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拓扑绝缘体中量子霍尔效应的研究进展

张帅 宋凤麒

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拓扑绝缘体中量子霍尔效应的研究进展

张帅, 宋凤麒

Research progress of quantum Hall effect in topological insulator

Zhang Shuai, Song Feng-Qi
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  • 三维拓扑绝缘体因其独特的物性备受研究人员关注, 而拓扑表面态的输运是探索其新奇物性的重要手段. 其中, 拓扑表面态的量子霍尔效应则是拓扑绝缘体输运研究的一个重要内容. 本文简要回顾了拓扑绝缘体中量子霍尔效应的实现与发展. 比较了拓扑表面态量子霍尔效应与其他体系的差别, 讨论了其材料体系的发展, 并介绍了其中的标度律行为. 之后详细回顾了实验上对拓扑表面态量子霍尔效应磁性近邻与栅压调控等方面的研究. 最后, 展望了拓扑绝缘体中量子霍尔态的研究前景, 希望能促进拓扑绝缘体的应用.
    Three-dimensional topological insulators (TIs) with gapless topological surface states (TSSs) have attracted considerable attention because of their unique properties. The transport of TSS is an essential means to explore the novel properties. The quantum Hall effect (QHE) of TSS is an important content in the study of topological insulator, for it is an important characteristic of the pure TSS transport. This paper briefly reviews the recent research progress of QHE in TIs. Firstly, we introduce the fundamental concepts of the QHE in TIs. In a three-dimensional TI, each TSS contributes to a half-integer QHE. An integer QHE should be observed due to the existence of top and bottom surface in TI. Then, we review the realization and development of QHE. With the optimization of TI materials, the QHE of TSS is observed in bulk-insulating TIs. Next, the phase transition and scaling law behavior of QHE in TIs are discussed. The dominance of electron-electron interaction of the TSS is revealed by the anomalous critical exponent. Also, the experimental studies of the magnetic proximity and gate voltage modulation of the QHE are reviewed in detail. Finally, the perspectives of QHE in TIs are discussed.
      通信作者: 宋凤麒, songfengqi@nju.edu.cn
    • 基金项目: 国家重点基础研究发展计划(批准号: 2022YFA1402404)和国家自然科学基金(批准号: 92161201, T2221003, 12025404, 11904166)资助的课题.
      Corresponding author: Song Feng-Qi, songfengqi@nju.edu.cn
    • Funds: Project supported by the National Key Research and Development Program of China (Grant No. 2022YFA1402404) and the National Natural Science Foundation of China (Grant Nos. 92161201, T2221003, 12025404, 11904166).
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    [3]

    Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801Google Scholar

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    Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802Google Scholar

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    Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757Google Scholar

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    König M, Wiedmann S, Brüne C, Roth A, Buhmann H, Molenkamp L W, Qi X L, Zhang S C 2007 Science 318 766Google Scholar

    [7]

    Moore J E, Balents L 2007 Phys. Rev. B 75 121306Google Scholar

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    Fu L, Kane C L 2007 Phys. Rev. B 76 045302.Google Scholar

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    Hsieh D, Qian D, Wray L, Xia Y Q, Hor Y S, Cava R J, Hasan M Z 2008 Nature 452 970Google Scholar

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    Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045Google Scholar

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    Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057Google Scholar

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    Ando Y 2013 J. Phys. Soc. Jpn. 82 102001Google Scholar

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    Moore J E 2010 Nature 464 194Google Scholar

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    Lu H Z, Shen S Q 2014 Proc. SPIE 9167 Spintronics VII 91672E

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    Bardarson J H, Moore J E 2013 Rep. Prog. Phys. 76 056501Google Scholar

    [16]

    Qu D X, Hor Y S, Xiong J, Cava R J, Ong N P 2010 Science 329 821Google Scholar

    [17]

    Li Z, Zhang S, Song F 2015 Acta Phys. Sin. 64 097202Google Scholar

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    Goerbig M O 2009 arXiv: 0909.1998

    [19]

    Tong D 2016 arXiv: 1606.06687

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    Nielsen H B, Ninomiya M 1981 Phys. Lett. B 105 219Google Scholar

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    Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V, Firsov A A 2005 Nature 438 197Google Scholar

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    Zhang Y, Tan Y W, Stormer H L, Kim P 2005 Nature 438 201Google Scholar

