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arxiv: 2605.21297 · v1 · pith:4QTSU3JEnew · submitted 2026-05-20 · ❄️ cond-mat.mes-hall · cond-mat.mtrl-sci

Designing Magnetic Topological Insulator Trilayers for Highly-Efficient Spin-Orbit Torque Switching

Pith reviewed 2026-05-21 03:49 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.mtrl-sci
keywords magnetic topological insulatorsspin-orbit torquequantum anomalous HallSrTiO3 substrateschemical potential asymmetrymagnetization switchingedge current chiralitytrilayer heterostructures
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The pith

Substrate-induced charging creates chemical potential asymmetry that governs efficient spin-orbit torque switching in magnetic topological insulator trilayers.

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

The paper establishes that growing magnetic topological insulator trilayers on heat-treated SrTiO3(111) substrates produces an interface charging effect. This charging shifts the chemical potentials differently in the top and bottom magnetic layers, which then dictates how spin-orbit torques reverse the magnetization and flip the chirality of the edge currents in the quantum anomalous Hall state. A sympathetic reader would care because the resulting asymmetry delivers high switching efficiency that can be adjusted by layer thicknesses, gate voltage, or an in-plane field. The work therefore supplies a concrete design route for using these materials in low-power devices that electrically control topological edge states.

Core claim

SOT-driven magnetization reversal and the associated switching of edge current chirality are governed by the SrTiO3(111) substrate-induced charging effect, which generates an asymmetric chemical-potential alignment between the top and bottom magnetic TI layers. The switching polarity and efficiency can be tuned through heterostructure design, gate voltage, and in-plane magnetic field, consistent with SOT symmetry. These results identify chemical potential asymmetry as the origin of the large SOT switching ratio.

What carries the argument

The SrTiO3(111) substrate-induced charging effect that generates asymmetric chemical-potential alignment between the top and bottom magnetic TI layers.

If this is right

  • Switching polarity and efficiency can be adjusted by changing the thicknesses of the magnetic TI layers in the trilayer stack.
  • Gate voltage and in-plane magnetic field provide additional knobs to tune the direction and magnitude of the magnetization reversal.
  • Chemical potential asymmetry is established as the mechanism behind the large observed SOT switching ratio.
  • The design supplies a practical route for electrically controlling edge current chirality in quantum anomalous Hall insulators.

Where Pith is reading between the lines

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

  • The same substrate-charging approach might be tested on other oxide surfaces to induce comparable potential shifts in different topological layers.
  • These trilayers could be combined with conventional semiconductors to create hybrid circuits that switch topological currents on demand.
  • Varying the growth temperature or surface treatment of the substrate offers a testable way to strengthen or weaken the charging effect and measure the resulting change in switching efficiency.

Load-bearing premise

The observed switching polarity and efficiency arise primarily from substrate-induced charging and the resulting chemical potential asymmetry rather than from interface defects, growth variations, or other unaccounted transport contributions.

What would settle it

A direct probe, such as angle-resolved photoemission or Kelvin probe force microscopy, that measures equal chemical potentials in the top and bottom layers while the high-efficiency SOT switching persists.

