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In this paper, we propose three plasmon-induced transparency(PIT) hybrid models based on the coupling of black phosphorus (BP) nanorods. By using the bright-bright mode coupling between BP nanorods with different lengths, and according to the weak hybrid effect after the detuning of each bright mode unit, we can trigger the single-band, dual-band and triple-band PIT effects. Secondly, by changing the relaxation rate of BP, the resonant frequency can be adjusted in each PIT model. When the relaxation rate of BP changes from small to large, the resonance frequencies of the transparent windows in those three PIT models all increase and the blue shifts occur. In a single-band PIT model, when the relaxation rate of BP changes from $0.8 \times {10^{ - 14}}$ to$1.4 \times {10^{ - 14}}\;{{\rm{cm}}^{ - 2}}$ , the resonant frequency of transparent window increases 7.5628 THz. In a dual-band PIT model, under the same change of relaxation rate, the resonant frequencies of the two transparent windows increase 6.8593 and 9.1457 THz, at the same time, and the resonant frequencies of the three transparent windows in triple-band PIT model increase 6.8593, 8.7939 and 11.2563 THz respectively. In addition to the resonant frequency, the transmittance at the dip frequency in each model gradually decreases, and the depth of depression gradually increases.Finally, the sensing characteristics of the single-band PIT model are further studied. When the refractive index of the background changes from small to large, the dip frequency and the resonant frequency of the transparent window will be significantly red-shifted. The change of frequency is approximately linear with refractive index. The model has a sensitivity of 6110.6 (nm/RIU) and a figure of merit of 7.39 (1/RIU) which is better than the same type of sensor. This model provides a theoretical reference for designing the multiband filtering and ultrasensitive sensors. [1] Alp A, Ahmet A, Hatice A 2011 Nano Lett. 11 1685
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[2] Li H J, Wang L L, Liu J Q 2013 Appl. Phys. Lett. 103 211104
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[3] Liu N, Langguth L, Weiss T 2009 Nat. Mater. 8 9
Google Scholar
[4] Papasimakis N, Fu Y H, Fedotov V A 2009 Appl. Phys. Lett. 94 211902
Google Scholar
[5] Safavi A H, Mayer T P, Ch an 2011 Nature 472 7341
[6] Qin M, Wang L L, Zhai X 2018 IEEE Photonics Technol. Lett. 30 11
Google Scholar
[7] Wang H, Wang X, Xia F 2014 Nano Lett. 14 6424
Google Scholar
[8] Buscema M, Groenendijk D J, Blanter S I 2014 Nano Lett. 14 3347
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[9] Deng Y, Luo Z, Conrad N J 2014 ACS Nano 8 8292
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[10] Buscema M, Groenendijk D J, Steele G A 2014 Nat. Commun. 5 4651
Google Scholar
[11] Engel M, Steiner M, Avouris P 2014 Nano Lett. 14 6414
Google Scholar
[12] Du Y, Ouyang C, Shi S 2010 J. Appl. Phys. 107 093718
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[13] Prytz O, Flage L E 2010 J. Phys. Condens. Matter 22 015502
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[14] Qiao J, Kong X, Hu Z 2014 Nat. Commun. 5 4475
Google Scholar
[15] Wang X, Lan S 2016 Adv. Opt. Photonics 8 618
Google Scholar
[16] Huang Y, Liu Y, Fang C Z 2020 Opt. Mater. Express 10 14
Google Scholar
[17] Fo Q Q, Ling P, Chen X Y 2018 IEEE Photonics J. 10 4500709
[18] Liu Z Z, Aydin K 2016 Nano Lett. 1 1
[19] Nong J P, Wei W, Wang W 2018 Opt. Express 26 1633
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[20] Liu C, Li H J, Xu H 2019 J. Phys. D: Appl. Phys. 52 120137
[21] Li H, Wang L, Xing H Z 2019 Opt. Mater. Express 8 352
[22] Jia Z P, Huang L, Su J B 2020 Appl. Phys. Express 13 104027
[23] Deng Z L, Fu T, Zheng B 2016 Appl. Phys. Lett. 108 081109
Google Scholar
[24] Deng Z L, Deng H, Zhuang X 2018 Nano Lett. 18 2885
Google Scholar
[25] Deng Z L, Jin M K, Ye X 2020 Adv. Funct. Mater. 1 1910610
[26] Cen H F, Wang F Q, Liang R S 2018 Opt. Commun. 420 78
Google Scholar
[27] Hu X G, Yuan S, Armghan A 2017 J. Phys. D Appl. Phys. 50 025301
Google Scholar
[28] Wang J, Jiang Y N 2017 Opt. Express 25 5206
Google Scholar
[29] Liu Y, Zhong R B, Lian Z 2018 Sci. Rep. 8 2828
Google Scholar
[30] Liu C, Li H J, Xu H 2020 New J. Phys. 1 111962
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图 1 单频段PIT模型结构图 (a) 三维空间结构图; (b) 二维平面结构图
Figure 1. Schematic diagram of single-band PIT model: (a) Three-dimensional space schematic; (b) two-dimensional plane schematic.