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    Heremans J P, Cava R J, Samarth N 2017 Nat. Rev. Mater. 2 17049Google Scholar

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    Ren Z, Taskin A A, Sasaki S, Segawa K, Ando Y 2011 Phys. Rev. B 84 165311Google Scholar

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    Arakane T, Sato T, Souma S, Kosaka K, Nakayama K, Komatsu M, Takahashi M, Ren Z, Segawa K, Ando Y 2012 Nat. Commun. 3 636Google Scholar

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    Brüne C, Liu C X, Novik E G, Hankiewicz E M, Buhmann H, Chen Y L, Qi X L, Shen Z X, Zhang S C, Molenkamp L W 2011 Phys. Rev. Lett. 106 126803Google Scholar

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    Xu Y, Miotkowski I, Liu C, Tian J, Nam H, Alidoust N, Hu J, Shih C K, Hasan M Z, Chen Y P 2014 Nat. Phys. 10 956Google Scholar

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    Yoshimi R, Tsukazaki A, Kozuka Y, Falson J, Takahashi K S, Checkelsky J G, Nagaosa N, Kawasaki M, Tokura Y 2015 Nat. Commun. 6 6627Google Scholar

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    Zou W, Wang W, Kou X, Lang M, Fan Y, Choi E S, Fedorov A V, Wang K, He L, Xu Y, Wang K L 2017 Appl. Phys. Lett. 110 212401Google Scholar

    [31]

    Koirala N, Brahlek M, Salehi M, Wu L, Dai J, Waugh J, Nummy T, Han M G, Moon J, Zhu Y, Dessau D, Wu W, Armitage N P, Oh S 2015 Nano Lett. 15 8245Google Scholar

    [32]

    Moon J, Koirala N, Salehi M, Zhang W, Wu W, Oh S, 2018 Nano Lett. 18 820Google Scholar

    [33]

    Koirala N, Salehi M, Moon J, Oh S 2019 Phys. Rev. B 100 085404Google Scholar

    [34]

    Kushwaha S K, Pletikosic´ I, Liang T, Gyenis A, Lapidus S H, Tian Y, Zhao H, Burch K S, Lin J, Wang W, Ji H, Fedorov A V, Yazdani A, Ong N P, Valla T, Cava R J 2016 Nat. Commun. 7 11456Google Scholar

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    Xie F, Zhang S, Liu Q, Xi C, Kang T T, Wang R, Wei B, Pan X C, Zhang M, Fei F, Wang X, Pi L, Yu G L, Wang B, Song F 2019 Phys. Rev. B 99 081113(RGoogle Scholar

    [36]

    Ichimura K, Matsushita S Y, Huynh K K, Tanigaki K 2019 Appl. Phys. Lett. 115 052104Google Scholar

    [37]

    Khmelnitskii D 1983 JETP Lett. 38 454

    [38]

    Murzin S S, Dorozhkin S I, Maude D K, Jansen A G M 2005 Phys. Rev. B 72 195317Google Scholar

    [39]

    Checkelsky J G, Yoshimi R, Tsukazaki A, Takahashi K S, Kozuka Y, Falson J, Kawasaki M, Tokura Y 2014 Nat. Phys. 10 731Google Scholar

    [40]

    Xu Y, Miotkowski I, Chen Y P 2016 Nat. Commun. 7 11434Google Scholar

    [41]

    Huckestein B 1995 Rev. Mod. Phys. 67 357Google Scholar

    [42]

    Sondhi S L, Girvin S M, Carini J P, Shahar D 1997 Rev. Mod. Phys. 69 315Google Scholar

    [43]

    Salehi M, Shapourian H, Rosen I T, Han M G, Moon J, Shibayev P, Jain D, Goldhaber-Gordon D, Oh S 2019 Adv. Mater. 31 1901091Google Scholar

    [44]

    Tang C, Chang C Z, Zhao G, Liu Y, Jiang Z, Liu C X, McCartney M R, Smith D J, Chen T, Moodera J S, Shi J 2017 Sci. Adv. 3 e1700307Google Scholar

    [45]

    Watanabe R, Yoshimi R, Kawamura M, Mogi M, Tsukazaki A, Yu X Z, Nakajima K, Takahashi K S, Kawasaki M, Tokura Y 2019 Appl. Phys. Lett. 115 102403Google Scholar

