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REVIEW 3 major objections 4 minor 134 references

Hole-doped germanium hosts a giant orbital Hall effect, larger than its spin Hall effect by four orders of magnitude.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-04 15:14 UTC pith:IG2DKPRN

load-bearing objection Modern-theory OHE in holes is new and testable, but the headline magnitude hinges on an OAM identification the paper leaves open. the 3 major comments →

arxiv 2509.20436 v3 pith:IG2DKPRN submitted 2025-09-24 cond-mat.mes-hall

Orbital Hall effect in spin-3/2 hole-doped semiconductors and its implications for orbitronics

James H. Cullen , Zhanning Wang , Dimitrie Culcer This is my paper
classification cond-mat.mes-hall
keywords orbital Hall effectorbitronicsgermaniumLuttinger modelspin-orbit couplingspin Hall effectorbital torquep-type semiconductors
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proposes p-type semiconductors, and germanium in particular, as strong orbitronic materials. It argues, using the Luttinger model with the spherical approximation plus the modern theory of orbital magnetisation, that holes in bulk Ge carry an intrinsic orbital Hall conductivity of order 10^3 (ℏ/e) Ω⁻¹ cm⁻¹. At a Fermi energy of 10 meV, the calculated orbital Hall conductivity is −3.1 e/nm, about 6000 times larger than the calculated spin Hall conductivity in the same system and larger than the bulk-state value in Bi₂Se₃. A reader should care because this makes Ge a clean platform for orbital torques: inversion symmetry forbids the Edelstein effects there, so the orbital Hall effect would be the dominant angular-momentum channel in a Ge/ferromagnet device.

Core claim

Within this model, the Berry curvature and the equilibrium orbital angular momentum are both finite at each wave vector even though their occupied-state integrals vanish by time-reversal symmetry. The orbital current is dominated by quantum corrections—especially the field-induced dipole/convection term Δj₁—rather than by the conventional term. The response vanishes in the limit γ̄→0, so, in the 4×4 hole model, spin-orbit coupling is the sole driver. The same formalism applied to Si, GaAs, InAs and InSb gives similarly large orbital Hall conductivities, with Ge singled out by its mobility and manufacturing compatibility. The paper frames this as the first modern-theory OHE calculation in an

What carries the argument

The Luttinger-Kohn Hamiltonian for spin-3/2 holes, with the spherical approximation, provides the band structure. The orbital angular momentum operator is the symmetrized modern-theory form L = (m/2)(r×v − v×r), with the bare electron mass (negative for holes). The nonzero per-k Berry curvature and OAM of the heavy- and light-hole bands feed a non-equilibrium density matrix obtained from the quantum Liouville equation, and the orbital current is evaluated with a gauge-invariant expression that includes quantum corrections beyond the conventional term. These corrections, dominated by Δj₁, are what turn a modest conventional response into a giant one.

Load-bearing premise

The central quantitative claim depends on identifying the physical orbital angular momentum with the modern-theory operator L=(m/2)(r×v − v×r), with negative bare mass for holes, and on the quantum-corrected orbital-current expression taken from earlier work; if the quantity that couples to a ferromagnet in a torque experiment is instead the atomic OAM, the giant OHE number is not the physical one.

What would settle it

Measure the orbital torque in a Ge/Co or Ge/Pt stack with bulk p-Ge and an in-plane current at low temperature and low doping, where Edelstein effects are forbidden. If the torque-per-current corresponds to an OHE two to four orders smaller than the calculated −3.1 e/nm at 10 meV—or of opposite sign, as the atomic-OAM ab initio calculation suggests—the central claim fails. A complementary check is to compute the OHE ab initio using the modern-theory OAM operator and including the quantum corrections; if that result differs from the Luttinger value by more than an order of magnitude, the effect

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • Bulk p-type Ge has an orbital Hall conductivity of order 10^3 (ℏ/e) Ω⁻¹ cm⁻¹, exceeding its spin Hall conductivity by about four orders of magnitude, making orbital current the dominant transverse channel.
  • Because spin- and orbital-Edelstein effects are symmetry-forbidden in bulk Ge, an orbital-torque experiment in a Ge/ferromagnet stack isolates OHE plus SHE, with OHE dominating for any orbital-to-spin conversion above roughly 0.2%.
  • The OHE in Ge exceeds the bulk-state OHE of Bi₂Se₃ for Fermi energies above 2 meV, and Ge offers higher mobility and silicon-compatible fabrication.
  • The OHE vanishes as γ̄→0, so in this model the entire orbital response is driven by spin-orbit coupling, and the quantum corrections rather than the conventional term set the magnitude.
  • Five common p-type semiconductors all show large orbital Hall conductivities; Ge ranks third in magnitude but first in hole mobility, making it the best practical candidate.

