Search

Article

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

Topological properties and orbital magnetism in twisted graphene systems

Liu Jian-Peng Dai Xi

Citation:

Topological properties and orbital magnetism in twisted graphene systems

Liu Jian-Peng, Dai Xi
PDF
HTML
Get Citation
  • We review and discuss the electronic structures, topological properties and orbital magnetism in twisted bilayer (TBG) and multilayer graphene systems. Moiré pattern is formed in twisted bilayer graphene due to the mutual twist of the two graphene layers. The moiré potential induced by the twist can generate opposite pseudo magnetic fields in the Moiré supercell, which are coupled with the Dirac fermions and generate two sets of pseudo Landau levels with opposite Chern numbers $\pm1$. The two flat bands for each valley each spin of TBG are equivalent to the two zeroth pseudo Landau levels with opposite Chern numbers and opposite sublattice polarizations. Such a pseudo-Landau-level representation has significant implications on the quantum anomalous Hall states observed at integer fillings of the flat bands in TBG at the magic angle. The origin of the magic angle can also be naturally explained by using the pseudo-Landau-level picture. We further discuss twisted multilayer graphene systems, and show that topological flat bands generally exist in the twisted multilayer graphene systems. These topological flat bands have nonzero valley Chern numbers, which can be described by a succinct formula under certain approxmations. These topological flat bands in twisted bilayer and multilayer graphene systems are associated with orbital magnetism. A valley polarized state in the twist graphene system is an orbital magnetic state with nontrivial current-loop pattern in the moiré supercell. The experimentally observed correlated insulating states at $\pm 1/2$ fillings and at charge neutrality point of magic-angle TBG can be valley polarized states, which are associated with compensating current loops and induce staggered orbital magnetizations on the moiré length scale. If $C_{2z}$ symmetry is broken due to the alignment of hexagonal boron nitride substrate, then a valley-polarized ground state would be a moiré orbital ferromagnetic state, which exhibits not only (quantum) anomalous Hall effect, but also novel magneto-optical and nonlinear optical responses.
      Corresponding author: Liu Jian-Peng, liujp@shanghaitech.edu.cn ; Dai Xi, daix@ust.hk
    • Funds: Project supported by the Hong Kong Research Grants Council Project, China (Grant No. GRF16300918
    [1]

    Lopes dos Santos J M B, Peres N M R, Castro Neto A H 2007 Phys. Rev. Lett. 99 256802Google Scholar

    [2]

    Mele E J 2010 Phys. Rev. B 81 161405Google Scholar

    [3]

    Trambly de Laissardiere G, Mayou D, Magaud L 2010 Nano Lett. 10 804Google Scholar

    [4]

    Bistritzer R, MacDonald A H 2011 Proc. Natl. Acad. Sci. 108 12233Google Scholar

    [5]

    Lopes dos Santos J M B, Peres N M R, Castro Neto A H 2012 Phys. Rev. B 86 155449Google Scholar

    [6]

    San-Jose P, González J, Guinea F 2012 Phys. Rev. Lett. 108 216802Google Scholar

    [7]

    Cao Y, Fatemi V, Demir A, et al. 2018 Nature 556 80Google Scholar

    [8]

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

    [9]

    Yankowitz M, Chen S, Polshyn H, Zhang Y, Watanabe K, Taniguchi T, Graf D, Young A F, Dean C R 2019 Science 363 1059Google Scholar

    [10]

    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

    [11]

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

    [12]

    Stepanov P, Das I, Lu X, Fahimniya A, Watanabe K, Taniguchi T, Koppens F H, Lischner J, Levitov L, Efetov D K 2019 arXiv preprint arXiv: 1911.09198

    [13]

    Po H C, Zou L, Vishwanath A, Senthil T 2018 Phys. Rev. X 8 031089

    [14]

    Koshino M, Yuan N F Q, Koretsune T, Ochi M, Kuroki K, Fu L 2018 Phys. Rev. X 8 031087

    [15]

    Kang J, Vafek O 2018 Phys. Rev. X 8 031088

    [16]

    Isobe H, Yuan N F Q, Fu L 2018 Phys. Rev. X 8 041041

    [17]

    Xu X Y, Law K T, Lee P A 2018 Phys. Rev. B 98 121406Google Scholar

    [18]