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    Yoshimi R, Yasuda K, Tsukazaki A, Takahashi K S, Nagaosa N, Kawasaki M, Tokura Y 2015 Nat. Commun. 6 8530Google Scholar

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    Zhang S, Pi L, Wang R, Yu G, Pan X C, Wei Z, Zhang J, Xi C, Bai Z, Fei F, Wang M, Liao J, Li Y, Wang X, Song F, Zhang Y, Wang B, Xing D, G. Wang G 2017 Nat. Commun. 8 977Google Scholar

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    Chong S K, Han K B, Nagaoka A, Tsuchikawa R, Liu R, Liu H, Vardeny Z V, Pesin D C, Lee C, Sparks T D, Deshpande V V 2018 Nano Lett. 18 8047Google Scholar

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    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 2022 Nat. Phys. 18 390Google Scholar

    [50]

    Li C, de Ronde B, Nikitin A, Huang Y, Golden M S, de Visser A, Brinkman A 2017 Phys. Rev. B 96 195427Google Scholar

    [51]

    Chong S K, Han K B, Sparks T D, Deshpande V V 2019 Phys. Rev. Lett. 123 036804Google Scholar

    [52]

    Banerjee A, Sundaresh A, Biswas S, Ganesan R, Sen D, Anil Kumar P S 2019 Nanoscale 11 5317Google Scholar

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    Xie F, Lian Z, Zhang S, Wang T, Miao S, Song Z, Ying Z, Pan X C, Long M, Zhang M, Fei F, Hu W, Yu G, Song F, Kang T T, Shi S F 2021 Nanotechnology 32 17LT01Google Scholar

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    Chong S K, Tsuchikawa R, Harmer J, Sparks T D, Deshpande V V 2020 ACS Nano 14 1158Google Scholar

    [55]

    Wang J, Gorini C, Richter K, Wang Z, Ando Y, Weiss D 2020 Nano Lett. 20 8493Google Scholar

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    Novoselov K S, Jiang Z, Zhang Y, Morozov S V, Stormer H L, Zeitler U, Maan J C, Boebinger G S, Kim P, Geim A K 2007 Science 315 1379Google Scholar

    [57]

    Morimoto T, Furusaki A, Nagaosa N 2015 Phys. Rev. Lett. 114 146803Google Scholar

    [58]

    Morimoto T, Furusaki A, Nagaosa N 2015 Phys. Rev. B 92 085113Google Scholar

    [59]

    Seradjeh B, Moore J E, Franz M 2009 Rev. Lett. 103 066402Google Scholar

    [60]

    Ilan R, de Juan F, Moore J E 2015 Phys. Rev. Lett. 115 096802Google Scholar

    [61]

    Seredinski A, Draelos A W, Arnault E G, Wei M T, Li H, Fleming T, Watanabe K, Taniguchi T, Amet F, Finkelstein G 2019 Sci. Adv. 5 eaaw8693Google Scholar

    [62]

    Qi X L, Hughes T L, Zhang S C 2011 Phys. Rev. B 83 1057Google Scholar

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    Liu J, Hsieh T H, Wei P, Duan W, Moodera J, Fu L 2014 Nat. Mater. 13 178Google Scholar

  • 图 1  量子霍尔效应与朗道能级 (a) 量子霍尔效应的典型输运特征; (b) 传统二维电子气在磁场下的朗道能级示意图; (c) 二维狄拉克费米子体系在磁场下的朗道能级示意图; (d) 传统二维电子气量子霍尔手性边缘态示意图; (e) 拓扑绝缘体上下表面量子霍尔手性边缘态示意图.

    Fig. 1.  Quantum Hall effect and Landau levels: (a) Transport characteristics of quantum Hall effect; (b) Landau level diagram of conventional two-dimensional electron gas in magnetic field; (c) Landau level diagram of two-dimensional Dirac fermion system in magnetic field; (d) diagram of quantum Hall chiral edge states in conventional two-dimensional electron gas; (e) diagram of quantum Hall chiral edge states of top and bottom surface states in topological insulator.

    图 2  拓扑绝缘体中的量子霍尔态 (a) Sn-Bi1.1Sb0.9Te2S器件中高质量的量子霍尔态[35]; (b)三维拓扑绝缘体中量子霍尔态的观测温度、厚度及体能隙间的关系[36]

    Fig. 2.  Quantum Hall states in topological insulators: (a) High-quality Quantum Hall states in Sn-Bi1.1Sb0.9Te2S device[35]; (b) relationship among temperature, thickness and energy gap of quantum Hall states in topological insulators [36].

