Interband optical conductivity in electromagnetic field modulated strained black phosphorene

Bai Wen-Qing; Yang Jiang-Tao; Yang Cui-Hong; Chen Yun-Yun

doi:10.7498/aps.73.20240445

Abstract

Black phosphorene (BP) has been widely investigated for its anisotropic and unique photoelectric properties. Strain, voltage and so on are commonly used to modulate the energy band structure and accordingly its photoelectric characteristics. In this study, we consider the energy band structure of BP in the vertical magnetic field, electric field, and in-plane/out-of-plane strains by using the tight-binding approximate Hamiltonian. The anisotropic frequency-dependent interband optical conductivity (IOC) of BP is investigated by using the Kubo formula in these modulation factors. Inherent asymmetry in band dispersion along the armchair (AC) direction and the zigzag (ZZ) direction leads to anisotropic IOC. The introduction of a vertical magnetic field induces band splitting, thereby generating multiple interband transition channels. In this case, the IOC along both the AC direction and the ZZ direction exhibits three peaks around the original peak position, and the magnitudes of the peaks are also modulated. With the increase of in-plane strain (from –20% to 20%), the band gap increases monotonically, and both the position and magnitude of the peaks vary with band gap changing. However, the band gap of BP undergoes a non-monotonic change under out-of-plane strain (from –20% to 20%), which is different from the change under in-plane strain. The band gap reaches a minimum value when a tensile strain of 12% is applied. Along the AC direction, the modulation of the IOC by in-plane strain is opposite to the modulation of out-of-plane strain (ε_z < 12%), indicating a competitive effect when triaxial strains are applied. Along the ZZ direction, in-plane strain primarily modulates the peak magnitude, while out-of-plane strain effectively modulates not only the peak position but also the peak magnitude obviously. The modulation of the IOC by forward and reverse electric fields are symmetrical. The coefficient for the peak position shift due to the vertical electric field is 1/2 in the AC direction and 1/10 in the ZZ direction. By integrating various modulation factors, we achieve versatile control over the energy band and IOC of BP, providing theoretical support for the application of BP in optoelectronic devices.

Keywords:

Authors and contacts

1.
School of Physics and Optoelectronic Engineering, Nanjing University of Information Science & Technology, Nanjing 210044, China
2.
Jiangsu Key Laboratory for Optoelectronic Detection of Atmosphere and Ocean, Nanjing University of Information Science & Technology, Nanjing 210044, China
3.
Jiangsu International Joint Laboratory on Meteorological Photonics and Optoelectronic Detection, Nanjing University of Information Science & Technology, Nanjing 210044, China

Corresponding author: Yang Cui-Hong, chyang@nuist.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11547030).