    Huang T, Zhang L, Ma T 2019 Sci. Bull. 64 310Google Scholar

    [19]

    Liu C C, Zhang L D, Chen W Q, Yang F 2018 Phys. Rev. Lett. 121 217001Google Scholar

    [20]

    Rademaker L, Mellado P 2018 Phys. Rev. B 98 235158Google Scholar

    [21]

    Venderbos J W F, Fernandes R M 2018 Phys. Rev. B 98 245103Google Scholar

    [22]

    Kang J, Vafek O 2019 Phys. Rev. Lett. 122 246401Google Scholar

    [23]

    Xie M, MacDonald A H 2020 Phys. Rev. Lett. 124 097601Google Scholar

    [24]

    Jian C M, Xu C 2018 arXiv preprint arXiv: 1810.03610

    [25]

    Bultinck N, Chatterjee S, Zaletel M P 2020 Phys. Rev. Lett. 124 166601Google Scholar

    [26]

    Zhang Y H, Mao D, Senthil T 2019 Phys. Rev. Res. 1 033126Google Scholar

    [27]

    Liu S, Khalaf E, Lee J Y, Vishwanath A 2019 arXiv preprint arXiv: 1905.07409

    [28]

    Wu F, Das Sarma S 2020 Phys. Rev. Lett. 124 046403Google Scholar

    [29]

    Chatterjee S, Bultinck N, Zaletel M P 2020 Phys. Rev. B 101 165141Google Scholar

    [30]

    Alavirad Y, Sau J D 2019 arXiv preprint arXiv: 1907.13633

    [31]

    Cécile R, Dong Z H, Zhang Y H, Senthil T 2020 Phys. Rev. Lett. 124 187601Google Scholar

    [32]

    Bultinck N, Khalaf E, Liu S, Chatterjee S, Vishwanath A, Zaletel M P 2019 arXiv preprent, arXiv: 1911.02045

    [33]

    Angeli M, Tosatti E, Fabrizio M 2019 Phys. Rev. X 9 041010

    [34]

    Zhang Y, Jiang K, Wang Z, Zhang F 2020 arXiv preprint arXiv: 2001.02476

    [35]

    Lu C, Zhang Y, Zhang Y, Zhang M, Liu C C, Gu Z C, Chen W Q, Yang F 2020 arXiv preprint arXiv: 2003.09513

    [36]

    Po H C, Zou L, Senthil T, Vishwanath A 2019 Phys. Rev. B 99 195455Google Scholar

    [37]

    Song Z, Wang Z, Shi W, Li G, Fang C, Bernevig B A 2019 Phys. Rev. Lett. 123 036401Google Scholar

    [38]

    Yuan N F Q, Fu L 2018 Phys. Rev. B 98 045103Google Scholar

    [39]

    Zou L, Po H C, Vishwanath A, Senthil T 2018 Phys. Rev. B 98 085435Google Scholar

    [40]

    Liu J, Dai X 2019 arXiv preprint arXiv: 1911.03760

    [41]

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

    [42]

    Liu X, Hao Z, Khalaf E, Lee J Y, Watanabe K, Taniguchi T, Vishwanath A, Kim P 2019 arXiv preprint arXiv: 1903.08130

    [43]

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

    [44]

    Chen G, Sharpe A L, Fox E J, et al. 2020 Nature 579 56Google Scholar

    [45]

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

    [46]

    Chen G, Jiang L, Wu S, et al. 2019 Nat. Phys. 15 237Google Scholar

    [47]

    Zhang Y H, Mao D, Cao Y, Jarillo-Herrero P, Senthil T 2019 Phys. Rev. B 99 075127Google Scholar

    [48]

    Liu J, Ma Z, Gao J, Dai X 2019 Phys. Rev. X 9 031021

    [49]

    Lee J Y, Khalaf E, Liu S, Liu X, Hao Z, Kim P, Vishwanath A 2019 Nat. Commun. 10 5333Google Scholar

    [50]

    Chebrolu N R, Chittari B L, Jung J 2019 Phys. Rev. B 99 235417Google Scholar

    [51]

    Cea T, Walet N R, Guinea F 2019 Nano Lett. 19 8683Google Scholar

    [52]

