Lattice relaxation and substrate effects of graphene moiré superlattice

Zhan Zhen; Zhang Ya-Lei; Yuan Sheng-Jun

doi:10.7498/aps.71.20220872

Abstract

When two two-dimensional (2D) materials with different lattice constants or with different rotation angles are superimposed, a moiré superlattice can be constructed. The electronic properties of the superlattice are strongly dependent on the stacking configuration, twist angle and substrate. For instance, theoretically, when the rotation angle of twisted bilayer graphene is reduced to a set of specific values, the so-called magic angles, flat bands appear near the charge neutrality, and the electron-electron interaction is significantly enhanced. The Mott insulator and unconventional superconductivity are detected in the twisted bilayer graphene with a twist angle around 1.1°. For a moiré pattern with a large enough periodicity, lattice relaxation caused by an interplay between van der Waals force and the in-plane elasticity force comes into being. The atomic relaxation forces atoms to deviate from their equilibrium positions, and thus making the system reconstructed. This review mainly focuses on the effects of the lattice relaxation and substrates on the electronic properties of the graphene superlattices. From both theoretical and experimental point of view, the lattice relaxation effects on the atomic structure and electronic properties of graphene-based superlattices, for example, the twisted bilayer graphene, twisted trilayer graphene, graphene-hexagonal boron nitride superlattice and twisted bilayer graphene-boron nitride superlattice are discussed. Finally, a summary and perspective of the investigation of the 2D material superlattice are presented.

Keywords:

Authors and contacts

1.
School of Physics and Technology, Wuhan University, Wuhan 430072, China
2.
Wuhan Institute of Quantum Technology, Wuhan 430206, China

Corresponding author: Yuan Sheng-Jun, s.yuan@whu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12174291, 12047543) and the National Key R&D Program of China (Grant No. 2018YFA0305800).

