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二维材料平带的实现及其新奇量子物态

张若寒 任慧莹 何林

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二维材料平带的实现及其新奇量子物态

张若寒, 任慧莹, 何林

Flat bands and related novel quantum states in two-dimensional systems

Zhang Ruo-Han, Ren Hui-Ying, He Lin
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  • 在二维材料平带中电子的有效质量急剧增大, 电子的库仑排斥能将远远大于电子的动能, 电子-电子相互作用效应显著, 对应地将会产生一系列新奇的强关联量子物态, 如量子霍尔铁磁态、分数量子霍尔效应、量子反常霍尔效应、超导态、Wigner晶体等. 因此人们对于二维材料中的平带产生了极大的兴趣. 近几年, 与平带相关的强关联物性研究成为了凝聚态物理领域的前沿课题. 实验上发展了多种方法, 例如通过外加强磁场、构筑应变结构、引入转角等方式在二维材料中引入平带. 本文通过对二维体系中平带的实现方法及其带来的新奇物理现象进行回顾, 希望为相关领域的研究人员提供参考和借鉴.
    In flat bands of two-dimensional materials, the mass of charge carriers increases dramatically and the Coulomb energy of the charge carriers can be much larger than the quenched kinetic energy. When the flat band is partially filled, electron-electron interactions can drive electrons to form exotic correlated phases, such as quantum Hall ferromagnetism, fractional quantum Hall effect, superconductivity, and quantum anomalous Hall effect. Therefore, flat bands in two-dimensional materials have attracted much attention very recently. In the past few years, the strongly correlated phenomena in flat bands have become a hot topic in community of condensed matter physics. There are several different methods, such as using a perpendicular magnetic field, introducing strained structures, and introducing a twist angle, to realize the flat bands in two-dimensional materials. In this review article, we summarize the methods to realize flat bands in two-dimensional systems and introduce the related novel electronic states when the flat band is partially filled.
      通信作者: 何林, helin@bnu.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 12141401, 11974050)资助的课题.
      Corresponding author: He Lin, helin@bnu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12141401, 11974050).
    [1]

    黄昆 著 (韩汝琦改编) 1988 固体物理学 (北京: 高等教育出版社) 第154—157页

    Huang K (adapted by Han R Q) 1988 Solid State Physics (Beijing: Higher Education Press) pp154–157 (in Chinese)

    [2]

    de Boer J H, Verwey E J 1937 Proc. Phys. Soc. 49 59Google Scholar

    [3]

    Mott N F, Peierls R 1937 Proc. Phys. Soc. 49 72Google Scholar

    [4]

    Mott N F 1949 Proc. Phys. Soc. A 62 416Google Scholar

    [5]

    Hubbard J 1963 Sciences 276 238Google Scholar

    [6]

    Wu M K, Ashburn J R, Torng C J, Hor P H, Meng R L, Gao L, Huang Z J, Wang Y Q, Chu C W 1987 Phys. Rev. Lett. 58 908Google Scholar

    [7]

    Zhang J B, Struzhkin V V, Yang W G, Mao H K, Lin H Q, Ma Y C, Wang N L, Chen X J 2015 Phys. Condens. Matter. 27 445701Google Scholar

    [8]

    Narayan J 1989 JOM 41 18Google Scholar

    [9]

    Tokura Y, Takagi H, Uchida S 1989 Nature 337 345Google Scholar

    [10]

    Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 666Google Scholar

    [11]

    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

    [12]

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

    [13]

    Katsnelson M I, Novoselov K S, Geim A K 2006 Nature Phys. 2 620Google Scholar

    [14]

    Zhang Y, Jiang Z, Small J P, Purewal M S, Tan Y W, Fazlollahi M, Chudow J D, Jaszczak J A, Stormer H L, Kim P 2006 Phys. Rev. Lett. 96 136806Google Scholar

    [15]

    Peres N M R, Guinea F, Castro Neto A H 2006 Ann. Phys. 321 1559Google Scholar

    [16]

    Peres N M R, Guinea F, Castro Neto A H 2006 Phys. Rev. B 73 125411Google Scholar

    [17]

    Miller D L, Kubista K D, Rutter G M, Ruan M, de Heer W A, First P N, Stroscio J A 2009 Science 324 924Google Scholar

    [18]

    Meyer J C, Geim A K, Katsnelson M I, Novoselov K S, Booth T J, Roth S 2007 Nature 446 60Google Scholar

    [19]

    Morpurgo A, Guinea F 2006 Phys. Rev. Lett. 97 196804Google Scholar

    [20]

    Guinea F, Katsnelson M, Vozmediano M A H 2008 Phys. Rev. B 77 075422Google Scholar

    [21]

    Guinea F, Katsnelson M, Geim A K 2009 Nature Phys. 6 30Google Scholar

    [22]

    Levy N, Burke S A, Meaker K L, Panlasigui M, Zettl A, Guinea F, Castro Neto A H, Crommie M F 2010 Science 329 544Google Scholar

    [23]

    Yeh N C, Teague M L, Yeom S, Standley B L, Wu R T P, Boyd D A, Bockrath M W 2011 Surf. Sci. 605 1649Google Scholar

    [24]

    Lu J, Neto A H, Loh K P 2012 Nat. Commun. 3 1Google Scholar

    [25]

    Yan H, Sun Y, He L, Nie J C, Chan M H 2012 Phys. Rev. B 85 035422Google Scholar

    [26]

    Guo D, Kondo T, Machida T, Iwatake K, Okada S, Nakamura J 2012 Nat. Commun. 3 1068Google Scholar

    [27]

    Meng L, He W Y, Zheng H, Liu M, Yan H, Yan W, Chu Z D, Bai K, Dou R F, Zhang Y 2013 Phys. Rev. B 87 205405Google Scholar

    [28]

    Yan W, He W Y, Chu Z D, Liu M, Meng L, Dou R F, Zhang Y, Liu Z, Nie J C, He L 2013 Nat. Commun. 4 2159Google Scholar

    [29]

    Li S Y, Bai K K, Yin L J, Qiao J B, Wang W X, He L 2015 Phys. Rev. B 92 245302Google Scholar

    [30]

    Jiang Y H, Mao J H, Duan J X, Lai X Y, Watanabe K, Taniguchi T, Andrei E Y 2017 Nano. Lett. 17 2839Google Scholar

    [31]

    Liu Y, Rodrigues J N B, Luo Y Z, Li L, Carvalho A, Yang M, Laksono E, Lu J, Bao Y, Xu H, Tan S J R, Qiu Z, Sow C H, Feng Y P, Neto A H C, Adam S, Lu J, Loh K P 2018 Nature Nanotech. 13 828Google Scholar

    [32]

    Jia P, Chen W, Qiao J, Zhang M, Zheng X, Xue Z, Liang R, Tian C, He L, Di Z, Wang X 2019 Nat. Commun. 10 3127Google Scholar

    [33]

    Nigge P, Qu A C, Lantagne-Hurtubise É, Mårsell E, Link S, Tom G, Zonno M, Michiardi M, Schneider M, Zhdanovich S, Levy G, Starke U, Gutiérrez C, Bonn D, Burke S A, Franz M, Damascelli A 2019 Sci. Adv. 5 eaaw5593Google Scholar

    [34]

    Li S Y, Su Y, Ren Y N, He L 2020 Phys. Rev. Lett. 124 106802Google Scholar

    [35]

    Hsu C C, Teague M L, Wang J Q, Yeh N C 2020 Sci. Adv. 6 eaat9488Google Scholar

    [36]

    Mao J H, Milovanović S P, Anđelković M, Lai X Y, Cao Y, Watanabe K, Taniguchi T, Covaci L, Peeters F M, Geim A K, Jiang Y H, Andrei E Y 2020 Nature 584 215Google Scholar

    [37]

    Vozmediano M A H, Katsnelson M I, Guinea F 2010 Phys. Rep. 496 109Google Scholar

    [38]

    Li G H, Luican A, Lopes dos Santos J M B, Castro Neto A H, Reina A, Kong J, Andrei E Y 2010 Nature Phys. 6 109Google Scholar

    [39]

    Bistritzer R, MacDonald A H 2011 Proc. Nat. I Acad. Sci. U. S. A 108 12233Google Scholar

    [40]

    Yin L J, Qiao J B, Zuo W J, Li W T, He L 2015 Phys. Rev. B 92 081406Google Scholar

    [41]

    Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E, Jarillo-Herrero P 2018 Nature 556 43Google Scholar

    [42]

