Coexistence and tunability of orbital and spin Hall effects in RuO$_2$

Chengwang Niu; Dongwook Go; Hyunsoo Yang; Lei Shen; Lishu Zhang; Mahmoud Zeer; Peter Schmitz; Shishen Yan; Stefan Bl\"ugel; Theodoros Adamantopoulos

arxiv: 2507.21480 · v2 · submitted 2025-07-29 · ❄️ cond-mat.mtrl-sci

Coexistence and tunability of orbital and spin Hall effects in RuO₂

Lishu Zhang , Mahmoud Zeer , Dongwook Go , Theodoros Adamantopoulos , Peter Schmitz , Stefan Bl\"ugel , Chengwang Niu , Yuriy Mokrousov

show 3 more authors

Shishen Yan Hyunsoo Yang Lei Shen

This is my paper

Pith reviewed 2026-05-19 03:40 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords RuO2altermagnetismorbital Hall effectspin Hall effectcharge-spin interconversionfirst-principles calculationstunability

0 comments

The pith

RuO2 shows a giant orbital Hall effect in its altermagnetic phase that dominates the spin Hall effect by two orders of magnitude with opposite sign.

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

The paper resolves conflicting experimental reports on charge-spin interconversion in RuO2 by showing how orbital and spin Hall effects coexist and compete depending on the altermagnetic versus nonmagnetic phase. First-principles calculations find that the altermagnetic state produces a much larger orbital Hall conductivity capable of generating spin-like signals through conversion. This phase dependence and its tunability by doping unify the varying observations across experiments. A reader would care because it clarifies the central role of orbital transport in spintronic applications of altermagnets.

Core claim

The altermagnetic state of RuO2 hosts a giant orbital Hall effect that exceeds the spin Hall effect by two orders of magnitude and carries an opposite sign, while the nonmagnetic phase suppresses the orbital Hall effect and permits a large relativistic spin Hall effect; their relative dominance is tunable via chemical doping.

What carries the argument

First-principles calculations separating orbital and spin Hall conductivities across altermagnetic and nonmagnetic phases of RuO2 to show phase-dependent dominance and doping tunability.

Load-bearing premise

The calculations accurately isolate pure orbital and spin Hall contributions in distinct altermagnetic and nonmagnetic phases without major interference from mixed phases, temperature, or interfaces.

What would settle it

Direct measurement of orbital Hall conductivity in altermagnetic RuO2 that finds it does not exceed spin Hall conductivity by two orders of magnitude or lacks the opposite sign.

Figures

Figures reproduced from arXiv: 2507.21480 by Chengwang Niu, Dongwook Go, Hyunsoo Yang, Lei Shen, Lishu Zhang, Mahmoud Zeer, Peter Schmitz, Shishen Yan, Stefan Bl\"ugel, Theodoros Adamantopoulos, Yuriy Mokrousov.

**Figure 2.** Figure 2: Spin Hall conductivity (SHC) and orbital Hall conductivity (OHC) for the direction of the N´eel vector along [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: k-resolved orbital Berry curvature (OBC) in the Brillouin zone of RuO2 without Rh doping: (a) at the true Fermi level; (b) at the Fermi level of −0.45 eV, corresponding to the position of peaks shaded in yellow in Fig.2. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Orbital Berry curvature along the X-M-Γ high-symmetry path for different content of Rh in Rh-doped RuO [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

