Theoretical studies on optical absorption in twisted bilayer graphene under vertical electric field

Zhou Chang; Gong Rui; Feng Xiao-Bo

doi:10.7498/aps.71.20211406

Abstract

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.

Keywords:

Authors and contacts

1.
School of Physics and Electronic Information, Yunnan Normal University, Kunming 650500, China
2.
Yunnan Key Laboratory of Opto-electronic Information Technology, Yunnan Normal University, Kunming 650500, China
3.
Key Laboratory of Advanced Technique & Preparation for Renewable Energy Materials, Ministry of Education, Yunnan Normal University, Kunming 650500, China

Corresponding author: Feng Xiao-Bo, fengxiaobo1220@gmail.com

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11304275, 11764047).

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References

[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
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[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
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[41]	McCann E, Fal’ko V I 2006 Phys. Rev. Lett. 96 086805 Google Scholar

Cited By

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 K^θ 为来自于上下两个单层石墨烯的狄拉克点

Figure 1. (a) Atomic structure for tBLG, θ is the twist angle between layers; (b) the first Brillouin region of tBLG, where K and K^θ are the Dirac points of the upper and lower monolayers.

DownLoad: Full-Size Img PowerPoint

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

Figure 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.

DownLoad: Full-Size Img PowerPoint

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

Figure 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.

DownLoad: Full-Size Img PowerPoint

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

Figure 4. Optical spectrums for tBLG around VHS with 5 different twisted angles: (a) Optical spectrums for interband transitions; (b) optical spectrums for intraband transitions.

DownLoad: Full-Size Img PowerPoint

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

Figure 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.

DownLoad: Full-Size Img PowerPoint

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°

Figure 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°.

DownLoad: Full-Size Img PowerPoint

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
U/meV	k₁/ΔK	k₂/ΔK	ΔE_α1/eV	ΔE_α2/eV	ΔE_β/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

DownLoad: CSV

[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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