    Khalaf E, Kruchkov A J, Tarnopolsky G, Vishwanath A 2019 Phys. Rev. B 100 085109Google Scholar

    [43]

    Carr S, Li C Y, Zhu Z Y, Kaxiras E, Sachdev S, Kruchkov A J 2020 Nano. Lett. 20 3030Google Scholar

    [44]

    Hao Z Y, Zimmerman A M, Ledwith P, Khalaf E, Najafabadi D H, Watanabe K, Taniguchi T, Vishwanath A, Kim P 2021 Science 371 1133Google Scholar

    [45]

    Liu X M, Hao Z Y, Khalaf E, Lee J Y, Ronen Y, Yoo H, Najafabadi D H, Watanabe K, Taniguchi T, Vishwanath A, Kim P 2020 Nature 583 221Google Scholar

    [46]

    Burg G W, Zhu J H, Taniguchi T, Watanabe K, MacDonald A H, Tutuc E 2019 Phys. Rev. Lett. 123 197702Google Scholar

    [47]

    Cao Y, Rodan-Legrain D, Rubies-Bigorda O, Park J M, Watanabe K, Taniguchi T, Jarillo-Herrero P 2020 Nature 583 215Google Scholar

    [48]

    Shen C, Chu Y B, Wu Q S, Li N, Wang S P, Zhao Y C, Tang J, Liu J Y, Tian J P, Watanabe K, Taniguchi T, Yang R, Meng Z Y, Shi D X, Yazyev O V, Zhang G Y 2020 Nature Phys. 16 520Google Scholar

    [49]

    Wu F, Lovorn T, Tutuc E, Macdonald A H 2018 Phys. Rev. Lett. 121 026402Google Scholar

    [50]

    Naik M H, Jain M 2018 Phys. Rev. Lett. 121 266401Google Scholar

    [51]

    Wu F, Lovorn T, Tutuc E, Martin I, Macdonald A H 2019 Phys. Rev. Lett. 122 86402Google Scholar

    [52]

    Ruiz-Tijerina D A, Fal’Ko V I 2019 Phys. Rev. B 99 125424Google Scholar

    [53]

    Wang L, Shih E M, Ghiotto A, Xian L, Rhodes D A, Tan C, Claassen M, Kennes D M, Bai Y S, Kim B, Watanabe K, Taniguchi T, Zhu X Y, Hone J, Rubio A, Pasupathy A N, Dean C R 2020 Nature Mater. 19 861Google Scholar

    [54]

    Zhang Z M, Wang Y M, Watanabe K, Taniguchi T, Ueno K, Tutuc E, LeRoy B J 2020 Nature Phys. 16 1093Google Scholar

    [55]

    Lui C H, Li Z, Mak K F, Cappelluti, Heinz T F 2011 Nature Phys. 7 944Google Scholar

    [56]

    Yin L J, Wang W X, Zhang Y, Ou Y Y, Zhang H T, Shen C Y, He L 2017 Phys. Rev. B 95 081402Google Scholar

    [57]

    Kumar A, Nandkishore R 2013 Phys. Rev. B 87 241108Google Scholar

    [58]

    Mazo V, Shimshoni E, Fertig H A 2011 Phys. Rev. B 84 045405Google Scholar

    [59]

    Xu R, Yin L J, Qiao J B, Bai K K, Nie J C, He L 2015 Phys. Rev. B 91 035410Google Scholar

    [60]

    Barlas Y, Cote R, Rondeau M 2012 Phys. Rev. Lett. 109 126804Google Scholar

    [61]

    Lee Y, Tran D, Myhro K, Velasco Jr J, Gillgren N, Poumirol J M, Smirnov D, Barlas Y, Lau C N 2016 Nano. Lett. 16 227Google Scholar

    [62]

    Mak K F, Shan J, Heinz T F 2010 Phys. Rev. Lett. 104 176404Google Scholar

    [63]

    Zhang L, Zhang Y, Camacho J, Khodas M, Zaliznyak I 2011 Nature Phys. 7 953Google Scholar

    [64]

    Lee Y, Che S, Velasco Jr J, Tran D, Baima J, Mauri F, Calandra M, Bockrath M, Lau C N 2019 arXiv: 1911.04450

    [65]

    Zhou H X, Xie T, Taniguchi T, Watanabe K, Young A K 2021 Nature 598 434Google Scholar

    [66]

    Zhou H X, Xie T, Ghazaryan A, Holder T, Ehrets J R, Spanton E M, Taniguchi T, Watanabe K, Berg E, Serbyn M, Young A F 2021 Nature 598 429Google Scholar

    [67]

    Guo H M, Franz M 2009 Phys. Rev. B 80 113102Google Scholar

    [68]

    Li Z, Zhuang J, Wang L, Feng H, Gao Q, Xu X, Hao W, Wang X, Zhang C, Wu K, Dou S X, Chen L, Hu Z, Du Y 2018 Sci. Adv. 4 eaau4511Google Scholar

    [69]

    Tang E, Mei J W, Wen X G 2011 Phys. Rev. Lett. 106 236802Google Scholar

    [70]

    Sun K, Gu Z C, Katsura H, Sarma S. Das 2011 Phys. Rev. Lett. 106 236803Google Scholar

    [71]

    Neupert T, Santos L, Chamon C, Mudry C 2011 Phys. Rev. Lett. 106 236804Google Scholar

    [72]

    Liu Z, Wang Z F, Mei J W, Wu Y S, Liu F 2013 Phys. Rev. Lett. 110 106804Google Scholar

    [73]

    Wang Z F, Liu Z, Liu F 2013 Nat. Commun. 4 1471Google Scholar

    [74]

    Wang Z F, Su N, Liu F 2013 Nano. Lett. 13 2842Google Scholar

    [75]

    Mazin I I, Jeschke H O, Lechermann F, Lee H, Fink M, Thomale R, Valentí R 2014 Nat. Commun. 5 4261Google Scholar

    [76]

    Xu G, Lian B, Zhang S C 2015 Phys. Rev. Lett. 115 186802Google Scholar

    [77]

    Ye L, Kang M, Liu J W, Cube F, Wicker C R, Suzuki T, Jozwiak C, Bostwick A, Rotenberg E, Bell D C, Fu L, Comin R, Checkelsky J G 2018 Nature 555 638Google Scholar

    [78]

    Ye L, Chan M K, McDonald R D, Graf D, Kang M, Liu J W, Suzuki T, Comin R, Fu L, Checkelsky J G 2019 Nat. Commun. 10 4870Google Scholar

    [79]

    Inoue H, Han M, Ye L, Suzuki T, Checkelsky J G 2019 Appl. Phys. Lett. 115 072403Google Scholar

    [80]

    Sales B C, Yan J Q, Meier W R, Christianson A D, Okamoto S, McGuire M A 2019 Phys. Rev. Mater. 3 114203Google Scholar

    [81]

    Liu Z H, Li M, Wang Q, Wang G W, Wen C H P, Jiang K, Lu X L, Yan S C, Huang Y B, Shen D W, Yin J X, Wang Z Q, Yin Z P, Lei H C, Wang S C 2020 Nat. Commun. 11 4002Google Scholar

    [82]

    Yin J X, Shumiya N, Mardanya S, Wang Q, Zhang S S, Tien H J, Multer D, Jiang Y X, Cheng G M, Yao N, Wu S F, Wu D S, Deng L Z, Ye Z P, He R, Chang G Q, Liu Z H, Jiang K, Wang Z Q, Neupert T, Agarwal A, Chang T R, Chu C W, Lei H C, Hasan M Z 2020 Nat. Commun. 11 4003Google Scholar

    [83]

    Kang M, Fang S, Ye L, Po H C, Denlinger J, Jozwiak C, Bostwick A, Rotenberg E, Kaxiras E, Checkelsky J G, Comin R 2020 Nat. Commun. 11 4004Google Scholar

    [84]

    Kang M, Ye L, Fang S, You J S, Levitan A, Han M, Facio J I, Jozwiak C, Bostwick A, Rotenberg E, Chan M K, McDonald R D, Graf D, Kaznatcheev K, Vescovo E, Bell D C, Kaxiras E, van den Brink J, Richter M, Ghimire M P, Checkelsky J G, Comin R 2020 Nature Mater. 19 163Google Scholar

    [85]

    Han M Y, Inoue H, Fang S, John C, Ye L, Chan M K, Graf D, Suzuki T, Ghimire P, Cho W J, Kaxiras E, Checkelsky J G 2021 Nat. Commun. 12 5345Google Scholar