read the original abstract

Spin-orbit torque (SOT) enables efficient electrical control of magnetization, offering a pathway towards low-power spintronic devices. Magnetic topological insulators (TIs), with spin-momentum-locked surface states and intrinsic ferromagnetism, provide a unique platform for realizing SOT switching of edge current chirality in quantum anomalous Hall (QAH) insulators. In this work, we employ molecular beam epitaxy to synthesize a series of magnetic TI trilayers with controlled layer thicknesses on heat-treated SrTiO3(111) substrates. Electrical transport measurements reveal that SOT-driven magnetization reversal and the associated switching of edge current chirality are governed by the SrTiO3(111) substrate-induced charging effect, which generates an asymmetric chemical-potential alignment between the top and bottom magnetic TI layers. Furthermore, we demonstrate that the switching polarity and efficiency can be tuned through heterostructure design, gate voltage, and in-plane magnetic field, consistent with SOT symmetry. These findings identify chemical potential asymmetry as the origin of the large SOT switching ratio in magnetic TI trilayers and establish a route for electrical control of edge current chirality in QAH insulators. This work advances the understanding of SOT switching mechanism in magnetic topological materials and paves the way for next-generation, energy-efficient QAH-based logic and memory devices.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript reports the MBE growth of magnetic topological insulator trilayers with controlled thicknesses on heat-treated SrTiO3(111) substrates. Electrical transport measurements are presented to show SOT-driven magnetization reversal and switching of edge current chirality in QAH insulators. The central claim is that these effects are governed by a substrate-induced charging effect that produces an asymmetric chemical-potential alignment between the top and bottom magnetic TI layers; switching polarity and efficiency are tuned via heterostructure design, gate voltage, and in-plane field, with chemical potential asymmetry identified as the origin of the large SOT switching ratio.

Significance. If the mechanism interpretation holds, the work would advance understanding of substrate effects on SOT in magnetic TIs and provide a design route for efficient electrical control of edge states in QAH systems, with potential relevance to low-power spintronic devices. The experimental use of thickness series together with gate and field tuning demonstrates systematic exploration of the parameter space.

major comments (2)
  1. [Transport measurements and mechanism interpretation] The attribution of switching polarity and efficiency to SrTiO3(111) substrate charging and the resulting chemical-potential asymmetry between top and bottom layers (abstract and mechanism discussion) is load-bearing for the central claim. The presented thickness series, gate-voltage, and in-plane-field data do not include control samples grown on non-polar substrates or with intentional interface modifications, leaving open contributions from growth-induced defects, strain, or doping gradients.
  2. [Discussion of SOT efficiency] Quantitative support for the claimed dominance of the chemical-potential asymmetry would require explicit modeling or extraction of the potential offset (e.g., from gate dependence) and comparison of switching ratios with error bars or device statistics to establish reproducibility beyond the observed trends.
minor comments (2)
  1. [Methods and figures] Clarify the exact layer sequence and notation for the trilayer stack in the methods and figure captions.
  2. [Experimental methods] Add details on substrate heat-treatment conditions and post-growth surface characterization to support the charging-effect premise.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and have revised the manuscript to strengthen the discussion of the proposed mechanism and to include additional quantitative analysis where feasible.

read point-by-point responses
  1. Referee: [Transport measurements and mechanism interpretation] The attribution of switching polarity and efficiency to SrTiO3(111) substrate charging and the resulting chemical-potential asymmetry between top and bottom layers (abstract and mechanism discussion) is load-bearing for the central claim. The presented thickness series, gate-voltage, and in-plane-field data do not include control samples grown on non-polar substrates or with intentional interface modifications, leaving open contributions from growth-induced defects, strain, or doping gradients.

    Authors: We acknowledge that experiments on non-polar substrates or with deliberate interface modifications would provide stronger isolation of the substrate charging effect. At the same time, the systematic reversal of switching polarity with changes in top versus bottom layer thickness, together with the gate-voltage tunability that aligns with expected chemical-potential shifts, is inconsistent with uniform strain or doping gradients across the heterostructure. We have added a dedicated paragraph in the discussion section that explicitly considers these alternative contributions and explains why the full dataset favors the substrate-induced asymmetry interpretation. revision: partial

  2. Referee: [Discussion of SOT efficiency] Quantitative support for the claimed dominance of the chemical-potential asymmetry would require explicit modeling or extraction of the potential offset (e.g., from gate dependence) and comparison of switching ratios with error bars or device statistics to establish reproducibility beyond the observed trends.