Where Pith is reading between the lines

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

  • If a ferromagnet detector couples to atomic-like OAM rather than the modern-theory operator, torque measurements in Ge should land near the smaller ab initio values, not at 10^3; a torque experiment would settle which definition is physical.
  • The 4×4 model omits the split-off band; a full 6×6 treatment could harden or soften the claim that OHE requires spin-orbit coupling, since without SOC the split-off band becomes degenerate with the heavy and light holes at k=0.
  • The same quantum-correction machinery could be carried into strained or two-dimensional hole gases, where confinement and strain change the Luttinger parameters and may shift the relative weight of the conventional term.
  • The comparison with topological insulators ignores surface/interface contributions, so in a real device the interface conversion efficiency, not just the bulk OHE, will decide whether the predicted angular-momentum current dominates.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper calculates the intrinsic orbital Hall conductivity (OHC) and spin Hall conductivity (SHC) for holes in bulk Ge and other zinc-blende semiconductors, using a spherical Luttinger Hamiltonian and the modern theory of orbital magnetisation. The central result is a large OHC of order -3.1 e/nm at a Fermi energy of 10 meV in Ge, which exceeds the SHC by roughly four orders of magnitude and exceeds the bulk OHC of Bi2Se3 by a factor of about two. The authors emphasize that the effect is dominated by quantum corrections to the orbital current operator, that it arises entirely from spin-orbit coupling within the 4x4 Luttinger model, and that Ge is a promising orbitronic platform because spin- and orbital-Edelstein effects are symmetry-forbidden. They also compare with a previous ab initio study (Ref. 76) and note an order-of-magnitude discrepancy, which they attribute to the use of atomic OAM operators in that work versus the modern-theory OAM in the present one.

Significance. If the central claim holds, the paper identifies a conventional semiconductor with a 'giant' orbital Hall effect, which would be an experimentally attractive and technologically relevant orbitronic material. The calculation is internally consistent, uses standard Luttinger parameters, and is not fitted to any target observable; the Fermi-energy dependence and material comparison are clear predictions. The emphasis on quantum corrections to the orbital current is also a useful contribution, as is the explicit statement that the modern-theory OAM can be large in an inversion-symmetric system. However, the quantitative claim depends on identifying the modern-theory OAM operator with the orbital angular momentum actually exchanged at a ferromagnetic interface, and the paper itself states that this relationship is not yet clarified. That uncertainty, together with the discrepancy with ab initio results based on atomic OAM operators, means the headline magnitude is not yet established as the physically relevant one.

