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层间扭转角度是对石墨烯物理性质宽波段可调谐的一个新参量. 本文采用2° < θ < 15°扭转角度下的连续近似模型, 获得了不同扭转角度双层石墨烯分别在有、无电场下的能带结构, 通过电子-光子相互作用跃迁速率, 计算模拟了范霍夫奇点附近电子带内跃迁和带间跃迁所引起的光学吸收谱. 结果表明, 在无外加电场时, 带间跃迁吸收峰的位置随着扭转角度的增大而发生从红外到可见光波段的蓝移, 且吸收系数增大, 带内跃迁的光学吸收系数相对于带间跃迁高出2个数量级; 而存在外加电场时, 两个范霍夫奇点在波矢空间的位置发生偏移, 带间跃迁吸收峰发生分裂, 且两个分裂的吸收峰位置随着电场强度的不断增大而反向行进. 上述研究结果对石墨烯材料在光电器件方面的应用有一定指导作用.The interlayer twist angle is an important parameter that can tune the physical properties of graphene in a wide wavelength range. In this paper, we employ an effective continuum model to calculate the band structure of twisted bilayer graphene with different twist angles in the presence and absence of vertical electric field. Based on the transition rate of the electron-photon interaction, we calculate and simulate the optical absorption spectra caused by the interband and intraband transitions around the van Hove singularities. The calculation results show that the optical absorption caused by the interband transitions occurs in the wavelength range from visible light to near-infrared while it appears in far-infrared for intraband transitions. The optical absorption coefficient of the intra-band transitions is almost two orders of magnitude larger than that of inter-band transitions. In the absence of an external electric field, as the twist angle increases, the absorption peak of the inter band transition moves from the infrared light band to the visible light band, but the resonant peak position of its intra-band transition does not change. At the same time, the absorption coefficient values corresponding to the above two transitions will increase. When an electric field is applied perpendicular to the twisted bilayer graphene, the symmetry of the initial band structure of bilayer graphene is destroyed, which results in the splitting of the absorption peaks associated the with interband transitions, and the distance between the two splitting peaks increases with the electric field intensity increasing; while the position and amplitude of the absorption peak associated with the intraband transition are completely unaffected by the applied electric field. The theoretical calculation results in this paper can provide the theoretical guidance for further applying twisted graphene to optoelectronic devices such as tunable dual-band filters.
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Keywords:
- optical absorption /
- bilayer graphene /
- twist angle /
- external electric field
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图 1 (a) tBLG的原子结构图, θ为层间扭转角度; (b) tBLG的第一布里渊区, K 和 Kθ 为来自于上下两个单层石墨烯的狄拉克点
Fig. 1. (a) Atomic structure for tBLG, θ is the twist angle between layers; (b) the first Brillouin region of tBLG, where K and Kθ are the Dirac points of the upper and lower monolayers.
图 2 无外电场时θ = 3.89°的tBLG三维能带结构图. K和K θ为两个狄拉克点, VHS1和VHS2为两个范霍夫奇点
Fig. 2. The 3D band structure of tBLG at θ = 3.89° without external electric field. K and K θ are two separate Dirac points, VHS1 and VHS2 are two van Hove singularities.
图 3 无外电场时四个扭转角度下tBLG的能带结构图 (a) θ = 3.89°; (b) θ = 5.09°; (c) θ = 7.34°; (d) θ = 9.43°. 其中红色(蓝色)箭头表示可能发生的带间(带内)跃迁, 蓝色虚线为费米面
Fig. 3. The energy band structure of TBG at four twisted angles without electric field: (a) θ = 3.89°; (b) θ = 5.09°; (c) θ = 7.34°; (d) θ = 9.43°. Red (blue) arrow is the possible interband (intraband) transitions, and the dashed blue line shows the Fermi level.
图 4 不同扭转角度下tBLG在范霍夫奇点附近的光学吸收谱 (a) 带间跃迁光学吸收谱; (b) 带内跃迁光学吸收谱
Fig. 4. Optical spectrums for tBLG around VHS with 5 different twisted angles: (a) Optical spectrums for interband transitions; (b) optical spectrums for intraband transitions.
