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