图 2 BP原子结构和介电常数示意图 (a) BP错列原子结构示意图; (b)
${\varepsilon _{xx}}$ 实部随${n_{\rm{s}}}$ 的变化规律; (c)${\varepsilon _{xx}}$ 虚部随${n_{\rm{s}}}$ 的变化规律; (d)${\varepsilon _{yy}}$ 实部随${n_{\rm{s}}}$ 的变化规律; (e)${\varepsilon _{yy}}$ 虚部随${n_{\rm{s}}}$ 的变化规律Figure 2. Schematic diagrams of BP atomic structure and dielectric constant: (a) The staggered atomic structure of BP; (b) variations of the real part of
${\varepsilon _{xx}}$ with different${n_{\rm{s}}}$ ; (c) variations of the imaginary part of${\varepsilon _{xx}}$ with different${n_{\rm{s}}}$ ; (d) variations of the real part of${\varepsilon _{yy}}$ with different${n_{\rm{s}}}$ ; (e) variations of the imaginary part of${\varepsilon _{yy}}$ with different${n_{\rm{s}}}$ .
图 3 单BP纳米棒谐振器、单频段PIT模型的透射曲线透射曲线
Figure 3. Transmission spectra of the sole BP rods array, and the single-band PIT model.
图 4 单频段PIT模型在dip A, dip B和peak的电场分布 (a) dip A; (b) dip B; (c) peak
Figure 4. Distributions of electric field of single-band PIT model at: (a) dip A; (b) dip B; (c) peak.
图 5
${n_{\rm{s}}}$ 由$0.8 \times {10^{ - 14}}\;{\rm{c}}{{\rm{m}}^{ - 2}}$ 变化到$1.4 \times {10^{ - 14}}\;{\rm{c}}{{\rm{m}}^{ - 2}}$ 时, 单频段PIT模型透射谱变化规律及其与RTO拟合结果比较Figure 5. Variations law of transmission spectrum of single-band PIT model and comparison with RTO fitting results when the
${n_{\rm{s}}}$ is changed from$0.8 \times {10^{ - 14}}$ to$1.4 \times {10^{ - 14}}\;{\rm{c}}{{\rm{m}}^{ - 2}}$ 
图 6 单频段PIT模型不同背景折射率下的透射窗对比
Figure 6. The variation of transmission windows with different background index in single-band PIT model.
图 7 单频段PIT模型中dip A, dip B和peak随背景折射率的变化规律
Figure 7. The variations of dip A, dip B and peak with different background index.
图 8 双频段和三频段PIT模型二维结构图 (a) 双频段模型二维结构图; (b) 三频段模型二维结构图
Figure 8. Two-dimensional plane schematic diagram of dual-band and triple-band PIT models: (a) Two-dimensional plane schematic of dual-band PIT model; (b) two-dimensional plane schematic of triple-band PIT model.
图 9 双频段和三频段PIT的透射谱曲线 (a) 双频段PIT的透射谱曲线; (b) 三频段PIT的透射谱曲线
Figure 9. Transmission spectra of dual-band PIT and triple-band models: (a) Transmission spectra of dual-band PIT model; (b) transmission spectra of triple-band PIT model.
图 10
${n_{\rm{s}}}$ 由$0.8 \times {10^{ - 14}}\;{\rm{c}}{{\rm{m}}^{ - 2}}$ 变化到$1.4 \times {10^{ - 14}}\;{\rm{c}}{{\rm{m}}^{ - 2}}$ 时双频段PIT模型透射谱变化规律Figure 10. Variation law of transmission spectrum of dual-band PIT model when the
${n_{\rm{s}}}$ is changed from$0.8 \times {10^{ - 14}}$ to$1.4 \times {10^{ - 14}}\;{\rm{c}}{{\rm{m}}^{ - 2}}$ .