major comments (3)
  1. [Sec. IV; Eq. (5); Table I] The central quantitative claim—OHC ≈ -3.1 e/nm, four orders of magnitude larger than the SHC—rests on the orbital-current expression in Eq. (5) built from the modern-theory OAM operator in Eq. (3). The paper explicitly states in Sec. IV that the relationship between this modern-theory OAM and the atomic OAM operator used in Ref. 76 'has not been clarified to date.' The ab initio result of Ref. 76, which uses atomic OAM, gives an OHC roughly an order of magnitude smaller and an opposite sign for the SHE. Since orbital torque experiments detect the orbital angular momentum that can be transferred to a ferromagnet, it is essential to establish which operator is the experimentally relevant one. This is not an internal inconsistency, but it is a load-bearing open question: if the atomic OAM is the physical observable, the headline 'giant OHE' and 'OHE ≫ SHE' claims fail quantitatively. I reco
  2. [Sec. III, last paragraph; Eq. (5)] The paper states that detailed derivations of the orbital/spin Hall conductivity expressions are in a Supplement, but no supplement is included with the preprint. Since Eq. (5) is the basis for the main result and is taken from the authors' own Ref. 88, a reader cannot independently check the derivation from the present manuscript alone. This is a reproducibility concern, especially because the quantum correction ∆σ is claimed to dominate the OHC. The authors should include the derivation in the main text or an accessible appendix, or at least reproduce the essential steps that justify applying Eq. (5) to the Luttinger model.
  3. [Sec. III, paragraph on split-off band; Sec. IV, 'spin-orbit coupling is required'] The conclusion that the OHE is entirely driven by spin-orbit coupling is stated as a general observation, but the authors themselves restrict it to their 4×4 model and acknowledge that the split-off band, ignored here, would become degenerate in the limit of zero spin-orbit coupling. This caveat is important for the interpretation of Fig. 5 and for any claim that OHE in Ge is impossible without spin-orbit coupling. The statement is defensible for the model considered, but the paper should more clearly separate model-specific conclusions from statements about the full 6×6 valence band structure.
minor comments (4)
  1. [Abstract and Sec. I/III] The abstract says the OHE 'exceeding the spin-Hall effect by 2-3 orders of magnitude,' while the introduction, Table I, and conclusions state 'four orders of magnitude.' For EF=10 meV, Table I gives a ratio of about 6200, which is closer to four orders. Please harmonize the wording.
  2. [Fig. 5 and Sec. III, 'Comparison with other hole systems'] The statement that the spherical approximation is 'generally invalid for silicon' but that the actual OHC should nevertheless be of similar magnitude is speculative. Since Si is included in the comparison, the authors should either present a caveat in the caption and text or remove Si from the quantitative comparison.
  3. [Sec. II C, after Eq. (4)] The intrinsic density-matrix expression in Eq. (4) is quoted as valid 'in the absence of disorder.' The paper later mentions that extrinsic disorder terms are ignored, but the relation to the Fermi-energy window (EF up to 10 meV) and realistic mobilities is not discussed. A brief estimate of the corresponding hole density and mean free path would help assess whether the intrinsic regime is experimentally relevant.
  4. [Sec. III, Table I] It would be useful to quote the Bi2Se3 value with the same units and to specify whether the value -1.2 e/nm is for the same orbital-current definition, since Ref. 53 uses the same formalism but readers may still find the comparison clearer with explicit definition of the transport axis and OAM polarization.

Circularity Check

0 steps flagged

No significant circularity: the OHE values are a new model application, not an output fitted or defined into existence; the main limitations are operator-identification and reproducibility, which are correctness risks rather than circularity.

full rationale

The derivation chain is not circular. Sec. II takes a standard spherical Luttinger-Kohn Hamiltonian (Eq. 1) with literature Luttinger parameters (γ1=13.38, γ̄=4.97), computes the Berry curvature and OAM per the stated modern-theory definitions (Eqs. 2-3), and evaluates the orbital current with the expression in Eq. 5. That expression is taken from the authors' own Ref. 88, but it is a previously derived formula for a definite observable, not a fit to the paper's target. Applying it to Ge holes produces the plotted σ_yz_x (Tab. I, Figs. 2, 5); no parameter is fitted to reproduce Table I and no prediction is a restatement of the input. The comparison with Bi2Se3 uses their own Ref. 53 value, which is a citation of prior independent/computational work, and the Ge number is computed here, so the comparative claim is not a constructional equivalence. The paper honestly flags the main ambiguity: its modern-theory OAM and the atomic-OAM basis of Ref. 76 'are clearly related and lead to the same observable' but 'their relationship has not been clarified to date' (Sec. IV). That is a physical/operator-identification concern, not circularity: the calculated observable is well defined by Eq. 5; whether it is the observable measured in orbital-torque experiments is an open question. Likewise, the omission of the Supplement ('detailed derivations... in the Supplement') and the caveats on split-off band and spherical approximation reduce reproducibility/confidence but do not turn the derivation into a self-fulfilling fit. Therefore score 0-2; I assign 2 because the central numerical formula and the comparative Bi2Se3 value both come from the authors' own prior papers, so external checking of Eq. 5 and Ref. 53 is needed, but this is not circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The calculation rests on standard k.p band parameters and the modern theory of orbital magnetization, but the quantitative outcome is sensitive to three choices internal to the authors' program: the modern-theory OAM operator, the quantum-corrected orbital current expression from Ref. 88, and the proper spin current definition. None of these is fitted to the target OHE value, so the ledger contains no target-fitted parameters, but the physical relevance of the chosen OAM operator is the main unresolved assumption.