    [86]

    Li M, Wang Q, Wang G W, Yuan Z H, Song W H, Lou R, Liu Z T, Huang Y B, Liu Z H, Lei H C, Yin Z P, Wang S C 2021 Nat. Commun. 12 3129Google Scholar

    [87]

    Liu Z H, Zhao N N, Li M, Yin Q W, Wang Q, Liu Z T, Shen D W, Huang Y B, Lei H C, Liu K, Wang S C 2021 Phys. Rev. B 104 115122Google Scholar

    [88]

    Nomura K, MacDonald A H 2006 Phys. Rev. Lett. 96 256602Google Scholar

    [89]

    Young A F, Dean C R, Wang L, Ren H, Cadden-Zimansky P, Watanabe K, Taniguchi T, Hone J, Shepard K L, Kim P 2012 Nature Phys. 8 550Google Scholar

    [90]

    Li S Y, Zhang Y, Yin L J, He L 2019 Phys. Rev. B 100 085437Google Scholar

    [91]

    Liu X M, Farahi G, Chiu C L, Papic Z, Watanabe K, Taniguchi T, Zaletel M P, Yazdani A 2021 Science 375 321Google Scholar

    [92]

    Young A F, Sanchez-Yamagishi J D, Hunt B, Choi S H, Watanabe K, Taniguchi T, Ashoori R C, Jarillo-Herrero P 2014 Nature 505 528Google Scholar

    [93]

    Veyrat L, Déprez C, Coissard A, Li X X, Gay F, Watanabe K, Taniguchi T, Han Z, Piot B A, Sellier H, Sacépé B 2020 Science 367 781Google Scholar

    [94]

    Aleiner I L, Kharzeev D E, Tsvelik A M 2007 Phys. Rev. B 76 195415Google Scholar

    [95]

    Basko D M, Aleiner I L 2008 Phys. Rev. B 77 041409Google Scholar

    [96]

    Kharitonov M 2012 Phys. Rev. B 85 155439Google Scholar

    [97]

    Tsui D C, Stormer H L, Gossard A C 1982 Phys. Rev. Lett. 48 1559Google Scholar

    [98]

    Du X, Skachko I, Duerr F, Luican A, Andrei E Y 2009 Nature 462 192Google Scholar

    [99]

    Bolotin K I, Ghahari F, Shulman M D, Stormer H L, Kim P 2009 Nature 462 196Google Scholar

    [100]

    Dean C R, Young A F, Cadden-Zimansky P, Wang L, Ren H, Watanabe K, Taniguchi T, Kim P, Hone J, Shepard K L 2011 Nature Phys. 7 693Google Scholar

    [101]

    Zibrov A A, Spanton E M, Zhou H, Kometter C, Taniguchi T, Watanabe K, Young A F 2018 Nature Phys. 14 930Google Scholar

    [102]

    González J 2008 Phys. Rev. B 78 205431Google Scholar

    [103]

    Keimer B, Kivelson S A, Norman M R, Uchida S, Zaanen J 2015 Nature 518 179Google Scholar

    [104]

    Oh M, Nuckolls K P, Wong D, Lee R L, Liu X M, Watanabe K, Taniguchi T, Yazdani A 2021 Nature 600 240Google Scholar

    [105]

    Chen G R, Sharpe A L, Gallagher P, Rosen I T, Fox E J, Jiang L, Lyu B, Li H Y, Watanabe K, Taniguchi T, Jung J, Shi Z W, Goldhaber-Gordon D, Zhang Y B, Wang F 2019 Nature 572 215Google Scholar

    [106]

    Chen G, Jiang L, Wu S, Lyu B, Li H, Chittari B L, Watanabe K, Taniguchi T, Shi Z, Jung J, Zhang Y, Wang F 2019 Nature Phys. 15 237Google Scholar

    [107]

    Chittari B L, Chen G R, Zhang Y B, Wang F, Jung J 2019 Phys. Rev. Lett. 122 016401Google Scholar

    [108]

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

    [109]

    Ando T, Fowler A B, Stern F 1982 Rev. Mod. Phys. 54 437Google Scholar

    [110]

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

    [111]

    Haldane F M 1988 Phys. Rev. Lett. 61 2015Google Scholar

    [112]

    Onoda M, Nagaosa N 2003 Phys. Rev. Lett. 90 206601Google Scholar

    [113]

    Nagaosa N, Sinova J, Onoda S, MacDonald A H, Ong N P 2010 Rev. Mod. Phys. 82 1539Google Scholar

    [114]

    Qi X L, Wu Y, Zhang S C 2006 Phys. Rev. B 74 085308Google Scholar

    [115]

    Liu C, Qi X, Dai X, Fang Z, Zhang S C 2008 Phys. Rev. Lett. 101 146802Google Scholar

    [116]

    Qi X L, Hughes T L, Zhang S C 2008 Phys. Rev. B 78 195424Google Scholar

    [117]

    Nomurak, Nagaosa N, 2011 Phys. Rev. Lett. 106 166802Google Scholar

    [118]

    Yu R, Zhang W, Zhang H J, Zhang S C, Dai X, Fang Z 2010 Science 329 61Google Scholar

    [119]

    Liu C X, Zhang H J, Yan B H, Qi X L, Frauenheim T, Dai X, Fang Z, Zhang S C 2010 Phys. Rev. B 81 041307(RGoogle Scholar

    [120]

    Zhang Y, He K, Chang C Z, Song C L, Wang L L, Chen X, Jia J F, Fang Z, Dai X, Shan W Y, Shen S Q, Niu Q, Qi X L, Zhang S C, Ma X C, Xue Q K 2010 Nature Phys. 6 584Google Scholar

    [121]

    Chang C Z, Zhao W W, Duk Y K, Zhang H J, Badih A A, Don H, Zhang S C, Liu C X, Moses H W C, Jagadeesh S M 2015 Nature Mater. 14 473Google Scholar

    [122]

    Chang C Z, Zhang J S, F X, Shen J, Zhang Z C, Guo M H, Kang Li, Ou Y B, Pang Wei, Wang L L, Ji Z Q, Feng Y, Ji S H, Chen X, Jia J F, Dai X, Fang Z, Zhang S C, He K, Wang Y Y, Lu L, Ma X C, Xue Q K 2013 Science 340 167Google Scholar

    [123]

    Sharpe A L, Fox E J, Barnard A W, Finney J, Watanabe K, Taniguchi T, Kastner M A, Goldhaber-Gordon D 2019 Science 365 605Google Scholar

    [124]

    Watanabe, Taniguchi T, Moon P, Koshino M, Jarillo-Herrero P, Ashoori R C 2013 Science 340 1427Google Scholar

    [125]

    Amet F, Williams J R, Watanabe K, Taniguchi T, Goldhaber-Gordon D, 2013 Phys. Rev. Lett. 110 216601Google Scholar

    [126]

    Serlin M, Tschirhart C L, Polshyn H, Zhang Y, Zhu J, Watanabe K, Taniguchi T, Balents L, Young A F 2019 Science 367 900Google Scholar

    [127]

    Chen G R, Sharpe A L, Fox E J, Zhang Y H, Wang S X, Jiang L L, Bosai Lyu, Li H Y, Watanabe K, Taniguchi T, Shi Z W, Senthil T, Goldhaber-Gordon D, Zhang Y B, Wang F 2020 Nature 579 56Google Scholar

    [128]

    Zhang Y H, Senthil T 2019 Phys. Rev. B 99 205150Google Scholar

    [129]

    Hohenadler M, Assaad F F 2013 J. Phys. Condens. Matter 25 143201Google Scholar

    [130]

    Witczak-Krempa W, Chen G, Kim Y B, Balents L 2014 Annu. Rev. Condens. Matter. Phys. 5 57Google Scholar

    [131]

    Pesin D, Balents L 2010 Nat. Phys. 6 376Google Scholar

    [132]

    Raghu S, Qi X L, Honerkamp C, Zhang S C 2008 Phys. Rev. Lett. 100 156401Google Scholar

    [133]

    Devakul T, Crépell V, Yang Z L Fu 2021 Nat. Commun. 12 6730Google Scholar

    [134]

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

    [135]

    Li T X, Jiang S W, Shen B W, Zhang Y, Li L Z, Tao Z, Devakul T, Watanabe K, Taniguchi T, Fu L, Shan J, Mak K F 2021 Nature 600 641Google Scholar

    [136]

