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Abstract
We numerically investigate plasmon-induced transparency (PIT) featuring anisotropic high-Q resonance in monolayer black phosphorus (BP) nanostrip trimer. Transparent windows can be observed in both armchair and zigzag directions due to the coupling between the constituent nanostrip elements. By dynamically adjusting the Fermi level (EF) of BP, we show that the anisotropic and wide-range tuned PIT can be achieved. The strong polarization dependence of BP nanostrip trimer has also been studied. Furthermore, we explore the number of induced transparent windows through varying the EF of left nanostrip in the two parallel strips. The results can be applied to optical filters as well as photonic devices for sensing and communication at mid-infrared region.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Electromagnetically induced transparency (EIT) is produced by the quantum coherence effect between atomic light excitation channels, which leads to the absorption of light at the atomic resonance frequency to decreases or even becomes completely transparent [1,2]. In 3-level atomic EIT system, the coupling between energy level (|2>) with dipole-allowed transition to the ground state (|1>) and metastable level (|3>) by a pumping beam leads to destructive interference between two pathways, namely, |1>-|2> and |1>-|2>-|3>-|2> [2,3]. EIT has attracted widespread attention due to its applications in slow light devices [4–7], enhanced nonlinear effects [8], label-free biological sensing [9], and many more. However, implementation of the EIT phenomenon is limited by experimental drawbacks such as the need for low temperature and stable high power laser systems. In this regards, EIT analogs including photonic crystal waveguides [10], coupled microresonators [11,12] and metamaterials [13–15] have been proposed in a variety of optical structures, which are stable and do not require harsh conditions. Plasmon induced transparency (PIT) nanostructures have also been investigated [16–18] because of their operability at room temperature and wide operational bandwidth. Zhang et al. [19] designed a π-shaped metal structure to explore PIT. The single strip is considered bright mode which is strongly coupled with the incident field. Two parallel metal strips act as dark mode and cannot be directly coupled to the incident light due to counterpropagating currents. Shi et al. [20] utilized graphene to construct π-shaped nanostructure, and realized flexible tuning of PIT by simply adjusting the Fermi level.
Nevertheless, the gapless nature of graphene constrain its actual application [21]. Therefore, it is necessary to find an alternative material for graphene. Black phosphorus (BP) as an emerging two-dimensional (2D) material is an excellent candidate. Unlike graphene, monolayer BP has a direct band gap of ~2 eV [22,23]. The direct bandgap decreases with the increase of BP layer number because of interlayer interaction, and the bandgap of bulk BP reaches ~0.3eV [23]. Besides, BP has unique anisotropy derived from the asymmetry of its crystal structure [24]. BP surface plasmon (SP) behaviors have been fully explored in nanoribbons [25,26], square arrays [27], nanodisks [28] and so on.
In this work, we have delineated the anisotropic PIT based on coupling between dark and bright plasmon modes in BP nanostrip-trimer. The transparent windows corresponding to localized plasmons along two crystal directions in BP can be flexibly tuned over a wide range by slightly adjusting the Fermi level of BP. Anisotropic and wider tuned transparent windows can be obtained compared to graphene PIT [20,29]. In addition, the multi-spectral channels of PIT transparency are enabled by changing the electrostatic doping in one of nanostrips within the dark mode component. These unique results hold great promise for BP-based anisotropic metamaterials and detecting nanostructures.
2. Structure and method
Figure 1(a) displays the staggered atomic structure of monolayer BP, in which armchair and zigzag crystal directions are marked with x and y axis following the routine treatment. The BP nanostrip trimer is composed of three nanostrips as shown in Figs. 1(b) and 1(c). Unless specified, the geometric parameters are set as l1 = l2 = 50 nm, w1 = w2 = 10 nm, p = 100 nm, d = 10 nm and g = 30 nm. In order to investigate the anisotropic PIT in nanostructure BP, numerical simulation is performed by using Lumerical FDTD solutions, and the BP sheet is treated as a thin layer of 10 nm. Auto non-uniform grid is applied throughout the simulation region, and precision mesh (2 nm) is added to the BP area. The periodic boundary conditions are set in the x and y directions, and the perfectly matched layer (PML) is applied in the z direction. We placed frequency-domain field and power monitor on the upper and lower parts of the structure to obtain reflection (R) and transmission (T). It should be noted that the reflection monitor must be on the back of the light source. The absorption can be expressed by a compact formula A = 1 - R - T.
Fig. 1 (a) Schematic of monolayer BP, where different crystal structures along the armchair (x) and zigzag (y) directions. (b) Schematic of the BP nanostrip trimer. The plane wave is incident perpendicularly on the structure. (c) Top view of BP nanostrip trimer.