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References

[1]	Novoselov K S, Geim A K, Morozov S V, et al. 2004 Science 306 666 Google Scholar
[2]	Manzeli S, Ovchinnikov D, Pasquier D, Yazyev O V, Kis A 2017 Nat. Rev. Mater. 2 17033 Google Scholar
[3]	Li L K, Yu Y J, Ye G J, Ge Q Q, Ou X D, Wu H, Feng D L, Chen X H, Zhang Y B 2014 Nat. Nanotechnol. 9 372 Google Scholar
[4]	Liu H, Neal A T, Zhu Z, Luo Z, Xu X F, Tománek D, Ye P D 2014 ACS Nano 8 4033 Google Scholar
[5]	Xia F N, Wang H, Jia Y C 2014 Nat. Commun. 5 4458 Google Scholar
[6]	Jain A, McGaughey A J H 2015 Sci. Rep. 5 8501 Google Scholar
[7]	Qiao J S, Kong X H, Hu Z X, Yang F, Ji W 2014 Nat. Commun. 5 4475 Google Scholar
[8]	Ezawa M 2014 New J. Phys. 16 115004 Google Scholar
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[10]	Li L K, Kim J, Jin C H, et al. 2017 Nat. Nanotechnol. 12 21 Google Scholar
[11]	Kim J, Baik S S, Ryu S H, Sohn Y, Park S, Park B G, Denlinger J, Yi Y, Choi H J, Kim K S 2015 Science 349 723 Google Scholar
[12]	Peng X H, Wei Q, Copple A 2014 Phys. Rev. B 90 085402 Google Scholar
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[14]	Fei R, Yang L 2014 Nano Lett. 14 2884 Google Scholar
[15]	Dai J, Zeng X C 2014 J. Phys. Chem. Lett. 5 1289 Google Scholar
[16]	Chen X L, Lu X B, Deng B C, et al. 2017 Nat. Commun. 8 1672 Google Scholar
[17]	Li L K, Yang F Y, Ye G J, et al. 2016 Nat. Nanotechnol. 11 593 Google Scholar
[18]	Pereira Jr J M, Katsnelson M I 2015 Phys. Rev. B 92 075437 Google Scholar
[19]	Zhou X Y, Lou W K, Zhai F, Chang K 2015 Phys. Rev. B 92 165405 Google Scholar
[20]	Phuong L T T, Phong T C, Yarmohammadi M 2020 Sci. Rep. 10 9201 Google Scholar
[21]	Keshtan M A M, Esmaeilzadeh M 2015 J. Phys. D: Appl. Phys. 48 485301 Google Scholar
[22]	Wang Y, Guo Y L, Wang Z K, et al. 2021 ACS Nano 15 12069 Google Scholar
[23]	Wang Y, Xu W, Fu L, et al. 2023 ACS Appl. Mater. Interfaces 15 54797 Google Scholar
[24]	Wang Y, Xu W, Yang D Y, et al. 2023 ACS Nano 17 24320 Google Scholar
[25]	Li P K, Appelbaum I 2014 Phys. Rev. B 90 115439 Google Scholar
[26]	Le P T T, Yarmohammadi M 2019 J. Magn. Magn. Mater. 491 165629 Google Scholar
[27]	Rudenko A N, Katsnelson M I 2014 Phys. Rev. B 89 201408 Google Scholar
[28]	Yang C H, Zhang J Y, Wang G X, Zhang C 2018 Phys. Rev. B 97 245408 Google Scholar
[29]	Jiang J W, Park H S 2015 Phys. Rev. B 91 235118 Google Scholar
[30]	Khang P D, Davoudiniya M, Phuong L T T, Phong T C, Yarmohammadi M 2019 Phys. Chem. Chem. Phys. 21 15133 Google Scholar
[31]	Yang C H, Zhang J Y, Wieser R, Xu W 2022 J. Phys. D: Appl. Phys. 55 085103 Google Scholar
[32]	Le P T T, Mirabbaszadeh K, Yarmohammadi M 2019 J. Appl. Phys. 125 193101 Google Scholar

Cited By

图 1 黑磷烯晶格结构示意图　(a) 三维; (b) 二维

Figure 1. Schematic diagram of lattice structure of BP: (a) Three-dimensional; (b) two-dimensional.

DownLoad: Full-Size Img PowerPoint

图 2 黑磷烯能带示意图　(a) 磁场下; (b) 电场下; (c) x轴应变下

Figure 2. Energy band structure of BP in the presence: (a) Magnetic field; (b) electric field; (c) x-axis strain.

DownLoad: Full-Size Img PowerPoint

图 3 同时施加磁场和应变下的黑磷烯带隙(虚线1.52对应的是本征带隙的位置)　(a) 施加x轴应变 ; (b) 施加y轴应变; (c) 施加z轴应变

Figure 3. Bandgap of BP in the presence of the magnetic field and strain applied along (Dashed line corresponds to the position of the intrinsic band gap): (a) x-axis; (b) y-axis; (c) z-axis.

DownLoad: Full-Size Img PowerPoint

图 4 同时施加电场和应变下的黑磷烯带隙(虚线1.52对应的是本征带隙的位置)　(a) 施加x轴应变; (b) 施加y轴应变; (c) 施加z轴应变

Figure 4. Bandgap of BP in the presence of the electric field and strain applied along (Dashed line corresponds to the position of the intrinsic band gap): (a) x-axis; (b) y-axis; (c) z-axis.