    Li T X, Jiang S W, Li L Z, Zhang Y, Kang K F, Zhu J C, Watanabe K, Taniguchi T, Chowdhury D, Fu L, Shan J, Mak K F 2021 Nature 597 350Google Scholar

    [137]

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

    [138]

    Tenasini G, Martino E, Ubrig N, Ghimire N J, Berger H, Zaharko O, Wu F C, Mitchell J F, Martin I, Forró L, Morpurgo A F 2020 Phys. Rev. Res. 2 023051Google Scholar

    [139]

    Zhao Y F, Zhang R X, Mei R B, Zhou L J, Yi H M, Zhang Y Q, Yu J B, Xiao R, Wang K, Samarth N, Moses H. W. Chan, Liu C X, Chang Z C 2020 Nature 588 419Google Scholar

    [140]

    Zhang F, Jung J, Gregory A F, Niu Q, MacDonald A H 2011 Phys. Rev. Lett. 106 156801Google Scholar

    [141]

    Zhang F 2015 Met. 210 9Google Scholar

    [142]

    Nandkishore R, Levitov L 2010 Phys. Rev. B 82 115431Google Scholar

    [143]

    Geisenhof F R, Winterer F, Seiler A M, Jakob Lenz, Xu T Y, Zhang F, Weitz R T 2021 Nature 598 53Google Scholar

    [144]

    Martin J, Feldman B E, Weitz R T, Allen M T, Yacoby A 2010 Phys. Rev. Lett. 105 256806Google Scholar

    [145]

    Velasco Jr J, Lee Y, Zhang F, Myhro K, Tran D, Deo M, Smirnov D, MacDonald A H, Lau C N 2014 Nat. Commun. 5 4550Google Scholar

    [146]

    Ki D K, Morpurgo A F 2013 Nano Lett. 13 5165Google Scholar

    [147]

    Wigner E 1934 Phys. Rev. 46 1002Google Scholar

    [148]

    Grimes C C, Adams G 1979 Phys. Rev. Lett. 42 795Google Scholar

    [149]

    Andrei E Y, Deville G, Glattli D C, Williams F I B, Paris E, Etienne B 1988 Phys. Rev. Lett. 60 2765Google Scholar

    [150]

    Meng K Ma, Villegas Rosales K A, Deng H, Chung Y J, Pfeiffer L N, West K W, Baldwin K W, Winkler R, Shayegan M 2020 Phys. Rev. Lett. 125 036601Google Scholar

    [151]

    Jang J, Hunt B M, Pfeiffer L N, West K W, Ashoori R C 2017 Nature Phys. 13 340Google Scholar

    [152]

    Jin C H, Tao Z, Li T X, Xu Y, Tang Y H, Zhu J C, Liu S, Watanabe K, Taniguchi T, Hone J C, Fu L, Shan J, Mak K F 2021 Nat. Mater. 20 940Google Scholar

    [153]

    Huang X, Wang T M, Miao S N, Wang C, Li Z P, Lian Z, Taniguchi T, Watanabe K, Okamoto S, Xiao D, Shi S F, Cui Y T 2021 Nat. Phys. 17 715Google Scholar

    [154]

    Kormányos A, Burkard G, Gmitra M, Fabian J, Zólyomi V, Drummond N D, Fal'ko V 2015 2 D Mater. 2 022001Google Scholar

    [155]

    Larentis S, Movva H C P, Fallahazad B, Kim K, Behroozi A, Taniguchi T, Watanabe K, Banerjee S K, Tutuc E 2018 Phys. Rev. B 97 201407(RGoogle Scholar

    [156]

    Tang Y H, Li L Z, Li T X, Xu Y, Liu S, Barmak K, Watanabe K, Taniguchi T, MacDonald A H, Shan J, Mak K F 2020 Nature 579 353Google Scholar

    [157]

    Shimazaki Y, Schwartz I, Watanabe K, Taniguchi T, Kroner M, Imamoğlu A 2020 Nature 580 472Google Scholar

    [158]

    Hubbard J 1978 Phys. Rev. B 17 494Google Scholar

    [159]

    Wu C, Bergman D, Balents L, Das Sarma S 2007 Phys. Rev. Lett. 99 070401Google Scholar

    [160]

    Hiraki K, Kanoda K 1998 Phys. Rev. Lett. 80 4737Google Scholar

    [161]

    Padhi B, Setty C, Phillips P W 2018 Nano Lett. 18 6175Google Scholar

    [162]

    Padhi B, Phillips P W 2019 Phys. Rev. B 99 205141Google Scholar

    [163]

    Regan E C, Wang D Q, Jin C H, Iqbal Bakti Utama M, Gao B N, Wei X, Zhao S H, Zhao W Y, Zhang Z C, Yumigeta K, Blei M, Carlström J D, Watanabe K, Taniguchi T, Tongay S, Crommie M, Zettl A, Wang F 2020 Nature 579 359Google Scholar

    [164]

    Imada M, Fujimori A, Tokura Y 1998 Rev. Mod. Phys. 70 1039Google Scholar

    [165]

    Xu Y, Liu S, Rhodes D A, Watanabe K, Taniguchi T, Hone J, Elser V, Mak K F, Shan J 2020 Nature 587 214Google Scholar

    [166]

    Li H Y, Li S W, Regan E C, Wang D Q, Zhao W Y, Kahn S, Yumigeta K, Blei M, Taniguchi T, Watanabe K, Tongay S, Zettl A, Crommie M F, Wang F 2021 Nature 597 650Google Scholar

    [167]

    Smoleński T, Dolgirev P E, Kuhlenkamp C, Alexander Popert, Shimazaki Yuya, Back P, Lu X B, Kroner M, Watanabe K, Taniguchi T, Esterlis I, Demler E, Imamoğlu A 2021 Nature 595 53Google Scholar

    [168]

    Zhou Y, Sung J H, Brutschea E, Esterlis I, Wang Y, Giovanni Scuri, Gelly R J, Heo H, Taniguchi T, Watanabe K, Zaránd G, Lukin M D, Kim P, Demler E, Park H 2021 Nature 595 48Google Scholar

    [169]

    Świerkowsk L, Neilson D, Szymański J 1991 Phys. Rev. Lett. 67 240Google Scholar

    [170]

    Platzman P M, Fukuyama H 1974 Phys. Rev. B 10 3150Google Scholar

    [171]

    Bonsall L, Maradudin A A 1977 Phys. Rev. B 15 1959Google Scholar

    [172]

    Cao Y, Fatemi V, Demir A, Fang S, Tomarken S L, Luo J Y, Sanchez-Yamagishi J D, Watanabe K, Taniguchi T, Kaxiras E, Ashoori R C, Jarillo-Herrero P 2018 Nature 556 80Google Scholar

    [173]

    Park J M, Cao Y, Watanabe K, Taniguchi T, Jarillo-Herrero P 2021 Nature 590 249Google Scholar

  • 图 1  (a)电子关联示意图[4]; (b) Hubbard近似下的金属-绝缘体相变示意图[5]

    Fig. 1.  (a) Schematic diagram of electronic correlation[4]; (b) schematic diagram of metal-insulator phase transition under Hubbard approximation[5].

    图 2  平带的实现方法 (a) 理论模拟的石墨烯在磁场中的能级(上图)及连续极限下B = 12 T时电子态密度随杂质浓度的变化(下图, 虚线表示朗道能级的位置)[16]; (b) 从0—6 T不同磁场下的朗道能级光谱[17] (曲线偏移以保证清晰度; 隧道设定点, ${V}_{\mathrm{B}}=350\;\mathrm{m}\mathrm{V}, I=400\;\mathrm{p}\mathrm{A}$); (c) ABC堆垛三层石墨烯的晶体结构紧密结合图(左图)和低能电子能带结构图(右图)[56]; (d) 在Kagome晶格中诱导平带的破坏性量子干涉示意图[68]

    Fig. 2.  Method of introducing flat bands: (a) Calculated energy levels of a graphene in a magnetic field (figure above) and the electronic density of states as a function of impurity concentration in the continuum limit for B = 12 T (figure bellow, the dashed line indicates the position of the Landau level)[16]; (b) Landau level spectra for various applied magnetic fields from 0–6 T (The curves are offset for clarity; tunneling set point, ${V}_{\mathrm{B}}=350\;\mathrm{m}\mathrm{V}, I=400\;\mathrm{p}\mathrm{A}$)[17]; (c) tight-binding diagrams (figure left) and the predicted band structure (figure right) of ABC trilayer graphene[56]; (d) schematic diagram of destructive quantum interference inducing a flat band in the Kagome lattice[68].

