pair of orthogonal 2D hole gases, so this may be a family effect.
  • Because the spin-resolved Hall response reverses as the in-plane field rotates past $\theta = \pi/4$, a device could use field angle rather than field sign as the control knob for a spin-polarized current; the paper does not propose this device explicitly.
  • The paper treats the hole density as an input parameter; a natural next step is to predict how the Hall plateaus move with gating or doping, which would make the effect testable in a field-effect geometry.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper reports first-principles DFT calculations for rutile CoF2 in its altermagnetic ground state, finding spin-split flat bands at the valence band edge along M-Γ-M'. From the DFT band structure, the authors construct a spin-resolved effective low-energy model in which each spin has quadratic dispersion in two directions and is exactly flat in the third. Applying a magnetic field in the xy plane, they argue that each spin forms a two-dimensional hole gas (in the xz and yz planes) with Landau levels, and they compute spin-dependent Hall conductivities, Hall resistances, and longitudinal transport as functions of field angle, magnitude, Fermi level, and carrier density. The central prediction is a spin-dependent quantum Hall effect tunable by field direction.

Significance. If validated, the work would identify a concrete altermagnetic bulk material with robust spin-split flat bands at the valence band edge and a route to spin-resolved 2D hole gases and quantum Hall physics without exfoliation or heterostructure engineering, with a clear falsifiable signature (angle-dependent spin-resolved Hall quantization). The effective model is transparent and the transport predictions are explicit. The main limitation is that the central QHE conclusion relies on exact one-dimensional flatness and a 2D reduction that are asserted rather than demonstrated quantitatively from the DFT bands; the Hubbard U is also chosen without a sensitivity analysis in the main text.

major comments (3)
  1. [III, Eq. (1) and surrounding text] The effective Hamiltonians in Eq. (1) assume exactly zero dispersion along one in-plane direction for each spin (no p_y term for spin-up and no p_x term for spin-down). The paper never quantifies the residual bandwidth along these directions from the DFT band structure. Using the fitted masses M0 = 0.1603 m_e and Mz = 0.2461 m_e, the Landau gap at B = 3.2 T is about 1.9 meV; any residual dispersion of this order would destroy the flatness of the Landau levels and the quantized Hall plateaus in Figs. 4 and 5. The authors should provide the DFT dispersion along the flat direction over the energy window relevant for hole doping and give a quantitative bound (e.g., bandwidth < ℏω_c), or the QHE conclusion is not supported.
  2. [III, after Eq. (4)] The treatment of the bulk flat bands as independent two-dimensional hole gases is not justified. The paper assumes a 2D carrier density n = 8×10^11 cm^-2 with no derivation from any doping level, gate voltage, or confinement potential. In a bulk 3D material with finite (even if small) dispersion along the third direction, the states form a 3D Fermi sea and a 2D sheet density is ill-defined. A concrete mechanism for realizing 2D hole gases in CoF2 (e.g., surface, interface, or heterostructure) is needed, or the approximation of decoupled 2D planes must be explicitly stated and validated.
  3. [II and III, Hubbard U choice] The choice U = 3 eV is justified only by the statement that the HSE06 gap is generally larger than reality. This is not a quantitative criterion. Since the flat bands and their position at the VBM are the core input to the model, the authors should demonstrate in the main text that the flat band survives with a bandwidth below the Landau gap for a reasonable range of U (or show the supplementary figure that supports this). Without such sensitivity analysis, the band-structure input is not sufficiently constrained.
minor comments (3)
  1. [III, Eq. (2)] The stated vector potential A↑ = (0, zBx, -xBy) has curl (-Bx, By, 0), not (Bx, By, 0) as claimed; although the A_y component does not enter H↑, the inconsistency with the stated field should be resolved, likely by a sign typo.
  2. [III, after Eq. (1)] The sentence 'doped holes have a huge effective mass and can hardly move in that direction... can be considered to be degenerate and used to make two-dimensional hole gases' is not a rigorous argument for 2D behavior; a finite mass in the third direction still leaves a 3D system unless confinement is introduced.
  3. [III, references to Supplementary] The manuscript refers to Figs. S1–S4 for key information such as the U-dependence and the additional band structure results; the main text should at least summarize the conclusions of these figures so that the robustness of the flat bands is established without requiring the reader to access the supplement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the DFT calculation, fitted effective model, and Landau-level/Chern transport results form a standard first-principles-to-model derivation chain with no input reused as output.