    Authors: We have now extracted a quantitative estimate of the chemical-potential offset directly from the gate-voltage dependence of the switching threshold and included a simple electrostatic model in the revised manuscript and supplementary information. We have also added error bars to the reported switching ratios and included data from multiple devices fabricated on different growth runs to document reproducibility of the observed trends. revision: yes

Circularity Check

0 steps flagged

No circularity in experimental observations and interpretation

full rationale

The manuscript is a purely experimental report on MBE-grown magnetic TI trilayers, transport measurements of SOT switching, and thickness/gate/field dependence. No equations, derivations, fitted parameters renamed as predictions, or self-citation chains appear in the provided text or abstract. The central attribution to SrTiO3(111) substrate charging is an interpretive inference from polarity and efficiency trends rather than a mathematical reduction to inputs by construction. The analysis therefore remains self-contained against external benchmarks with no load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions of thin-film growth quality and the reliability of magnetotransport as a probe of magnetization reversal. No numerical free parameters, new particles, or ad-hoc entities are introduced in the abstract.

axioms (2)
  • domain assumption Molecular beam epitaxy produces uniform magnetic TI trilayers with thickness control sufficient to isolate top-bottom asymmetry effects.
    Invoked to justify the synthesis method and layer design.
  • domain assumption Electrical transport signatures directly reflect SOT-driven magnetization reversal and edge current chirality without dominant parasitic contributions.
    Central to linking measured switching to the proposed chemical potential mechanism.

pith-pipeline@v0.9.0 · 5796 in / 1409 out tokens · 50660 ms · 2026-05-21T03:49:26.546991+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages

  1. [1]

    M.; Jungwirth, T.; Sinova, J.; Thiaville, A.; Garello, K.; Gambardella, P

    Manchon, A.; Železný, J.; Miron, I. M.; Jungwirth, T.; Sinova, J.; Thiaville, A.; Garello, K.; Gambardella, P. Current-induced spin-orbit torques in ferromagnetic and antiferromagnetic systems. Rev. Mod. Phys. 2019, 91, 035004

  2. [2]

    L.; Polshyn, H.; Zhang, Y .; Zhu, J.; Watanabe, K.; Taniguchi, T.; Balents, L.; Young, A

    Serlin, M.; Tschirhart, C. L.; Polshyn, H.; Zhang, Y .; Zhu, J.; Watanabe, K.; Taniguchi, T.; Balents, L.; Young, A. F. Intrinsic quantized anomalous Hall effect in a moiré heterostructure. Science 2020, 367, 900-903

  3. [3]

    D.; Emori, S.; Hoffmann, A.; Åkerman, J.; Roy, K.; Wang, J

    Shao, Q.; Li, P.; Liu, L.; Yang, H.; Fukami, S.; Razavi, A.; Wu, H.; Wang, K.; Freimuth, F.; Mokrousov, Y .; Stiles, M. D.; Emori, S.; Hoffmann, A.; Åkerman, J.; Roy, K.; Wang, J. P.; Yang, S. H.; Garello, K.; Zhang, W. Roadmap of Spin –Orbit Torques. IEEE Trans. Magn. 2021, 57, 1-39

  4. [4]

    Spin- orbit torques: Materials, mechanisms, performances, and potential applications

    Song, C.; Zhang, R.; Liao, L.; Zhou, Y .; Zhou, X.; Chen, R.; You, Y .; Chen, X.; Pan, F. Spin- orbit torques: Materials, mechanisms, performances, and potential applications. Prog. Mater. Sci. 2021, 118, 100761

  5. [5]

    L.; Redekop, E.; Li, L.; Li, T.; Jiang, S.; Arp, T.; Sheekey, O.; Taniguchi, T.; Watanabe, K.; Huber, M

    Tschirhart, C. L.; Redekop, E.; Li, L.; Li, T.; Jiang, S.; Arp, T.; Sheekey, O.; Taniguchi, T.; Watanabe, K.; Huber, M. E.; Mak, K. F.; Shan, J.; Young, A. F. Intrinsic spin Hall torque in a moiré Chern magnet. Nat. Phys. 2023, 19, 807-813

  6. [6]

    Yuan, W.; Zhou, L.-J.; Yang, K.; Zhao, Y .-F.; Zhang, R.; Yan, Z.; Zhuo, D.; Mei, R.; Wang, Y .; Yi, H.; Chan, M. H. W.; Kayyalha, M.; Liu, C.-X.; Chang, C.-Z. Electrical switching of the edge current chirality in quantum anomalous Hall insulators. Nat. Mater. 2024, 23, 58- 64