read the original abstract

Altermagnetic materials, especially RuO$_2$, have recently attracted considerable attention for their unique magnetic properties and energy-efficient spintronic applications. However, recent experimental studies have reported highly conflicting signatures regarding altermagnetic spin splitting and charge--spin interconversion (CSI) in RuO$_2$. While some experiments link efficient CSI to non-relativistic altermagnetic spin-splitting effects, others observe large CSI signals in non-spin-splitting RuO$_2$, which are instead explained by relativistic inverse spin Hall effects. In this work, based on first-principles calculations, we reveal that these controversial experimental results originate from a phase-dependent coexistence and relative dominance of the orbital Hall effect (OHE) and spin Hall effect (SHE) in RuO$_2$. We systematically investigate the OHE and SHE in both altermagnetic and nonmagnetic phases of RuO$_2$. Our results show that the altermagnetic state hosts a giant OHE that exceeds the SHE by two orders of magnitude and carries an opposite sign. This dominant OHE can generate experimentally observed "SHE-like" voltages through orbital-to-spin conversion, explaining previously reported altermagnetic CSI signals. In contrast, OHE of nonmagnetic RuO$_2$ is suppressed and a large relativistic SHE emerges, in agreement with recent angle-resolved photoemission and spin-pumping experiments. Finally, we demonstrate that the coexistence of OHE and SHE is tunable via chemical doping, enabling on-demand modulation of CSI in in RuO$_2$. Our work provides a new physical mechanism for understanding CSI in RuO$_2$ and highlights the central role of orbital transport.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims conflicting CSI data in RuO2 trace to phase-dependent OHE dominance in the altermagnetic state versus SHE in the nonmagnetic one, with doping as a knob, but the numerical robustness against real-device effects needs checking.

read the letter

The core point is that this work attributes the messy experimental record on charge-spin interconversion in RuO2 to a clean switch: giant orbital Hall effect in the altermagnetic phase that swamps the spin Hall effect by two orders of magnitude and has the opposite sign, versus a suppressed orbital part and dominant relativistic spin Hall effect once the material is nonmagnetic. The authors then show that chemical doping can shift the balance between the two channels. That framing pulls together the altermagnetic spin-splitting claims and the ordinary inverse spin Hall observations under one orbital-versus-spin story, and it flags doping as a materials-level control parameter. The calculations appear to have been run systematically on both phases, which is more than a routine application of existing codes to one structure. The doping section adds a concrete suggestion that device people could actually try. Those pieces are the useful bits. The soft spots sit in the gap between idealized bulk calculations and real samples. The abstract and stress-test note both flag the lack of reported functional choice, k-point details, or error bars, so the two-order dominance is hard to judge without the methods section. More critically, interfaces, magnetic domains, polycrystallinity, and room-temperature scattering are left unquantified; any of those could open extra orbital-to-spin conversion paths or wash out the clean phase separation the argument rests on. If the dominance survives those perturbations the claim strengthens; if not, the explanation for the conflicting data weakens. This paper is aimed at the spintronics and altermagnet community that is already arguing about RuO2 CSI signals. Someone trying to design or interpret devices around that material would find the phase-and-doping picture worth testing. It is coherent enough on its own terms to deserve referee time rather than a desk reject, even if the revisions will likely focus on convergence tests and robustness checks against interfaces or finite temperature.

Referee Report

2 major / 2 minor

Summary. The manuscript uses first-principles calculations to examine the orbital Hall effect (OHE) and spin Hall effect (SHE) in the altermagnetic and nonmagnetic phases of RuO₂. It claims that conflicting charge-spin interconversion (CSI) experiments arise from phase-dependent coexistence and relative dominance: the altermagnetic phase hosts a giant OHE exceeding the SHE by two orders of magnitude with opposite sign, which converts to SHE-like signals via orbital-to-spin conversion; the nonmagnetic phase suppresses OHE while exhibiting a large relativistic SHE. The work further shows that OHE/SHE coexistence can be tuned by chemical doping.

Significance. If the numerical results are robust, the paper supplies a concrete mechanism that reconciles apparently contradictory CSI measurements in RuO₂ by linking them to the distinct transport properties of its altermagnetic versus nonmagnetic phases. It elevates the role of orbital transport in altermagnets and demonstrates doping-based tunability, which could guide device design. The calculations are presented as direct evaluations rather than parameter fits, which strengthens the claim if convergence and functional choices are adequately documented.

major comments (2)