FullText HTML : translate this paragraph

References

[1]	Geim A K, Grigorieva I V 2013 Nature 499 419 Google Scholar
[2]	Chang C, Chen W, Chen Y, et al. 2021 Acta Phys. Chim. Sin. 37 2108017 Google Scholar
[3]	Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 666 Google Scholar
[4]	Bistritzer R, MacDonald A H 2011 Proc. Natl. Acad. Sci. USA 108 12233 Google Scholar
[5]	Yan W, Liu M, Dou R F, Meng L, Feng L, Chu Z D, Zhang Y, Liu Z, Nie J C, He L 2012 Phys. Rev. Lett. 109 126801 Google Scholar
[6]	Cao Y, Fatemi V, Demir A, et al. 2018 Nature 556 80 Google Scholar
[7]	Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E, Jarillo-Herrero P 2018 Nature 556 43 Google Scholar
[8]	Polshyn H, Zhu J, Kumar M A, Zhang Y, Yang F, Tschirhart C L, Serlin M, Watanabe K, Taniguchi T, MacDonald A H, Young A F 2020 Nature 588 66 Google Scholar
[9]	Shen C, Ying J, Liu L, Liu J, Li N, Wang S, Tang J, Zhao Y, Chu Y, Watanabe K, Taniguchi T, Yang R, Shi D, Qu F, Lu L, Yang W, Zhang G 2021 Chin. Phys. Lett. 38 047301 Google Scholar
[10]	Yankowitz M, Chen S, Polshyn H, Zhang Y, Watanabe K, Taniguchi T, Graf D, Young A F, Dean C R 2019 Science 363 1059 Google Scholar
[11]	Zheng Z, Ma Q, Bi Z, et al. 2020 Nature 588 71 Google Scholar
[12]	Sharpe A L, Fox E J, Barnard A W, Finney J, Watanabe K, Taniguchi T, Kastner M A, Goldhaber-Gordon D 2019 Science 365 605 Google Scholar
[13]	Serlin M, Tschirhart C, Polshyn H, Zhang Y, Zhu J, Watanabe K, Taniguchi T, Balents L, Young A 2020 Science 367 900 Google Scholar
[14]	Carr S, Massatt D, Fang S, Cazeaux P, Luskin M, Kaxiras E 2017 Phys. Rev. B 95 075420 Google Scholar
[15]	Gargiulo F, Yazyev O V 2017 2D Mater. 5 015019
[16]	van Wijk M, Schuring A, Katsnelson M, Fasolino A 2015 2D Mater. 2 034010
[17]	Nam N N T, Koshino M 2017 Phys. Rev. B 96 075311 Google Scholar
[18]	Yoo H, Engelke R, Carr S, et al. 2019 Nat. Mater. 18 448 Google Scholar
[19]	Shi H, Zhan Z, Qi Z, Huang K, Veen E V, Silva-Guillen J A, Zhang R, Li P, Xie K, Ji H, Katsnelson M I, Yuan S, Qin S, Zhang Z 2020 Nat. Commun. 11 371 Google Scholar
[20]	Huang S, Kim K, Efimkin D K, Lovorn T, Taniguchi T, Watanabe K, MacDonald A H, Tutuc E, LeRoy B J 2018 Phys. Rev. Lett. 121 037702 Google Scholar
[21]	Oh M, Nuckolls K P, Wong D, Lee R L, Liu X, Watanabe K, Taniguchi T, Yazdani A 2021 Nature 600 240 Google Scholar
[22]	Stepanov P, Das I, Lu X, Fahimniya A, Watanabe K, Taniguchi T, Koppens F H L, Lischner J, Levitov L, Efetov D K 2020 Nature 583 375 Google Scholar
[23]	Artaud A, Magaud L, Le Quang T, Guisset V, David P, Chapelier C, Coraux J 2016 Sci. Rep. 6 1 Google Scholar
[24]	Huder L, Artaud A, Le Quang T, de Laissardiere G T, Jansen A G M, Lapertot G, Chapelier C, Renard V T 2018 Phys. Rev. Lett. 120 156405 Google Scholar
[25]	Dos Santos J L, Peres N, Neto A C 2007 Phys. Rev. Lett. 99 256802 Google Scholar
[26]	Koshino M 2015 New J. Phys. 17 015014 Google Scholar
[27]	Koshino M, Yuan N F, Koretsune T, Ochi M, Kuroki K, Fu L 2018 Phys. Rev. X 8 031087
[28]	Popov A M, Lebedeva I V, Knizhnik A A, Lozovik Y E, Potapkin B V 2011 Phys. Rev. B 84 045404 Google Scholar
[29]	Lebedeva I V, Knizhnik A A, Popov A M, Lozovik Y E, Potapkin B V 2011 Phys. Chem. Chem. Phys. 13 5687 Google Scholar
[30]	Gould T, Lebegue S, Dobson J F 2013 J Phys. Condens. Matter. 25 445010 Google Scholar
[31]	Slater J C, Koster G F 1954 Phys. Rev. 94 1498 Google Scholar
[32]	Trambly de Laissardière G, Mayou D, Magaud L 2010 Nano Lett. 10 804 Google Scholar
[33]	Yuan S, De Raedt H, Katsnelson M I 2010 Phys. Rev. B 82 115448 Google Scholar
[34]	Yuan S, Roldán R, Katsnelson M I 2011 Phys. Rev. B 84 035439 Google Scholar
[35]	Logemann R, Reijnders K, Tudorovskiy T, Katsnelson M, Yuan S 2015 Phys. Rev. B 91 045420 Google Scholar
[36]	Hams A, De Raedt H 2000 Phys. Rev. E 62 4365 Google Scholar
[37]	Li Y, Zhan Z, Li Y, Yuan S 2022 arXiv:2209.00806 [cond-mat.mtrl-sci]
[38]	Brihuega I, Mallet P, Gonzalez-Herrero H, et al. 2012 Phys. Rev. Lett. 109 196802 Google Scholar
[39]	De Laissardiere G T, Mayou D, Magaud L 2012 Phys. Rev. B 86 125413 Google Scholar
[40]	Guinea F, Walet N R 2019 Phys. Rev. B 99 205134 Google Scholar
[41]	Slotman G J, van Wijk M M, Zhao P L, Fasolino A, Katsnelson M I, Yuan S 2015 Phys. Rev. Lett. 115 186801 Google Scholar
[42]	Gadelha A C, Ohlberg D A A, Rabelo C, et al. 2021 Nature 590 405 Google Scholar
[43]	Kuang X H, Zhan Z, Yuan S J 2021 Phys. Rev. B 103 115431 Google Scholar
[44]	Ju L, Shi Z W, Nair N, Lv Y C, Jin C H, Velasco J, Ojeda-Aristizabal C, Bechtel H A, Martin M C, Zettl A, Analytis J, Wang F 2015 Nature 520 650 Google Scholar
[45]	Sunku S, Ni G, Jiang B Y, Yoo H, Sternbach A, McLeod A, Stauber T, Xiong L, Taniguchi T, Watanabe K 2018 Science 362 1153 Google Scholar
[46]	Zhang F, MacDonald A H, Mele E J 2013 Proc. Natl. Acad. Sci. USA 110 10546 Google Scholar
[47]	Xu S G, Berdyugin A I, Kumaravadivel P, et al. 2019 Nat. Commun. 10 4008 Google Scholar
[48]	Verbakel J, Yao Q, Sotthewes K, Zandvliet H 2021 Phys. Rev. B 103 165134 Google Scholar
[49]	Lu X, Lian B, Chaudhary G, Piot B A, Romagnoli G, Watanabe K, Taniguchi T, Poggio M, MacDonald A H, Bernevig B A, Efetov D K 2021 Proc. Natl. Acad. Sci. USA 118 e2100006118 Google Scholar
[50]	Nguyen V H, Paszko D, Lamparski M, Van Troeye B, Meunier V, Charlier J C 2021 2D Mater. 8 035046
[51]	Plimpton S 1995 J. Comput. Phys. 117 1 Google Scholar
[52]	Los J H, Fasolino A 2003 Phys. Rev. B 68 024107 Google Scholar
[53]	Kolmogorov A N, Crespi V H 2005 Phys. Rev. B 71 235415 Google Scholar
[54]	Zhou H, Xie T, Taniguchi T, Watanabe K, Young A F 2021 Nature 598 434 Google Scholar
[55]	Zhou H, 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 598 429 Google Scholar
[56]	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 Nat. Phys. 15 237 Google Scholar
[57]	Chen G, Sharpe A L, Gallagher P, et al. 2019 Nature 572 215 Google Scholar
[58]	Carr S, Li C, Zhu Z, Kaxiras E, Sachdev S, Kruchkov A 2020 Nano Lett. 20 3030 Google Scholar
[59]	Mora C, Regnault N, Bernevig B A 2019 Phys. Rev. Lett. 123 026402 Google Scholar
[60]	Zhu Z, Carr S, Massatt D, Luskin M, Kaxiras E 2020 Phys. Rev. Lett. 125 116404 Google Scholar
[61]	Ma Z, Li S, Zheng Y W, Xiao M M, Jiang H, Gao J H, Xie X 2021 Sci. Bull. 66 18 Google Scholar
[62]	Haddadi F, Wu Q, Kruchkov A J, Yazyev O V 2020 Nano Lett. 20 2410 Google Scholar
[63]	Rubio-Verdú C, Turkel S, Song Y, et al. 2022 Nat. Phys. 18 196 Google Scholar
[64]	Wu Z, Zhan Z, Yuan S 2021 Sci. China Phys. Mech. Astron. 64 267811 Google Scholar
[65]	Park J M, Cao Y, Watanabe K, Taniguchi T, Jarillo-Herrero P 2021 Nature 590 249 Google Scholar
[66]	Cao Y, Park J M, Watanabe K, Taniguchi T, Jarillo-Herrero P 2021 Nature 595 526 Google Scholar
[67]	Hao Z, Zimmerman A, Ledwith P, Khalaf E, Najafabadi D H, Watanabe K, Taniguchi T, Vishwanath A, Kim P 2021 Science 371 1133 Google Scholar
[68]	Zhang X, Tsai K T, Zhu Z, Ren W, Luo Y, Carr S, Luskin M, Kaxiras E, Wang K 2021 Phys. Rev. Lett. 127 166802 Google Scholar
[69]	Turkel S, Swann J, Zhu Z, Christos M, Watanabe K, Taniguchi T, Sachdev S, Scheurer M S, Kaxiras E, Dean C R, Pasupathy A N 2022 Science 376 193 Google Scholar
[70]	Chen S, He M, Zhang Y H, Hsieh V, Fei Z, Watanabe K, Taniguchi T, Cobden D H, Xu X, Dean C R, Yankowitz M 2021 Nat. Phys. 17 374 Google Scholar
[71]	Xu S, Al Ezzi M M, Balakrishnan N, et al. 2021 Nat. Phys. 17 619 Google Scholar
[72]	Wu Z, Kuang X, Zhan Z, Yuan S 2021 Phys. Rev. B 104 205104 Google Scholar
[73]	Carr S, Fang S, Jarillo-Herrero P, Kaxiras E 2018 Phys. Rev. B 98 085144 Google Scholar
[74]	Kretinin A V, Cao Y, Tu J S, et al. 2014 Nano Lett. 14 3270 Google Scholar
[75]	Giovannetti G, Khomyakov P A, Brocks G, Karpan V M, van den Brink J, Kelly P J 2008 Phys. Rev. Lett. 101 026803 Google Scholar
[76]	Dean C R, Young A F, Meric I, Lee C, Wang L, Sorgenfrei S, Watanabe K, Taniguchi T, Kim P, Shepard K L, Hone J 2010 Nat. Nanotechnol. 5 722 Google Scholar
[77]	Moon P, Koshino M 2014 Phys. Rev. B 90 155406 Google Scholar
[78]	San-Jose P, Gutiérrez-Rubio A, Sturla M, Guinea F 2014 Phys. Rev. B 90 075428 Google Scholar
[79]	Van Wijk M, Schuring A, Katsnelson M, Fasolino A 2014 Phys. Rev. Lett. 113 135504 Google Scholar
[80]	Ponomarenko L A, Gorbachev R V, Yu G L, et al. 2013 Nature 497 594 Google Scholar
[81]	Dean C R, Wang L, Maher P, et al. 2013 Nature 497 598 Google Scholar
[82]	Hunt B, Sanchez-Yamagishi J D, Young A F, Yankowitz M, LeRoy B J, Watanabe K, Taniguchi T, Moon P, Koshino M, Jarillo-Herrero P, Ashoori R C 2013 Science 340 1427 Google Scholar
[83]	Barrier J, Kumaravadivel P, Krishna Kumar R, et al. 2020 Nat. Commun. 11 5756 Google Scholar
[84]	Wang L, Zihlmann S, Liu M H, Makk P, Watanabe K, Taniguchi T, Baumgartner A, Schönenberger C 2019 Nano Lett. 19 2371 Google Scholar
[85]	Wang Z, Wang Y B, Yin J, et al. 2019 Sci. Adv. 5 eaay8897 Google Scholar
[86]	Anđelković M, Milovanović S P, Covaci L, Peeters F M 2020 Nano Lett. 20 979 Google Scholar
[87]	Yang Y, Li J, Yin J, Xu S, Mullan C, Taniguchi T, Watanabe K, Geim A K, Novoselov K S, Mishchenko A 2020 Sci. Adv. 6 eabd3655 Google Scholar
[88]	Jung J, DaSilva A M, MacDonald A H, Adam S 2015 Nat. Commun. 6 6308 Google Scholar
[89]	Chen Z G, Shi Z, Yang W, Lu X, Lai Y, Yan H, Wang F, Zhang G, Li Z 2014 Nat. Commun. 5 4461 Google Scholar
[90]	Wang E, Lu X, Ding S, Yao W, Yan M, Wan G, Deng K, Wang S, Chen G, Ma L, Jung J, Fedorov A V, Zhang Y, Zhang G, Zhou S 2016 Nat. Phys. 12 1111 Google Scholar
[91]	Han T, Yang J, Zhang Q, Wang L, Watanabe K, Taniguchi T, McEuen P L, Ju L 2021 Phys. Rev. Lett. 126 146402 Google Scholar
[92]	Kim H, Leconte N, Chittari B L, Watanabe K, Taniguchi T, MacDonald A H, Jung J, Jung S 2018 Nano Lett. 18 7732 Google Scholar
[93]	Bultinck N, Chatterjee S, Zaletel M P 2020 Phys. Rev. Lett. 124 166601 Google Scholar
[94]	Zhang Y H, Mao D, Senthil T 2019 Phys. Rev. Res. 1 033126 Google Scholar
[95]	Liu X, Wang Z, Watanabe K, Taniguchi T, Vafek O, Li J 2021 Science 371 1261 Google Scholar
[96]	Brenner D W, Shenderova O A, Harrison J A, Stuart S J, Ni B, Sinnott S B 2002 J. Condens. Matter. Phys. 14 783 Google Scholar
[97]	Ochoa H, Castro E V, Katsnelson M, Guinea F 2011 Phys. Rev. B 83 235416 Google Scholar
[98]	Xue J, Sanchez-Yamagishi J, Bulmash D, Jacquod P, Deshpande A, Watanabe K, Taniguchi T, Jarillo-Herrero P, LeRoy B J 2011 Nat. Mater. 10 282 Google Scholar
[99]	Yankowitz M, Xue J, Cormode D, Sanchez-Yamagishi J D, Watanabe K, Taniguchi T, Jarillo-Herrero P, Jacquod P, LeRoy B J 2012 Nat. Phys. 8 382 Google Scholar
[100]	Wallbank J, Patel A, Mucha-Kruczyński M, Geim A, Fal'Ko V 2013 Phys. Rev. B 87 245408 Google Scholar
[101]	Park C H, Yang L, Son Y W, Cohen M L, Louie S G 2008 Phys. Rev. Lett. 101 126804 Google Scholar
[102]	Long M, Pantaleón P A, Zhan Z, Guinea F, Silva-Guillén J Á, Yuan S 2022 npj Comput. Mater. 8 1 Google Scholar
[103]	Cea T, Pantaleón P A, Guinea F 2020 Phys. Rev. B 102 155136 Google Scholar
[104]	Mao D, Senthil T 2021 Phys. Rev. B 103 115110 Google Scholar
[105]	Shin J, Park Y, Chittari B L, Sun J H, Jung J 2021 Phys. Rev. B 103 075423 Google Scholar
[106]	Ramires A, Lado J L 2019 Phys. Rev. B 99 245118 Google Scholar
[107]	Zhang Z, Wang Y, Watanabe K, Taniguchi T, Ueno K, Tutuc E, LeRoy B J 2020 Nat. Phys. 16 1093 Google Scholar
[108]	Zhan Z, Zhang Y, Lv P, Zhong H, Yu G, Guinea F, Silva-Guillén J Á, Yuan S 2020 Phys. Rev. B 102 241106 Google Scholar
[109]	Naik M H, Jain M 2018 Phys. Rev. Lett. 121 266401 Google Scholar
[110]	Xu Y, Liu S, Rhodes D A, Watanabe K, Taniguchi T, Hone J, Elser V, Mak K F, Shan J 2020 Nature 587 214 Google Scholar
[111]	Li T, Jiang S, Shen B, Zhang Y, Li L, Tao Z, Devakul T, Watanabe K, Taniguchi T, Fu L, Shan J, Mak K F 2021 Nature 600 641 Google Scholar
[112]	Li T, Jiang S, Li L, Zhang Y, Kang K, Zhu J, Watanabe K, Taniguchi T, Chowdhury D, Fu L, Shan J, Mak K F 2021 Nature 597 350 Google Scholar
[113]	Ghiotto A, Shih E M, Pereira G, Rhodes D A, Kim B, Zang J, Millis A J, Watanabe K, Taniguchi T, Hone J C, Wang L, Dean C R, Pasupathy A N 2021 Nature 597 345 Google Scholar
[114]	Weston A, Castanon E G, Enaldiev V, et al. 2022 Nat. Nanotechnol. 17 390 Google Scholar

