Dynamically Tunable Plasmon-Induced Transparency in Parallel Black Phosphorus Nanoribbons

In this paper, we investigate the dynamically tunable plasmon-induced transparency (PIT) effects in parallel black phosphorus nanoribbons (BPNRs). The proposed structures consist of several parallel BPNRs having different lengths in one transverse period. The periods of all proposed structures are set as P=500nm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P = 500 \, nm$$\end{document}. To our best knowledge, these proposed structures have not been reported in published papers. It is the first time that simulates PIT effects by this way. The results show that the BPNRs can be regarded as bright modes. Single-band, double-band, triple-band, and multi-band PIT effects based on the bright-bright mode coupling between parallel BPNRs are achieved. The physical mechanism of the single-band model can be explained theoretically by the radiating two-oscillator (RTO) model. Due to the heavier effective mass in the zigzag (ZZ) direction of the BP, the frequencies of the transparent peaks are shifted to lower frequencies when the placement directions of BPNRs are changed from the X-direction to the Y-direction. Furthermore, the resonant frequencies of the transparent windows in each model can be tuned by changing the relaxation rates of the BPNRs. The frequencies of the transparent windows are blue-shifted as the relaxation rates are increased. Finally, the corresponding sensors based on single-band PIT effect show high sensitivities of 7.35 THz/RIU. Our study has potential applications for improving the design of multiple-band filters, sensors, and on–off switcher.


Introduction
Plasmon-induced transparency (PIT), an electromagnetically induced transparency (EIT)-like optical effect in metamaterials, is caused by the coupling between the bright mode and dark mode or the bright mode and bright mode [1]. This plasma phenomenon significantly changes the optical responses of the system and causes the appearance of a transparent window in the transmission spectrum. In recent years, PIT has been widely applied in the fields of optical filtering [2], sensing [3], slow light, and light conversion [4][5][6]. However, in practical applications, an active modulation of the transparent windows is necessary to control the optical responses dynamically and potentially expand the range of applications [7]. Because of the difficulties of active modulation in transparent windows, traditional metallic materials have been replaced by novel two-dimensional (2D) crystal metamaterials.
In recent years, black phosphorus (BP) has attracted extensive attentions as an alternative 2D material due to its excellent optical and electrical performances. Unlike graphene, the atoms in the BP material have a wrinkled hexagonal honeycomb structure, as shown in Fig. 1a [8,9]. Therefore, BP possesses larger conductivity in the armchair (AC) direction than in the zigzag (ZZ) direction, resulting in an anisotropic plasmonic response in the AC and ZZ directions [10,11]. Furthermore, the band gap of BP is related to the thickness of the material. The band gaps of bulk BP and monolayer BP are approximately 0.3 eV and 2.0 eV, respectively [12,13]. Because of the unique physical features of BP and the development of intelligent algorithms [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32], the researches on tunable PIT effects by using BP have become a new research highlight [33][34][35]. For example, in 2021, Wu et al. realized a tunable single-band PIT effect based on the destructive interference between the bright and dark modes by using monolayer BP metamaterial [36]. Han et al. achieved a single-band PIT effect with large tunability and high Q-factor by using graphene and BP hybrid system [37]. Moreover, Chen et al. also designed a single-band PIT model in a monolayer BP metasurface based on near-field coupling between two bright modes [38]. However, in the previous literature, utilizing parallel black phosphorus nanoribbons (BPNRs) to achieve tunable PIT effects has not been reported. In this study, we achieved single-band, double-band, triple-band, and multiband PIT effects based on bright-bright mode coupling by using parallel BPNRs with different lengths. To our knowledge, it is the first time to simulate the sing-band and multiband tunable PIT effects by this way.
In the section of theoretical methods and single-band PIT model, firstly, we will discuss the transmission spectra of the single-band PIT model and show the distributions of electric field intensity at the transmission dips and peak. Secondly, we will analyze the tunability of the single-band PIT model and explain the sing-band PIT effect theoretically by the RTO model. Finally, we will evaluate the application potential of the single-band PIT model as a refractive index sensor and investigate the relationships between transmission spectra and polarization angle of the incident light in single-band model.