图 11
${n_{\rm{s}}}$ 由$0.8 \times {10^{ - 14}}\;{\rm{c}}{{\rm{m}}^{ - 2}}$ 变化到$1.4 \times {10^{ - 14}}\;{\rm{c}}{{\rm{m}}^{ - 2}}$ 时三频段PIT模型透射谱变化规律Figure 11. Variation law of transmission spectrum of triple-band PIT model when the
${n_{\rm{s}}}$ is changed from$0.8 \times {10^{ - 14}}$ to$1.4 \times {10^{ - 14}}\;{\rm{c}}{{\rm{m}}^{ - 2}}$ .
图 12 双频段PIT模型在 (a) dip A, (b) dip B, (c) dip C, (d) peak I和(e) peak II的电场分布
Figure 12. Distribution of electric field of dual-band PIT model at (a) dip A, (b) dip B, (c) dip C, (d) peak I nd (e) peak II.
图 13 三频段PIT模型在 (a) dip A, (b) dip B, (c) dip C, (d) dip D, (e) peak I, (f) peak II和(g) peak III的电场分布
Figure 13. Distributions of electric field of dual-band PIT model at (a) dip A, (b) dip B, (c) dip C, (d) dip D, (e) peak I, (f) peak II and (g) peak III.
表 1 单频段PIT模型中dip A, dip B和peak的优值系数和灵敏度大小
Table 1. FOM and Sensitivity of dip A, dip B, and peak in the single-band PIT model.
FOM/RIU–1 Sensitivity/nm·RIU–1 dip A 2.385 6110.6 dip B 3.08 4018.7 peak 7.39 4806.8 
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[1] Alp A, Ahmet A, Hatice A 2011 Nano Lett. 11 1685
Google Scholar
[2] Li H J, Wang L L, Liu J Q 2013 Appl. Phys. Lett. 103 211104
Google Scholar
[3] Liu N, Langguth L, Weiss T 2009 Nat. Mater. 8 9
Google Scholar
[4] Papasimakis N, Fu Y H, Fedotov V A 2009 Appl. Phys. Lett. 94 211902
Google Scholar
[5] Safavi A H, Mayer T P, Ch an 2011 Nature 472 7341
[6] Qin M, Wang L L, Zhai X 2018 IEEE Photonics Technol. Lett. 30 11
Google Scholar
[7] Wang H, Wang X, Xia F 2014 Nano Lett. 14 6424
Google Scholar
[8] Buscema M, Groenendijk D J, Blanter S I 2014 Nano Lett. 14 3347
Google Scholar
[9] Deng Y, Luo Z, Conrad N J 2014 ACS Nano 8 8292
Google Scholar
[10] Buscema M, Groenendijk D J, Steele G A 2014 Nat. Commun. 5 4651
Google Scholar
[11] Engel M, Steiner M, Avouris P 2014 Nano Lett. 14 6414
Google Scholar
[12] Du Y, Ouyang C, Shi S 2010 J. Appl. Phys. 107 093718
Google Scholar
[13] Prytz O, Flage L E 2010 J. Phys. Condens. Matter 22 015502
Google Scholar
[14] Qiao J, Kong X, Hu Z 2014 Nat. Commun. 5 4475
Google Scholar
[15] Wang X, Lan S 2016 Adv. Opt. Photonics 8 618
Google Scholar
[16] Huang Y, Liu Y, Fang C Z 2020 Opt. Mater. Express 10 14
Google Scholar
[17] Fo Q Q, Ling P, Chen X Y 2018 IEEE Photonics J. 10 4500709
[18] Liu Z Z, Aydin K 2016 Nano Lett. 1 1
[19] Nong J P, Wei W, Wang W 2018 Opt. Express 26 1633
Google Scholar
[20] Liu C, Li H J, Xu H 2019 J. Phys. D: Appl. Phys. 52 120137
[21] Li H, Wang L, Xing H Z 2019 Opt. Mater. Express 8 352
[22] Jia Z P, Huang L, Su J B 2020 Appl. Phys. Express 13 104027
[23] Deng Z L, Fu T, Zheng B 2016 Appl. Phys. Lett. 108 081109
Google Scholar
[24] Deng Z L, Deng H, Zhuang X 2018 Nano Lett. 18 2885
Google Scholar
[25] Deng Z L, Jin M K, Ye X 2020 Adv. Funct. Mater. 1 1910610
[26] Cen H F, Wang F Q, Liang R S 2018 Opt. Commun. 420 78
Google Scholar
[27] Hu X G, Yuan S, Armghan A 2017 J. Phys. D Appl. Phys. 50 025301
Google Scholar
[28] Wang J, Jiang Y N 2017 Opt. Express 25 5206
Google Scholar
[29] Liu Y, Zhong R B, Lian Z 2018 Sci. Rep. 8 2828
Google Scholar
[30] Liu C, Li H J, Xu H 2020 New J. Phys. 1 111962
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