Cited By

图 2 (a) $ \theta =0.48^{\circ} $的TBG的STM形貌图 (100 nm $ \times $ 100 nm, 超晶格格矢$ L1\approx L2\approx L3\approx 29.6 $ nm, STM图在V = 100 mV和I_t = 1.0 nA下采集); (b) 超晶格结构中AA和AB区域的STS, 两条实(虚)线表示探测不同AA (AB)位置的STS, 证明实验数据的可重复性; (c) TBG非弛豫结构(上)和弛豫结构(下)中AA和AB区域的LDOS; (d) 形变势能V的分布; (e) 赝磁场B的分布; (f) $ \theta =0.48^{\circ} $的TBG理论计算模型图; (g) 路径M-N-P上原子在平面内(|Δd|)和z方向上(|Δz|)的位移; (h), (i) 原子分别在平面内(h)和z方向上(i)位移的实空间分布图^[19]

Figure 2. (a) STM topography image (100 nm $ \times $ 100 nm) for TBG with $ \theta =0.48^{\circ} $ (The three moiré wavelengths are $ L1\approx L2\approx L3\approx 29.6 $ nm, sample bias $ V=100 $ mV, tunneling current $ {I}_{\mathrm{t}}=1.0 $ nA); (b) logarithmic dI/dV spectra measured at AA and AB regions (The two solid/dashed lines were taken at different AA/AB regions to show reproducibility, and curves are vertically shifted for clarity); (c) calculated LDOS in the AA and AB regions for deformed (upper) and rigidly twisted (bottom) cases; (d) calculated local potential V; (e) calculated pseudo-magnetic fields B; (f) schematic model of the moiré pattern of TBG with $ \theta =0.48^{\circ} $; (g) absolute magnitude of different in-plane atomic displacements and out-of-plane displacements for the deformed system along the path M-N-P; (h), (i) maps of the absolute magnitude of the in-plane atomic displacement |Δd| (h) and out-of-plane displacement |Δz| (i) in deformed systems^[19].