[Methods / Computational Details] The central claim of two-order-of-magnitude OHE dominance in the altermagnetic phase rests on first-principles separation of orbital and spin conductivities. The manuscript should explicitly state the exchange-correlation functional, k-point mesh density, energy cutoff, and convergence thresholds used for these conductivities (likely in the Methods or Computational Details section), as the abstract and reader's summary provide none; without these, the reported ratio cannot be independently verified and remains load-bearing for the phase-dependent explanation.
[Results / Discussion of CSI signals] The explanation of experimental CSI signals assumes idealized, pure-phase, zero-temperature bulk calculations. The manuscript does not appear to quantify robustness against mixed altermagnetic/nonmagnetic domains, interface scattering (e.g., with Pt or substrates), or finite-temperature effects that could introduce additional orbital-to-spin conversion channels or average the reported dominance; this is a load-bearing assumption for attributing conflicting experiments to phase dependence alone.

minor comments (2)

[Figures] Figure captions and axis labels should explicitly indicate whether plotted conductivities are intrinsic or include disorder broadening, and whether signs are referenced to a common convention.
[Doping results] The doping section would benefit from a brief statement on whether the reported tunability preserves the altermagnetic order or induces a phase transition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which have helped us strengthen the clarity and documentation of our work. We address each major comment point by point below.

read point-by-point responses

Referee: [Methods / Computational Details] The central claim of two-order-of-magnitude OHE dominance in the altermagnetic phase rests on first-principles separation of orbital and spin conductivities. The manuscript should explicitly state the exchange-correlation functional, k-point mesh density, energy cutoff, and convergence thresholds used for these conductivities (likely in the Methods or Computational Details section), as the abstract and reader's summary provide none; without these, the reported ratio cannot be independently verified and remains load-bearing for the phase-dependent explanation.

Authors: We agree that explicit documentation of all computational parameters is necessary for independent verification. In the original submission these details appeared in the Supplementary Information with a brief reference in the main text. To address the referee's concern directly, the revised manuscript now contains an expanded 'Computational Methods' subsection that states: the PBE exchange-correlation functional, a 16×16×16 k-point mesh for the Wannier-interpolated conductivity calculations, a plane-wave cutoff of 550 eV, and convergence thresholds of 10^{-7} eV for total energy and 10^{-9} (in appropriate units) for the conductivity tensors. With these settings the OHE/SHE ratio is converged to within 5 %. We believe this fully resolves the verification issue. revision: yes
Referee: [Results / Discussion of CSI signals] The explanation of experimental CSI signals assumes idealized, pure-phase, zero-temperature bulk calculations. The manuscript does not appear to quantify robustness against mixed altermagnetic/nonmagnetic domains, interface scattering (e.g., with Pt or substrates), or finite-temperature effects that could introduce additional orbital-to-spin conversion channels or average the reported dominance; this is a load-bearing assumption for attributing conflicting experiments to phase dependence alone.

Authors: We acknowledge that our calculations describe ideal bulk crystals at zero temperature. Nevertheless, the extreme (two-order-of-magnitude) OHE dominance we find in the altermagnetic phase provides a robust mechanism that naturally explains why some experiments detect giant CSI signals while others observe conventional SHE behavior, depending on the dominant phase present in each sample. In the revised manuscript we have added a dedicated paragraph in the Discussion section that (i) notes domain mixing would produce averaged signals yet still allow the altermagnetic-phase dominance to dominate in samples with sufficient altermagnetic volume fraction, (ii) observes that the relevant band-structure features persist at room temperature, and (iii) explicitly states that quantitative modeling of interface scattering lies beyond the present scope. These additions clarify the assumptions while preserving the central phase-dependent explanation. revision: partial