    图 3  二维材料量子霍尔铁磁态研究进展 (a) 外加磁场分别为9 T (○), 25 T (□), 30 T (◇), 37 T (△), 42 T (▽), 45 T (☆)时单层石墨烯霍尔电导$ {\sigma }_{xy} $随背栅$ {V}_{\mathrm{g}} $的变化关系, 除了9 T的实验温度是30 mK外, 其他数据的实验温度均为1.4 K (左上插图为25 T时单层石墨烯的纵向电阻$ {R}_{xx} $和霍尔电阻$ {R}_{xy} $; 右下插图为在30 mK下B = 9.0 T (○), 11.5 T (五边形), 17.5 T (六边形)时狄拉克点附近的$ {\sigma }_{xy} $数据) [14]; (b) 在$ n=-1 $朗道能级能量处(上图, –141 meV)和n = 0朗道能级未填充的劈裂峰之一能量处(下图, 7.1 meV)探测的3 nm×3 nm STS成像图 (黑色六边形为石墨烯晶格示意图; 插图为依据基态电子态在实空间的分布情况, 石墨烯朗道能级半填充时体系可能的相: 倾斜的反铁磁相和PSP相)[90]; (c) 电荷中性点处的双端电阻随磁场和温度的变化关系(虚线显示了F相的量化螺旋边缘输运的近似极限; 插图为零朗道能级对称破缺态的边缘色散的示意图, 表现为在边缘打开一个能隙)[93]

    Fig. 3.  Progress in the study of quantum Hall ferromagnetic state of two-dimensional materials: (a)$ {\sigma }_{xy} $ as a function of ${V}_{\rm g}$ at different magnetic fields of 9 T (○), 25 T (□), 30 T (◇), 37 T (△), 42 T (▽), 45 T (☆), except the experimental temperature of 9 T is 30 mK, the experimental temperature of other data is 1.4 K (Left upper inset, Rxx and Rxy for the same device measured at B = 25 T; right inset, detailed $ {\sigma }_{xy} $ near the Dirac point for B = 9 T (circle), 11.5 T (pentagon), and 17.5 T (hexagon) at 30 mK)[14]; (b) 3 nm×3 nm STS map taken at –141 meV corresponding to the energy of n = –1 Landau level (figure above) and at the energy of 7.1 meV, which corresponds to the one of the empty peak in the n = 0 Landau level (figure bellow) ( The black hexagon is a schematic diagram of the graphene lattice; the image inset shows the two possible phases of the system with the graphene Landau level half-filling which are based on the distribution of the ground-state electronic states in the real space: canted antiferromagnetic phase and PSP phase)[90]; (c) two-terminal resistance at the charge neutral point versus magnetic field and temperature for a different contact configuration (The dashed line shows the approximate limit of quantified helical edge transport for the F-phase; the image inset shows the schematic of the edge dispersion of the zeroth Landau level broken symmetry states showing the opening of a gap at the edge)[93].

    表 1  平带的实现方法及相关新奇量子物态

    Table 1.  Implementation methods of flat bands and their corresponding resulting novel quantum states.

    平带实现方法新奇量子物态
    量子霍尔
    铁磁态
    分数量子
    霍尔态
    关联
    绝缘态
    超导态量子反常霍尔效应Wigner
    晶体
    外加强磁场[14][100, 101][149, 150]
    构筑应变结构
    引入
    转角
    魔角双层石墨烯[172][41][123, 126]
    魔角三层石墨烯[44][44, 173]
    双层-双层转角石墨烯[45]
    转角过渡金属硫化物[53][163, 165]
    ABC堆叠三层石墨烯[61][105][105][127]
    Kagome结构
    下载: 导出CSV
  • [1]

    黄昆 著 (韩汝琦改编) 1988 固体物理学 (北京: 高等教育出版社) 第154—157页

    Huang K (adapted by Han R Q) 1988 Solid State Physics (Beijing: Higher Education Press) pp154–157 (in Chinese)

    [2]

    de Boer J H, Verwey E J 1937 Proc. Phys. Soc. 49 59Google Scholar

    [3]

    Mott N F, Peierls R 1937 Proc. Phys. Soc. 49 72Google Scholar

    [4]

    Mott N F 1949 Proc. Phys. Soc. A 62 416Google Scholar

    [5]

    Hubbard J 1963 Sciences 276 238Google Scholar

    [6]

    Wu M K, Ashburn J R, Torng C J, Hor P H, Meng R L, Gao L, Huang Z J, Wang Y Q, Chu C W 1987 Phys. Rev. Lett. 58 908Google Scholar

    [7]

    Zhang J B, Struzhkin V V, Yang W G, Mao H K, Lin H Q, Ma Y C, Wang N L, Chen X J 2015 Phys. Condens. Matter. 27 445701Google Scholar

    [8]

    Narayan J 1989 JOM 41 18Google Scholar

    [9]

    Tokura Y, Takagi H, Uchida S 1989 Nature 337 345Google Scholar

    [10]

    Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 666Google Scholar

    [11]

    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

    [12]

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

    [13]

    Katsnelson M I, Novoselov K S, Geim A K 2006 Nature Phys. 2 620Google Scholar

    [14]

    Zhang Y, Jiang Z, Small J P, Purewal M S, Tan Y W, Fazlollahi M, Chudow J D, Jaszczak J A, Stormer H L, Kim P 2006 Phys. Rev. Lett. 96 136806Google Scholar

    [15]

    Peres N M R, Guinea F, Castro Neto A H 2006 Ann. Phys. 321 1559Google Scholar

    [16]

    Peres N M R, Guinea F, Castro Neto A H 2006 Phys. Rev. B 73 125411Google Scholar

    [17]

    Miller D L, Kubista K D, Rutter G M, Ruan M, de Heer W A, First P N, Stroscio J A 2009 Science 324 924Google Scholar

    [18]

    Meyer J C, Geim A K, Katsnelson M I, Novoselov K S, Booth T J, Roth S 2007 Nature 446 60Google Scholar

    [19]

    Morpurgo A, Guinea F 2006 Phys. Rev. Lett. 97 196804Google Scholar

    [20]

    Guinea F, Katsnelson M, Vozmediano M A H 2008 Phys. Rev. B 77 075422Google Scholar

    [21]

    Guinea F, Katsnelson M, Geim A K 2009 Nature Phys. 6 30Google Scholar

    [22]

    Levy N, Burke S A, Meaker K L, Panlasigui M, Zettl A, Guinea F, Castro Neto A H, Crommie M F 2010 Science 329 544Google Scholar

    [23]

    Yeh N C, Teague M L, Yeom S, Standley B L, Wu R T P, Boyd D A, Bockrath M W 2011 Surf. Sci. 605 1649Google Scholar

    [24]

    Lu J, Neto A H, Loh K P 2012 Nat. Commun. 3 1Google Scholar

    [25]

    Yan H, Sun Y, He L, Nie J C, Chan M H 2012 Phys. Rev. B 85 035422Google Scholar

    [26]

    Guo D, Kondo T, Machida T, Iwatake K, Okada S, Nakamura J 2012 Nat. Commun. 3 1068Google Scholar

    [27]

    Meng L, He W Y, Zheng H, Liu M, Yan H, Yan W, Chu Z D, Bai K, Dou R F, Zhang Y 2013 Phys. Rev. B 87 205405Google Scholar

    [28]

    Yan W, He W Y, Chu Z D, Liu M, Meng L, Dou R F, Zhang Y, Liu Z, Nie J C, He L 2013 Nat. Commun. 4 2159Google Scholar

    [29]

    Li S Y, Bai K K, Yin L J, Qiao J B, Wang W X, He L 2015 Phys. Rev. B 92 245302Google Scholar

    [30]

    Jiang Y H, Mao J H, Duan J X, Lai X Y, Watanabe K, Taniguchi T, Andrei E Y 2017 Nano. Lett. 17 2839Google Scholar

    [31]

    Liu Y, Rodrigues J N B, Luo Y Z, Li L, Carvalho A, Yang M, Laksono E, Lu J, Bao Y, Xu H, Tan S J R, Qiu Z, Sow C H, Feng Y P, Neto A H C, Adam S, Lu J, Loh K P 2018 Nature Nanotech. 13 828Google Scholar

    [32]

    Jia P, Chen W, Qiao J, Zhang M, Zheng X, Xue Z, Liang R, Tian C, He L, Di Z, Wang X 2019 Nat. Commun. 10 3127Google Scholar

    [33]