DownLoad: Full-Size Img PowerPoint

图 5 AC方向的带间光电导率在不同x轴应变条件下随入射光能量的变化　(a), (c) 不同磁场下的光电导实部; (b), (d) 不同磁场下的光电导虚部. 黑色、红色和蓝色曲线分别表示无应变、压缩应变和拉伸应变下的结果

Figure 5. Interband optical conductivity along the AC direction as a function of the incident photon energy at different x-axial strain: (a), (c) Real part under different magnetic fields; (b), (d) the imaginary part under different magnetic fields. Black, red, and blue curves represent the results with no strain, compressive strain, and tensile strain, respectively.

DownLoad: Full-Size Img PowerPoint

图 8 带间光电导率实部在不同z轴应变条件下随入射光能量的变化　(a), (c)不同磁场下AC方向的结果; (b), (d)不同磁场下ZZ方向的结果

Figure 8. Real part of the interband optical conductivity as a function of the incident photon energy at different z-axial strain: (a), (c) Results along the AC direction under different magnetic fields; (b), (d) results along the ZZ direction under different magnetic fields.

DownLoad: Full-Size Img PowerPoint

图 7 带间光电导率实部在不同y轴应变条件下随入射光能量的变化　(a), (c) 不同磁场下AC方向的结果; (b), (d)不同磁场下ZZ方向的结果

Figure 7. Real part of the interband optical conductivity as a function of the incident photon energy at different y-axial strain: (a), (c) Results along the AC direction under different magnetic fields; (b), (d) the results along the ZZ direction under different magnetic fields.

DownLoad: Full-Size Img PowerPoint

图 6 ZZ方向的带间光电导率在不同x轴应变条件下随入射光能量的变化　(a), (c) 不同磁场下的光电导实部; (b), (d) 不同磁场下的光电导虚部

Figure 6. Interband optical conductivity along the ZZ direction as a function of the incident photon energy at different x-axial strain: (a), (c) Real part under different magnetic fields; (b), (d) the imaginary part under different magnetic fields.

DownLoad: Full-Size Img PowerPoint

图 9 带间光电导率实部在三轴应变条件下随入射光能量的变化　(a), (c)不同磁场下AC方向的结果; (b), (d)不同磁场下ZZ方向的结果

Figure 9. Real part of the interband optical conductivity as a function of the incident photon energy at different triaxial strains: (a), (c) Results along the AC direction under different magnetic fields; (b), (d) the results along the ZZ direction under different magnetic fields.

DownLoad: Full-Size Img PowerPoint

图 10 带间光电导率实部在三轴应变条件下随入射光能量的变化　(a), (c) 不同电场下AC方向的结果; (b), (d) 不同电场下ZZ方向的结果

Figure 10. Real part of the interband optical conductivity as a function of the incident photon energy at different triaxial strains: (a), (c) Results along the AC direction under different electric fields; (b), (d) the results along the ZZ direction under different electric fields.

DownLoad: Full-Size Img PowerPoint

DownLoad: Full-Size Img PowerPoint

表 1 黑磷烯的跳变参数与晶格常数^[27,28]

Table 1. Hopping energy and lattice parameter of BP^[27,28].

t_i 具体数值/eV a_i 具体数值/Å

t₁ –1.22 a₁ 1.41763

t₂ 3.665 a₂ 0.79732

t₃ –0.205 a₃ 3.01227

t₄ –0.105 a₄ 2.21468

t₅ –0.055 a₅ 3.63258

DownLoad: CSV

表 2 黑磷烯的应变系数^[29,30]

Table 2. Strain coefficient of BP^[29,30].

γ $ \alpha _{1}^\gamma $ $ \alpha _{2}^\gamma $ $ \alpha _{3}^\gamma $ $ \alpha _{4}^\gamma $ $ \alpha _{5}^\gamma $

x 0.4460 0.0992 0.7505 0.3976 0.7530
y 0.5571 0 0.2461 0.2280 0
z 0 0.9052 0 0.3722 0.2538