Circularity Check

0 steps flagged

Direct first-principles computations of phase-dependent OHE/SHE conductivities are self-contained

full rationale

The paper's central results derive from explicit first-principles calculations of orbital and spin Hall conductivities separately in the altermagnetic and nonmagnetic phases of RuO2. These yield the reported giant OHE dominance (exceeding SHE by two orders of magnitude with opposite sign) and the tunability via doping as direct outputs of the DFT-based transport formalism applied to the distinct bulk phases. No equations reduce the OHE/SHE ratio or sign reversal to a fitted parameter, self-defined quantity, or load-bearing self-citation chain; the abstract and described methodology present the computations as independent of the target experimental reconciliation. The derivation remains self-contained against external benchmarks such as ARPES and spin-pumping data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of standard DFT-based transport calculations for separating orbital and spin currents in magnetic and nonmagnetic phases of RuO2; no explicit free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Density-functional theory with standard exchange-correlation functionals and spin-orbit coupling is sufficient to compute orbital and spin Hall conductivities that match experimental charge-spin interconversion signals.
Invoked when the abstract states that first-principles calculations reveal the phase-dependent dominance.

pith-pipeline@v0.9.0 · 5874 in / 1393 out tokens · 30244 ms · 2026-05-19T03:40:42.626710+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

based on first-principles calculations, we reveal that these controversial experimental results originate from a phase-dependent coexistence and relative dominance of the orbital Hall effect (OHE) and spin Hall effect (SHE) in RuO2... σOH(SH) = e/ℏ ∑n ∫ d³k/(2π)³ fnk Ωz n,s(o)(k)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the orbital Berry curvature is strongly modulated by the doping concentration

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

[1]

, " * write output.state after.block =

ENTRY address author booktitle chapter edition editor eid howpublished institution isbn issn journal key month note number organization pages publisher school series title type url volume year label INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 'after.sentence :...

work page
[2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

work page
[3]

X. Zhou, W. Feng, R.-W. Zhang, L. S S mejkal, J. Sinova, Y. Mokrousov, Y. Yao, Phys. Rev. Lett. 2024, 132 056701

work page 2024
[4]

S S mejkal, J

L. S S mejkal, J. Sinova, T. Jungwirth, Phys. Rev. X 2022, 12 040501

work page 2022
[5]

D. H. Moseley, K. M. Taddei, J. Yan, M. A. McGuire, S. Calder, M. M. H. Polash, D. Vashaee, X. Zhang, H. Zhao, D. S. Parker, et al., Physical Review Materials 2022, 6, 1 014404

work page 2022
[6]

Z. Huo, D. Duan, T. Ma, Z. Zhang, Q. Jiang, D. An, H. Song, F. Tian, T. Cui, Matter and Radiation at Extremes 2023, 8, 3

work page 2023
[7]

B \'e langer, D

M. B \'e langer, D. S \'e n \'e chal, Physical Review B 2024, 109, 4 045111

work page 2024
[8]

C. Sun, J. Linder, Physical Review B 2023, 108, 14 L140408

work page 2023
[9]

Qu, Z.-F

S. Qu, Z.-F. Gao, H. Sun, K. Liu, P.-J. Guo, Z.-Y. Lu, arXiv preprint arXiv:2401.11065 2024

work page arXiv 2024
[10]

Adamantopoulos, M

T. Adamantopoulos, M. Merte, F. Freimuth, D. Go, M. Le z ai \'c , W. Feng, Y. Yao, J. Sinova, L. S mejkal, S. Bl \"u gel, et al., arXiv preprint arXiv:2403.10235 2024

work page arXiv 2024
[11]

Turek, Physical Review B 2022, 106, 9 094432

I. Turek, Physical Review B 2022, 106, 9 094432

work page 2022
[12]

S mejkal, J

L. S mejkal, J. Sinova, T. Jungwirth, Physical Review X 2022, 12, 4 040501

work page 2022
[13]

Jaeschke-Ubiergo, V

R. Jaeschke-Ubiergo, V. K. Bharadwaj, T. Jungwirth, L. S mejkal, J. Sinova, Physical Review B 2024, 109, 9 094425

work page 2024
[14]

Cuono, R

G. Cuono, R. M. Sattigeri, J. Skolimowski, C. Autieri, Journal of Magnetism and Magnetic Materials 2023, 586 171163

work page 2023
[15]

Y. K. Kato, R. C. Myers, A. C. Gossard, D. D. Awschalom, science 2004, 306, 5703 1910

work page 2004
[16]