full rationale

The central derivation chain is linear. DFT (PAW/PBEsol+U) is used to establish the altermagnetic spin-split flat bands and the [001] easy axis; this is a first-principles calculation, not an output of the transport model. The effective Hamiltonian in Eq. (1) is obtained by fitting M0 and Mz to the DFT bands ('The M0 and Mz can be obtained by fitting the curves of the band structure near the valence band edge'), and the Landau-level energies, Chern numbers, and Hall conductivities in Eqs. (2)-(3) and Figs. 3-5 are mathematical consequences of that fitted Hamiltonian plus standard Landau-level/Chern theory, with the Chern-number formula cited to the independent reference [72]. No Hall conductance or resistance value is used to adjust the model, so the transport prediction is not a re-fit of its own input. The assumed 2D density n=8e11 cm^-2 and the neglect of residual flat-band dispersion are external assumptions and potential quantitative risks, but they do not make any equation both an input and an output. The only apparent self-citation ([57], containing B. Liu) is cited as background on CoF2 properties and is not load-bearing. The paper is therefore self-contained against external benchmarks in the sense that the QHE statement is derived, not assumed.

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

The predictive content reduces to a quadratic two-band model whose masses are fitted to DFT; the Landau-level and Chern-number results are exact consequences of that model. The main unverified inputs are the hand-picked U, the exact flatness assumption, the assumed 2D density, and the neglect of magnetic-field-induced changes to the magnetic order.

free parameters (5)
  • Hubbard U = 3 eV
    Chosen by hand as 'the most appropriate value' for the band description after comparing HSE06 and PBE+U (Sec. II); the band gap and flat bands depend on this choice.
  • Hole effective mass M0 = 0.1603 me
    Fitted to the DFT valence band curvature along the dispersive in-plane direction (Sec. III); it sets the Landau level spacing for both spins.
  • Hole effective mass Mz = 0.2461 me
    Fitted to the DFT valence band curvature along z (Sec. III); it enters the Landau level frequencies and all transport formulas.
  • 2D carrier density n = 8e11 cm^-2
    Assumed for the Hall conductivity and resistance plots (Fig. 4); it is not derived from the bulk band structure.
  • Relaxation time tau = 5e-11 s
    Assumed for the longitudinal conductivity and resistance plots (Fig. 4); no material-specific estimate is provided.
assumptions (5)
  • domain assumption DFT within PBEsol+U with U=3 eV gives an accurate description of the CoF2 ground state and band edges.
    The choice U=3 eV is justified only by qualitative comparison with HSE06 and by the belief that HSE06 overestimates the gap (Sec. II); the flat bands are claimed robust across methods, but the main calculations use this hand-picked U.
  • ad hoc to paper The valence band edge near the VBM is exactly parabolic in two dispersive directions and exactly flat in the third direction, as captured by Eq. (1).
    Eq. (1) omits the p_y term for spin up and the p_x term for spin down; the paper provides no numerical bound on the residual dispersion in the flat direction, which would broaden Landau levels.
  • ad hoc to paper The bulk flat bands can be treated as independent two-dimensional hole gases with a well-defined 2D carrier density.
    The paper reduces the 3D band structure to two planar low-energy Hamiltonians without deriving a confinement potential or a per-layer density; n=8e11 cm^-2 is assumed in Fig. 4.
  • domain assumption An in-plane magnetic field leaves the [001] collinear magnetic order unchanged, and Zeeman and spin-flop effects can be neglected.
    The paper states the magnetocrystalline energy is large, but the calculated anisotropy differences are only 0.2 to 1.2 meV per cell, comparable to Zeeman energies at the 3 to 4 T fields used.
  • standard math Independent spin channels allow different gauge choices for the vector potential.
    For single-particle spin sectors this is formally allowed, but the paper's spin-up gauge A_up=(0,zBx,-xBy) has curl (-Bx,By,0), not (Bx,By,0), so the notation is inconsistent even though the erroneous component drops out.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Spin-split flat bands at the band edge and two-dimensional hole gases towards quantum Hall effect in altermagnetic CoF$_2$." pith.science (2026). https://pith.science/paper/2WVXN447

@misc{pith2026241116188,
  author       = {Pith},
  title        = {Pith review of: Spin-split flat bands at the band edge and two-dimensional hole gases towards quantum Hall effect in altermagnetic CoF$_2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2WVXN447}},
  note         = {Machine review of arXiv:2411.16188}
}
abstract