图 5 不同外加电场下tBLG的能带结构图(θ = 3.89°) (a) U = 0 meV; (b) U = 40 meV; (c) U = 60 meV; (d) U = 100 meV. 其中箭头表示可能发生的带间或者带内跃迁, 蓝色虚线为费米面, 黑色虚线表示范霍夫奇点的位置
Fig. 5. The energy band structure of tBLG at θ = 3.89° with or without electric field: (a) U = 0 meV; (b) U = 40 meV (c) U = 60 meV; (d) U = 100 meV. The arrows are the possible interband (intraband) transitions, and the dashed blue line shows the Fermi level, and the dashed black lines are the positions of VHS.
图 6 不同电场强度下tBLG范霍夫奇点附近的光学吸收谱, 三种扭转角度下的带间跃迁吸收谱(左列)和带内跃迁吸收谱(右列) (a), (b) θ = 3.89°; (c), (d) θ = 6.01°; (e), (f) θ = 9.43°
Fig. 6. Optical spectra for tBLG around VHS with different perpendicular electric field intensities, and optical spectra for interband transitions (left column) and Optical spectrums for intraband transitions (right column) under three twist angles: (a), (b) θ = 3.89°; (c), (d) θ = 6.01°; (e), (f) θ = 9.43°.
表 1 不同电场强度下tBLG能带结构几个关键点的位置及带间能级差(θ = 3.89°)
Table 1. Positions of key points and energy level differences (θ = 3.89°) in the band structure of tBLG under different electric field intensities.
U/meV VHS奇点位置 带间跃迁能级差 带内跃迁能级差 k1/ΔK k2/ΔK ΔEα1/eV ΔEα2/eV ΔEβ/eV 0 0 0 0.7550 0.7550 0.11 20 –0.015 0.015 0.7511 0.7588 0.1101 40 –0.030 0.030 0.7306 0.7692 0.1101 60 –0.040 0.040 0.7289 0.7810 0.11 80 –0.055 0.055 0.7124 0.7976 0.11 100 –0.065 0.065 0.6977 0.8124 0.11 
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[1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Gregorieva I V, Firsov A A 2004 Science 306 666
Google Scholar
[2] Castro Neto A H, Guinea F, Peres N M R, Novoselov K S, Geim A K 2009 Rev. Mod. Phys. 81 109
Google Scholar
[3] Zhou S Y, Gweon G H, Graf J, Fedorov A V, Spataru C D, Diehl R D, Kopelevich Y, Lee D H, Louie S G, Lanzara A 2006 Nat. Phys. 2 595
Google Scholar
[4] Mak K F, Sfeir M Y, Wu Y, Lui C H, Misewich J A, Heinz T F 2008 Phys. Rev. Lett. 101 196405
Google Scholar
[5] Stauber T, Peres N, Geim A K 2008 Phys. Rev. B 78 085432
Google Scholar
[6] Rutter G M, Crain J N, Guisinger N P, Li T, First P N, Stroscio J A 2007 Science 317 219
Google Scholar
[7] Geim A K, Grigorieva I V 2013 Nature 499 419
Google Scholar
[8] Dai S, Xiang Y, Srolovitz D J 2016 Nano Lett. 16 5923
Google Scholar
[9] Li G, Luican A, Santos J, Neto A, Reina A, Kong J, Andrei E Y 2009 Nature 6 109
Google Scholar
[10] Yan W, Liu M X, Dou R F, Meng L, Feng L, Chu Z D, Zhang Y F 2012 Phys. Rev. Lett. 109 126801
Google Scholar
[11] Bistritzer R, MacDonald A H 2011 Proc. Natl. Acad. Sci. U. S. A. 108 12233
Google Scholar
[12] Bistritzer R, MacDonald A H 2011 Phys. Rev. B 84 035440
Google Scholar
[13] Moon P, Koshino M 2012 Phys. Rev. B 85 195458
Google Scholar
[14] Cao Y, Fatemi V, Fang S, Watanabe K, Taniguchi T, Kaxiras E, Jarillo-Herrero P 2018 Nature 556 43