    Nigge P, Qu A C, Lantagne-Hurtubise É, Mårsell E, Link S, Tom G, Zonno M, Michiardi M, Schneider M, Zhdanovich S, Levy G, Starke U, Gutiérrez C, Bonn D, Burke S A, Franz M, Damascelli A 2019 Sci. Adv. 5 eaaw5593Google Scholar

    [34]

    Li S Y, Su Y, Ren Y N, He L 2020 Phys. Rev. Lett. 124 106802Google Scholar

    [35]

    Hsu C C, Teague M L, Wang J Q, Yeh N C 2020 Sci. Adv. 6 eaat9488Google Scholar

    [36]

    Mao J H, Milovanović S P, Anđelković M, Lai X Y, Cao Y, Watanabe K, Taniguchi T, Covaci L, Peeters F M, Geim A K, Jiang Y H, Andrei E Y 2020 Nature 584 215Google Scholar

    [37]

    Vozmediano M A H, Katsnelson M I, Guinea F 2010 Phys. Rep. 496 109Google Scholar

    [38]

    Li G H, Luican A, Lopes dos Santos J M B, Castro Neto A H, Reina A, Kong J, Andrei E Y 2010 Nature Phys. 6 109Google Scholar

    [39]

    Bistritzer R, MacDonald A H 2011 Proc. Nat. I Acad. Sci. U. S. A 108 12233Google Scholar

    [40]

    Yin L J, Qiao J B, Zuo W J, Li W T, He L 2015 Phys. Rev. B 92 081406Google Scholar

    [41]

    Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E, Jarillo-Herrero P 2018 Nature 556 43Google Scholar

    [42]

    Khalaf E, Kruchkov A J, Tarnopolsky G, Vishwanath A 2019 Phys. Rev. B 100 085109Google Scholar

    [43]

    Carr S, Li C Y, Zhu Z Y, Kaxiras E, Sachdev S, Kruchkov A J 2020 Nano. Lett. 20 3030Google Scholar

    [44]

    Hao Z Y, Zimmerman A M, Ledwith P, Khalaf E, Najafabadi D H, Watanabe K, Taniguchi T, Vishwanath A, Kim P 2021 Science 371 1133Google Scholar

    [45]

    Liu X M, Hao Z Y, Khalaf E, Lee J Y, Ronen Y, Yoo H, Najafabadi D H, Watanabe K, Taniguchi T, Vishwanath A, Kim P 2020 Nature 583 221Google Scholar

    [46]

    Burg G W, Zhu J H, Taniguchi T, Watanabe K, MacDonald A H, Tutuc E 2019 Phys. Rev. Lett. 123 197702Google Scholar

    [47]

    Cao Y, Rodan-Legrain D, Rubies-Bigorda O, Park J M, Watanabe K, Taniguchi T, Jarillo-Herrero P 2020 Nature 583 215Google Scholar

    [48]

    Shen C, Chu Y B, Wu Q S, Li N, Wang S P, Zhao Y C, Tang J, Liu J Y, Tian J P, Watanabe K, Taniguchi T, Yang R, Meng Z Y, Shi D X, Yazyev O V, Zhang G Y 2020 Nature Phys. 16 520Google Scholar

    [49]

    Wu F, Lovorn T, Tutuc E, Macdonald A H 2018 Phys. Rev. Lett. 121 026402Google Scholar

    [50]

    Naik M H, Jain M 2018 Phys. Rev. Lett. 121 266401Google Scholar

    [51]

    Wu F, Lovorn T, Tutuc E, Martin I, Macdonald A H 2019 Phys. Rev. Lett. 122 86402Google Scholar

    [52]

    Ruiz-Tijerina D A, Fal’Ko V I 2019 Phys. Rev. B 99 125424Google Scholar

    [53]

    Wang L, Shih E M, Ghiotto A, Xian L, Rhodes D A, Tan C, Claassen M, Kennes D M, Bai Y S, Kim B, Watanabe K, Taniguchi T, Zhu X Y, Hone J, Rubio A, Pasupathy A N, Dean C R 2020 Nature Mater. 19 861Google Scholar

    [54]

    Zhang Z M, Wang Y M, Watanabe K, Taniguchi T, Ueno K, Tutuc E, LeRoy B J 2020 Nature Phys. 16 1093Google Scholar

    [55]

    Lui C H, Li Z, Mak K F, Cappelluti, Heinz T F 2011 Nature Phys. 7 944Google Scholar

    [56]

    Yin L J, Wang W X, Zhang Y, Ou Y Y, Zhang H T, Shen C Y, He L 2017 Phys. Rev. B 95 081402Google Scholar

    [57]

    Kumar A, Nandkishore R 2013 Phys. Rev. B 87 241108Google Scholar

    [58]

    Mazo V, Shimshoni E, Fertig H A 2011 Phys. Rev. B 84 045405Google Scholar

    [59]

    Xu R, Yin L J, Qiao J B, Bai K K, Nie J C, He L 2015 Phys. Rev. B 91 035410Google Scholar

    [60]

    Barlas Y, Cote R, Rondeau M 2012 Phys. Rev. Lett. 109 126804Google Scholar

    [61]

    Lee Y, Tran D, Myhro K, Velasco Jr J, Gillgren N, Poumirol J M, Smirnov D, Barlas Y, Lau C N 2016 Nano. Lett. 16 227Google Scholar

    [62]

    Mak K F, Shan J, Heinz T F 2010 Phys. Rev. Lett. 104 176404Google Scholar

    [63]

    Zhang L, Zhang Y, Camacho J, Khodas M, Zaliznyak I 2011 Nature Phys. 7 953Google Scholar

    [64]

    Lee Y, Che S, Velasco Jr J, Tran D, Baima J, Mauri F, Calandra M, Bockrath M, Lau C N 2019 arXiv: 1911.04450

    [65]

    Zhou H X, Xie T, Taniguchi T, Watanabe K, Young A K 2021 Nature 598 434Google Scholar

    [66]

    Zhou H X, Xie T, Ghazaryan A, Holder T, Ehrets J R, Spanton E M, Taniguchi T, Watanabe K, Berg E, Serbyn M, Young A F 2021 Nature 598 429Google Scholar

    [67]

    Guo H M, Franz M 2009 Phys. Rev. B 80 113102Google Scholar

    [68]

    Li Z, Zhuang J, Wang L, Feng H, Gao Q, Xu X, Hao W, Wang X, Zhang C, Wu K, Dou S X, Chen L, Hu Z, Du Y 2018 Sci. Adv. 4 eaau4511Google Scholar

    [69]

    Tang E, Mei J W, Wen X G 2011 Phys. Rev. Lett. 106 236802Google Scholar

    [70]

    Sun K, Gu Z C, Katsura H, Sarma S. Das 2011 Phys. Rev. Lett. 106 236803Google Scholar

    [71]

    Neupert T, Santos L, Chamon C, Mudry C 2011 Phys. Rev. Lett. 106 236804Google Scholar

    [72]

    Liu Z, Wang Z F, Mei J W, Wu Y S, Liu F 2013 Phys. Rev. Lett. 110 106804Google Scholar

    [73]

    Wang Z F, Liu Z, Liu F 2013 Nat. Commun. 4 1471Google Scholar

    [74]

    Wang Z F, Su N, Liu F 2013 Nano. Lett. 13 2842Google Scholar

    [75]

    Mazin I I, Jeschke H O, Lechermann F, Lee H, Fink M, Thomale R, Valentí R 2014 Nat. Commun. 5 4261Google Scholar

    [76]

    Xu G, Lian B, Zhang S C 2015 Phys. Rev. Lett. 115 186802Google Scholar

    [77]

    Ye L, Kang M, Liu J W, Cube F, Wicker C R, Suzuki T, Jozwiak C, Bostwick A, Rotenberg E, Bell D C, Fu L, Comin R, Checkelsky J G 2018 Nature 555 638Google Scholar

    [78]

    Ye L, Chan M K, McDonald R D, Graf D, Kang M, Liu J W, Suzuki T, Comin R, Fu L, Checkelsky J G 2019 Nat. Commun. 10 4870Google Scholar

    [79]

    Inoue H, Han M, Ye L, Suzuki T, Checkelsky J G 2019 Appl. Phys. Lett. 115 072403Google Scholar

    [80]

    Sales B C, Yan J Q, Meier W R, Christianson A D, Okamoto S, McGuire M A 2019 Phys. Rev. Mater. 3 114203Google Scholar

    [81]