Altermagnetic phase is recently found as a new magnetic phase in addition to the conventional collinear spin orders, and great efforts have been made to explore novel effects and potential applications in such materials. Here, we show that there are robust altermagnetic spin-split flat bands near the valence band edge in rutile CoF$_2$ through first-principles investigation. It is uncovered that the magnetic moments can remain in the z axis because of the magnetocrystalline energy due to the spin-orbits coupling and the spin orientation can be made more stable by magnetic field applied in the xy plane. We describe the spin-dependent band structure (including the flat bands) near the Fermi level by a spin-resolved effective low-energy model, and reveal that they can host spin-dependent two-dimensional hole gases. Importantly, we find spin-dependent quantum Hall effects in the two-dimensional hole gases by applying the magnetic field in the xy plane, and then explore the dependence of Hall conductivity and Hall resistance on the Fermi level and the magnetic field (both magnitude and direction) and related longitudinal carrier transport properties.

Figures

Figures reproduced from arXiv: 2411.16188 by the authors.

Figure 1
Figure 1. FIG. 1: (a) The structures of rutile CoF [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The electronic band structures of the CoF [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: (a) The direction ( [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: FIG. 5: (a) The spin-dependent Hall resistance ( [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

75 extracted references · 35 canonical work pages

  1. [1]

    directions, and present the SOC band structures near the valence and conduction band edges in Fig. 2. It is clear that the flat bands remain at the valence band edge (VBM) when the magnetic moments are in the [001] direction. When the magnetic moments are made differ- ent from the [001] direction, there will be no flat bands at the Γ point and there will ...

    work page

  2. [2]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12, 031042 (2022)

    2022

  3. [3]

    Mazin (The PRX Editors), Phys

    I. Mazin (The PRX Editors), Phys. Rev. X 12, 040002 (2022)

    2022

  4. [4]

    Litvin and W

    D. Litvin and W. Opechowski, Physica 76, 538 (1974)

    work page 1974

  5. [5]

    H.-Y. Ma, M. Hu, N. Li, J. Liu, W. Yao, J.-F. Jia, and J. Liu, Nature Communications 12, 2846 (2021)

    2021

  6. [6]

    Z. Xiao, J. Zhao, Y. Li, R. Shindou, and Z.-D. Song, Phys. Rev. X 14, 031037 (2024)

    2024

  7. [7]

    D. B. Litvin, Acta Crystallographica Section A 33, 279 (1977)

    1977

  8. [8]

    Jiang, Z

    Y. Jiang, Z. Song, T. Zhu, Z. Fang, H. Weng, Z.-X. Liu, J. Yang, and C. Fang, Phys. Rev. X 14, 031039 (2024)

    2024

Show all 75 references
  1. [9]

    X. Chen, J. Ren, Y. Zhu, Y. Yu, A. Zhang, P. Liu, J. Li, Y. Liu, C. Li, and Q. Liu, Phys. Rev. X 14, 031038 (2024)

    2024

  2. [10]

    L. Han, X. Fu, R. Peng, X. Cheng, J. Dai, L. Liu, Y. Li, Y. Zhang, W. Zhu, H. Bai, Y. Zhou, S. Liang, C. Chen, Q. Wang, X. Chen, L. Yang, Y. Zhang, C. Song, J. Liu, and F. Pan, Sci. Adv. 10, eadn0479 (2024)

    2024

  3. [11]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12, 040501 (2022)

    2022

  4. [12]

    Reimers, L

    S. Reimers, L. Odenbreit, L. ˇSmejkal, V. N. Strocov, P. Constantinou, A. B. Hellenes, R. Jaeschke Ubiergo, W. H. Campos, V. K. Bharadwaj, A. Chakraborty, T. Denneulin, W. Shi, R. E. Dunin-Borkowski, S. Das, M. Kl¨ aui, J. Sinova, and M. Jourdan, Nature Communi- cations 15, 21...

    2024

  5. [13]

    Reichlova, R

    H. Reichlova, R. Lopes Seeger, R. Gonz´ alez-Hern´ andez, I. Kounta, R. Schlitz, D. Kriegner, P. Ritzinger, M. Lammel, M. Leivisk¨ a, A. Birk Hellenes, K. Olejn ´ ık, V. Petˇ riˇ cek, P. Doleˇ zal, L. Horak, E. Schmoranzerova, A. Badura, S. Bertaina, A. Thomas, V. Baltz, L. Mi...