The anisotropic dielectric function of monolayer BP can be described by Drude model as [30]
where j represents the x and y directions,is the relative permittivity [24,31], is the plasma resonance frequency and is the collision frequency. For n-doped BP, where is the electron density [32], mj is the effective mass of electron in both directions [33] and τj (relaxation time) is determined by the carrier mobility μ (τj = μmj/e). Here, μ = 1000 cm2/(V·s) is adopted as retrieved from the experimental measurement [23].3. Results and discussion
The real and imaginary parts of the permittivity for BP along the x and y directions at various Fermi level are plotted in Fig. 2. As shown, for both directions, the real part of ɛxx (Re(ɛxx)) and ɛyy (Re(ɛyy)) is always negative, and the imaginary part (Im(ɛxx) and Im(ɛyy)) is positive. Besides, Re(ɛxx) is smaller than Re(ɛyy) and Im(ɛxx) is larger than Im(ɛyy). With the increase of EF, Re(ɛxx) and Re(ɛyy) become smaller, however, Im(ɛxx) and Im(ɛyy) become larger. This explains why the transmission peaks of the PIT become sharper with increasing Fermi energy in later essays.
Fig. 2 The dependence of the permittivity of BP in both directions on Fermi level. (a, c) Spectra of the real part of permittivity ɛxx and ɛyy of BP at EF = 0.10 eV, 0.15 eV and 0.20 eV. (b, d) Spectra of the imaginary part of permittivity ɛxx and ɛyy of BP at EF = 0.10 eV, 0.15 eV and 0.20 eV.
Localized surface plasmons (LSPs) in nanostructured BP array can be easily excited by external radiation to form localized electric dipoles oscillation. The simulated absorption and transmission spectra of individual BP nanostrip (EF = 0.20 eV) in related with plasmons along different crystal directions are shown in Figs. 3(a) and 3(b). For x direction, LSP resonance peak at 5.45 μm can be observed, but the resonance wavelength along the y direction red-shifts to 14.90 μm. The above phenomenon is reasonably attributed to the anisotropic effective mass of the BP [25]. For periodic BP arrays, the plasmon wave vector approximately follows q ~π/w [34], which determines the geometric parameters at the desired resonance. The illustrations in Figs. 3(a) and 3(b) display the corresponding electric field distribution at resonance peak. It can be seen that the electric field is strongly localized at both ends of the nanostrip due to the induced electric dipoles, which strongly hints a radiative component in the atomic system.
Fig. 3 (a, b) Absorption and transmission spectra of individual BP nanostrip corresponding to the plasmon along the armchair and zigzag directions. The insets show the electric field distributions at transmission dips. (c, d) Absorption and transmission spectra of BP nanostrip trimer, and the corresponding lattice directions are shown in the inset, E is the direction of incident electric field, (c) is related with the incident electric polarized along the x crystal axis (armchair direction), (d) is related with the incident electric field polarized along the y crystal axis (zigzag direction). (e, f) Corresponding electric field and charge distributions at A (5.32 μm), B (5.56 μm), C (5.78 μm), D (13.93 μm), E (14.78 μm) and F (15.61 μm) in Figs. 2(c) and 2(d).
The absorption and transmission spectra of BP nanostrip trimer along the x and y crystal directions are demonstrated in Figs. 3(c) and 3(d), respectively. Unlike graphene PIT [20], we can observe BP-PIT spectra with different transparent wavelengths in both directions. For x direction, the transparent peak emerges at 5.56 μm with two dips positioned at 5.32 μm and 5.78 μm. While for y direction, the transparent point (14.78 μm) and the transmission valleys (13.93 μm and 15.61 μm) shift to longer wavelengths due to the heavier effective mass along the zigzag direction. The corresponding electric field and charge distributions at A, B, C, D, E and F are shown in Figs. 3(e) and 3(f). The dips at A, C, D, and F are mainly caused by the dipole oscillation of bright plasmon mode element. For the transparent points at B and E, the plane wave can directly excite the LSPR in the bright element while the cross-distributed quadruple mode in the dark element cannot be aroused. However, the anti-parallel quadruple mode can be indirectly excited by the coupling between the lateral and longitudinal nanostrips. Eventually, the electric field in the bright element is suppressed due to the destructive interference between bright and dark elements, resulting in the transparent window.