    图 3  量子霍尔态的重整化群流 (a)半圆形标度的重整化群流; (b)拓扑绝缘体中栅压驱动的重整化群流[40]

    Fig. 3.  Renormalization group flow of quantum Hall states: (a) Semi-circle renormalization group flow; (b) gate-driven renormalization group flow in a topological insulator[40].

    图 4  拓扑绝缘体量子霍尔态的标度律行为[35] (a)量子霍尔平台在不同温度下的转变, 上图为霍尔电阻的转变, 下图为纵向电阻的转变; (b)量子霍尔平台转变中的标度律, 上图为dRxy/dB随温度的关系, 下图为ΔB随温度的关系

    Fig. 4.  Scaling law of quantum Hall states in topological insulators[35]: (a) Quantum Hall plateaus transition, Hall resistance (upper) and longitudinal resistance (lower); (b) scaling law in plateau transition region, relationship between dRxy/dB and temperature (upper) and relationship between ΔB and temperature (lower).

    图 5  磁性修饰与近邻下的量子霍尔态. 磁性团簇修饰前(a)和磁性团簇修饰后(b)的量子霍尔态的重整化群流[47]; h-BN作为介电层(c)和Cr2Ge2Te6作为介电层(d)的量子霍尔态[48]

    Fig. 5.  Quantum Hall states tuned by magnetic modification and proximity. Renormalization group flow of quantum Hall States before (a) and after (b) magnetic cluster modification[47]; quantum Hall states with h-BN as the dielectric layer (c) and Cr2Ge2Te6 as the dielectric layer (d)[48].

    图 6  拓扑表面态量子霍尔效应的双栅调控 (a)—(c)随厚度减小, 量子霍尔态在双栅调控下的不同行为[40,50,51]

    Fig. 6.  Dual-gate-tuned quantum Hall states of topological surface states: (a)–(c) different behavior of quantum Hall states with decreasing thickness[40,50,51].

    表 1  不同拓扑绝缘体中量子霍尔态的特征

    Table 1.  Properties of quantum Hall state in topological insulators.

    MethodMaterial*T@B@v = 1Thickness/nmReference

    Exfoliated
    五元Sn-(Bi, Sb)2(Te, S)32 K@ 8 T3×103Ichimura等[36]
    20 K@ 12 T82Xie等[35]
    四元 (Bi, Sb)2(Te, Se)310 K@ 31 T80Xu等[28]

    MBE

    三元
    (Bi, Sb)2Te30.04 K@ 14 T8Yoshimi等[29]
    Ca-Bi2Se30.3 K@ 25 T8Moon等[32]
    Ti-Sb2Te30.35 K@30 T10Salehi等[43]
    二元 Bi2Se320 K@35 T8Koirala等[33]
    注: “*”表示报道的观测到ν = 1量子霍尔平台(且要求此时Rxx < 0.04 h/e2)对应的最高温度和对应该温度的最低磁场大小.
    下载: 导出CSV
  • [1]

    Klitzing K, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494Google Scholar

    [2]

    Thouless D J, Kohmoto K, Nightingale M P, den Nijs M 1982 Phys. Rev. Lett. 49 405Google Scholar

    [3]

    Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801Google Scholar

    [4]

    Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802Google Scholar

    [5]

    Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757Google Scholar

    [6]

    König M, Wiedmann S, Brüne C, Roth A, Buhmann H, Molenkamp L W, Qi X L, Zhang S C 2007 Science 318 766Google Scholar

    [7]

    Moore J E, Balents L 2007 Phys. Rev. B 75 121306Google Scholar

    [8]

    Fu L, Kane C L 2007 Phys. Rev. B 76 045302.Google Scholar

    [9]

    Hsieh D, Qian D, Wray L, Xia Y Q, Hor Y S, Cava R J, Hasan M Z 2008 Nature 452 970Google Scholar

    [10]

    Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045Google Scholar

    [11]

    Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057Google Scholar

    [12]

    Ando Y 2013 J. Phys. Soc. Jpn. 82 102001Google Scholar

    [13]

    Moore J E 2010 Nature 464 194Google Scholar

    [14]

    Lu H Z, Shen S Q 2014 Proc. SPIE 9167 Spintronics VII 91672E

    [15]