DownLoad: Full-Size Img PowerPoint

图 1 $ \theta =6.01^{\circ} $时TBG结构示意图(黑色实线框表示莫尔超晶格的原胞; 圆圈表示4种高对称堆垛结构, 分别为AA堆垛(红色)、AB堆垛(蓝色)、DW堆垛(紫色)和BA堆垛(绿色))

Figure 1. Schematics of the atomic configuration of TBG with$ \theta =6.01^{\circ} $ (The moiré supercell is outlined in black line. High-symmetry stacking regions of AA, AB, DW and BA are marked by the red, blue, purple and green circles, respectively).

DownLoad: Full-Size Img PowerPoint

图 3 (a)—(c) $ \theta =1.05^{\circ}\mathrm{ }\left(\mathrm{a}\right) $, $ \theta =0.53^{\circ}\mathrm{ }\left(\mathrm{b}\right) $和$ \theta =0.35^{\circ}\mathrm{ }\left(\mathrm{c}\right) $的面内应变张量$ \mathit{u}\left(\mathit{r}\right) $(白色箭头表示原子在平面内的位移矢量; 彩色条表示原子在面内的旋转角度$\Delta \theta =\nabla \times {\boldsymbol u }$, 正值表示顺时针旋转; 莫尔结构的晶胞由黑色边框标记; 3个高对称区域分别是AA, AB和DW); (d)—(f) $ \theta =1.05^{\circ} $ (d), $ \theta =0.53^{\circ} $ (e)和$ \theta =0.35^{\circ} $ (f)的层间距(ILS)在实空间的分布

Figure 3. (a)–(c) In-plane strain $ \mathit{u}\left(\mathit{r}\right) $ in twisted bilayer graphene with (a) $ \theta =1.05^{\circ} $, (b) $ \theta =0.53^{\circ} $ and (c) $ \theta =0.35^{\circ} $ (The in-plane displacements are visualized with white arrows; the color data denotes the local value of the in-plane twist of the atoms with respect to their original position ($\Delta \theta =\nabla \times {\boldsymbol u})$, and the positive values indicate counterclockwise rotation. The moiré supercell is outlined in black, and the high-symmetry stacking regions of AA, AB and DW are illustrated); (d)–(f) the interlayer spacing of TBG with (d) $ \theta =1.05^{\circ} $, (e) $ \theta =0.53^{\circ} $ and (f) $ \theta =0.35^{\circ} $.

DownLoad: Full-Size Img PowerPoint

图 4 (a), (d), (g) 三类魔角的DOS分布; (b), (e), (h) 三类魔角非弛豫体系中不同能量下LDOS在实空间的分布; (c), (f), (i) 三类魔角弛豫体系中不同能量下LDOS在实空间的分布. 其中(a)—(c)$ \theta =1.05^{\circ} $; (d)—(f) $ \theta =0.53^{\circ} $; (g)—(i) $ \theta =0.35^{\circ} $

Figure 4. (a), (d), (g) DOS distributions of three types of magic angles; (b), (e), (h) LDOS distributions in real space at different energies in non-relaxation systems of three types of magic angle non-relaxation systems; (c), (f), (i) LDOS distributions in real space at different energies in relaxation systems of three types of magic angle. The rotation angle θ is 1.05° (a)–(c), 0.53° (d)–(f), 0.35° (g)–(i).

DownLoad: Full-Size Img PowerPoint

图 5 (a) tTLG-AÃA-6.01体系的侧视图(上)和俯视图(下); (b), (c) 分别为tTLG-AÃA-1.35体系和tTLG-ÃAB-1.05体系中原子在z方向的位移Δz; (d)—(f) tTLG-AÃA-1.35体系中DOS (d)和不同能量点LDOS (e), (f)在实空间的分布; (g)—(i) tTLG-ÃAB-1.05体系中DOS (g)和不同能量点LDOS (h), (i)在实空间的分布^[64]

Figure 5. (a) Side (upper) and top (lower) views of tTLG-AÃA-6.01; (b), (c) the displacement Δz of atoms in the z direction for tTLG-AÃA-1.35 and tTLG-ÃAB-1.05, respectively; (d)–(f) distribution of DOS (d) and LDOS at different energy points in real space (e), (f) of tTLG-AÃA-1.35. (g)–(i) distribution of DOS (g) and LDOS at different energy points in real space (h), (i) of tTLG-ÃAB-1.05^[64].

DownLoad: Full-Size Img PowerPoint

图 6 (a), (c) tTLG-AÃA-1.89体系分别在常压和4 GPa高压条件下原子在z方向的位移Δz; (b), (e) 体系在常压条件下的能带、DOS (b)和范霍夫奇点处LDOS在实空间的分布(e); (d), (f) 体系在4 GPa高压条件下的能带、DOS (d)和范霍夫奇点处LDOS在实空间的分布(f)^[72]

Figure 6. (a), (c) Out-of-plane displacement Δz of relaxed tTLG-AÃA-1.89 without and with 4 GPa vertical pressure, respectively; (b), (e) the band structure, DOS (b) and LDOS mappings of van Hove singularities (e) of tTLG-AÃA-1.89 under ambient pressure; (d), (f) the band structure, DOS (d) and LDOS mappings of van Hove singularities (f) of tTLG-AÃA-1.89 with 4 GPa pressure^[72].

DownLoad: Full-Size Img PowerPoint

图 7 (a), (b) 石墨烯/氮化硼之间转角$ \theta =0^{\circ} $的体系在晶格弛豫后的紧束缚模型参数(λ = 13.8 nm; 从左到右分别为在位能$ {V}_{D} $和碳原子最近邻的跃迁振幅t1, t2, t3, 彩色条的单位是t = 2.7 eV); (c) 非弛豫和弛豫结构石墨烯的DOS分布; (d) 不同$ \theta $下的DOS (箭头表示超晶格狄拉克点随转角的减小而向高能部分移动; 当$ \theta =1.85^{\circ} $, 超晶格狄拉克点消失; 相应的莫尔长度分别为 λ = 13.8, 11.9, 6.7 nm); (e) 转角$ \theta =0^{\circ} $时不同能量下的准本征态在实空间的分布 (左侧和右侧分别是A子晶格和B子晶格的准本征态; 对于接近超晶格狄拉克点的能量, 可以形成一个清晰的莫尔条纹^[41])

Figure 7. (a), (b) Modified tight-binding parameters for a relaxed sample of graphene on hBN with θ = 0° (λ = 13.8 nm; from left to right, the on-site potential $ {V}_{D} $ and the hopping parameters t1, t2, and t3. The color bars are in units of t = 2.7 eV); (c) DOS distributions of unrelaxed and relaxed graphene; (d) DOS for different angles θ (As indicated by the arrows, superlattice Dirac point moves towards the high-energy part with the decreasing of the turning angle; the superlattice Dirac point disappears at $ \theta =1.85^{\circ} $. The corresponding moiré lengths λ = 13.8, 11.9, 6.7 nm, respectively); (e) amplitude of the quasi eigenstates for different energies in real space for θ = 0° (The left-hand panels show sublattice A and the right-hand panels show sublattice B. For energies closer to the extra Dirac cones, a clear moiré pattern can be distinguished)^[41].