In the section of design and result of a tunable double-band PIT model, firstly, we will discuss the transmission spectra of the double-band PIT model and show the distributions of electric field intensity at the three transmission dips and two transparent peaks. Secondly, we will show the transmission spectra of the proposed double-band PIT model for different relaxation rates. Finally, we will discuss the tunable multi-band PIT effects based on the single-band and double-band models.

Theoretical Methods and Single-Band PIT Model
The PIT effect based on bright-bright mode coupling can be explained theoretically by the radiating two-oscillator (RTO) model [39,40].
The first bright mode resonator is described by p 1 (t) . Its resonant frequency is 1 , and its damping factor is 1 . The second bright mode resonator is described as p 2 (t) . Its resonant frequency and damping factor are 2 and 2 , respectively. Ω 2 exp(j ) is the complex coupling coefficient, and is the phase shift between the two bright mode resonators.
By assuming p 1 (t)=P 1 exp(−j t) , p 2 (t)=P 2 exp(−j t) , and f 1 (t)=f 2 (t) = f exp(−j t) , Eqs. (1) and (2) in the frequency domain can be solved: Under the approximation of a thin structure, the physical relationship between the electric current density J and the surface conductivity e can be expressed as: where n s and E s denote the average electron density and spatially averaged electric field, respectively. If f ∝ E s , the surface conductivity e can be solved by: Once the surface conductivity is determined, the transmission coefficient of the metamaterial is: where Z 0 is the wave impedance of the external wave. Finally, the transmissivity of the metamaterial can be determined by fitting the corresponding |T| 2 .
The permittivity of a 2D BP layer can be characterized as a diagonal tensor [41]: where xx , yy , and zz denote the dispersion elements in the X-, Y-, and Z-directions, which can be expressed as where r is the relative permittivity, which is 5.76 for the monolayer BP. 0 is free space permittivity. d represents the thickness of the BP layer. is the frequency of incident light. ii is the surface conductivity, and zz ≅ 0 . Based on the classical Drude model, the conductivity ii of BP can be expressed as [9]: where ℏ is the reduced Planck constant, and denotes the electron doping, which is 10 meV [42]. D ii is the Drude weight, which is defined as: where e is the electron charge. n s represents the relaxation rate. m ii is the carrier effective mass in the X-and Y-directions, respectively, and can be described as: For the monolayer BP with a scale length a = 0.223 nm , the parameter values in Eq. (12) are = 4a∕ eVm ,, c = ℏ 2 ∕(0.4m 0 ) and v c = ℏ 2 ∕(0.7m 0 ) ; the band gap Δ = 2 eV , and for a standard electron rest mass, m 0 = 9.10938 × 10 −31 kg [43,44]. Figure 1a displays the staggered atomic structure of the monolayer BP. The X-axis and Y-axis represent the AC and ZZ crystal directions following the routine treatment. Figure 1b-e shows the real and the imaginary parts of the permittivity of BP with different relaxation rates. When the relaxation rates increase from 0.2 × 10 14 cm −2 to 1.4 × 10 14 cm −2 , Re( xx ) and Re( yy ) are negative, whereas Im( xx ) and Im( yy ) are positive. Re( xx ) is smaller than Re( yy ) , and Im( xx ) is larger than Im( yy ) . Most importantly, as the relaxation rates of BP increase, Re( xx ) and Re( yy ) decrease, and Im( xx ) and Im( yy ) increase. Therefore, the responses of the proposed PIT models in this study can be controlled by adjusting the relaxation rates of the BP.
The three-dimensional space schematics of single-band model are shown in Fig. 2a, b. These proposed PIT structures haven't been reported in published papers yet. The dielectric layer is assumed to be a non-dispersive dielectric with a refractive index of 1.6 [33]. In Fig. 2a, the parallel BPNRs are placed along the X-direction, and the lengths of them are assumed to be infinite in Y-direction. In Fig. 2b, the placement direction of parallel BPNRs is changed to the Y-direction, and the lengths of them are assumed to be infinite in X-direction. The two-dimensional perspective side views and plane schematics of Fig. 2a, b are shown in Fig. 2c, d, respectively. In Fig. 2c, the AC direction of BP is along X-axis. In the Fig. 2d, the AC direction of BP is along Y-axis.