free parameters (2)
  • Fermi energy for headline comparison E_F = 10 meV = 10 meV
    The magnitude of the OHE and the comparison with Bi2Se3 in Table I are quoted at E_F = 10 meV. The Fermi energy is swept in the figures, so the 'giant' framing depends on a chosen doping level, but no fitting to experimental data is performed.
  • Spherical-averaged Luttinger parameter gamma_bar = (gamma2+gamma3)/2 = 4.97 for Ge (with gamma1 = 13.38)
    The spherical approximation replaces the cubic Luttinger parameters gamma2 and gamma3 with their average, enabling a clean analytical expression for the orbital current. This is a hand-chosen modeling reduction that affects the quantitative result and is acknowledged to be invalid for Si.
axioms (5)
  • domain assumption The spherical Luttinger-Kohn Hamiltonian (Eq. 1) describes the valence band of Ge in the relevant doping range.
    Used throughout Secs. II and III; the spherical approximation is stated to work well for Ge but is not exact, and the authors note it is generally invalid for Si.
  • standard math The gauge-invariant orbital current expression Eq. (5), including the quantum corrections Delta j1-3, is the correct form of the intrinsic orbital current.
    Eq. (5) is taken from the authors' prior work Ref. [88] and its derivation is deferred to the Supplement; the quantum corrections dominate the Ge result.
  • domain assumption The orbital angular momentum observable is the modern-theory operator L = (m/2)(r x v - v x r) with the bare electron mass, using the negative electron mass for holes (Eq. 3).
    Central to the whole calculation; the independent ab initio study [76] using atomic OAM operators obtains an order-of-magnitude smaller OHE, so this axiom is load-bearing.
  • domain assumption The proper spin current expression Eq. (6) from Refs. [89,90] determines the SHE used in the OHE/SHE comparison.
    The claim that OHE exceeds SHE by orders of magnitude uses this specific spin current definition; the authors note a separate expression in the literature yields the same order of magnitude.
  • domain assumption Disorder and extrinsic contributions to the OHE are negligible, so the clean-system linear response density matrix Eq. (4) suffices.
    The authors state 'we ignore extrinsic terms due to disorder and instead focus on intrinsic terms'; standard for intrinsic Hall calculations but potentially important in real Ge.

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Cite this review

Pith. "Pith review of Orbital Hall effect in spin-3/2 hole-doped semiconductors and its implications for orbitronics." pith.science (2026). https://pith.science/paper/IG2DKPRN

@misc{pith2026250920436,
  author       = {Pith},
  title        = {Pith review of: Orbital Hall effect in spin-3/2 hole-doped semiconductors and its implications for orbitronics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IG2DKPRN}},
  note         = {Machine review of arXiv:2509.20436}
}
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read the original abstract

State-of-the-art magnetic devices rely on faster, more efficient memory elements. A major recent advance is the discovery of orbital torques, which use the orbital angular momentum of Bloch electrons to switch the magnetisation of an adjacent ferromagnet, motivating the search for orbitronic materials with strong orbital responses, exemplified by the orbital Hall effect (OHE). Here we propose $p$-type semiconductors, with a focus on Ge, as orbitronic platforms. We demonstrate that bulk holes in five common semiconductors exhibit a large orbital Hall conductivity of order $10^3 (\hbar/e)\Omega^{-1}$cm$^{-1}$, exceeding the spin-Hall effect by 2-3 orders of magnitude. The calculation is performed within the framework of the modern theory of orbital magnetisation, while incorporating recently-discovered quantum corrections to the OHE. Moreover, we argue that bulk $p$-type Ge and Si serve as ideal testbeds for the orbital torque resulting from a charge current, since the spin- and orbital-Edelstein effects are forbidden by symmetry. Our results provide a blueprint for producing strong orbital torques in magnetic devices with $p$-type semiconductors, guiding experimental work in this direction.