DownLoad: Full-Size Img PowerPoint

图 8 (a) TBG/hBN的结构示意图; (b) TBG/hBN的俯视图和高对称堆垛结构; (c) 不同$ {\theta }_{\mathrm{b}\mathrm{o}\mathrm{t}} $ 体系的面内形变u(r)和面内转角$\Delta \theta =\nabla \times {\boldsymbol u}$, 白色箭头是原子的面内位移; (d)—(f) 不同$ {\theta }_{\mathrm{b}\mathrm{o}\mathrm{t}} $体系的能带图和DOS (彩色条表示每个谷$ \langle{{\widehat{V}}_{z}}\rangle $的带, 如果属于K谷状态(红色), 则$\langle{{\widehat{V}}_{z}}\rangle \approx 1$, 如果属于K'谷状态(蓝色), 则$\langle{{\widehat{V}}_{z}}\rangle \approx -1$); (g), (h) 晶格重构引起的形变势能$ {V}_{D} $与赝磁场${\boldsymbol B}=\nabla \times {\boldsymbol A}$ (红色箭头代表矢势A(r), TBG的转角固定为$ 1.05^{\circ} $^[102])

Figure 8. (a) Schematic of the atomic configuration of TBG/hBN; (b) top view and high-symmetry stacking regions of the atomic configuration of TBG/hBN; (c) in-plane strain u(r) and in-plane rotation angle $\Delta \theta =\nabla \times {\boldsymbol u}$ with varying $ {\theta }_{\mathrm{b}\mathrm{o}\mathrm{t}} $ (The in-plane displacements are visualized with white arrows); (d)–(f) band structure and DOS of TBG/hBN with different $ {\theta }_{\mathrm{b}\mathrm{o}\mathrm{t}} $ (The color bar denotes the band for each valley $ \langle{{\widehat{V}}_{z}}\rangle $ with $ \langle{{\widehat{V}}_{z}}\rangle\approx 1 $ if a state belongs to valley K and $ \langle{{\widehat{V}}_{z}}\rangle\approx -1 $ if a state belongs to valley K'); (g) the deformation potential $ {V}_{D} $ and (h) pseudo-magnetic field ${\boldsymbol B}=\nabla \times {\boldsymbol A}$ induced by lattice relaxations in the TBG/hBN with $ {\theta }_{\mathrm{b}\mathrm{o}\mathrm{t}}=0.53^{\circ} $ (The vector field A(r) is visualized with red arrows in (h) TBG is fixed to $ {\theta }_{\mathrm{t}\mathrm{b}\mathrm{g}}=1.05^{\circ} $ in all cases)^[102].

DownLoad: Full-Size Img PowerPoint

图 9 (a) 三明治结构hBN/TBG/hBN的形貌图和高对称堆垛结构; (b)—(d) 不同$ {\theta }_{\mathrm{t}\mathrm{o}\mathrm{p}} $和$ {\theta }_{\mathrm{b}\mathrm{o}\mathrm{t}} $组成体系的(从左到右)能带图、面内转角、形变势能和赝磁场 (TBG的转角固定为$ {\theta }_{\mathrm{t}\mathrm{b}\mathrm{g}}=1.05^{\circ} $)^[102]

Figure 9. (a) Schematic structure of the hBN/TBG/hBN system and the different high-symmetry stackings in the superlattice; (b)–(d) panels from left to right display the band structure, in-plane twist of the atoms with respect to their original position, scalar potential and pseudo-magnetic field of the systems with different $ {\theta }_{\mathrm{t}\mathrm{o}\mathrm{p}} $ and $ {\theta }_{\mathrm{b}\mathrm{o}\mathrm{t}} $ (TBG is fixed to $ 1.05^{\circ} $^[102]).