The BP plasmon behaviors can be actively control by the EF determined across the gate voltage or doping [26,27]. As shown in Figs. 4(a) and 4(b), with the EF increasing (0.05 eV to 0.2 eV), PIT phenomenon becomes more pronounced and transmission peaks become sharper (i.e., higher quality factors). Figures 4(c) and 4(d) show the transmission map along the x and y directions as a function of EF, respectively. Obviously, the transparent windows can be tuned quickly and accurately over a wide range as shown by the black dash line. For x direction case, the transparent dot is shifted from 10.8 μm to 5.6 μm. Likewise, for y direction, the transmission peak is shifted from 28.6 μm to 14.7 μm. BP-PIT has an anisotropic and wider tuned transparent band in comparison with graphene PIT [20,35,36]. In addition, compared with metal PIT [37,38], the implementation of BP-PIT tuning simply requires gate voltage or doping without the need for complex geometric parameter variations, which enormously reduces the complexity of the fabrication process.
Fig. 4 (a, b) Transmission spectra of the BP-PIT structure when EF is 0.10 eV, 0.15 eV, and 0.20 eV of two directions. (c, d) Numerical transmission map as a function of EF along x and y directions. The change in the induced transparent point with EF is indicated by the black dash line.
We also simulated the transmission spectra at different polarization angles (θ) as shown in Fig. 5. The transmission spectra have strong polarization dependence. With θ increases, the coupling between bright and dark elements is weakened. When θ = 90°, only the electric dipole in the vertical direction of the dark element is excited, and the transmission spectrum exhibits a single peak. Note that for y direction, peak 1 is generated by dipole of dark element along the vertical direction, while the coupling between the bright element and the dark element still exists, so double peak appears.
Fig. 5 (a, b) Transmission spectra of the BP-PIT structure when polarization angle (θ) is 0°, 30°, 60° and 90° of two directions.
Next, the transmission spectra along the x and y directions have been performed by varying the EF of dark element for fixed EF = 0.10 eV of radiative element as shown in Figs. 6(a) and 6(b), respectively. As described above, strong coupling between the bright mode and dark mode results in the appearance of PIT windows. The dispersion relation of strong coupling in hybrid system can be express as [26,39]
where ω ± is the resonance frequencies of the hybrid system, ωradiaj and ωdarkj are the resonance frequency of the radiative element and dark element, respectively.is coupling frequency which represents the coupling strength between the radiative and dark modes.Fig. 6 (a, b) Transmission spectra of two directions with fixed EF (0.10 eV) of radiative element and various EF of dark element. (c, d) Theoretical and emulational dispersion relationships for the hybridization between bright element and dark element in both directions.
We extracted the resonance wavelength at the transmission dip and compared it with the theoretical curve, as shown in Fig. 6(c) and 6(d). Apparently, anti-crossing behaviors due to the hybridization can be observed and the numerical results are well matched to the theoretical curves.
To further explore the PIT in the BP trimer, the EF in the left nanostrip of dark element is tuned and the transmission spectra are presented in Figs. 7(a) and 7(b), respectively. As demonstrated, two distinct transparent windows are exhibited in transmission spectra of the nanostrip trimer. We have carried out a detailed analysis of the electric field distribution corresponding to the transparent points A (14.00 μm), B (14.42 μm), C (14.76 μm), D (15.60 μm) and E (16.27 μm) of the transmission spectrum (red line) along the y direction at EF = 0.17 eV as shown in Fig. 7(c). It can be seen that the radiative element is still within an electric dipole, whereas the dark element becomes two asymmetric dipoles. This is equivalent to the effect of two different dipole dark modes on the dipole radiation of the bright mode. The coupling strength of each vertical nanostrip to the radiative element is different, and the more complex coherent superposition among them results in the appearance of two transparent windows.
Fig. 7 Transmission spectra when the EF in one of the nanostrips (i. e. the dark element) changes, (a) corresponding to the incident electric field parallel with the armchair direction of bright element, and (b) corresponding to the incident field parallel with the zigzag direction of bright element, respectively. (c) The charge and electric field intensity distribution at A, B, C, D and E.
4. Conclusion
In summary, we constructed a BP nanostrip trimer to theoretically investigate the anisotropic PIT caused by hybridization. The lateral BP nanostrip acts as a bright element, while the other two nanostrips perpendicular to it act as dark element, and the strong coupling between them results in a transparent window. In particular, by varying the Fermi level of BP, the PIT can achieve flexible adjustments in both armchair and zigzag crystal directions. Additionally, the number of induced transparent windows can be varied by adjusting the Fermi level in one of the nanostrips in the dark element, i.e., changing the symmetry. These results may provide an extremely important guidance for developing optical switches, biosensing, and slow light devices.
Funding
State Key Program for Basic Research of China (Nos. 2013CB632705, 2011CB922004, 2017YFA0305500); National Natural Science Foundation of China (Nos. 10990104, 11334008, 61405230, 61290301, 61376102, 11274225, and 61675222); and the Youth Innovation Promotion Association (CAS).
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