Figures

Figures reproduced from arXiv: 2509.20436 by Dimitrie Culcer, James H. Cullen, Zhanning Wang.

Figure 1
Figure 1. Figure 1: Dispersion for holes in Ge. In this figure the bands [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The orbital Hall conductivity vs the Fermi energy in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Diagram of the orbital Hall torque in a Ge/Co het [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The orbital Hall conductivity vs Fermi energy for [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

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Reference graph

Works this paper leans on

134 extracted references · 7 linked inside Pith

  1. [1]

    Das, Nature Physics 19, 1085 (2023)

    D. Das, Nature Physics 19, 1085 (2023)

    2023

  2. [2]

    R. B. Atencia, A. Agarwal, and D. Culcer, Advances in Physics: X 9, 2371972 (2024), https://doi.org/10.1080/23746149.2024.2371972

    arXiv 2024

  3. [3]

    D. Go, D. Jo, H.-W. Lee, M. Kl¨ aui, and Y. Mokrousov, Europhysics Letters 135, 37001 (2021)

    2021

  4. [4]

    P. Wang, F. Chen, Y. Yang, S. Hu, Y. Li, W. Wang, D. Zhang, and Y. Jiang, Advanced Electronic Materials , 2400554 (2024)

    2024

  5. [5]

    J. Kim, D. Go, H. Tsai, D. Jo, K. Kondou, H.-W. Lee, and Y. Otani, Phys. Rev. B 103, L020407 (2021)

    2021

  6. [6]

    X. Chen, Y. Liu, G. Yang, H. Shi, C. Hu, M. Li, and H. Zeng, Nature Communications 9, 2569 (2018)

    2018

  7. [7]

    Go and H.-W

    D. Go and H.-W. Lee, Phys. Rev. Res. 2, 013177 (2020)

    2020

  8. [8]

    D. Lee, D. Go, H.-J. Park, W. Jeong, H.-W. Ko, D. Yun, D. Jo, S. Lee, G. Go, J. H. Oh, K.-J. Kim, B.-G. Park, B.-C. Min, H. C. Koo, H.-W. Lee, O. Lee, and K.-J. Lee, Nature Communications 12, 6710 (2021)

    2021

  9. [9]

    Z. C. Zheng, Q. X. Guo, D. Jo, D. Go, L. H. Wang, H. C. Chen, W. Yin, X. M. Wang, G. H. Yu, W. He, H.-W. Lee, J. Teng, and T. Zhu, Phys. Rev. Res. 2, 013127 (2020)

    2020

  10. [11]

    Lee, M.-G

    S. Lee, M.-G. Kang, D. Go, D. Kim, J.-H. Kang, T. Lee, G.-H. Lee, J. Kang, N. J. Lee, Y. Mokrousov, S. Kim, K.-J. Kim, K.-J. Lee, and B.-G. Park, Communications Physics 4, 234 (2021)

    2021

  11. [12]

    S. Ding, A. Ross, D. Go, L. Baldrati, Z. Ren, F. Freimuth, S. Becker, F. Kammerbauer, J. Yang, G. Jakob, Y. Mokrousov, and M. Kl¨ aui, Phys. Rev. Lett. 125, 177201 (2020)

    2020

  12. [13]

    Sala and P

    G. Sala and P. Gambardella, Phys. Rev. Res. 4, 033037 (2022)

    2022

  13. [14]

    Hayashi, D

    H. Hayashi, D. Jo, D. Go, T. Gao, S. Haku, Y. Mokrousov, H.-W. Lee, and K. Ando, Communica- tions Physics 6, 32 (2023)

    2023

  14. [15]

    T. Li, L. Liu, X. Li, X. Zhao, H. An, and K. Ando, Nano Letters 23, 7174 (2023), pMID: 37466330

    2023

  15. [16]

    J. Xiao, Y. Liu, and B. Yan, Memorial Volume for Shoucheng Zhang (2021) Chap. Chapter 13, pp. 353– 364

    2021

  16. [17]

    Tokura, K

    Y. Tokura, K. Yasuda, and A. Tsukazaki, Nature Re- views Physics 1, 126 (2019)

    2019

  17. [18]