DownLoad: Full-Size Img PowerPoint

[1]	Geim A K, Grigorieva I V 2013 Nature 499 419 Google Scholar
[2]	Chang C, Chen W, Chen Y, et al. 2021 Acta Phys. Chim. Sin. 37 2108017 Google Scholar
[3]	Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 666 Google Scholar
[4]	Bistritzer R, MacDonald A H 2011 Proc. Natl. Acad. Sci. USA 108 12233 Google Scholar
[5]	Yan W, Liu M, Dou R F, Meng L, Feng L, Chu Z D, Zhang Y, Liu Z, Nie J C, He L 2012 Phys. Rev. Lett. 109 126801 Google Scholar
[6]	Cao Y, Fatemi V, Demir A, et al. 2018 Nature 556 80 Google Scholar
[7]	Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E, Jarillo-Herrero P 2018 Nature 556 43 Google Scholar
[8]	Polshyn H, Zhu J, Kumar M A, Zhang Y, Yang F, Tschirhart C L, Serlin M, Watanabe K, Taniguchi T, MacDonald A H, Young A F 2020 Nature 588 66 Google Scholar
[9]	Shen C, Ying J, Liu L, Liu J, Li N, Wang S, Tang J, Zhao Y, Chu Y, Watanabe K, Taniguchi T, Yang R, Shi D, Qu F, Lu L, Yang W, Zhang G 2021 Chin. Phys. Lett. 38 047301 Google Scholar
[10]	Yankowitz M, Chen S, Polshyn H, Zhang Y, Watanabe K, Taniguchi T, Graf D, Young A F, Dean C R 2019 Science 363 1059 Google Scholar
[11]	Zheng Z, Ma Q, Bi Z, et al. 2020 Nature 588 71 Google Scholar
[12]	Sharpe A L, Fox E J, Barnard A W, Finney J, Watanabe K, Taniguchi T, Kastner M A, Goldhaber-Gordon D 2019 Science 365 605 Google Scholar
[13]	Serlin M, Tschirhart C, Polshyn H, Zhang Y, Zhu J, Watanabe K, Taniguchi T, Balents L, Young A 2020 Science 367 900 Google Scholar
[14]	Carr S, Massatt D, Fang S, Cazeaux P, Luskin M, Kaxiras E 2017 Phys. Rev. B 95 075420 Google Scholar
[15]	Gargiulo F, Yazyev O V 2017 2D Mater. 5 015019
[16]	van Wijk M, Schuring A, Katsnelson M, Fasolino A 2015 2D Mater. 2 034010
[17]	Nam N N T, Koshino M 2017 Phys. Rev. B 96 075311 Google Scholar
[18]	Yoo H, Engelke R, Carr S, et al. 2019 Nat. Mater. 18 448 Google Scholar
[19]	Shi H, Zhan Z, Qi Z, Huang K, Veen E V, Silva-Guillen J A, Zhang R, Li P, Xie K, Ji H, Katsnelson M I, Yuan S, Qin S, Zhang Z 2020 Nat. Commun. 11 371 Google Scholar
[20]	Huang S, Kim K, Efimkin D K, Lovorn T, Taniguchi T, Watanabe K, MacDonald A H, Tutuc E, LeRoy B J 2018 Phys. Rev. Lett. 121 037702 Google Scholar
[21]	Oh M, Nuckolls K P, Wong D, Lee R L, Liu X, Watanabe K, Taniguchi T, Yazdani A 2021 Nature 600 240 Google Scholar
[22]	Stepanov P, Das I, Lu X, Fahimniya A, Watanabe K, Taniguchi T, Koppens F H L, Lischner J, Levitov L, Efetov D K 2020 Nature 583 375 Google Scholar
[23]	Artaud A, Magaud L, Le Quang T, Guisset V, David P, Chapelier C, Coraux J 2016 Sci. Rep. 6 1 Google Scholar
[24]	Huder L, Artaud A, Le Quang T, de Laissardiere G T, Jansen A G M, Lapertot G, Chapelier C, Renard V T 2018 Phys. Rev. Lett. 120 156405 Google Scholar
[25]	Dos Santos J L, Peres N, Neto A C 2007 Phys. Rev. Lett. 99 256802 Google Scholar
[26]	Koshino M 2015 New J. Phys. 17 015014 Google Scholar
[27]	Koshino M, Yuan N F, Koretsune T, Ochi M, Kuroki K, Fu L 2018 Phys. Rev. X 8 031087
[28]	Popov A M, Lebedeva I V, Knizhnik A A, Lozovik Y E, Potapkin B V 2011 Phys. Rev. B 84 045404 Google Scholar
[29]	Lebedeva I V, Knizhnik A A, Popov A M, Lozovik Y E, Potapkin B V 2011 Phys. Chem. Chem. Phys. 13 5687 Google Scholar
[30]	Gould T, Lebegue S, Dobson J F 2013 J Phys. Condens. Matter. 25 445010 Google Scholar
[31]	Slater J C, Koster G F 1954 Phys. Rev. 94 1498 Google Scholar
[32]	Trambly de Laissardière G, Mayou D, Magaud L 2010 Nano Lett. 10 804 Google Scholar
[33]	Yuan S, De Raedt H, Katsnelson M I 2010 Phys. Rev. B 82 115448 Google Scholar
[34]	Yuan S, Roldán R, Katsnelson M I 2011 Phys. Rev. B 84 035439 Google Scholar
[35]	Logemann R, Reijnders K, Tudorovskiy T, Katsnelson M, Yuan S 2015 Phys. Rev. B 91 045420 Google Scholar
[36]	Hams A, De Raedt H 2000 Phys. Rev. E 62 4365 Google Scholar
[37]	Li Y, Zhan Z, Li Y, Yuan S 2022 arXiv:2209.00806 [cond-mat.mtrl-sci]
[38]	Brihuega I, Mallet P, Gonzalez-Herrero H, et al. 2012 Phys. Rev. Lett. 109 196802 Google Scholar
[39]	De Laissardiere G T, Mayou D, Magaud L 2012 Phys. Rev. B 86 125413 Google Scholar
[40]	Guinea F, Walet N R 2019 Phys. Rev. B 99 205134 Google Scholar
[41]	Slotman G J, van Wijk M M, Zhao P L, Fasolino A, Katsnelson M I, Yuan S 2015 Phys. Rev. Lett. 115 186801 Google Scholar
[42]	Gadelha A C, Ohlberg D A A, Rabelo C, et al. 2021 Nature 590 405 Google Scholar
[43]	Kuang X H, Zhan Z, Yuan S J 2021 Phys. Rev. B 103 115431 Google Scholar
[44]	Ju L, Shi Z W, Nair N, Lv Y C, Jin C H, Velasco J, Ojeda-Aristizabal C, Bechtel H A, Martin M C, Zettl A, Analytis J, Wang F 2015 Nature 520 650 Google Scholar
[45]	Sunku S, Ni G, Jiang B Y, Yoo H, Sternbach A, McLeod A, Stauber T, Xiong L, Taniguchi T, Watanabe K 2018 Science 362 1153 Google Scholar
[46]	Zhang F, MacDonald A H, Mele E J 2013 Proc. Natl. Acad. Sci. USA 110 10546 Google Scholar
[47]	Xu S G, Berdyugin A I, Kumaravadivel P, et al. 2019 Nat. Commun. 10 4008 Google Scholar
[48]	Verbakel J, Yao Q, Sotthewes K, Zandvliet H 2021 Phys. Rev. B 103 165134 Google Scholar
[49]	Lu X, Lian B, Chaudhary G, Piot B A, Romagnoli G, Watanabe K, Taniguchi T, Poggio M, MacDonald A H, Bernevig B A, Efetov D K 2021 Proc. Natl. Acad. Sci. USA 118 e2100006118 Google Scholar
[50]	Nguyen V H, Paszko D, Lamparski M, Van Troeye B, Meunier V, Charlier J C 2021 2D Mater. 8 035046
[51]	Plimpton S 1995 J. Comput. Phys. 117 1 Google Scholar
[52]	Los J H, Fasolino A 2003 Phys. Rev. B 68 024107 Google Scholar
[53]	Kolmogorov A N, Crespi V H 2005 Phys. Rev. B 71 235415 Google Scholar
[54]	Zhou H, Xie T, Taniguchi T, Watanabe K, Young A F 2021 Nature 598 434 Google Scholar
[55]	Zhou H, 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 598 429 Google Scholar
[56]	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 Nat. Phys. 15 237 Google Scholar
[57]	Chen G, Sharpe A L, Gallagher P, et al. 2019 Nature 572 215 Google Scholar
[58]	Carr S, Li C, Zhu Z, Kaxiras E, Sachdev S, Kruchkov A 2020 Nano Lett. 20 3030 Google Scholar
[59]	Mora C, Regnault N, Bernevig B A 2019 Phys. Rev. Lett. 123 026402 Google Scholar