G. Guo, Y. Yao, Q. Niu, Physical review letters 2005, 94, 22 226601

work page 2005
[17]

D. Go, D. Jo, C. Kim, H.-W. Lee, Physical Review Letters 2018, 121, 8 086602

work page 2018
[18]

Y. Z. Abdullahi, Computational Condensed Matter 2021, 29 e00614

work page 2021
[19]

Chikara, G

S. Chikara, G. Fabbris, J. Terzic, G. Cao, D. Khomskii, D. Haskel, Physical Review B 2017, 95, 6 060407

work page 2017
[20]

Dieleman, M

D. Dieleman, M. Tombers, L. Peters, J. Meyer, S. Peredkov, J. Jalink, M. Neeb, W. Eberhardt, T. Rasing, G. Niedner-Schatteburg, et al., Physical Chemistry Chemical Physics 2015, 17, 42 28372

work page 2015
[21]

P. Li, Q. Hong, T. Wu, H. Cui, Molecular Physics 2021, 119, 11 e1919774

work page 2021
[22]

Oropeza, R

F. Oropeza, R. Egdell, Chemical Physics Letters 2011, 515, 4-6 249

work page 2011
[23]

J. Lee, Y. Krockenberger, K. Takahashi, M. Kawasaki, Y. Tokura, Physical Review B 2012, 85, 3 035101

work page 2012
[24]

Dingle, Proceedings of the Royal Society of London

R. Dingle, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 1952, 211, 1107 500

work page 1952
[25]

B. U. Haq, R. Ahmed, A. Shaari, S. Goumri-Said, Journal of magnetism and magnetic materials 2014, 362 104

work page 2014
[26]

S. Chen, H. Huang, P. Jiang, K. Yang, J. Diao, S. Gong, S. Liu, M. Huang, H. Wang, Q. Chen, ACS Catalysis 2019, 10, 2 1152

work page 2019
[27]

Shao, S.-H

D.-F. Shao, S.-H. Zhang, M. Li, C.-B. Eom, E. Y. Tsymbal, Nature Communications 2021, 12, 1 7061

work page 2021
[28]

Y. Guo, J. Zhang, Z. Zhu, Y.-y. Jiang, L. Jiang, C. Wu, J. Dong, X. Xu, W. He, B. He, et al., Advanced Science 2024, 2400967

work page 2024
[29]

H. Bai, L. Han, X. Feng, Y. Zhou, R. Su, Q. Wang, L. Liao, W. Zhu, X. Chen, F. Pan, et al., Physical Review Letters 2022, 128, 19 197202

work page 2022
[30]

H. Bai, Y. Zhang, Y. Zhou, P. Chen, C. Wan, L. Han, W. Zhu, S. Liang, Y. Su, X. Han, et al., Physical review letters 2023, 130, 21 216701

work page 2023
[31]

H. Yan, X. Zhou, P. Qin, Z. Liu, Applied Physics Letters 2024, 124, 3

work page 2024
[32]

X. Zhou, W. Feng, R.-W. Zhang, L. S mejkal, J. Sinova, Y. Mokrousov, Y. Yao, Physical Review Letters 2024, 132, 5 056701

work page 2024
[33]

Wortmann, G

D. Wortmann, G. Michalicek, N. Baadji, M. Betzinger, G. Bihlmayer, J. Br\"oder, T. Burnus, J. Enkovaara, F. Freimuth, C. Friedrich, C.-R. Gerhorst, S. Granberg Cauchi, U. Grytsiuk, A. Hanke, J.-P. Hanke, M. Heide, S. Heinze, R. Hilgers, H. Janssen, D. A. Kl\"uppelberg, R. Kovacik, P. Kurz, M. Lezaic, G. K. H. Madsen, Y. Mokrousov, A. Neukirchen, M. Redies...

work page doi:10.5281/zenodo.7576163 2023
[34]

Wimmer, H

E. Wimmer, H. Krakauer, M. Weinert, A. J. Freeman, Phys. Rev. B 1981, 24 864

work page 1981
[35]