    Bardarson J H, Moore J E 2013 Rep. Prog. Phys. 76 056501Google Scholar

    [16]

    Qu D X, Hor Y S, Xiong J, Cava R J, Ong N P 2010 Science 329 821Google Scholar

    [17]

    Li Z, Zhang S, Song F 2015 Acta Phys. Sin. 64 097202Google Scholar

    [18]

    Goerbig M O 2009 arXiv: 0909.1998

    [19]

    Tong D 2016 arXiv: 1606.06687

    [20]

    Nielsen H B, Ninomiya M 1981 Phys. Lett. B 105 219Google Scholar

    [21]

    Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V, Firsov A A 2005 Nature 438 197Google Scholar

    [22]

    Zhang Y, Tan Y W, Stormer H L, Kim P 2005 Nature 438 201Google Scholar

    [23]

    Zhang H, Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2009 Nat. Phys. 5 438Google Scholar

    [24]

    Heremans J P, Cava R J, Samarth N 2017 Nat. Rev. Mater. 2 17049Google Scholar

    [25]

    Ren Z, Taskin A A, Sasaki S, Segawa K, Ando Y 2011 Phys. Rev. B 84 165311Google Scholar

    [26]

    Arakane T, Sato T, Souma S, Kosaka K, Nakayama K, Komatsu M, Takahashi M, Ren Z, Segawa K, Ando Y 2012 Nat. Commun. 3 636Google Scholar

    [27]

    Brüne C, Liu C X, Novik E G, Hankiewicz E M, Buhmann H, Chen Y L, Qi X L, Shen Z X, Zhang S C, Molenkamp L W 2011 Phys. Rev. Lett. 106 126803Google Scholar

    [28]

    Xu Y, Miotkowski I, Liu C, Tian J, Nam H, Alidoust N, Hu J, Shih C K, Hasan M Z, Chen Y P 2014 Nat. Phys. 10 956Google Scholar

    [29]

    Yoshimi R, Tsukazaki A, Kozuka Y, Falson J, Takahashi K S, Checkelsky J G, Nagaosa N, Kawasaki M, Tokura Y 2015 Nat. Commun. 6 6627Google Scholar

    [30]

    Zou W, Wang W, Kou X, Lang M, Fan Y, Choi E S, Fedorov A V, Wang K, He L, Xu Y, Wang K L 2017 Appl. Phys. Lett. 110 212401Google Scholar

    [31]

    Koirala N, Brahlek M, Salehi M, Wu L, Dai J, Waugh J, Nummy T, Han M G, Moon J, Zhu Y, Dessau D, Wu W, Armitage N P, Oh S 2015 Nano Lett. 15 8245Google Scholar

    [32]

    Moon J, Koirala N, Salehi M, Zhang W, Wu W, Oh S, 2018 Nano Lett. 18 820Google Scholar

    [33]

    Koirala N, Salehi M, Moon J, Oh S 2019 Phys. Rev. B 100 085404Google Scholar

    [34]

    Kushwaha S K, Pletikosic´ I, Liang T, Gyenis A, Lapidus S H, Tian Y, Zhao H, Burch K S, Lin J, Wang W, Ji H, Fedorov A V, Yazdani A, Ong N P, Valla T, Cava R J 2016 Nat. Commun. 7 11456Google Scholar

    [35]

    Xie F, Zhang S, Liu Q, Xi C, Kang T T, Wang R, Wei B, Pan X C, Zhang M, Fei F, Wang X, Pi L, Yu G L, Wang B, Song F 2019 Phys. Rev. B 99 081113(RGoogle Scholar

    [36]

    Ichimura K, Matsushita S Y, Huynh K K, Tanigaki K 2019 Appl. Phys. Lett. 115 052104Google Scholar

    [37]

    Khmelnitskii D 1983 JETP Lett. 38 454

    [38]

    Murzin S S, Dorozhkin S I, Maude D K, Jansen A G M 2005 Phys. Rev. B 72 195317Google Scholar

    [39]

    Checkelsky J G, Yoshimi R, Tsukazaki A, Takahashi K S, Kozuka Y, Falson J, Kawasaki M, Tokura Y 2014 Nat. Phys. 10 731Google Scholar

    [40]

    Xu Y, Miotkowski I, Chen Y P 2016 Nat. Commun. 7 11434Google Scholar

    [41]

    Huckestein B 1995 Rev. Mod. Phys. 67 357Google Scholar

    [42]