    Liu Z H, Li M, Wang Q, Wang G W, Wen C H P, Jiang K, Lu X L, Yan S C, Huang Y B, Shen D W, Yin J X, Wang Z Q, Yin Z P, Lei H C, Wang S C 2020 Nat. Commun. 11 4002Google Scholar

    [82]

    Yin J X, Shumiya N, Mardanya S, Wang Q, Zhang S S, Tien H J, Multer D, Jiang Y X, Cheng G M, Yao N, Wu S F, Wu D S, Deng L Z, Ye Z P, He R, Chang G Q, Liu Z H, Jiang K, Wang Z Q, Neupert T, Agarwal A, Chang T R, Chu C W, Lei H C, Hasan M Z 2020 Nat. Commun. 11 4003Google Scholar

    [83]

    Kang M, Fang S, Ye L, Po H C, Denlinger J, Jozwiak C, Bostwick A, Rotenberg E, Kaxiras E, Checkelsky J G, Comin R 2020 Nat. Commun. 11 4004Google Scholar

    [84]

    Kang M, Ye L, Fang S, You J S, Levitan A, Han M, Facio J I, Jozwiak C, Bostwick A, Rotenberg E, Chan M K, McDonald R D, Graf D, Kaznatcheev K, Vescovo E, Bell D C, Kaxiras E, van den Brink J, Richter M, Ghimire M P, Checkelsky J G, Comin R 2020 Nature Mater. 19 163Google Scholar

    [85]

    Han M Y, Inoue H, Fang S, John C, Ye L, Chan M K, Graf D, Suzuki T, Ghimire P, Cho W J, Kaxiras E, Checkelsky J G 2021 Nat. Commun. 12 5345Google Scholar

    [86]

    Li M, Wang Q, Wang G W, Yuan Z H, Song W H, Lou R, Liu Z T, Huang Y B, Liu Z H, Lei H C, Yin Z P, Wang S C 2021 Nat. Commun. 12 3129Google Scholar

    [87]

    Liu Z H, Zhao N N, Li M, Yin Q W, Wang Q, Liu Z T, Shen D W, Huang Y B, Lei H C, Liu K, Wang S C 2021 Phys. Rev. B 104 115122Google Scholar

    [88]

    Nomura K, MacDonald A H 2006 Phys. Rev. Lett. 96 256602Google Scholar

    [89]

    Young A F, Dean C R, Wang L, Ren H, Cadden-Zimansky P, Watanabe K, Taniguchi T, Hone J, Shepard K L, Kim P 2012 Nature Phys. 8 550Google Scholar

    [90]

    Li S Y, Zhang Y, Yin L J, He L 2019 Phys. Rev. B 100 085437Google Scholar

    [91]

    Liu X M, Farahi G, Chiu C L, Papic Z, Watanabe K, Taniguchi T, Zaletel M P, Yazdani A 2021 Science 375 321Google Scholar

    [92]

    Young A F, Sanchez-Yamagishi J D, Hunt B, Choi S H, Watanabe K, Taniguchi T, Ashoori R C, Jarillo-Herrero P 2014 Nature 505 528Google Scholar

    [93]

    Veyrat L, Déprez C, Coissard A, Li X X, Gay F, Watanabe K, Taniguchi T, Han Z, Piot B A, Sellier H, Sacépé B 2020 Science 367 781Google Scholar

    [94]

    Aleiner I L, Kharzeev D E, Tsvelik A M 2007 Phys. Rev. B 76 195415Google Scholar

    [95]

    Basko D M, Aleiner I L 2008 Phys. Rev. B 77 041409Google Scholar

    [96]

    Kharitonov M 2012 Phys. Rev. B 85 155439Google Scholar

    [97]

    Tsui D C, Stormer H L, Gossard A C 1982 Phys. Rev. Lett. 48 1559Google Scholar

    [98]

    Du X, Skachko I, Duerr F, Luican A, Andrei E Y 2009 Nature 462 192Google Scholar

    [99]

    Bolotin K I, Ghahari F, Shulman M D, Stormer H L, Kim P 2009 Nature 462 196Google Scholar

    [100]

    Dean C R, Young A F, Cadden-Zimansky P, Wang L, Ren H, Watanabe K, Taniguchi T, Kim P, Hone J, Shepard K L 2011 Nature Phys. 7 693Google Scholar

    [101]

    Zibrov A A, Spanton E M, Zhou H, Kometter C, Taniguchi T, Watanabe K, Young A F 2018 Nature Phys. 14 930Google Scholar

    [102]

    González J 2008 Phys. Rev. B 78 205431Google Scholar

    [103]

    Keimer B, Kivelson S A, Norman M R, Uchida S, Zaanen J 2015 Nature 518 179Google Scholar

    [104]

    Oh M, Nuckolls K P, Wong D, Lee R L, Liu X M, Watanabe K, Taniguchi T, Yazdani A 2021 Nature 600 240Google Scholar

    [105]

    Chen G R, Sharpe A L, Gallagher P, Rosen I T, Fox E J, Jiang L, Lyu B, Li H Y, Watanabe K, Taniguchi T, Jung J, Shi Z W, Goldhaber-Gordon D, Zhang Y B, Wang F 2019 Nature 572 215Google Scholar

    [106]

    Chen G, Jiang L, Wu S, Lyu B, Li H, Chittari B L, Watanabe K, Taniguchi T, Shi Z, Jung J, Zhang Y, Wang F 2019 Nature Phys. 15 237Google Scholar

    [107]

    Chittari B L, Chen G R, Zhang Y B, Wang F, Jung J 2019 Phys. Rev. Lett. 122 016401Google Scholar

    [108]

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

    [109]

    Ando T, Fowler A B, Stern F 1982 Rev. Mod. Phys. 54 437Google Scholar

    [110]

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

    [111]

    Haldane F M 1988 Phys. Rev. Lett. 61 2015Google Scholar

    [112]

    Onoda M, Nagaosa N 2003 Phys. Rev. Lett. 90 206601Google Scholar

    [113]

    Nagaosa N, Sinova J, Onoda S, MacDonald A H, Ong N P 2010 Rev. Mod. Phys. 82 1539Google Scholar

    [114]

    Qi X L, Wu Y, Zhang S C 2006 Phys. Rev. B 74 085308Google Scholar

    [115]

    Liu C, Qi X, Dai X, Fang Z, Zhang S C 2008 Phys. Rev. Lett. 101 146802Google Scholar

    [116]

    Qi X L, Hughes T L, Zhang S C 2008 Phys. Rev. B 78 195424Google Scholar

    [117]

    Nomurak, Nagaosa N, 2011 Phys. Rev. Lett. 106 166802Google Scholar

    [118]

    Yu R, Zhang W, Zhang H J, Zhang S C, Dai X, Fang Z 2010 Science 329 61Google Scholar

    [119]

    Liu C X, Zhang H J, Yan B H, Qi X L, Frauenheim T, Dai X, Fang Z, Zhang S C 2010 Phys. Rev. B 81 041307(RGoogle Scholar

    [120]

    Zhang Y, He K, Chang C Z, Song C L, Wang L L, Chen X, Jia J F, Fang Z, Dai X, Shan W Y, Shen S Q, Niu Q, Qi X L, Zhang S C, Ma X C, Xue Q K 2010 Nature Phys. 6 584Google Scholar

    [121]

    Chang C Z, Zhao W W, Duk Y K, Zhang H J, Badih A A, Don H, Zhang S C, Liu C X, Moses H W C, Jagadeesh S M 2015 Nature Mater. 14 473Google Scholar

    [122]

    Chang C Z, Zhang J S, F X, Shen J, Zhang Z C, Guo M H, Kang Li, Ou Y B, Pang Wei, Wang L L, Ji Z Q, Feng Y, Ji S H, Chen X, Jia J F, Dai X, Fang Z, Zhang S C, He K, Wang Y Y, Lu L, Ma X C, Xue Q K 2013 Science 340 167Google Scholar

    [123]

    Sharpe A L, Fox E J, Barnard A W, Finney J, Watanabe K, Taniguchi T, Kastner M A, Goldhaber-Gordon D 2019 Science 365 605Google Scholar

    [124]

    Watanabe, Taniguchi T, Moon P, Koshino M, Jarillo-Herrero P, Ashoori R C 2013 Science 340 1427Google Scholar

    [125]

    Amet F, Williams J R, Watanabe K, Taniguchi T, Goldhaber-Gordon D, 2013 Phys. Rev. Lett. 110 216601Google Scholar

    [126]