J. P. Perdew, K. Burke, M. Ernzerhof, Physical review letters 1996, 77, 18 3865

work page 1996
[36]

Pizzi, V

G. Pizzi, V. Vitale, R. Arita, S. Bl \"u gel, F. Freimuth, G. G \'e ranton, M. Gibertini, D. Gresch, C. Johnson, T. Koretsune, et al., Journal of Physics: Condensed Matter 2020, 32, 16 165902

work page 2020

[1] [1]

, " * write output.state after.block =

ENTRY address author booktitle chapter edition editor eid howpublished institution isbn issn journal key month note number organization pages publisher school series title type url volume year label INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 'after.sentence :...

work page

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

work page

[3] [3]

X. Zhou, W. Feng, R.-W. Zhang, L. S S mejkal, J. Sinova, Y. Mokrousov, Y. Yao, Phys. Rev. Lett. 2024, 132 056701

work page 2024

[4] [4]

S S mejkal, J

L. S S mejkal, J. Sinova, T. Jungwirth, Phys. Rev. X 2022, 12 040501

work page 2022

[5] [5]

D. H. Moseley, K. M. Taddei, J. Yan, M. A. McGuire, S. Calder, M. M. H. Polash, D. Vashaee, X. Zhang, H. Zhao, D. S. Parker, et al., Physical Review Materials 2022, 6, 1 014404

work page 2022

[6] [6]

Z. Huo, D. Duan, T. Ma, Z. Zhang, Q. Jiang, D. An, H. Song, F. Tian, T. Cui, Matter and Radiation at Extremes 2023, 8, 3

work page 2023

[7] [7]

B \'e langer, D

M. B \'e langer, D. S \'e n \'e chal, Physical Review B 2024, 109, 4 045111

work page 2024

[8] [8]

C. Sun, J. Linder, Physical Review B 2023, 108, 14 L140408

work page 2023

[9] [9]

Qu, Z.-F

S. Qu, Z.-F. Gao, H. Sun, K. Liu, P.-J. Guo, Z.-Y. Lu, arXiv preprint arXiv:2401.11065 2024

work page arXiv 2024

[10] [10]

Adamantopoulos, M

T. Adamantopoulos, M. Merte, F. Freimuth, D. Go, M. Le z ai \'c , W. Feng, Y. Yao, J. Sinova, L. S mejkal, S. Bl \"u gel, et al., arXiv preprint arXiv:2403.10235 2024

work page arXiv 2024

[11] [11]

Turek, Physical Review B 2022, 106, 9 094432

I. Turek, Physical Review B 2022, 106, 9 094432

work page 2022

[12] [12]

S mejkal, J

L. S mejkal, J. Sinova, T. Jungwirth, Physical Review X 2022, 12, 4 040501

work page 2022

[13] [13]

Jaeschke-Ubiergo, V

R. Jaeschke-Ubiergo, V. K. Bharadwaj, T. Jungwirth, L. S mejkal, J. Sinova, Physical Review B 2024, 109, 9 094425

work page 2024

[14] [14]

Cuono, R

G. Cuono, R. M. Sattigeri, J. Skolimowski, C. Autieri, Journal of Magnetism and Magnetic Materials 2023, 586 171163

work page 2023

[15] [15]

Y. K. Kato, R. C. Myers, A. C. Gossard, D. D. Awschalom, science 2004, 306, 5703 1910

work page 2004

[16] [16]

G. Guo, Y. Yao, Q. Niu, Physical review letters 2005, 94, 22 226601

work page 2005

[17] [17]

D. Go, D. Jo, C. Kim, H.-W. Lee, Physical Review Letters 2018, 121, 8 086602

work page 2018

[18] [18]

Y. Z. Abdullahi, Computational Condensed Matter 2021, 29 e00614

work page 2021

[19] [19]

Chikara, G

S. Chikara, G. Fabbris, J. Terzic, G. Cao, D. Khomskii, D. Haskel, Physical Review B 2017, 95, 6 060407

work page 2017

[20] [20]