Google Scholar
[15] Liu J P, Dai X 2020 NPJ Comput. Mater. 6 57
Google Scholar
[16] Yang F, Song W, Meng F, Luo F, Luo S, Lin S, Gong Z, Cao J, Bernard E, Chan E, Yang L, Yao J 2020 Matter 3 1361
Google Scholar
[17] Deng B, Ma C, Wang Q, Yuan S, Watanabe K, Taniguchi T, Zhang F, Xia F 2020 Nat. Photonics 14 549
Google Scholar
[18] Yin J, Wang H, Peng H, Tan Z, Liao L, Lin L, Sun X, Koh A L, Chen Y, Peng H, Liu Z 2016 Nat. Commun. 7 10699
Google Scholar
[19] Ha S, Park N H, Kim H, Shin J, Choi J, Park S, Moon J, Chae K, Jung J, Lee J, Yoo Y, Park J, Ahn K J, Yeom D 2021 Light-Sci. Appl. 10 19
Google Scholar
[20] Yu K, Luan N V, Kim T, Jeon J, Kim J, Moon P, Lee Y H, Choi E J 2019 Phys. Rev. B 99 241405
Google Scholar
[21] Moon P, Koshino M 2013 Phys. Rev. B 87 205404
Google Scholar
[22] Moon P, Son Y, Koshino M 2014 Phys. Rev. B 90 155427
Google Scholar
[23] Tabert C Jand Nicol E J 2013 Phys. Rev. B 87 121402
Google Scholar
[24] Stauber T, San-Jose P, Brey L 2013 New J. Phys. 15 113050
Google Scholar
[25] McCann E 2006 Phys. Rev. B 74 161403
Google Scholar
[26] McCann E, Abergel D S L, Fal’ko V I 2007 Solid State Commun. 143 110
Google Scholar
[27] Aoki M, Amawashi H 2010 Solid State Commun. 142 123
Google Scholar
[28] Lu C L, Chang C P, Huang Y C, Chen R B, Lin M L 2006 Phys. Rev. B 73 144427
Google Scholar
[29] Brihuega I, Mallet P, González-Herrero H, Laissardière G T D, Ugeda M M, Magaud L, Gómez-Rodríguez J M, Ynduráin F, Veuillen J Y 2012 Phys. Rev. Lett. 109 196802
Google Scholar
[30] Yan W, Meng L, Liu M, Qiao J B, Chu Z D, Dou R F, Liu Z, Nie J C, Naugle D G, He L 2014 Phys. Rev. B 90 115402
Google Scholar
[31] Lopes dos Santos J M B, Peres N M R, Castro Neto A H 2007 Phys. Rev. Lett. 99 256802
Google Scholar
[32] Gail R D, Goerbig M O, Guinea F, Montambaux G, Neto A H C 2011 Phys. Rev. B 84 045436
Google Scholar
[33] Koshino M 2015 New J. Phys. 17 015014
Google Scholar
[34] Yin L, Qiao J, Wang W, Zuo W, Yan W, Xu R, Dou R, Nie J, He L 2015 Phys. Rev. B 92 201408
Google Scholar
[35] San-Jose P, Prada E 2013 Phys. Rev. B 88 121408
Google Scholar
[36] Morell E S, Correa J D, Vargas P, Pacheco M, Barticevic Z 2010 Phys. Rev. B 82 121407
Google Scholar
[37] Shallcross S, Sharma S, Pankratov O 2013 Phys. Rev. B 87 245403
Google Scholar
[38] Laissardière G T D, Mayou D, Magaud L 2010 Nano Lett. 10 804
Google Scholar
[39] Luican A, Li G, Reina A, Kong J, Nair R R, Novoselov K S, Geim A K, Andrei E Y 2011 Phys. Rev. Lett. 106 126802
Google Scholar
[40] Mele E J 2012 J. Phys. D: Appl. Phys. 45 154004
Google Scholar
[41] McCann E, Fal’ko V I 2006 Phys. Rev. Lett. 96 086805
Google Scholar
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