DownLoad: CSV

[1]	Novoselov K S, Geim A K, Morozov S V, et al. 2004 Science 306 666 Google Scholar
[2]	Manzeli S, Ovchinnikov D, Pasquier D, Yazyev O V, Kis A 2017 Nat. Rev. Mater. 2 17033 Google Scholar
[3]	Li L K, Yu Y J, Ye G J, Ge Q Q, Ou X D, Wu H, Feng D L, Chen X H, Zhang Y B 2014 Nat. Nanotechnol. 9 372 Google Scholar
[4]	Liu H, Neal A T, Zhu Z, Luo Z, Xu X F, Tománek D, Ye P D 2014 ACS Nano 8 4033 Google Scholar
[5]	Xia F N, Wang H, Jia Y C 2014 Nat. Commun. 5 4458 Google Scholar
[6]	Jain A, McGaughey A J H 2015 Sci. Rep. 5 8501 Google Scholar
[7]	Qiao J S, Kong X H, Hu Z X, Yang F, Ji W 2014 Nat. Commun. 5 4475 Google Scholar
[8]	Ezawa M 2014 New J. Phys. 16 115004 Google Scholar
[9]	Tran V, Soklaski R, Liang Y F, Yang L 2014 Phys. Rev. B 89 235319 Google Scholar
[10]	Li L K, Kim J, Jin C H, et al. 2017 Nat. Nanotechnol. 12 21 Google Scholar
[11]	Kim J, Baik S S, Ryu S H, Sohn Y, Park S, Park B G, Denlinger J, Yi Y, Choi H J, Kim K S 2015 Science 349 723 Google Scholar
[12]	Peng X H, Wei Q, Copple A 2014 Phys. Rev. B 90 085402 Google Scholar
[13]	Rodin A S, Carvalho A, Castro Neto A H 2014 Phys. Rev. Lett. 112 176801 Google Scholar
[14]	Fei R, Yang L 2014 Nano Lett. 14 2884 Google Scholar
[15]	Dai J, Zeng X C 2014 J. Phys. Chem. Lett. 5 1289 Google Scholar
[16]	Chen X L, Lu X B, Deng B C, et al. 2017 Nat. Commun. 8 1672 Google Scholar
[17]	Li L K, Yang F Y, Ye G J, et al. 2016 Nat. Nanotechnol. 11 593 Google Scholar
[18]	Pereira Jr J M, Katsnelson M I 2015 Phys. Rev. B 92 075437 Google Scholar
[19]	Zhou X Y, Lou W K, Zhai F, Chang K 2015 Phys. Rev. B 92 165405 Google Scholar
[20]	Phuong L T T, Phong T C, Yarmohammadi M 2020 Sci. Rep. 10 9201 Google Scholar
[21]	Keshtan M A M, Esmaeilzadeh M 2015 J. Phys. D: Appl. Phys. 48 485301 Google Scholar
[22]	Wang Y, Guo Y L, Wang Z K, et al. 2021 ACS Nano 15 12069 Google Scholar
[23]	Wang Y, Xu W, Fu L, et al. 2023 ACS Appl. Mater. Interfaces 15 54797 Google Scholar
[24]	Wang Y, Xu W, Yang D Y, et al. 2023 ACS Nano 17 24320 Google Scholar
[25]	Li P K, Appelbaum I 2014 Phys. Rev. B 90 115439 Google Scholar
[26]	Le P T T, Yarmohammadi M 2019 J. Magn. Magn. Mater. 491 165629 Google Scholar
[27]	Rudenko A N, Katsnelson M I 2014 Phys. Rev. B 89 201408 Google Scholar
[28]	Yang C H, Zhang J Y, Wang G X, Zhang C 2018 Phys. Rev. B 97 245408 Google Scholar
[29]	Jiang J W, Park H S 2015 Phys. Rev. B 91 235118 Google Scholar
[30]	Khang P D, Davoudiniya M, Phuong L T T, Phong T C, Yarmohammadi M 2019 Phys. Chem. Chem. Phys. 21 15133 Google Scholar
[31]	Yang C H, Zhang J Y, Wieser R, Xu W 2022 J. Phys. D: Appl. Phys. 55 085103 Google Scholar
[32]	Le P T T, Mirabbaszadeh K, Yarmohammadi M 2019 J. Appl. Phys. 125 193101 Google Scholar

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