The period of a transverse in the structure is P = 500 nm . It consists of four BPNRs in one transverse period. The geometric parameters of a pair of BPNRs in the first layer are the same as W 1 = 200 nm . The geometric parameters of a pair of BPNRs in the second layer are W 2 = 120 nm . The left and right intervals of twin BPNRs in both layers are d 1 = 60 nm . The distance between twin BPNRs in two layers is d 2 = 1 m . In the simulation, the thickness of BP is set to 1 nm.
The numerical calculations are performed by using the finite-difference time-domain (FDTD) method. The twounit cells have periodic boundary conditions in the X-and Y-directions and perfectly matched layer (PML) absorbing boundary conditions in the Z-direction. The surface plasmon resonances are excited by incident transverse magnetic (TM) waves in the model.   Figure 3a shows the transmission spectrum of Figs. 2a and 3b shows the transmission spectrum of Fig. 2b. As a reference, the transmission spectra of BPNRs in the two layers under the same E-field polarization are also given.

Results and Discussions
As shown in Fig. 3a, b, regardless of the placement directions of BPNRs are in X-axis or Y-axis direction, all the single-band models in Fig. 2a, b present a distinct absorption peak. Meanwhile, all the BPNRs in first and second layers exhibit typical Lorentz line-shaped resonances; thus, the BPNRs can be regarded as bright modes that are excited by the incident light. Moreover, it is observed that the transmission dips of the PIT spectra are close to the initial resonant modes of the BPNRs. Therefore, based on the bright-bright mode coupling between the BPNRs in first and second layers, the model produces a single-band PIT effect that can be explained theoretically by the RTO model depicted in Fig. 5.
Furthermore, because of the instinct anisotropic structure of 2D BP, the permittivity along the AC and ZZ directions is different. Hence, the optical responses of the BP are also different in the AC and ZZ directions. As shown in Fig. 3a, when the placement directions of the BPNRs are in the X-axis direction, a transmission peak occurs at 28.0905 THz, with two dips at 22.2111 THz and 33.5176 THz. However, as shown in Fig. 3b, when the placement directions of the BPNRs are in the Y-axis direction, the peak (18.5930 THz) and the transmission   [34]. In the following content, we mainly discuss the results of single-band model in Fig. 2a.
In order to investigate the electromagnetic mechanism of the single-band PIT model in Figs. 2a and 4 shows the distributions of electric field intensity at the frequencies of dip I, dip II, and peak in Fig. 3a.
As shown in Fig. 4a, b, and d at dip I, only the BPNRs in the first layer are excited by the incident wave and produce a dipole, whereas BPNRs in the second layer are only weakly excited. In contrast, at dip II, only the BPNRs in the second layer are excited strongly by the incident light and produce a dipole, whereas BPNRs in the first layer are weakly excited. At the peak, the BPNRs in both layers are excited simultaneously to produce two dipoles. Therefore, a single-band PIT effect is produced by the bright-bright mode coupling between the BPNRs in first and second layers.
For the purpose of analyzing the tunability of the singleband PIT model in Figs. 2a and 5 depicts the transmission spectra of the single-band PIT model at different relaxation rates and the results fitted by the RTO model.
As described above, the PIT effect based on the brightbright mode coupling can be explained theoretically by the RTO model. As shown in Fig. 5, the numerical results calculated by the FDTD method at different relaxation rates are consistent with the theoretical results fitted by the RTO model, confirming that the single-band PIT effect is produced by bright-bright mode coupling between the twin BPNRs in first and second layers.
Furthermore, according to the perturbation theory [45,46], Δ will be greater than 0 if Δ is less than 0, indicating that the dips and peak frequencies will be increased as the relaxation rates rise. As a result, when the relaxation rates of the BPNRs are changed from 0.8 × 10 14 cm −2 to 1.4 × 10 14 cm −2 , the frequencies of dip I increase from 19.9 to 26.3 THz. The frequencies of dip II increase from 30.1 to 39.4 THz. Meanwhile, as the frequencies of dip I and dip II increase, the frequencies of the transparent peak also increase from 25.6 to 33.1 THz.