    Koshino M 2019 Phys. Rev. B 99 235406Google Scholar

    [53]

    Ma Z, Li S, Zheng Y W, Xiao M M, Jiang H, Gao J H, Xie X 2019 arXiv preprint arXiv: 1905.00622

    [54]

    Haddadi F, Wu Q, Kruchkov A J, Yazyev O V 2020 Nano Lett. 20 2410Google Scholar

    [55]

    Li S Y, Zhang Y, Ren Y N, Liu J, Dai X, He L 2019 arXiv preprint arXiv: 1912.13133

    [56]

    Liu J, Dai X 2020 npj Comput. Mater. 6 57Google Scholar

    [57]

    de Gail R, Goerbig M O, Guinea F, Montambaux G, Castro Neto A H 2011 Phys. Rev. B 84 045436Google Scholar

    [58]

    Uchida K, Furuya S, Iwata J I, Oshiyama A 2014 Phys. Rev. B 90 155451Google Scholar

    [59]

    Liu J, Liu J, Dai X 2019 Phys. Rev. B 99 155415Google Scholar

    [60]

    Jung J, Raoux A, Qiao Z, MacDonald A H 2014 Phys. Rev. B 89 205414Google Scholar

    [61]

    Moon P, Koshino M 2014 Phys. Rev. B 90 155406Google Scholar

    [62]

    Zhang Y H, Mao D, Senthil T 2019 arXiv preprint arXiv: 1901.08209

    [63]

    Ahn J, Park S, Yang B J 2019 Phys. Rev. X 9 021013

    [64]

    Yu R, Qi X L, Bernevig A, Fang Z, Dai X 2011 Phys. Rev. B 84 075119Google Scholar

    [65]

    Soluyanov A A, Vanderbilt D 2011 Phys. Rev. B 83 235401Google Scholar

    [66]

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

    [67]

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

    [68]

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

    [69]

    Castro Neto A H, Guinea F, Peres N M R, Novoselov K S, Geim A K 2009 Rev. Mod. Phys. 81 109Google Scholar

    [70]

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

    [71]

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

    [72]

    McCann E, Koshino M 2013 Rep. Prog. Phys. 76 056503Google Scholar

    [73]

    Thonhauser T, Ceresoli D, Vanderbilt D, Resta R 2005 Phys. Rev. Lett. 95 137205Google Scholar

    [74]

    Ceresoli D, Thonhauser T, Vanderbilt D, Resta R 2006 Phys. Rev. B 74 024408Google Scholar

    [75]

    Shi J, Vignale G, Xiao D, Niu Q 2007 Phys. Rev. Lett. 99 197202Google Scholar

    [76]

    Choi Y, Kemmer J, Peng Y, et al. 2019 Nat. Phys. 15 1174Google Scholar

    [77]

    Jiang Y, Lai X, Watanabe K, Taniguchi T, Haule K, Mao J, Andrei E Y 2019 Nature 573 91Google Scholar

    [78]

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

    [79]

    Fukumura T, Toyosaki H, Ueno K, Nakano M, Yamasaki T, Kawasaki M 2007 Jpn. J. Appl. Phys. 46 L642Google Scholar

    [80]

    Sarma S D, Pinczuk A 2008 Perspectives in Quantum Hall Effects: Novel Quantum Liquids in Low-dimensional Semiconductor Structures (Hoboken: John Wiley & Sons)

    [81]

    Wu F, Lovorn T, Tutuc E, Martin I, MacDonald A H 2019 Phys. Rev. Lett. 122 086402Google Scholar

    [82]

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

    [83]

    Wang L, Shih E M, Ghiotto A, et al. 2019 arXiv preprint arXiv: 1910.12147

    [84]

    Regan E C, Wang D, Jin C, et al. 2020 Nature 579 359Google Scholar

    [85]

    An L, Cai X, Huang M, Wu Z, Lin J, Ying Z, Ye Z, Feng X, Wang N 2019 arXiv preprint arXiv: 1907.03966

    [86]

    Zhang Y, Yuan N F, Fu L 2019 arXiv preprint arXiv: 1910.14061

    [87]

    Lian B, Liu Z, Zhang Y, Wang J 2020 Phys. Rev. Lett. 124 126402Google Scholar

    [88]