    2024

  6. [14]

    Z. Zhou, X. Cheng, M. Hu, R. Chu, H. Bai, L. Han, J. Liu, F. Pan, and C. Song, Nature 638, 645 (2025)

    2025

  7. [15]

    Urata, W

    T. Urata, W. Hattori, and H. Ikuta, Phys. Rev. Mater. 8, 084412 (2024)

    2024

  8. [16]

    Krempask´ y, L

    J. Krempask´ y, L. ˇSmejkal, S. W. D’Souza, M. Ha- jlaoui, G. Springholz, K. Uhl ´ ıˇ rov´ a, F. Alarab, P. C. Constantinou, V. Strocov, D. Usanov, W. R. Pudelko, R. Gonz´ alez-Hern´ andez, A. Birk Hellenes, Z. Jansa, H. Reichlov´ a, Z. ˇSob´ aˇ n, R. D. Gonzalez Betancourt, P...

    2024

  9. [17]

    Y. Wu, L. Deng, X. Yin, J. Tong, F. Tian, and X. Zhang, Nano Letters 24, 10534 (2024), pMID: 39145607

    2024

  10. [18]

    Osumi, S

    T. Osumi, S. Souma, T. Aoyama, K. Yamauchi, A. Honma, K. Nakayama, T. Takahashi, K. Ohgushi, and T. Sato, Phys. Rev. B 109, 115102 (2024)

    2024

  11. [19]

    S. Lee, S. Lee, S. Jung, J. Jung, D. Kim, Y. Lee, B. Seok, J. Kim, B. G. Park, L. ˇSmejkal, C.-J. Kang, and C. Kim, Phys. Rev. Lett. 132, 036702 (2024)

    2024

  12. [20]

    Y.-P. Zhu, X. Chen, X.-R. Liu, Y. Liu, P. Liu, H. Zha, G. Qu, C. Hong, J. Li, Z. Jiang, X.-M. Ma, Y.-J. Hao, M.-Y. Zhu, W. Liu, M. Zeng, S. Jayaram, M. Lenger, J. Ding, S. Mo, K. Tanaka, M. Arita, Z. Liu, M. Ye, D. Shen, J. Wrachtrup, Y. Huang, R.-H. He, S. Qiao, Q. Liu, and C...

    2024

  13. [21]

    O. J. Amin, A. D. Din, E. Golias, Y. Niu, A. Za- kharov, S. C. Fromage, C. J. B. Fields, S. L. Heywood, R. B. Cousins, F. Maccherozzi, J. Krempasky, J. H. Dil, D. Kriegner, B. Kiraly, R. P. Campion, A. W. Rushforth, K. W. Edmonds, S. S. Dhesi, L. Smejkal, T. Jungwirth, and P. ...

    2024

  14. [22]

    Hiraishi, H

    M. Hiraishi, H. Okabe, A. Koda, R. Kadono, T. Muroi, D. Hirai, and Z. Hiroi, Phys. Rev. Lett. 132, 166702 (2024)

    2024

  15. [23]

    H. Bai, Y. C. Zhang, Y. J. Zhou, P. Chen, C. H. Wan, L. Han, W. X. Zhu, S. X. Liang, Y. C. Su, X. F. Han, F. Pan, and C. Song, Phys. Rev. Lett. 130, 216701 (2023)

    2023

  16. [24]

    Smolyanyuk, I

    A. Smolyanyuk, I. I. Mazin, L. Garcia-Gassull, and R. Valent ´ ı, Phys. Rev. B109, 134424 (2024)

    2024

  17. [25]

    Z. Q. Wang, Z. Q. Li, L. Sun, Z. Y. Zhang, K. He, H. Niu, J. Cheng, M. Yang, X. Yang, G. Chen, Z. Yuan, H. F. Ding, and B. F. Miao, Phys. Rev. Lett.133, 046701 (2024)

    2024

  18. [26]

    Adamantopoulos, M

    T. Adamantopoulos, M. Merte, F. Freimuth, D. Go, L. Zhang, M. Leˇ zai´ c, W. Feng, Y. Yao, J. Sinova, L. ˇSmejkal, S. Bl¨ ugel, and Y. Mokrousov, npj Spintronics 2, 46 (2024)

    2024

  19. [27]

    Z. Lin, D. Chen, W. Lu, X. Liang, S. Feng, K. Yamagami, J. Osiecki, M. Leandersson, B. Thiagarajan, J. Liu, C. Felser, and J. Ma, Observation of giant spin splitting and d-wave spin texture in room temperature altermag- net ruo2 (2024), arXiv:2402.04995 [cond-mat.mtrl-sci]