Therefore, the proposed single-band PIT model can be dynamically tuned by changing the relaxation rates of the twin BPNRs in first and second layers without changing the structure.
To evaluate the application potentials of the single-band PIT model as a refractive index sensor, Fig. 6 shows the sensing performances of the model for different refractive indices of the surrounding medium. The relaxation rates of BPNRs in both layers are n s = 1.0 × 10 14 cm −2 .
As shown in Fig. 6a, as the refractive index increases from 1.0 to 1.8, the dip I and peak exhibit a significant red-shift. The relationships between the refractive index and the frequencies of the dip I and peak are approximately linear (Fig. 6b).
The performance of a sensor usually can be evaluated by the sensitivity ( S = Δ ∕Δn ). The sensitivity of dip I is The relationships between transmission spectra and polarization angle of the incident light. The polarization angle is defined by the illustration 7.35 (THz/RIU), which is higher than the sensors in papers [47][48][49][50][51]. The sensitivity of dip II is only 0.29 (THz/RIU) because the BPNRs in second layer are in the dielectric layer, and the medium surrounding them is almost unchanged as the refractive indices of the surrounding medium increase.
Finally, the relationships between transmission spectra and polarization angle of the incident light in single-band model are investigated. The polarization angle of the incident light shown in Fig. 7a is assumed to be the angle between the incident electric field direction and the X-axis direction.
As shown in Fig. 7b, when the is changed from 0 。 to 90 。 , the resonant frequencies at transmission dips and peak remain basically unchanged, but the transmittance values at transmission dips increase, and the PIT effect is gradually weaken. As shown in Fig. 7a, when the < 40 • , the PIT effect is evident, and when the > 60 • , the PIT effect is disappeared.

Design and Result of a Tunable Double-Band PIT Model
The plane schematics of tunable double-band PIT model are shown in Fig. 8a, b. These proposed PIT structures haven't been reported in published papers too. The dielectric layer is also assumed to be a non-dispersive dielectric with a refractive index of 1.6. In Fig. 8a, the parallel BPNRs are placed along the X-direction, and the lengths of them are assumed to be infinite in Y-direction. In Fig. 8b, the parallel BPNRs are placed along the Y-direction, and the lengths of them are assumed to be infinite in X-direction. In Fig. 8a, the AC direction of BP is along X-axis. In the Fig. 8b, the AC direction of BP is along Y-axis. The two units can be obtained by adding a pair of BPNRs with different lengths based on the single-band PIT model shown in Fig. 2.
The period of a transverse in the structure is also P = 500 nm . It is composed of six BPNRs in a period. The geometric parameters of a pair of BPNRs in the first layer are W 1 = 200 nm . The lengths of two pairs of BPNRs in the second and third layers are W 2 = 160 nm and W 3 = 120 nm , respectively. The left and right intervals of twin BPNRs in both layers are d 1 = 60 nm . The distance between twin BPNRs in each layer is d 2 = 1 m . The thickness of BP is also set to 1 nm . The incident light configuration is the same as in the single-band PIT model. Figure 9 shows the transmission spectra of the doubleband PIT model. The relaxation rates of twin BPNRs in both layers are also n s = 1.0 × 10 14 cm −2 . Figure 9a shows transmission spectrum of Fig. 8a, and the transmission spectrum of Fig. 8b is shown in Fig. 9b. As a reference, the transmission spectra of the BPNRs in three layers under same E-field polarization are also shown.
In Fig. 9a, b, two distinct transmission peaks are observed. Similar to the results in Fig. 3, all BPNRs produce Lorentz line-shaped resonances; therefore, all of them can also be regarded as bright modes. The transmission dips in the PIT spectra are also close to the initial resonant modes of the BPNRs. Thus, similar to the single-band PIT effect, the double-band PIT effect is achieved by the bright-bright mode coupling between the BPNRs in the first, second, and third layers. Meanwhile, because of the heavier effective mass in the ZZ directions, the transmission peaks and valleys have also shifted to lower frequencies when the placement directions of BPNRs are changed. In the next, we mainly discuss the results of the double-band model shown in Fig. 8a. Figure 10 shows the distributions of electric field intensity at the frequencies of the three transmission dips and two transparent peaks in Fig. 9a.