[60]	Zhu Z, Carr S, Massatt D, Luskin M, Kaxiras E 2020 Phys. Rev. Lett. 125 116404 Google Scholar
[61]	Ma Z, Li S, Zheng Y W, Xiao M M, Jiang H, Gao J H, Xie X 2021 Sci. Bull. 66 18 Google Scholar
[62]	Haddadi F, Wu Q, Kruchkov A J, Yazyev O V 2020 Nano Lett. 20 2410 Google Scholar
[63]	Rubio-Verdú C, Turkel S, Song Y, et al. 2022 Nat. Phys. 18 196 Google Scholar
[64]	Wu Z, Zhan Z, Yuan S 2021 Sci. China Phys. Mech. Astron. 64 267811 Google Scholar
[65]	Park J M, Cao Y, Watanabe K, Taniguchi T, Jarillo-Herrero P 2021 Nature 590 249 Google Scholar
[66]	Cao Y, Park J M, Watanabe K, Taniguchi T, Jarillo-Herrero P 2021 Nature 595 526 Google Scholar
[67]	Hao Z, Zimmerman A, Ledwith P, Khalaf E, Najafabadi D H, Watanabe K, Taniguchi T, Vishwanath A, Kim P 2021 Science 371 1133 Google Scholar
[68]	Zhang X, Tsai K T, Zhu Z, Ren W, Luo Y, Carr S, Luskin M, Kaxiras E, Wang K 2021 Phys. Rev. Lett. 127 166802 Google Scholar
[69]	Turkel S, Swann J, Zhu Z, Christos M, Watanabe K, Taniguchi T, Sachdev S, Scheurer M S, Kaxiras E, Dean C R, Pasupathy A N 2022 Science 376 193 Google Scholar
[70]	Chen S, He M, Zhang Y H, Hsieh V, Fei Z, Watanabe K, Taniguchi T, Cobden D H, Xu X, Dean C R, Yankowitz M 2021 Nat. Phys. 17 374 Google Scholar
[71]	Xu S, Al Ezzi M M, Balakrishnan N, et al. 2021 Nat. Phys. 17 619 Google Scholar
[72]	Wu Z, Kuang X, Zhan Z, Yuan S 2021 Phys. Rev. B 104 205104 Google Scholar
[73]	Carr S, Fang S, Jarillo-Herrero P, Kaxiras E 2018 Phys. Rev. B 98 085144 Google Scholar
[74]	Kretinin A V, Cao Y, Tu J S, et al. 2014 Nano Lett. 14 3270 Google Scholar
[75]	Giovannetti G, Khomyakov P A, Brocks G, Karpan V M, van den Brink J, Kelly P J 2008 Phys. Rev. Lett. 101 026803 Google Scholar
[76]	Dean C R, Young A F, Meric I, Lee C, Wang L, Sorgenfrei S, Watanabe K, Taniguchi T, Kim P, Shepard K L, Hone J 2010 Nat. Nanotechnol. 5 722 Google Scholar
[77]	Moon P, Koshino M 2014 Phys. Rev. B 90 155406 Google Scholar
[78]	San-Jose P, Gutiérrez-Rubio A, Sturla M, Guinea F 2014 Phys. Rev. B 90 075428 Google Scholar
[79]	Van Wijk M, Schuring A, Katsnelson M, Fasolino A 2014 Phys. Rev. Lett. 113 135504 Google Scholar
[80]	Ponomarenko L A, Gorbachev R V, Yu G L, et al. 2013 Nature 497 594 Google Scholar
[81]	Dean C R, Wang L, Maher P, et al. 2013 Nature 497 598 Google Scholar
[82]	Hunt B, Sanchez-Yamagishi J D, Young A F, Yankowitz M, LeRoy B J, Watanabe K, Taniguchi T, Moon P, Koshino M, Jarillo-Herrero P, Ashoori R C 2013 Science 340 1427 Google Scholar
[83]	Barrier J, Kumaravadivel P, Krishna Kumar R, et al. 2020 Nat. Commun. 11 5756 Google Scholar
[84]	Wang L, Zihlmann S, Liu M H, Makk P, Watanabe K, Taniguchi T, Baumgartner A, Schönenberger C 2019 Nano Lett. 19 2371 Google Scholar
[85]	Wang Z, Wang Y B, Yin J, et al. 2019 Sci. Adv. 5 eaay8897 Google Scholar
[86]	Anđelković M, Milovanović S P, Covaci L, Peeters F M 2020 Nano Lett. 20 979 Google Scholar
[87]	Yang Y, Li J, Yin J, Xu S, Mullan C, Taniguchi T, Watanabe K, Geim A K, Novoselov K S, Mishchenko A 2020 Sci. Adv. 6 eabd3655 Google Scholar
[88]	Jung J, DaSilva A M, MacDonald A H, Adam S 2015 Nat. Commun. 6 6308 Google Scholar
[89]	Chen Z G, Shi Z, Yang W, Lu X, Lai Y, Yan H, Wang F, Zhang G, Li Z 2014 Nat. Commun. 5 4461 Google Scholar
[90]	Wang E, Lu X, Ding S, Yao W, Yan M, Wan G, Deng K, Wang S, Chen G, Ma L, Jung J, Fedorov A V, Zhang Y, Zhang G, Zhou S 2016 Nat. Phys. 12 1111 Google Scholar
[91]	Han T, Yang J, Zhang Q, Wang L, Watanabe K, Taniguchi T, McEuen P L, Ju L 2021 Phys. Rev. Lett. 126 146402 Google Scholar
[92]	Kim H, Leconte N, Chittari B L, Watanabe K, Taniguchi T, MacDonald A H, Jung J, Jung S 2018 Nano Lett. 18 7732 Google Scholar
[93]	Bultinck N, Chatterjee S, Zaletel M P 2020 Phys. Rev. Lett. 124 166601 Google Scholar
[94]	Zhang Y H, Mao D, Senthil T 2019 Phys. Rev. Res. 1 033126 Google Scholar
[95]	Liu X, Wang Z, Watanabe K, Taniguchi T, Vafek O, Li J 2021 Science 371 1261 Google Scholar
[96]	Brenner D W, Shenderova O A, Harrison J A, Stuart S J, Ni B, Sinnott S B 2002 J. Condens. Matter. Phys. 14 783 Google Scholar
[97]	Ochoa H, Castro E V, Katsnelson M, Guinea F 2011 Phys. Rev. B 83 235416 Google Scholar
[98]	Xue J, Sanchez-Yamagishi J, Bulmash D, Jacquod P, Deshpande A, Watanabe K, Taniguchi T, Jarillo-Herrero P, LeRoy B J 2011 Nat. Mater. 10 282 Google Scholar
[99]	Yankowitz M, Xue J, Cormode D, Sanchez-Yamagishi J D, Watanabe K, Taniguchi T, Jarillo-Herrero P, Jacquod P, LeRoy B J 2012 Nat. Phys. 8 382 Google Scholar
[100]	Wallbank J, Patel A, Mucha-Kruczyński M, Geim A, Fal'Ko V 2013 Phys. Rev. B 87 245408 Google Scholar
[101]	Park C H, Yang L, Son Y W, Cohen M L, Louie S G 2008 Phys. Rev. Lett. 101 126804 Google Scholar
[102]	Long M, Pantaleón P A, Zhan Z, Guinea F, Silva-Guillén J Á, Yuan S 2022 npj Comput. Mater. 8 1 Google Scholar
[103]	Cea T, Pantaleón P A, Guinea F 2020 Phys. Rev. B 102 155136 Google Scholar
[104]	Mao D, Senthil T 2021 Phys. Rev. B 103 115110 Google Scholar
[105]	Shin J, Park Y, Chittari B L, Sun J H, Jung J 2021 Phys. Rev. B 103 075423 Google Scholar
[106]	Ramires A, Lado J L 2019 Phys. Rev. B 99 245118 Google Scholar
[107]	Zhang Z, Wang Y, Watanabe K, Taniguchi T, Ueno K, Tutuc E, LeRoy B J 2020 Nat. Phys. 16 1093 Google Scholar
[108]	Zhan Z, Zhang Y, Lv P, Zhong H, Yu G, Guinea F, Silva-Guillén J Á, Yuan S 2020 Phys. Rev. B 102 241106 Google Scholar
[109]	Naik M H, Jain M 2018 Phys. Rev. Lett. 121 266401 Google Scholar
[110]	Xu Y, Liu S, Rhodes D A, Watanabe K, Taniguchi T, Hone J, Elser V, Mak K F, Shan J 2020 Nature 587 214 Google Scholar
[111]	Li T, Jiang S, Shen B, Zhang Y, Li L, Tao Z, Devakul T, Watanabe K, Taniguchi T, Fu L, Shan J, Mak K F 2021 Nature 600 641 Google Scholar
[112]	Li T, Jiang S, Li L, Zhang Y, Kang K, Zhu J, Watanabe K, Taniguchi T, Chowdhury D, Fu L, Shan J, Mak K F 2021 Nature 597 350 Google Scholar
[113]	Ghiotto A, Shih E M, Pereira G, Rhodes D A, Kim B, Zang J, Millis A J, Watanabe K, Taniguchi T, Hone J C, Wang L, Dean C R, Pasupathy A N 2021 Nature 597 345 Google Scholar
[114]	Weston A, Castanon E G, Enaldiev V, et al. 2022 Nat. Nanotechnol. 17 390 Google Scholar