    Hejazi K, Luo Z X, Balents L 2020 Proc. Natl. Acad. Sci. 117 10721Google Scholar

  • 图 1  (a)转角双层石墨烯的摩尔条纹示意图, 插图展示两层石墨烯在不同区域层间距离的褶皱起伏; (b) 转角双层-双层石墨烯体系的示意图; (c)转角石墨烯体系的摩尔布里渊区示意图

    Figure 1.  (a) Schematic illustration of the moiré pattern in twisted bilayer graphene, the inserted shows the wrinkles of the graphene for different layer distances; (b) schematic illustration of twisted double bilayer graphene system; (c) moiré Brillouin zone of twisted graphene systems.

    图 2  (a)转角双层石墨烯在魔角时的能带; (b)转角双层石墨烯中K谷两条平带的威尔逊圈 (以$2{\text{π}}$为单位)

    Figure 2.  (a) Band structures of twisted bilayer graphene at the magic angle; (b)Wilson loops of the two flat bands from the K valley of twisted bilayer graphene (in $2{\text{π}}$).

    图 3  (a) 转角双层-双层石墨烯在$1.24 ^{\circ}$时的能带; (b) AB-AB堆垛的转角双层-双层石墨烯在$1.24 ^{\circ}$K谷第一个导带陈数随转角$\theta$和垂直偏压$U_{\rm{d}}$的变化

    Figure 3.  (a) Band structures of twisted double bilayer graphene at $\theta=1.24^{\circ}$; (b) the Chern number of the first conduction band for the K valley of AB-AB stacked twist-ed double bilayer graphene vs. the twist angle $\theta$ and vertical potential drop $U_{\rm{d}}$.

    图 4  魔角双层石墨烯平带对应的实空间电流密度分布 (a) K谷, $\varDelta_{\rm M}=0$; (b) $K'$谷, $\varDelta_{\rm M}=0$; (c) K谷, $\varDelta_{\rm M}=15\, {\rm{meV}}$; (d) $K'$谷, $\varDelta_{\rm M}=15\, {\rm{meV}}$, 图中黑色箭头代表电流方向, 颜色编码表示电流诱导的磁场强度, 单位为T

    Figure 4.  Real-space current-density distribution contributed by the flat bands of magic-angle twisted bilayer graphene: (a) K valley, $\varDelta_{\rm M}=0$; (b) $K'$ valley, $\varDelta_{\rm M}=0$; (c) K valley, $\varDelta_{\rm M}=15\, {\rm{meV}}$; (d) $K'$ valley, $\varDelta_{\rm M}=15\, {\rm{meV}}$. The black arrows indicate the dire-ctions of the current density, and the color coding indicates the magnetic field induced by the current in unites of Tesla.

    图 5  (a)转角双层石墨烯在魔角时K谷平带贡献的轨道磁矩($M_ {\rm {orb}}$)随化学势 ($\varepsilon_{\rm{F}}$)的变化; (b)转角双层-双层石墨烯在$\theta=1.24^ {\circ}$K谷平带贡献的轨道磁矩随化学势的变化

    Figure 5.  (a) The orbital magnetization contributed by the flat bands of the K valley of twisted bilayer graphene at the magic angle; (b) the orbital magnetization contributed by the flat bands of K valley for twisted double bilayer graphene at $\theta=1.24^{\circ}$.

    表 1  由赝朗道能级图像推算出的魔角

    Table 1.  Magic angles derived from the pseudo Landau-level picture.

    n12345
    文献[4]1.05°0.50°0.35°0.24°0.20°
    (7)式1.05°0.52°0.35°0.26°0.21°
    DownLoad: CSV
  • [1]

    Lopes dos Santos J M B, Peres N M R, Castro Neto A H 2007 Phys. Rev. Lett. 99 256802Google Scholar

    [2]

    Mele E J 2010 Phys. Rev. B 81 161405Google Scholar

    [3]

    Trambly de Laissardiere G, Mayou D, Magaud L 2010 Nano Lett. 10 804Google Scholar

    [4]

    Bistritzer R, MacDonald A H 2011 Proc. Natl. Acad. Sci. 108 12233Google Scholar

    [5]