As shown in Fig. 10a-c, since the dips I-III are close to the initial resonant modes of BPNRs in the first, second, and third layers, respectively, therefore, at dip I, the electric field has high values at the edge and end of the BPNRs in the first layer, exhibiting a typical electric dipole mode. At dip II, only the BPNRs in the second layer are excited by the incident wave to produce a dipole. At dip III, the electric field has high values around the BPNRs in the third layer, producing a dipole mode. Furthermore, at peak A, the BPNRs in the first and second layers are excited simultaneously to produce two dipoles, and at peak B, the BPNRs in the second and third layers are excited simultaneously to produce two dipoles. Therefore, similar to  Fig. 12b the single-band PIT effect, the double-band PIT effect is produced by the bright-bright mode coupling between the BPNRs in the first, second, and third layers. Figure 11 shows the transmission spectra of the proposed double-band PIT model for different relaxation rates. As the relaxation rates of the BPNRs are increased, the frequencies of the three dips and two peaks in Fig. 9a are also blue-shifted.
When the relaxation rates of the BPNRs are changed from 0.8 × 10 14

Tunable Multi-Band PIT Effects
The triple-band and four-band PIT effects can be achieved by continuing to add the layers of twin BPNRs having different lengths based on the double-band model. Figure 12 shows the transmission spectra of the tripleband PIT model. The relaxation rates of BPNRs are still 1.0 × 10 14 cm 2 . The lengths of twin BPNRs in each layer are shown in Fig. 12b. The placement directions of BPNRs in Fig. 12a are along X-axis direction, and the placement directions of BPNRs in Fig. 12b are along Y-axis. Figure 13 shows the variation laws of transmission spectra of triple-band PIT model when the n s increases from 0.8 × 10 −14 cm −2 to 1.4 × 10 −14 cm −2 . The BPNRs are placed along X-axis direction. Figure 14 shows the transmission spectra of the four-band PIT model. In Fig. 14a, the BPNRs are placed along X-direction. In Fig. 14b, the BPNRs are placed along Y-direction.  Fig. 14b Obviously, the physical mechanism and variation laws of transmission spectra in triple-band and four-band models are the same as the single-band and double-band models.

Conclusions
In summary, dynamically tunable single-band, double-band, and multi-band PIT effects in parallel BPNRs are proposed due to the bright-bright mode coupling between BPNRs with different lengths. The physical mechanism of single-band can be explained by RTO model. Secondly, the dynamically tunable behavior is realized by changing the relaxation rates of the BPNPs. As the relaxation rates are increased, the resonant frequencies of the transmission peaks in each model are blue-shifted. The phenomena can be explained by the perturbation theory. Thirdly, because of the heavier effective mass in the ZZ direction, the frequencies of the transmission peaks and valleys are shifted to lower frequencies when the placement directions of BPNRs are changed from the X-direction to the Y-direction. Fourthly, as the refractive index increases from 1.0 to 1.8, the frequencies of the transmission peaks and valley in the single-band PIT model exhibit a significant red-shift, and the sensitivity can reach as high as 7.35 THz/ RIU. Finally, the relationships between transmission spectra and polarization angle of the incident light in single-band model are investigated. When the < 40 • , the PIT effect is evident, and when the > 60 • , the PIT effect is disappeared. The results of this study have potential applications for improving the designs of multiple-band filters sensors and on-off switcher.
Acknowledgements The authors would like to thank the reviewers for their thoughtful comments.
Author Contribution All authors contributed to the design and implementation of the research, analysis of the results, and the manuscript's preparation.
Funding This work was supported in part by National Natural Science Foundation of China (NSFC) (61461052, 11564044, 61863035).
Data Availability All data generated or used during the study appear in the submitted article.

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