    Sondhi S L, Girvin S M, Carini J P, Shahar D 1997 Rev. Mod. Phys. 69 315Google Scholar

    [43]

    Salehi M, Shapourian H, Rosen I T, Han M G, Moon J, Shibayev P, Jain D, Goldhaber-Gordon D, Oh S 2019 Adv. Mater. 31 1901091Google Scholar

    [44]

    Tang C, Chang C Z, Zhao G, Liu Y, Jiang Z, Liu C X, McCartney M R, Smith D J, Chen T, Moodera J S, Shi J 2017 Sci. Adv. 3 e1700307Google Scholar

    [45]

    Watanabe R, Yoshimi R, Kawamura M, Mogi M, Tsukazaki A, Yu X Z, Nakajima K, Takahashi K S, Kawasaki M, Tokura Y 2019 Appl. Phys. Lett. 115 102403Google Scholar

    [46]

    Yoshimi R, Yasuda K, Tsukazaki A, Takahashi K S, Nagaosa N, Kawasaki M, Tokura Y 2015 Nat. Commun. 6 8530Google Scholar

    [47]

    Zhang S, Pi L, Wang R, Yu G, Pan X C, Wei Z, Zhang J, Xi C, Bai Z, Fei F, Wang M, Liao J, Li Y, Wang X, Song F, Zhang Y, Wang B, Xing D, G. Wang G 2017 Nat. Commun. 8 977Google Scholar

    [48]

    Chong S K, Han K B, Nagaoka A, Tsuchikawa R, Liu R, Liu H, Vardeny Z V, Pesin D C, Lee C, Sparks T D, Deshpande V V 2018 Nano Lett. 18 8047Google Scholar

    [49]

    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 2022 Nat. Phys. 18 390Google Scholar

    [50]

    Li C, de Ronde B, Nikitin A, Huang Y, Golden M S, de Visser A, Brinkman A 2017 Phys. Rev. B 96 195427Google Scholar

    [51]

    Chong S K, Han K B, Sparks T D, Deshpande V V 2019 Phys. Rev. Lett. 123 036804Google Scholar

    [52]

    Banerjee A, Sundaresh A, Biswas S, Ganesan R, Sen D, Anil Kumar P S 2019 Nanoscale 11 5317Google Scholar

    [53]

    Xie F, Lian Z, Zhang S, Wang T, Miao S, Song Z, Ying Z, Pan X C, Long M, Zhang M, Fei F, Hu W, Yu G, Song F, Kang T T, Shi S F 2021 Nanotechnology 32 17LT01Google Scholar

    [54]

    Chong S K, Tsuchikawa R, Harmer J, Sparks T D, Deshpande V V 2020 ACS Nano 14 1158Google Scholar

    [55]

    Wang J, Gorini C, Richter K, Wang Z, Ando Y, Weiss D 2020 Nano Lett. 20 8493Google Scholar

    [56]

    Novoselov K S, Jiang Z, Zhang Y, Morozov S V, Stormer H L, Zeitler U, Maan J C, Boebinger G S, Kim P, Geim A K 2007 Science 315 1379Google Scholar

    [57]

    Morimoto T, Furusaki A, Nagaosa N 2015 Phys. Rev. Lett. 114 146803Google Scholar

    [58]

    Morimoto T, Furusaki A, Nagaosa N 2015 Phys. Rev. B 92 085113Google Scholar

    [59]

    Seradjeh B, Moore J E, Franz M 2009 Rev. Lett. 103 066402Google Scholar

    [60]

    Ilan R, de Juan F, Moore J E 2015 Phys. Rev. Lett. 115 096802Google Scholar

    [61]

    Seredinski A, Draelos A W, Arnault E G, Wei M T, Li H, Fleming T, Watanabe K, Taniguchi T, Amet F, Finkelstein G 2019 Sci. Adv. 5 eaaw8693Google Scholar

    [62]

    Qi X L, Hughes T L, Zhang S C 2011 Phys. Rev. B 83 1057Google Scholar

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    Liu J, Hsieh T H, Wei P, Duan W, Moodera J, Fu L 2014 Nat. Mater. 13 178Google Scholar

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出版历程
  • 收稿日期:  2023-04-30
  • 修回日期:  2023-07-04
  • 上网日期:  2023-07-18
  • 刊出日期:  2023-09-05

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