  7. [7]

    T.; Oleg, N

    Evgeny, Y . T.; Oleg, N. M.; Patrick, R. L. Spin -dependent tunnelling in magnetic tunnel junctions. J. Phys. Condens. Matter. 2003, 15, R109

  8. [8]

    B.; Ralph, D

    Myers, E. B.; Ralph, D. C.; Katine, J. A.; Louie, R. N.; Buhrman, R. A. Current -Induced Switching of Domains in Magnetic Multilayer Devices. Science 1999, 285, 867-870

  9. [9]

    A.; Carey, M

    Mangin, S.; Ravelosona, D.; Katine, J. A.; Carey, M. J.; Terris, B. D.; Fullerton, E. E. Current-induced magnetization reversal in nanopillars with perpendicular anisotropy. Nat. 17 Mater. 2006, 5, 210-215

  10. [10]

    D.; Endo, M.; Kanai, S.; Hayakawa, J.; Matsukura, F.; Ohno, H

    Ikeda, S.; Miura, K.; Yamamoto, H.; Mizunuma, K.; Gan, H. D.; Endo, M.; Kanai, S.; Hayakawa, J.; Matsukura, F.; Ohno, H. A perpendicular-anisotropy CoFeB–MgO magnetic tunnel junction. Nat. Mater. 2010, 9, 721-724

  11. [11]

    D.; Ohno, H

    Brataas, A.; Kent, A. D.; Ohno, H. Current-induced torques in magnetic materials. Nat. Mater. 2012, 11, 372-381

  12. [12]

    C.; Buhrman, R

    Liu, L.; Moriyama, T.; Ralph, D. C.; Buhrman, R. A. Spin-Torque Ferromagnetic Resonance Induced by the Spin Hall Effect. Phys. Rev. Lett. 2011, 106, 036601

  13. [13]

    J.; Gudmundsen, T

    Liu, L.; Lee, O. J.; Gudmundsen, T. J.; Ralph, D. C.; Buhrman, R. A. Current -Induced Switching of Perpendicularly Magnetized Magnetic Layers Using Spin Torque from the Spin Hall Effect. Phys. Rev. Lett. 2012, 109, 096602

  14. [14]

    W.; Ralph, D

    Liu, L.; Pai, C.-F.; Li, Y .; Tseng, H. W.; Ralph, D. C.; Buhrman, R. A. Spin-Torque Switching with the Giant Spin Hall Effect of Tantalum. Science 2012, 336, 555-558

  15. [15]

    Recent progress on controlling spin -orbit torques by materials design

    Ji, G.; Zhang, Y .; Chai, Y .; Nan, T. Recent progress on controlling spin -orbit torques by materials design. npj Spintronics 2024, 2, 56

  16. [16]

    M.; Garello, K.; Gaudin, G.; Zermatten, P

    Miron, I. M.; Garello, K.; Gaudin, G.; Zermatten, P. -J.; Costache, M. V .; Auffret, S.; Bandiera, S.; Rodmacq, B.; Schuhl, A.; Gambardella, P. Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection. Nature 2011, 476, 189-193

  17. [17]

    A.; Finley, J.; Samarth, N.; Liu, L

    Han, J.; Richardella, A.; Siddiqui, S. A.; Finley, J.; Samarth, N.; Liu, L. Room-Temperature Spin-Orbit Torque Switching Induced by a Topological Insulator. Phys. Rev. Lett. 2017, 119, 077702

  18. [18]

    A.; Che, X.; Huang, L.; Dai, B.; Wong, K.; Han, X.; Wang, K

    Wu, H.; Zhang, P.; Deng, P.; Lan, Q.; Pan, Q.; Razavi, S. A.; Che, X.; Huang, L.; Dai, B.; Wong, K.; Han, X.; Wang, K. L. Room -Temperature Spin-Orbit Torque from Topological Surface States. Phys. Rev. Lett. 2019, 123, 207205