[1]	Liu Zhao. Fractionalized topological states in moiré superlattices. Acta Physica Sinica, 2024, 73(20): 207303. doi: 10.7498/aps.73.20241029
[2]	Guo Rui-Ping, Yu Hong-Yi. Position- and momentum-dependent interlayer couplings in two-dimensional semiconductor moiré superlattices. Acta Physica Sinica, 2023, 72(2): 027302. doi: 10.7498/aps.72.20222046
[3]	Wu Ze-Fei, Huang Mei-Zhen, Wang Ning. Nonlinear Hall effects in two-dimensional moiré superlattices. Acta Physica Sinica, 2023, 72(23): 237301. doi: 10.7498/aps.72.20231324
[4]	Li Ting-Xin. Recent experimental research progress of two-dimensional van der Waals semiconductor moiré superlattices. Acta Physica Sinica, 2022, 71(12): 127309. doi: 10.7498/aps.71.20220347
[5]	Xiao Mei-Xia, Leng Hao, Song Hai-Yang, Wang Lei, Yao Ting-Zhen, He Cheng. Effects of organic molecule adsorption and substrate on electronic structure of germanene. Acta Physica Sinica, 2021, 70(6): 063101. doi: 10.7498/aps.70.20201657
[6]	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
[7]	Li Xiao-Bing, Lu Wei-Bing, Liu Zhen-Guo, Chen Hao. Dynamic beam-steering in wide angle range based on tunable graphene metasurface. Acta Physica Sinica, 2018, 67(18): 184101. doi: 10.7498/aps.67.20180592
[8]	Wang Yue, Leng Yan-Bing, Wang Li, Dong Lian-He, Liu Shun-Rui, Wang Jun, Sun Yan-Jun. Tunable grapheme amplitude based broadband electromagnetically-induced-transparency-like metamaterial. Acta Physica Sinica, 2018, 67(9): 097801. doi: 10.7498/aps.67.20180114
[9]	Yan Xin, Liang Lan-Ju, Zhang Zhang, Yang Mao-Sheng, Wei De-Quan, Wang Meng, Li Yuan-Ping, Lü Yi-Ying, Zhang Xing-Fang, Ding Xin, Yao Jian-Quan. Dynamic multifunctional control of terahertz beam based on graphene coding metamaterial. Acta Physica Sinica, 2018, 67(11): 118102. doi: 10.7498/aps.67.20180125
[10]	Chen Cai-Yun, Liu Jin-Xing, Zhang Xiao-Min, Li Jin-Long, Ren Ling-Ling, Dong Guo-Cai. Coverage measurement of graphene film on metallic substrate using scanning electron microscopy. Acta Physica Sinica, 2018, 67(7): 076802. doi: 10.7498/aps.67.20172654
[11]	Zhang Yin, Feng Yi-Jun, Jiang Tian, Cao Jie, Zhao Jun-Ming, Zhu Bo. Graphene based tunable metasurface for terahertz scattering manipulation. Acta Physica Sinica, 2017, 66(20): 204101. doi: 10.7498/aps.66.204101
[12]	Guo Hui, Lu Hong-Liang, Huang Li, Wang Xue-Yan, Lin Xiao, Wang Ye-Liang, Du Shi-Xuan, Gao Hong-Jun. Intercalation and its mechanism of high quality large area graphene on metal substrate. Acta Physica Sinica, 2017, 66(21): 216803. doi: 10.7498/aps.66.216803
[13]	Gu Ji-Wei, Wang Jin-Cheng, Wang Zhi-Jun, Li Jun-Jie, Guo Can, Tang Sai. Phase-field crystal modelling the nucleation processes of graphene structures on different substrates. Acta Physica Sinica, 2017, 66(21): 216101. doi: 10.7498/aps.66.216101
[14]	Zhang Hui-Yun, Huang Xiao-Yan, Chen Qi, Ding Chun-Feng, Li Tong-Tong, Lü Huan-Huan, Xu Shi-Lin, Zhang Xiao, Zhang Yu-Ping, Yao Jian-Quan. Tunable terahertz absorber based on complementary graphene meta-surface. Acta Physica Sinica, 2016, 65(1): 018101. doi: 10.7498/aps.65.018101
[15]	Jin Qin, Dong Hai-Ming, Han Kui, Wang Xue-Feng. Ultrafast dynamic optical properties of graphene. Acta Physica Sinica, 2015, 64(23): 237801. doi: 10.7498/aps.64.237801
[16]	Lu Xiao-Bo, Zhang Guang-Yu. Graphene/h-BN Moiré superlattice. Acta Physica Sinica, 2015, 64(7): 077305. doi: 10.7498/aps.64.077305
[17]	Wang Lang, Feng Wei, Yang Lian-Qiao, Zhang Jian-Hua. The pre-treatment of copper for graphene synthesis. Acta Physica Sinica, 2014, 63(17): 176801. doi: 10.7498/aps.63.176801
[18]	Huang Xiang-Qian, Lin Chen-Fang, Yin Xiu-Li, Zhao Ru-Guang, Wang En-Ge, Hu Zong-Hai. Hydrogen adsorption on one-dimensional graphene superlattices. Acta Physica Sinica, 2014, 63(19): 197301. doi: 10.7498/aps.63.197301
[19]	Han Tong-Wei, He Peng-Fei. Molecular dynamics simulation of relaxation properties of graphene sheets. Acta Physica Sinica, 2010, 59(5): 3408-3413. doi: 10.7498/aps.59.3408
[20]	CHEN GUANG-HUA, DENG JIN-XIANG, ZHANG SHENG-JUN, SONG XUE-MEI, WANG BO, YAN HUI. INFLUENCE OF SUBSTRATES ON THE FORMATION OF c-BN THIN FILMS. Acta Physica Sinica, 2001, 50(1): 83-87. doi: 10.7498/aps.50.83

Catalog

Vol. 71, Issue 18 - 2022-09-20

Metrics

Abstract views: 8229
PDF Downloads: 373
Cited By: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板