    2024 arXiv

  20. [28]

    Y. Guo, H. Liu, O. Janson, I. C. Fulga, J. van den Brink, and J. I. Facio, Materials Today Physics 32, 100991 (2023)

    2023

  21. [29]

    J. Liu, J. Zhan, T. Li, J. Liu, S. Cheng, Y. Shi, L. Deng, M. Zhang, C. Li, J. Ding, Q. Jiang, M. Ye, Z. Liu, Z. Jiang, S. Wang, Q. Li, Y. Xie, Y. Wang, S. Qiao, J. Wen, Y. Sun, and D. Shen, Phys. Rev. Lett. 133, 176401 (2024)

    2024

  22. [30]

    Milivojevi´ c, M

    M. Milivojevi´ c, M. Orozovi´ c, S. Picozzi, M. Gmitra, and S. Stavri´ c, 2D Materials11, 035025 (2024)

    2024

  23. [31]

    Bhowal and N

    S. Bhowal and N. A. Spaldin, Phys. Rev. X 14, 011019 (2024)

    2024

  24. [32]

    ˇSmejkal, A

    L. ˇSmejkal, A. B. Hellenes, R. Gonz´ alez-Hern´ andez, J. Sinova, and T. Jungwirth, Phys. Rev. X 12, 011028 (2022)

    2022

  25. [33]

    Gonz´ alez-Hern´ andez, L.ˇSmejkal, K

    R. Gonz´ alez-Hern´ andez, L.ˇSmejkal, K. V´ yborn´ y, Y. Ya- hagi, J. Sinova, T. c. v. Jungwirth, and J. ˇZelezn´ y, Phys. Rev. Lett. 126, 127701 (2021)

    2021

  26. [34]

    A. Bose, N. J. Schreiber, R. Jain, D.-F. Shao, H. P. Nair, J. Sun, X. S. Zhang, D. A. Muller, E. Y. Tsym- bal, D. G. Schlom, and D. C. Ralph, Nature Electronics 5, 267 (2022)

    2022

  27. [35]

    Shao, S.-H

    D.-F. Shao, S.-H. Zhang, M. Li, C.-B. Eom, and E. Y. Tsymbal, Nature Communications 12, 7061 (2021)

    2021

  28. [36]

    M. Naka, S. Hayami, H. Kusunose, Y. Yanagi, Y. Mo- tome, and H. Seo, Nature Communications 10, 4305 (2019)

    2019

  29. [37]

    H. Bai, L. Han, X. Y. Feng, Y. J. Zhou, R. X. Su, Q. Wang, L. Y. Liao, W. X. Zhu, X. Z. Chen, F. Pan, 7 X. L. Fan, and C. Song, Phys. Rev. Lett. 128, 197202 (2022)

    2022

  30. [38]

    ˇSmejkal, R

    L. ˇSmejkal, R. Gonz´ alez-Hern´ andez, T. Jungwirth, and J. Sinova, Science Advances 6, eaaz8809 (2020)

    2020

  31. [39]

    M. Naka, Y. Motome, and H. Seo, Phys. Rev. B 103, 125114 (2021)

    2021

  32. [40]

    Samanta, D.-F

    K. Samanta, D.-F. Shao, and E. Y. Tsymbal, Nano Let- ters 25, 3150 (2025)

    2025

  33. [41]

    R. He, D. Wang, N. Luo, J. Zeng, K.-Q. Chen, and L.-M. Tang, Phys. Rev. Lett. 130, 046401 (2023)

    2023

  34. [42]

    A. Bose, S. Vadnais, and A. Paramekanti, Phys. Rev. B 110, 205120 (2024)

    2024

  35. [43]

    Chakraborty and A

    D. Chakraborty and A. M. Black-Schaffer, Phys. Rev. B 110, L060508 (2024)

    2024

  36. [44]

    M. Wei, L. Xiang, F. Xu, L. Zhang, G. Tang, and J. Wang, Phys. Rev. B 109, L201404 (2024)

    2024

  37. [45]

    H. G. Giil, B. Brekke, J. Linder, and A. Brataas, Phys. Rev. B 110, L140506 (2024)

    2024

  38. [46]

    H. G. Giil and J. Linder, Phys. Rev. B 109, 134511 (2024)

    2024

  39. [47]