  19. [19]

    S.; Kawasaki, M.; Tokura, Y

    Mogi, M.; Yasuda, K.; Fujimura, R.; Yoshimi, R.; Ogawa, N.; Tsukazaki, A.; Kawamura, M.; Takahashi, K. S.; Kawasaki, M.; Tokura, Y . Current-induced switching of proximity-induced ferromagnetic surface states in a topological insulator. Nat. Commun. 2021, 12, 1404. 18

  20. [20]

    Magnetic topological insulators

    Tokura, Y .; Yasuda, K.; Tsukazaki, A. Magnetic topological insulators. Nat. Rev. Phys. 2019, 1, 126-143

  21. [21]

    K.; Zhang, H.; Huang, H.; Qiu, G.; Ren, Y .; Li, Y .; Yang, H.-Y .; Yang, T.-H.; Dong, X.; Dai, B.; Qu, T.; Shu, Q.; Pan, Q.; Zhang, P.; Xue, F.; Li, J.; Davydov, A

    Tai, L.; He, H.; Chong, S. K.; Zhang, H.; Huang, H.; Qiu, G.; Ren, Y .; Li, Y .; Yang, H.-Y .; Yang, T.-H.; Dong, X.; Dai, B.; Qu, T.; Shu, Q.; Pan, Q.; Zhang, P.; Xue, F.; Li, J.; Davydov, A. V .; Wang, K. L. Giant Hall Switching by Surface-State-Mediated Spin-Orbit Torque in a Hard Ferromagnetic Topological Insulator. Adv. Mater. 2024, 36, 2406772

  22. [22]

    M.; Guimaraes, M

    MacNeill, D.; Stiehl, G. M.; Guimaraes, M. H. D.; Buhrman, R. A.; Park, J.; Ralph, D. C. Control of spin–orbit torques through crystal symmetry in WTe2/ferromagnet bilayers. Nat. Phys. 2017, 13, 300-305

  23. [23]

    H.; Muzzio, R.; Zhang, H.; Zhu, M.; Gobbo, J.; Yuan, S.; Weber, D.; Rao, R.; Li, J.; Edgar, J

    Kao, I. H.; Muzzio, R.; Zhang, H.; Zhu, M.; Gobbo, J.; Yuan, S.; Weber, D.; Rao, R.; Li, J.; Edgar, J. H.; Goldberger, J. E.; Yan, J.; Mandrus, D. G.; Hwang, J.; Cheng, R.; Katoch, J.; Singh, S. Deterministic switching of a perpendicularly polarized magne t using unconventional spin–orbit torques in WTe2. Nat. Mater. 2022, 21, 1029-1034

  24. [24]

    Z.; Kane, C

    Hasan, M. Z.; Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 2010, 82, 3045-3067

  25. [25]

    Topological insulators and superconductors

    Qi, X.-L.; Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 2011, 83, 1057-1110

  26. [26]

    R.; Lee, J

    Mellnik, A. R.; Lee, J. S.; Richardella, A.; Grab, J. L.; Mintun, P. J.; Fischer, M. H.; Vaezi, A.; Manchon, A.; Kim, E. A.; Samarth, N.; Ralph, D. C. Spin-transfer torque generated by a topological insulator. Nature 2014, 511, 449-451

  27. [27]

    -L.; Chen, X.; Dai, X.; Fang, Z.; Qi, X.-L.; Zhang, S.-C.; Wang, Y .; He, K.; Ma, X.-C.; Xue, Q.-K

    Chang, C.-Z.; Zhang, J.; Liu, M.; Zhang, Z.; Feng, X.; Li, K.; Wang, L. -L.; Chen, X.; Dai, X.; Fang, Z.; Qi, X.-L.; Zhang, S.-C.; Wang, Y .; He, K.; Ma, X.-C.; Xue, Q.-K. Thin Films of Magnetically Doped Topological Insulator with Carrier -Independent Long -Range Ferromagnetic Order. Adv. Mater. 2013, 25, 1065-1070