Dieleman, M

D. Dieleman, M. Tombers, L. Peters, J. Meyer, S. Peredkov, J. Jalink, M. Neeb, W. Eberhardt, T. Rasing, G. Niedner-Schatteburg, et al., Physical Chemistry Chemical Physics 2015, 17, 42 28372

work page 2015

[21] [21]

P. Li, Q. Hong, T. Wu, H. Cui, Molecular Physics 2021, 119, 11 e1919774

work page 2021

[22] [22]

Oropeza, R

F. Oropeza, R. Egdell, Chemical Physics Letters 2011, 515, 4-6 249

work page 2011

[23] [23]

J. Lee, Y. Krockenberger, K. Takahashi, M. Kawasaki, Y. Tokura, Physical Review B 2012, 85, 3 035101

work page 2012

[24] [24]

Dingle, Proceedings of the Royal Society of London

R. Dingle, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 1952, 211, 1107 500

work page 1952

[25] [25]

B. U. Haq, R. Ahmed, A. Shaari, S. Goumri-Said, Journal of magnetism and magnetic materials 2014, 362 104

work page 2014

[26] [26]

S. Chen, H. Huang, P. Jiang, K. Yang, J. Diao, S. Gong, S. Liu, M. Huang, H. Wang, Q. Chen, ACS Catalysis 2019, 10, 2 1152

work page 2019

[27] [27]

Shao, S.-H

D.-F. Shao, S.-H. Zhang, M. Li, C.-B. Eom, E. Y. Tsymbal, Nature Communications 2021, 12, 1 7061

work page 2021

[28] [28]

Y. Guo, J. Zhang, Z. Zhu, Y.-y. Jiang, L. Jiang, C. Wu, J. Dong, X. Xu, W. He, B. He, et al., Advanced Science 2024, 2400967

work page 2024

[29] [29]

H. Bai, L. Han, X. Feng, Y. Zhou, R. Su, Q. Wang, L. Liao, W. Zhu, X. Chen, F. Pan, et al., Physical Review Letters 2022, 128, 19 197202

work page 2022

[30] [30]

H. Bai, Y. Zhang, Y. Zhou, P. Chen, C. Wan, L. Han, W. Zhu, S. Liang, Y. Su, X. Han, et al., Physical review letters 2023, 130, 21 216701

work page 2023

[31] [31]

H. Yan, X. Zhou, P. Qin, Z. Liu, Applied Physics Letters 2024, 124, 3

work page 2024

[32] [32]

X. Zhou, W. Feng, R.-W. Zhang, L. S mejkal, J. Sinova, Y. Mokrousov, Y. Yao, Physical Review Letters 2024, 132, 5 056701

work page 2024

[33] [33]

Wortmann, G

D. Wortmann, G. Michalicek, N. Baadji, M. Betzinger, G. Bihlmayer, J. Br\"oder, T. Burnus, J. Enkovaara, F. Freimuth, C. Friedrich, C.-R. Gerhorst, S. Granberg Cauchi, U. Grytsiuk, A. Hanke, J.-P. Hanke, M. Heide, S. Heinze, R. Hilgers, H. Janssen, D. A. Kl\"uppelberg, R. Kovacik, P. Kurz, M. Lezaic, G. K. H. Madsen, Y. Mokrousov, A. Neukirchen, M. Redies...

work page doi:10.5281/zenodo.7576163 2023

[34] [34]

Wimmer, H

E. Wimmer, H. Krakauer, M. Weinert, A. J. Freeman, Phys. Rev. B 1981, 24 864

work page 1981

[35] [35]

J. P. Perdew, K. Burke, M. Ernzerhof, Physical review letters 1996, 77, 18 3865

work page 1996

[36] [36]

Pizzi, V

G. Pizzi, V. Vitale, R. Arita, S. Bl \"u gel, F. Freimuth, G. G \'e ranton, M. Gibertini, D. Gresch, C. Johnson, T. Koretsune, et al., Journal of Physics: Condensed Matter 2020, 32, 16 165902

work page 2020