    Lopes dos Santos J M B, Peres N M R, Castro Neto A H 2012 Phys. Rev. B 86 155449Google Scholar

    [6]

    San-Jose P, González J, Guinea F 2012 Phys. Rev. Lett. 108 216802Google Scholar

    [7]

    Cao Y, Fatemi V, Demir A, et al. 2018 Nature 556 80Google Scholar

    [8]

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

    [9]

    Yankowitz M, Chen S, Polshyn H, Zhang Y, Watanabe K, Taniguchi T, Graf D, Young A F, Dean C R 2019 Science 363 1059Google Scholar

    [10]

    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

    [11]

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

    [12]

    Stepanov P, Das I, Lu X, Fahimniya A, Watanabe K, Taniguchi T, Koppens F H, Lischner J, Levitov L, Efetov D K 2019 arXiv preprint arXiv: 1911.09198

    [13]

    Po H C, Zou L, Vishwanath A, Senthil T 2018 Phys. Rev. X 8 031089

    [14]

    Koshino M, Yuan N F Q, Koretsune T, Ochi M, Kuroki K, Fu L 2018 Phys. Rev. X 8 031087

    [15]

    Kang J, Vafek O 2018 Phys. Rev. X 8 031088

    [16]

    Isobe H, Yuan N F Q, Fu L 2018 Phys. Rev. X 8 041041

    [17]

    Xu X Y, Law K T, Lee P A 2018 Phys. Rev. B 98 121406Google Scholar

    [18]

    Huang T, Zhang L, Ma T 2019 Sci. Bull. 64 310Google Scholar

    [19]

    Liu C C, Zhang L D, Chen W Q, Yang F 2018 Phys. Rev. Lett. 121 217001Google Scholar

    [20]

    Rademaker L, Mellado P 2018 Phys. Rev. B 98 235158Google Scholar

    [21]

    Venderbos J W F, Fernandes R M 2018 Phys. Rev. B 98 245103Google Scholar

    [22]

    Kang J, Vafek O 2019 Phys. Rev. Lett. 122 246401Google Scholar

    [23]

    Xie M, MacDonald A H 2020 Phys. Rev. Lett. 124 097601Google Scholar

    [24]

    Jian C M, Xu C 2018 arXiv preprint arXiv: 1810.03610

    [25]

    Bultinck N, Chatterjee S, Zaletel M P 2020 Phys. Rev. Lett. 124 166601Google Scholar

    [26]

    Zhang Y H, Mao D, Senthil T 2019 Phys. Rev. Res. 1 033126Google Scholar

    [27]

    Liu S, Khalaf E, Lee J Y, Vishwanath A 2019 arXiv preprint arXiv: 1905.07409

    [28]

    Wu F, Das Sarma S 2020 Phys. Rev. Lett. 124 046403Google Scholar

    [29]

    Chatterjee S, Bultinck N, Zaletel M P 2020 Phys. Rev. B 101 165141Google Scholar

    [30]

    Alavirad Y, Sau J D 2019 arXiv preprint arXiv: 1907.13633

    [31]

    Cécile R, Dong Z H, Zhang Y H, Senthil T 2020 Phys. Rev. Lett. 124 187601Google Scholar

    [32]

    Bultinck N, Khalaf E, Liu S, Chatterjee S, Vishwanath A, Zaletel M P 2019 arXiv preprent, arXiv: 1911.02045

    [33]

    Angeli M, Tosatti E, Fabrizio M 2019 Phys. Rev. X 9 041010

    [34]

    Zhang Y, Jiang K, Wang Z, Zhang F 2020 arXiv preprint arXiv: 2001.02476

    [35]

    Lu C, Zhang Y, Zhang Y, Zhang M, Liu C C, Gu Z C, Chen W Q, Yang F 2020 arXiv preprint arXiv: 2003.09513

    [36]

    Po H C, Zou L, Senthil T, Vishwanath A 2019 Phys. Rev. B 99 195455Google Scholar

    [37]

    Song Z, Wang Z, Shi W, Li G, Fang C, Bernevig B A 2019 Phys. Rev. Lett. 123 036401Google Scholar

    [38]

    Yuan N F Q, Fu L 2018 Phys. Rev. B 98 045103Google Scholar

    [39]