    Mæland, B

    K. Mæland, B. Brekke, and A. Sudbø, Phys. Rev. B 109, 134515 (2024)

    2024

  40. [48]

    Cheng and Q.-F

    Q. Cheng and Q.-F. Sun, Phys. Rev. B 109, 024517 (2024)

    2024

  41. [49]

    Zhang, L.-H

    S.-B. Zhang, L.-H. Hu, and T. Neupert, Nature Commu- nications 15, 1801 (2024)

    2024

  42. [50]

    C. Sun, A. Brataas, and J. Linder, Phys. Rev. B 108, 054511 (2023)

    2023

  43. [51]

    Brekke, A

    B. Brekke, A. Brataas, and A. Sudbø, Phys. Rev. B 108, 224421 (2023)

    2023

  44. [52]

    G. A. Slack, Phys. Rev. 122, 1451 (1961)

    1961

  45. [53]

    J. W. Stout and S. A. Reed, Journal of the American Chemical Society 76, 5279 (1954)

    1954

  46. [54]

    J. P. Van Der Ziel and H. J. Guggenheim, Phys. Rev. 166, 479 (1968)

    1968

  47. [55]

    M. E. Lines, Phys. Rev. 137, A982 (1965)

    1965

  48. [56]

    S. J. Allen and H. J. Guggenheim, Journal of Applied Physics 42, 1657 (1971)

    1971

  49. [57]

    Ishikawa and T

    A. Ishikawa and T. Moriya, Journal of the Physical So- ciety of Japan 30, 117 (1971)

    1971

  50. [58]

    J. A. Barreda-Arg¨ ueso, S. L´ opez-Moreno, M. N. Sanz- Ortiz, F. Aguado, R. Valiente, J. Gonz´ alez, F. Rodr ´ ıguez, A. H. Romero, A. Mu˜ noz, L. Nataf, and F. Baudelet, Phys. Rev. B 88, 214108 (2013)

    2013

  51. [59]

    A. S. Disa, M. Fechner, T. F. Nova, B. Liu, M. F¨ orst, D. Prabhakaran, P. G. Radaelli, and A. Cavalleri, Nature Physics 16, 937 (2020)

    2020

  52. [60]

    P. E. Bl¨ ochl, Phys. Rev. B50, 17953 (1994)

    1994

  53. [61]

    Strempfer, U

    J. Strempfer, U. R¨ utt, S. P. Bayrakci, T. Br¨ uckel, and W. Jauch, Phys. Rev. B 69, 014417 (2004)

    2004

  54. [62]

    J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996)

    1996

  55. [63]

    Kresse and J

    G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993)

    1993

  56. [64]

    H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976)

    1976

  57. [65]

    J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Phys. Rev. Lett. 100, 136406 (2008)

    2008

  58. [66]

    J. Heyd, G. E. Scuseria, and M. Ernzerhof, The Journal of Chemical Physics 118, 8207 (2003)

    2003

  59. [67]

    S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Phys. Rev. B 57, 1505 (1998)

    1998

  60. [68]

    N´ eel, Proc

    L. N´ eel, Proc. Phys. Soc. A65, 869 (1952)

    1952

  61. [69]

    N´ eel, Ann

    L. N´ eel, Ann. Phys. (Paris)11, 232 (1936)

    1936

  62. [70]

    A. N. Bogdanov, A. V. Zhuravlev, and U. K. Roessler, Phys. Rev. B 75, 094425 (2007)

    2007

  63. [71]

    N. J. Poulis, J. van den Handel, J. Ubbink, J. A. Poulis, and C. J. Gorter, Phys. Rev. 82, 552 (1951)

    1951

  64. [72]

    Ozawa and B

    T. Ozawa and B. Mera, Phys. Rev. B104, 045103 (2021)

    2021

  65. [73]

    Baltz, A

    V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Rev. Mod. Phys.90, 015005 (2018)

    2018

  66. [75]

    H. Lang, S. Zhang, and Z. Liu, Phys. Rev. B 94, 235306 (2016)

    2016

  67. [110]

    Several methods including HSE06[66], PBE+U and PBEsol+U[63, 65] are used for comprehensive evalua- tion of the band structure

    axises are higher than the ground state by 0.2meV and 1.2meV per unit cell, respectively. Several methods including HSE06[66], PBE+U and PBEsol+U[63, 65] are used for comprehensive evalua- tion of the band structure. The results with different U values and HSE06 method are pre...

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.