    D. Go, D. Jo, K.-W. Kim, S. Lee, M.-G. Kang, B.-G. Park, S. Bl¨ ugel, H.-W. Lee, and Y. Mokrousov, Physical review letters 130, 246701 (2023)

    2023

  18. [19]

    N. H. Aase, E. W. Hodt, K. B. Hallberg, A. Sudbø, and J. Linder, Phys. Rev. B 112, 014409 (2025)

    2025

  19. [20]

    Sakai and H

    A. Sakai and H. Kohno, Physical Review B 89, 165307 (2014)

    2014

  20. [21]

    Yasuda, A

    K. Yasuda, A. Tsukazaki, R. Yoshimi, K. Kondou, K. Takahashi, Y. Otani, M. Kawasaki, and Y. Tokura, Physical review letters 119, 137204 (2017)

    2017

  21. [22]

    Tatara, H

    G. Tatara, H. Kohno, J. Shibata, Y. Lemaho, and K.-J. Lee, Journal of the Physical Society of Japan76, 054707 (2007)

    2007

  22. [23]

    Kohno, G

    H. Kohno, G. Tatara, and J. Shibata, Journal of the Physical Society of Japan 75, 113706 (2006)

    2006

  23. [24]

    Belashchenko, A

    K. Belashchenko, A. A. Kovalev, and M. Van Schilf- gaarde, Physical Review Materials 3, 011401 (2019)

    2019

  24. [25]

    Manchon, J

    A. Manchon, J. ˇZelezn´ y, I. M. Miron, T. Jungwirth, J. Sinova, A. Thiaville, K. Garello, and P. Gambardella, Rev. Mod. Phys. 91, 035004 (2019)

    2019

  25. [26]

    B. K. Nikoli´ c, K. Dolui, M. D. Petrovi´ c, P. Plech´ aˇ c, T. Markussen, and K. Stokbro, Handbook of Materials Modeling: Applications: Current and Emerging Mate- rials , 499 (2020)

    2020

  26. [27]

    Gambardella and I

    P. Gambardella and I. M. Miron, Philosophical Trans- actions of the Royal Society A: Mathematical, Physical and Engineering Sciences 369, 3175 (2011)

    2011

  27. [28]

    Qaiumzadeh, R

    A. Qaiumzadeh, R. A. Duine, and M. Titov, Phys. Rev. B 92, 014402 (2015)

    2015

  28. [29]

    R. A. Duine, A. S. N´ u˜ nez, J. Sinova, and A. H. Mac- Donald, Phys. Rev. B 75, 214420 (2007)

    2007

  29. [30]

    Brataas, A

    A. Brataas, A. D. Kent, and H. Ohno, Nature materials 11, 372 (2012)

    2012

  30. [31]

    K. M. D. Hals and A. Brataas, Phys. Rev. B 88, 085423 (2013)

    2013

  31. [32]

    K. M. D. Hals and A. Brataas, Phys. Rev. B 91, 214401 (2015)

    2015

  32. [33]

    B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Phys. Rev. Lett. 95, 066601 (2005)

    2005

  33. [34]

    Tanaka, H

    T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. Hi- rashima, K. Yamada, and J. Inoue, Phys. Rev. B 77, 165117 (2008)

    2008

  34. [35]

    Kontani, T

    H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada, and J. Inoue, Phys. Rev. Lett. 102, 016601 (2009)

    2009

  35. [36]

    Y.-G. Choi, D. Jo, K.-H. Ko, D. Go, K.-H. Kim, H. G. Park, C. Kim, B.-C. Min, G.-M. Choi, and H.-W. Lee, Nature 619, 52 (2023)

    2023

  36. [37]

    Liu and D

    H. Liu and D. Culcer, Phys. Rev. Lett. 132, 186302 8 (2024)

    2024

  37. [38]

    A. Pezo, D. Garc ´ ıa Ovalle, and A. Manchon, Phys. Rev. B 106, 104414 (2022)

    2022

  38. [39]

    Salemi and P

    L. Salemi and P. M. Oppeneer, Phys. Rev. Mater. 6, 095001 (2022)

    2022

  39. [40]