    Zou L, Po H C, Vishwanath A, Senthil T 2018 Phys. Rev. B 98 085435Google Scholar

    [40]

    Liu J, Dai X 2019 arXiv preprint arXiv: 1911.03760

    [41]

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

    [42]

    Liu X, Hao Z, Khalaf E, Lee J Y, Watanabe K, Taniguchi T, Vishwanath A, Kim P 2019 arXiv preprint arXiv: 1903.08130

    [43]

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

    [44]

    Chen G, Sharpe A L, Fox E J, et al. 2020 Nature 579 56Google Scholar

    [45]

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

    [46]

    Chen G, Jiang L, Wu S, et al. 2019 Nat. Phys. 15 237Google Scholar

    [47]

    Zhang Y H, Mao D, Cao Y, Jarillo-Herrero P, Senthil T 2019 Phys. Rev. B 99 075127Google Scholar

    [48]

    Liu J, Ma Z, Gao J, Dai X 2019 Phys. Rev. X 9 031021

    [49]

    Lee J Y, Khalaf E, Liu S, Liu X, Hao Z, Kim P, Vishwanath A 2019 Nat. Commun. 10 5333Google Scholar

    [50]

    Chebrolu N R, Chittari B L, Jung J 2019 Phys. Rev. B 99 235417Google Scholar

    [51]

    Cea T, Walet N R, Guinea F 2019 Nano Lett. 19 8683Google Scholar

    [52]

    Koshino M 2019 Phys. Rev. B 99 235406Google Scholar

    [53]

    Ma Z, Li S, Zheng Y W, Xiao M M, Jiang H, Gao J H, Xie X 2019 arXiv preprint arXiv: 1905.00622

    [54]

    Haddadi F, Wu Q, Kruchkov A J, Yazyev O V 2020 Nano Lett. 20 2410Google Scholar

    [55]

    Li S Y, Zhang Y, Ren Y N, Liu J, Dai X, He L 2019 arXiv preprint arXiv: 1912.13133

    [56]

    Liu J, Dai X 2020 npj Comput. Mater. 6 57Google Scholar

    [57]

    de Gail R, Goerbig M O, Guinea F, Montambaux G, Castro Neto A H 2011 Phys. Rev. B 84 045436Google Scholar

    [58]

    Uchida K, Furuya S, Iwata J I, Oshiyama A 2014 Phys. Rev. B 90 155451Google Scholar

    [59]

    Liu J, Liu J, Dai X 2019 Phys. Rev. B 99 155415Google Scholar

    [60]

    Jung J, Raoux A, Qiao Z, MacDonald A H 2014 Phys. Rev. B 89 205414Google Scholar

    [61]

    Moon P, Koshino M 2014 Phys. Rev. B 90 155406Google Scholar

    [62]

    Zhang Y H, Mao D, Senthil T 2019 arXiv preprint arXiv: 1901.08209

    [63]

    Ahn J, Park S, Yang B J 2019 Phys. Rev. X 9 021013

    [64]

    Yu R, Qi X L, Bernevig A, Fang Z, Dai X 2011 Phys. Rev. B 84 075119Google Scholar

    [65]

    Soluyanov A A, Vanderbilt D 2011 Phys. Rev. B 83 235401Google Scholar

    [66]

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

    [67]

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

    [68]

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

    [69]

    Castro Neto A H, Guinea F, Peres N M R, Novoselov K S, Geim A K 2009 Rev. Mod. Phys. 81 109Google Scholar

    [70]

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

    [71]

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

    [72]

    McCann E, Koshino M 2013 Rep. Prog. Phys. 76 056503Google Scholar

    [73]

    Thonhauser T, Ceresoli D, Vanderbilt D, Resta R 2005 Phys. Rev. Lett. 95 137205Google Scholar

    [74]

    Ceresoli D, Thonhauser T, Vanderbilt D, Resta R 2006 Phys. Rev. B 74 024408Google Scholar

    [75]

    Shi J, Vignale G, Xiao D, Niu Q 2007 Phys. Rev. Lett. 99 197202Google Scholar

    [76]

    Choi Y, Kemmer J, Peng Y, et al. 2019 Nat. Phys. 15 1174Google Scholar

    [77]

    Jiang Y, Lai X, Watanabe K, Taniguchi T, Haule K, Mao J, Andrei E Y 2019 Nature 573 91Google Scholar