    Serlin M, Tschirhart C L, Polshyn H, Zhang Y, Zhu J, Watanabe K, Taniguchi T, Balents L, Young A F 2019 Science 367 900Google Scholar

    [127]

    Chen G R, Sharpe A L, Fox E J, Zhang Y H, Wang S X, Jiang L L, Bosai Lyu, Li H Y, Watanabe K, Taniguchi T, Shi Z W, Senthil T, Goldhaber-Gordon D, Zhang Y B, Wang F 2020 Nature 579 56Google Scholar

    [128]

    Zhang Y H, Senthil T 2019 Phys. Rev. B 99 205150Google Scholar

    [129]

    Hohenadler M, Assaad F F 2013 J. Phys. Condens. Matter 25 143201Google Scholar

    [130]

    Witczak-Krempa W, Chen G, Kim Y B, Balents L 2014 Annu. Rev. Condens. Matter. Phys. 5 57Google Scholar

    [131]

    Pesin D, Balents L 2010 Nat. Phys. 6 376Google Scholar

    [132]

    Raghu S, Qi X L, Honerkamp C, Zhang S C 2008 Phys. Rev. Lett. 100 156401Google Scholar

    [133]

    Devakul T, Crépell V, Yang Z L Fu 2021 Nat. Commun. 12 6730Google Scholar

    [134]

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

    [135]

    Li T X, Jiang S W, Shen B W, Zhang Y, Li L Z, Tao Z, Devakul T, Watanabe K, Taniguchi T, Fu L, Shan J, Mak K F 2021 Nature 600 641Google Scholar

    [136]

    Li T X, Jiang S W, Li L Z, Zhang Y, Kang K F, Zhu J C, Watanabe K, Taniguchi T, Chowdhury D, Fu L, Shan J, Mak K F 2021 Nature 597 350Google Scholar

    [137]

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

    [138]

    Tenasini G, Martino E, Ubrig N, Ghimire N J, Berger H, Zaharko O, Wu F C, Mitchell J F, Martin I, Forró L, Morpurgo A F 2020 Phys. Rev. Res. 2 023051Google Scholar

    [139]

    Zhao Y F, Zhang R X, Mei R B, Zhou L J, Yi H M, Zhang Y Q, Yu J B, Xiao R, Wang K, Samarth N, Moses H. W. Chan, Liu C X, Chang Z C 2020 Nature 588 419Google Scholar

    [140]

    Zhang F, Jung J, Gregory A F, Niu Q, MacDonald A H 2011 Phys. Rev. Lett. 106 156801Google Scholar

    [141]

    Zhang F 2015 Met. 210 9Google Scholar

    [142]

    Nandkishore R, Levitov L 2010 Phys. Rev. B 82 115431Google Scholar

    [143]

    Geisenhof F R, Winterer F, Seiler A M, Jakob Lenz, Xu T Y, Zhang F, Weitz R T 2021 Nature 598 53Google Scholar

    [144]

    Martin J, Feldman B E, Weitz R T, Allen M T, Yacoby A 2010 Phys. Rev. Lett. 105 256806Google Scholar

    [145]

    Velasco Jr J, Lee Y, Zhang F, Myhro K, Tran D, Deo M, Smirnov D, MacDonald A H, Lau C N 2014 Nat. Commun. 5 4550Google Scholar

    [146]

    Ki D K, Morpurgo A F 2013 Nano Lett. 13 5165Google Scholar

    [147]

    Wigner E 1934 Phys. Rev. 46 1002Google Scholar

    [148]

    Grimes C C, Adams G 1979 Phys. Rev. Lett. 42 795Google Scholar

    [149]

    Andrei E Y, Deville G, Glattli D C, Williams F I B, Paris E, Etienne B 1988 Phys. Rev. Lett. 60 2765Google Scholar

    [150]

    Meng K Ma, Villegas Rosales K A, Deng H, Chung Y J, Pfeiffer L N, West K W, Baldwin K W, Winkler R, Shayegan M 2020 Phys. Rev. Lett. 125 036601Google Scholar

    [151]

    Jang J, Hunt B M, Pfeiffer L N, West K W, Ashoori R C 2017 Nature Phys. 13 340Google Scholar

    [152]

    Jin C H, Tao Z, Li T X, Xu Y, Tang Y H, Zhu J C, Liu S, Watanabe K, Taniguchi T, Hone J C, Fu L, Shan J, Mak K F 2021 Nat. Mater. 20 940Google Scholar

    [153]

    Huang X, Wang T M, Miao S N, Wang C, Li Z P, Lian Z, Taniguchi T, Watanabe K, Okamoto S, Xiao D, Shi S F, Cui Y T 2021 Nat. Phys. 17 715Google Scholar

    [154]

    Kormányos A, Burkard G, Gmitra M, Fabian J, Zólyomi V, Drummond N D, Fal'ko V 2015 2 D Mater. 2 022001Google Scholar

    [155]

    Larentis S, Movva H C P, Fallahazad B, Kim K, Behroozi A, Taniguchi T, Watanabe K, Banerjee S K, Tutuc E 2018 Phys. Rev. B 97 201407(RGoogle Scholar

    [156]

    Tang Y H, Li L Z, Li T X, Xu Y, Liu S, Barmak K, Watanabe K, Taniguchi T, MacDonald A H, Shan J, Mak K F 2020 Nature 579 353Google Scholar

    [157]

    Shimazaki Y, Schwartz I, Watanabe K, Taniguchi T, Kroner M, Imamoğlu A 2020 Nature 580 472Google Scholar

    [158]

    Hubbard J 1978 Phys. Rev. B 17 494Google Scholar

    [159]

    Wu C, Bergman D, Balents L, Das Sarma S 2007 Phys. Rev. Lett. 99 070401Google Scholar

    [160]

    Hiraki K, Kanoda K 1998 Phys. Rev. Lett. 80 4737Google Scholar

    [161]

    Padhi B, Setty C, Phillips P W 2018 Nano Lett. 18 6175Google Scholar

    [162]

    Padhi B, Phillips P W 2019 Phys. Rev. B 99 205141Google Scholar

    [163]

    Regan E C, Wang D Q, Jin C H, Iqbal Bakti Utama M, Gao B N, Wei X, Zhao S H, Zhao W Y, Zhang Z C, Yumigeta K, Blei M, Carlström J D, Watanabe K, Taniguchi T, Tongay S, Crommie M, Zettl A, Wang F 2020 Nature 579 359Google Scholar

    [164]

    Imada M, Fujimori A, Tokura Y 1998 Rev. Mod. Phys. 70 1039Google Scholar

    [165]

    Xu Y, Liu S, Rhodes D A, Watanabe K, Taniguchi T, Hone J, Elser V, Mak K F, Shan J 2020 Nature 587 214Google Scholar

    [166]

    Li H Y, Li S W, Regan E C, Wang D Q, Zhao W Y, Kahn S, Yumigeta K, Blei M, Taniguchi T, Watanabe K, Tongay S, Zettl A, Crommie M F, Wang F 2021 Nature 597 650Google Scholar

    [167]

    Smoleński T, Dolgirev P E, Kuhlenkamp C, Alexander Popert, Shimazaki Yuya, Back P, Lu X B, Kroner M, Watanabe K, Taniguchi T, Esterlis I, Demler E, Imamoğlu A 2021 Nature 595 53Google Scholar

    [168]

    Zhou Y, Sung J H, Brutschea E, Esterlis I, Wang Y, Giovanni Scuri, Gelly R J, Heo H, Taniguchi T, Watanabe K, Zaránd G, Lukin M D, Kim P, Demler E, Park H 2021 Nature 595 48Google Scholar

    [169]

    Świerkowsk L, Neilson D, Szymański J 1991 Phys. Rev. Lett. 67 240Google Scholar

    [170]

    Platzman P M, Fukuyama H 1974 Phys. Rev. B 10 3150Google Scholar

    [171]

    Bonsall L, Maradudin A A 1977 Phys. Rev. B 15 1959Google Scholar

    [172]

    Cao Y, Fatemi V, Demir A, Fang S, Tomarken S L, Luo J Y, Sanchez-Yamagishi J D, Watanabe K, Taniguchi T, Kaxiras E, Ashoori R C, Jarillo-Herrero P 2018 Nature 556 80Google Scholar

    [173]

    Park J M, Cao Y, Watanabe K, Taniguchi T, Jarillo-Herrero P 2021 Nature 590 249Google Scholar

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  • 收稿日期:  2022-02-01
  • 修回日期:  2022-02-28
  • 上网日期:  2022-06-06
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