    T. P. Cysne, M. Costa, L. M. Canonico, M. B. Nardelli, R. B. Muniz, and T. G. Rappoport, Phys. Rev. Lett. 126, 056601 (2021)

    2021

  40. [41]

    L. M. Canonico, T. P. Cysne, A. Molina-Sanchez, R. B. Muniz, and T. G. Rappoport, Phys. Rev. B 101, 161409 (2020)

    2020

  41. [42]

    S. R. Luis M. Canonico, Jose H. Garc ´ ıa, arXiv:2404.01739 (2024)

    Pith/arXiv arXiv 2024

  42. [43]

    P. Wang, Z. Feng, Y. Yang, D. Zhang, Q. Liu, Z. Xu, Z. Jia, Y. Wu, G. Yu, X. Xu, and Y. Jiang, npj Quantum Materials 8, 28 (2023)

    2023

  43. [44]

    Lyalin, S

    I. Lyalin, S. Alikhah, M. Berritta, P. M. Oppeneer, and R. K. Kawakami, Phys. Rev. Lett. 131, 156702 (2023)

    2023

  44. [45]

    Dutta and A

    S. Dutta and A. A. Tulapurkar, Phys. Rev. B 106, 184406 (2022)

    2022

  45. [46]

    G. Sala, H. Wang, W. Legrand, and P. Gambardella, Phys. Rev. Lett. 131, 156703 (2023)

    2023

  46. [47]

    Zhang, H

    J. Zhang, H. Xie, X. Zhang, Z. Yan, Y. Zhai, J. Chi, H. Xu, Y. Zuo, and L. Xi, Applied Physics Letters 121, 172405 (2022)

    2022

  47. [48]

    B. Bony, S. Krishnia, Y. Xu, S. Collin, A. Fert, J.-M. George, M. Viret, V. Cros, and H. Jaffr` es, arXiv e-prints , arXiv (2025)

    2025

  48. [49]

    El Hamdi, J.-Y

    A. El Hamdi, J.-Y. Chauleau, M. Boselli, C. Thibault, C. Gorini, A. Smogunov, C. Barreteau, S. Gariglio, J.- M. Triscone, and M. Viret, Nature Physics 19, 1855 (2023)

    2023

  49. [50]

    D. B. Fonseca, L. L. A. Pereira, and A. L. R. Barbosa, Phys. Rev. B 108, 245105 (2023)

    2023

  50. [51]

    Shinada, A

    K. Shinada, A. Kofuji, and R. Peters, Phys. Rev. B 107, 094106 (2023)

    2023

  51. [52]

    L. M. Canonico, T. P. Cysne, T. G. Rappoport, and R. Muniz, Physical Review B 101, 075429 (2020)

    2020

  52. [53]

    J. H. Cullen, H. Liu, and D. Culcer, npj Spintronics 3, 22 (2025)

    2025

  53. [54]

    Kazantsev, A

    A. Kazantsev, A. Mills, E. O’Neill, H. Sun, G. Vignale, and A. Principi, Phys. Rev. Lett. 132, 106301 (2024)

    2024

  54. [55]

    H. Sun, A. Kazantsev, A. Principi, and G. Vignale, Phys. Rev. B 111, 075432 (2025)

    2025

  55. [56]

    Sun and G

    H. Sun and G. Vignale, arXiv preprint arXiv:2408.02887 (2024)

    Pith/arXiv arXiv 2024

  56. [57]

    Bhowal and G

    S. Bhowal and G. Vignale, Phys. Rev. B 103, 195309 (2021)

    2021

  57. [58]

    T. P. Cysne, S. Bhowal, G. Vignale, and T. G. Rap- poport, Phys. Rev. B 105, 195421 (2022)

    2022

  58. [59]

    Veneri, T

    A. Veneri, T. G. Rappoport, and A. Ferreira, Phys. Rev. Lett. 134, 136201 (2025)

    2025

  59. [60]

    D. Go, D. Jo, C. Kim, and H.-W. Lee, Phys. Rev. Lett. 121, 086602 (2018)

    2018

  60. [61]

    D. Jo, D. Go, and H.-W. Lee, Phys. Rev. B 98, 214405 (2018)

    2018

  61. [62]