    [78]

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

    [79]

    Fukumura T, Toyosaki H, Ueno K, Nakano M, Yamasaki T, Kawasaki M 2007 Jpn. J. Appl. Phys. 46 L642Google Scholar

    [80]

    Sarma S D, Pinczuk A 2008 Perspectives in Quantum Hall Effects: Novel Quantum Liquids in Low-dimensional Semiconductor Structures (Hoboken: John Wiley & Sons)

    [81]

    Wu F, Lovorn T, Tutuc E, Martin I, MacDonald A H 2019 Phys. Rev. Lett. 122 086402Google Scholar

    [82]

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

    [83]

    Wang L, Shih E M, Ghiotto A, et al. 2019 arXiv preprint arXiv: 1910.12147

    [84]

    Regan E C, Wang D, Jin C, et al. 2020 Nature 579 359Google Scholar

    [85]

    An L, Cai X, Huang M, Wu Z, Lin J, Ying Z, Ye Z, Feng X, Wang N 2019 arXiv preprint arXiv: 1907.03966

    [86]

    Zhang Y, Yuan N F, Fu L 2019 arXiv preprint arXiv: 1910.14061

    [87]

    Lian B, Liu Z, Zhang Y, Wang J 2020 Phys. Rev. Lett. 124 126402Google Scholar

    [88]