  28. [28]

    -C.; Xue, Q

    Chang, C.-Z.; Zhang, J.; Feng, X.; Shen, J.; Zhang, Z.; Guo, M.; Li, K.; Ou, Y .; Wei, P.; Wang, L.-L.; Ji, Z.-Q.; Feng, Y .; Ji, S.; Chen, X.; Jia, J.; Dai, X.; Fang, Z.; Zhang, S.-C.; He, K.; Wang, Y .; Lu, L.; Ma, X. -C.; Xue, Q. -K. Experimental Observatio n of the Quantum 19 Anomalous Hall Effect in a Magnetic Topological Insulator. Science 2013, 340...

  29. [29]

    Y .; Zhang, H.; Assaf, B

    Chang, C.-Z.; Zhao, W.; Kim, D. Y .; Zhang, H.; Assaf, B. A.; Heiman, D.; Zhang, S.-C.; Liu, C.; Chan, M. H. W.; Moodera, J. S. High-precision realization of robust quantum anomalous Hall state in a hard ferromagnetic topological insulator. Nat. Mater. 2015, 14, 473-477

  30. [30]

    Chang, C.-Z.; Liu, C.-X.; MacDonald, A. H. Colloquium: Quantum anomalous Hall effect. Rev. Mod. Phys. 2023, 95, 011002

  31. [31]

    Enhancing the Quantum Anomalous Hall Effect by Magnetic Codoping in a Topological Insulator

    Ou, Y .; Liu, C.; Jiang, G.; Feng, Y .; Zhao, D.; Wu, W.; Wang, X.-X.; Li, W.; Song, C.; Wang, L.-L.; Wang, W.; Wu, W.; Wang, Y .; He, K.; Ma, X.-C.; Xue, Q.-K. Enhancing the Quantum Anomalous Hall Effect by Magnetic Codoping in a Topological Insulator. Adv. Mater. 2018, 30, 1703062

  32. [32]

    G.; Yoshimi, R.; Tsukazaki, A.; Takahashi, K

    Checkelsky, J. G.; Yoshimi, R.; Tsukazaki, A.; Takahashi, K. S.; Kozuka, Y .; Falson, J.; Kawasaki, M.; Tokura, Y . Trajectory of the anomalous Hall effect towards the quantized state in a ferromagnetic topological insulator. Nat. Phys. 2014, 10, 731-736

  33. [33]

    -T.; Fan, Y .; Pan, L.; Lang, M.; Jiang, Y .; Shao, Q.; Nie, T.; Murata, K.; Tang, J.; Wang, Y .; He, L.; Lee, T.-K.; Lee, W.-L.; Wang, K

    Kou, X.; Guo, S. -T.; Fan, Y .; Pan, L.; Lang, M.; Jiang, Y .; Shao, Q.; Nie, T.; Murata, K.; Tang, J.; Wang, Y .; He, L.; Lee, T.-K.; Lee, W.-L.; Wang, K. L. Scale -Invariant Quantum Anomalous Hall Effect in Magnetic Topological Insulators beyond the Two -Dimensional Limit. Phys. Rev. Lett. 2014, 113, 137201

  34. [34]

    S.; Kawasaki, M.; Tokura, Y

    Mogi, M.; Yoshimi, R.; Tsukazaki, A.; Yasuda, K.; Kozuka, Y .; Takahashi, K. S.; Kawasaki, M.; Tokura, Y . Magnetic modulation doping in topological insulators toward higher - temperature quantum anomalous Hall effect. Appl. Phys. Lett. 2015, 107

  35. [35]

    J.; Yang, F.; Liu, J.; Guan, T.; Qu, F

    Chen, J.; Qin, H. J.; Yang, F.; Liu, J.; Guan, T.; Qu, F. M.; Zhang, G. H.; Shi, J. R.; Xie, X. C.; Yang, C. L.; Wu, K. H.; Li, Y . Q.; Lu, L. Gate-V oltage Control of Chemical Potential and Weak Antilocalization in Bi2Se3. Phys. Rev. Lett. 2010, 105, 176602