    J. M. Lee, Physical Review B 112, 054441 (2025)

    2025

  62. [63]

    Santos, J

    E. Santos, J. Abr˜ ao, J. Costa, J. Santos, G. Rodrigues- Junior, J. Mendes, and A. Azevedo, Physical Review Applied 22, 064071 (2024)

    2024

  63. [64]

    S. R. Park, J. Han, C. Kim, Y. Y. Koh, C. Kim, H. Lee, H. J. Choi, J. H. Han, K. D. Lee, N. J. Hur, M. Arita, K. Shimada, H. Namatame, and M. Taniguchi, Phys. Rev. Lett. 108, 046805 (2012)

    2012

  64. [65]

    J. Shi, G. Vignale, D. Xiao, and Q. Niu, Phys. Rev. Lett. 99, 197202 (2007)

    2007

  65. [66]

    Thonhauser, D

    T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta, Phys. Rev. Lett. 95, 137205 (2005)

    2005

  66. [67]

    Vanderbilt, Berry Phases in Electronic Struc- ture Theory: Electric Polarization, Orbital Magnetiza- tion and Topological Insulators(Cambridge University Press, 2018)

    D. Vanderbilt, Berry Phases in Electronic Struc- ture Theory: Electric Polarization, Orbital Magnetiza- tion and Topological Insulators(Cambridge University Press, 2018)

    2018

  67. [68]

    Resta, Phys

    R. Resta, Phys. Rev. Res. 2, 023139 (2020)

    2020

  68. [69]

    Ceresoli, T

    D. Ceresoli, T. Thonhauser, D. Vanderbilt, and R. Resta, Phys. Rev. B 74, 024408 (2006)

    2006

  69. [70]

    Thonhauser, International Journal of Modern Physics B 25, 1429 (2011)

    T. Thonhauser, International Journal of Modern Physics B 25, 1429 (2011)

    2011

  70. [71]

    J. Voss, I. Ado, and M. Titov, arXiv preprint arXiv:2405.11979 (2024)

    Pith/arXiv arXiv 2024

  71. [72]

    S. Ding, Z. Liang, D. Go, C. Yun, M. Xue, Z. Liu, S. Becker, W. Yang, H. Du, C. Wang, Y. Yang, G. Jakob, M. Kl¨ aui, Y. Mokrousov, and J. Yang, Phys. Rev. Lett. 128, 067201 (2022)

    2022

  72. [73]

    J. M. Lee, M. J. Park, and H.-W. Lee, Physical Review B 110, 134436 (2024)

    2024

  73. [74]

    C. Xiao, H. Liu, J. Zhao, S. A. Yang, and Q. Niu, Phys. Rev. B 103, 045401 (2021)

    2021

  74. [75]

    Pezo, J.-M

    A. Pezo, J.-M. George, and H. Jaffr` es, Physical Review Research 6, 043332 (2024)

    2024

  75. [76]

    Baek and H.-W

    I. Baek and H.-W. Lee, Phys. Rev. B 104, 245204 (2021)

    2021

  76. [77]

    Malashevich, I

    A. Malashevich, I. Souza, S. Coh, and D. Vanderbilt, New Journal of Physics 12, 053032 (2010)

    2010

  77. [78]

    L. E. Stehouwer, C. X. Yu, B. van Straaten, A. Tosato, V. John, D. Degli Esposti, A. Elsayed, D. Costa, S. D. Oosterhout, N. W. Hendrickx, et al., Nature Materials , 1 (2025)

    2025

  78. [79]

    L. E. Stehouwer, A. Tosato, D. Degli Esposti, D. Costa, M. Veldhorst, A. Sammak, and G. Scappucci, Applied Physics Letters 123 (2023)

    2023

  79. [80]

    Scappucci, C

    G. Scappucci, C. Kloeffel, F. A. Zwanenburg, D. Loss, M. Myronov, J.-J. Zhang, S. De Franceschi, G. Kat- saros, and M. Veldhorst, Nature Reviews Materials 6, 926 (2021)

    2021

  80. [81]

    Y. Fang, P. Philippopoulos, D. Culcer, W. Coish, and S. Chesi, Materials for Quantum Technology 3, 012003 (2023)

    2023

Showing first 80 references.