    Hejazi K, Luo Z X, Balents L 2020 Proc. Natl. Acad. Sci. 117 10721Google Scholar

  • [1] Qin Zhi-Jie, Zhang Hui-Qing, Zhang Guang-Ping, Ren Jun-Feng, Wang Chuan-Kui, Hu Gui-Chao, Qiu Shuai. Theoretical study of introducing spin into nonmagnetic graphene-based single-molecule junction by edge modifications. Acta Physica Sinica, 2023, 72(13): 138504. doi: 10.7498/aps.72.20230267
    [2] Wang Huan, He Chun-Juan, Xu Sheng, Wang Yi-Yan, Zeng Xiang-Yu, Lin Jun-Fa, Wang Xiao-Yan, Gong Jing, Ma Xiao-Ping, Han Kun, Wang Yi-Ting, Xia Tian-Long. Single crystal growth of topological semimetals and magnetic topological materials. Acta Physica Sinica, 2023, 72(3): 038103. doi: 10.7498/aps.72.20221574
    [3] Liu Yi-Jun, Chen Yi-Wei, Zhu Yu-Jian, Huang Yan, An Dong-Dong, Li Qing-Xin, Gan Qi-Kang, Zhu Wang, Song Jun-Wei, Wang Kai-Yuan, Wei Ling-Nan, Zong Qi-Jun, Liu Shuo-Han, Li Shi-Wei, Liu Zhi, Zhang Qi, Xu Ying-Hai, Cao Xin-Yu, Yang Ao, Wang Hao-Lin, Yang Bing, Andy Shen, Yu Ge-Liang, Wang Lei. Isospin polarized Chern insulator state of C = 4 in twisted double bilayer graphene. Acta Physica Sinica, 2023, 72(14): 147303. doi: 10.7498/aps.72.20230497
    [4] Bai Zhan-Bin, Wang Rui, Zhou Ya-Zhou, Wu Tian-Ru, Ge Jian-Lei, Li Jing, Qin Yu-Yuan, Fei Fu-Cong, Cao Lu, Wang Xue-Feng, Wang Xin-Ran, Zhang Shuai, Sun Li-Ling, Song You, Song Feng-Qi. Selective enhancement of Kane Mele-type spin-orbit interaction in graphene. Acta Physica Sinica, 2022, 71(6): 067202. doi: 10.7498/aps.71.20211815
    [5] Zheng Yu-Qiang, Wang Shi-Yong. Delocalized magnetism in low-dimensional graphene system. Acta Physica Sinica, 2022, 71(18): 188101. doi: 10.7498/aps.71.20220895
    [6] Mimicing the Kane-Mele type spin orbit interaction by spin-flexual phonon coupling in graphene devices. Acta Physica Sinica, 2021, (): . doi: 10.7498/aps.70.20211815
    [7] Cai Xiao-Xiao, Luo Guo-Yu, Li Zhi-Qiang, He Yan. Optical conductivity of twisted bilayer graphene under heterostrain. Acta Physica Sinica, 2021, 70(18): 187301. doi: 10.7498/aps.70.20210110
    [8] Liu Yu-Ting, He Wen-Yu, Liu Jun-Wei, Shao Qi-Ming. Berry curvature-induced emerging magnetic response in two-dimensional materials. Acta Physica Sinica, 2021, 70(12): 127303. doi: 10.7498/aps.70.20202132
    [9] Xu Xian-Da, Zhao Lei, Sun Wei-Feng. First-principles on the energy band mechanism for modifying conduction property of graphene nanomeshes. Acta Physica Sinica, 2020, 69(4): 047101. doi: 10.7498/aps.69.20190657
    [10] Xi Xiang, Ye Kang-Ping, Wu Rui-Xin. Influence of bias magnetic field direction on band structure of magnetic photonic crystal and its role in constructing topological edge states. Acta Physica Sinica, 2020, 69(15): 154102. doi: 10.7498/aps.69.20200198
    [11] Lü Xin-Yu, Li Zhi-Qiang. Topological properties of graphene moiré superlattice systems and recent optical studies. Acta Physica Sinica, 2019, 68(22): 220303. doi: 10.7498/aps.68.20191317
    [12] Zhang Hua-Lin, Sun Lin, Wang Ding. Electromagnetic properties of zigzag graphene nanoribbons with single-row line defect. Acta Physica Sinica, 2016, 65(1): 016101. doi: 10.7498/aps.65.016101
    [13] Chen Dong-Hai, Yang Mou, Duan Hou-Jian, Wang Rui-Qiang. Electronic transport properties of graphene pn junctions with spin-orbit coupling. Acta Physica Sinica, 2015, 64(9): 097201. doi: 10.7498/aps.64.097201
    [14] Xu Lei, Dai Zhen-Hong, Sui Peng-Fei, Wang Wei-Tian, Sun Yu-Ming. Electronic and magnetic properties of fluorinated graphene sheets with divacancy substitutional doping. Acta Physica Sinica, 2014, 63(18): 186101. doi: 10.7498/aps.63.186101
    [15] Gao Tan-Hua. Magnetic and electronic properties of fluorographene sheet with foreign atom substitutions. Acta Physica Sinica, 2014, 63(4): 046102. doi: 10.7498/aps.63.046102
    [16] Jin Feng, Zhang Zhen-Hua, Wang Cheng-Zhi, Deng Xiao-Qing, Fan Zhi-Qiang. Twisting effects on energy band structures and transmission behaviors of graphene nanoribbons. Acta Physica Sinica, 2013, 62(3): 036103. doi: 10.7498/aps.62.036103
    [17] Lin Qi, Chen Yu-Hang, Wu Jian-Bao, Kong Zong-Min. Effect of N-doping on band structure and transport property of zigzag graphene nanoribbons. Acta Physica Sinica, 2011, 60(9): 097103. doi: 10.7498/aps.60.097103
    [18] Gao Shuang-Hong, Ren Zhao-Yu, Guo Ping, Zheng Ji-Ming, Du Gong-He, Wan Li-Juan, Zheng Lin-Lin. Magnetic properties and excited states of thegraphene quantum dots. Acta Physica Sinica, 2011, 60(4): 047105. doi: 10.7498/aps.60.047105
    [19] Wang Xue-Mei, Liu Hong. Band structures of zigzag graphene nanoribbons. Acta Physica Sinica, 2011, 60(4): 047102. doi: 10.7498/aps.60.047102
    [20] Wang Yong-Long, Pan Hong-Zhe, Xu Ming, Chen Li, Sun Yuan-Yuan. Electronic structure and magnetism of single-layer trigonal graphene quantum dots with zigzag edges. Acta Physica Sinica, 2010, 59(9): 6443-6449. doi: 10.7498/aps.59.6443
Metrics
  • Abstract views:  14029
  • PDF Downloads:  1274
  • Cited By: 0
Publishing process
  • Received Date:  07 April 2020
  • Accepted Date:  19 June 2020
  • Available Online:  10 July 2020
  • Published Online:  20 July 2020

/

返回文章
返回