  36. [36]

    -J.; Yi, H.; Zhang, Y .-Q.; Yu, J.; Xiao, R.; Wang, K.; Samarth, N.; Chan, M

    Zhao, Y .-F.; Zhang, R.; Mei, R.; Zhou, L. -J.; Yi, H.; Zhang, Y .-Q.; Yu, J.; Xiao, R.; Wang, K.; Samarth, N.; Chan, M. H. W.; Liu, C. -X.; Chang, C. -Z. Tuning the Chern number in quantum anomalous Hall insulators. Nature 2020, 588, 419-423

  37. [37]

    Zhao, Y .-F.; Zhang, R.; Zhou, L.-J.; Mei, R.; Yan, Z.-J.; Chan, M. H. W.; Liu, C.-X.; Chang, 20 C.-Z. Zero Magnetic Field Plateau Phase Transition in Higher Chern Number Quantum Anomalous Hall Insulators. Phys. Rev. Lett. 2022, 128, 216801

  38. [38]

    Zhou, L.-J.; Zhuo, D.; Mei, R.; Zhao, Y .-F.; Yang, K.; Zhang, R.; Yan, Z.; Tay, H.; Chan, M. H. W.; Liu, C. -X.; Chang, C. -Z. Interlayer coupling induced quantum phase transition in quantum anomalous Hall multilayers. Phys. Rev. B 2025, 111, L201304

  39. [39]

    Zhou, L.-J.; Mei, R.; Zhao, Y .-F.; Zhang, R.; Zhuo, D.; Yan, Z.-J.; Yuan, W.; Kayyalha, M.; Chan, M. H. W.; Liu, C. -X.; Chang, C. -Z. Confinement -Induced Chiral Edge Channel Interaction in Quantum Anomalous Hall Insulators. Phys. Rev. Lett. 2023, 130, 086201

  40. [40]

    S.; Morimoto, T.; Nagaosa, N.; Kawasaki, M.; Takahashi, Y .; Tokura, Y

    Mogi, M.; Okamura, Y .; Kawamura, M.; Yoshimi, R.; Yasuda, K.; Tsukazaki, A.; Takahashi, K. S.; Morimoto, T.; Nagaosa, N.; Kawasaki, M.; Takahashi, Y .; Tokura, Y . Experimental signature of the parity anomaly in a semi -magnetic topological insulator. Nat. Phys. 2022, 18, 390-394

  41. [41]

    Edelstein, V . M. Spin polarization of conduction electrons induced by electric current in two- dimensional asymmetric electron systems. Solid State Commun. 1990, 73, 233-235

  42. [42]

    -Z.; Zhao, W.; Li, J.; Jain, J

    Chang, C. -Z.; Zhao, W.; Li, J.; Jain, J. K.; Liu, C.; Moodera, J. S.; Chan, M. H. W. Observation of the Quantum Anomalous Hall Insulator to Anderson Insulator Quantum Phase Transition and its Scaling Behavior. Phys. Rev. Lett. 2016, 117, 126802

  43. [43]

    N.; Wang, K

    Kou, X.; He, L.; Lang, M.; Fan, Y .; Wong, K.; Jiang, Y .; Nie, T.; Jiang, W.; Upadhyaya, P.; Xing, Z.; Wang, Y .; Xiu, F.; Schwartz, R. N.; Wang, K. L. Manipulating Surface -Related Ferromagnetism in Modulation -Doped Topological Insulators. Nano Lett. 2013, 13, 4587- 4593

  44. [44]

    S.; Nagaosa, N.; Kawasaki, M.; Tokura, Y

    Yoshimi, R.; Yasuda, K.; Tsukazaki, A.; Takahashi, K. S.; Nagaosa, N.; Kawasaki, M.; Tokura, Y . Quantum Hall states stabilized in semi-magnetic bilayers of topological insulators. Nat. Commun. 2015, 6, 8530