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Abstract
We propose, both numerically and theoretically, a uniform model to investigate the plasmonically induced transparency effect in plasmonic metamaterial consisting of dual-layer spatially separated borophene nanoribbons array. The dynamic transfer properties of light between two borophene resonators can be effectively described by the proposed model, with which we can distinguish and connect the direct and indirect coupling schemes in the metamaterial system. By adjusting the electron density and separation of two borophene ribbons, the proposed metamaterials enable a narrow band in the near-infrared region to reach high transmission. It provides a new, to the best of our knowledge, platform for optoelectronic integrated high-performance devices in the communication band.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Borophene, a monolayer of boron, has been successfully synthesized by molecular beam epitaxy in 2015 [1]. It has been predicted to applied in nanophotonics and optoelectronics in the visible and near-infrared region for it’s high electron density (∼1019 m-2) [2–4]. Recently, introducing borophene into the metamaterials which enable it to support high confinement surface plasmon polaritons (SPPs) in near-infrared has attracted amounts of attention [5]. Nong et al. proposed a coupled resonant hybridization of an anisotropic borophene localized plasmonic (BLP) and a delocalized guided plasmonic (DGP) mode which enable manipulation in the near-infrared region [6].
Electromagnetically induced transparency (EIT) in atomic system is a quantum phenomenon due to the energy destructive interference between different excitation routes, giving rise to a narrowband transparency window in the absorption band [7,8]. It’s demanding requirements of experiment, such as low-temperature environments and stable gas lasers, catalyze various researches for mimicking EIT systems. SPPs enable strong electric field enhancement upon a nanoscale particle, namely strong dispersion, providing a new path for surmounting the classical diffraction limit [9,10] and manipulating the light at subwavelength scale [11]. With some specific conditions of structural arrangement, a plasmonic system can achieve optical transparency in narrow band which also known as the plasmon-induced transparency (PIT) [12–15]. Such the EIT-like effect promises huge potential applications in the optoelectronics realm such as ultra-compact optoelectronic sensors [16–18] and optical switches [19–21] in deep subwavelength spatial regions.
Exploiting the interactions among plasmons in micro structure is usually bring about novel optical responses [22–24]. Zhang et al. [25] first used nanoplasmonic cells as the radiative (bright) elements and subradiant (dark) elements to investigate the spectral splitting in metamaterial. Liu et al. used the radiation field model to demonstrate a EIT-like effect at the Drude damping limit in a stacked metamaterial [26]. Those configurations mentioned above exhibit a dominated coupling between two nano-resonators, that the bright mode with radiatively broadened linewidth directly spectrally effect an underlying dark mode, strongly coupling in a finite zone. Another coupling scheme of adding medium between the resonators provides an indirect, periodic coupling path which can extend to infinity, ideally. Such the indirect coupling scheme is hinge on the optical path difference inside the medium. Lu et al. [27] introduced a bus-waveguide structure aperture-side cascaded by multi nanodisk resonators, which establish the phase coupling between parallel excitation pathway of the resonators. What the transparency mechanism of an indirect coupling system is vary similar to the Fabry-Perot (F-P) oscillator in the wave optics. That the classical optical principle how to guide the studies of optical features and dynamically characteristics in metamaterials inspires researchers greatly [28–33]. However, up to now, the theoretical studies which employing a unified method to discuss the direct and indirect coupled PIT phenomenon in metamaterials system have rarely been reported.
In this paper, we propose a uniform model combining the temporal coupled theory (CMT) and transfer matrix method (TMM) to universally describe direct and indirect PIT effect in borophene-based metamaterial which can be used as a F-P oscillator. It is found that plasmonic coupling and spectral characteristics indicate a significant difference when varying the separation distance between two borophene ribbons. In addition, the filed distribution around the borophene ribbon is also investigated as an vital role in the process of plasmonic coupling. Owing to the high carrier density of borophene, the PIT effect can be actively optimized to the near-infrared band. Numerical calculations and theoretical results show good consistence. Our result promises to open an avenue for chip-integrated devices and boron-based photonics in the communication band.
2. Structure and method
Borophene metamaterial is shown schematically in Fig. 1(a). Each unit cell consists of dual-layer borophene ribbons embedded in a dielectric (SiO2) spacer. The specific geometrical parameters P, W, and h represent the period of borophene metamaterials, the width of ribbons, and the vertical separation distance between two ribbons along z-axis, respectively. On the condition of normal incidence, a linear polarized light along the x-axis direction excites the localized plasmonic mode upon the ribbons. The cross-section schematic is shown in Fig. 1(b), illustrating the transport properties in proposed borophene metamaterial. The temporal coupled-mode theory (CMT) [34] as model for the plasmon mode on ribbon-resonator with simplified consideration, ignoring the dispersion and loss of medium, can well describe the steady mode amplitudes ai
Fig. 1. (a) Schematic of the plasmonic borophene metamaterial. (b) The cross-section schematic used for theoretical model. (c) Transmission spectra of this system for the plasmons vibrate along the x (top) and y (bottom) directions with definitions of the parameters: the widths of ribbons W are set to 30 nm, the carrier densities of two nanoribbons are set to 3.7×1019 m-2 and 4.3×1019 m-2, respectively, and separation h is 538 nm.
The plasmonic mode is periodically oscillating in form as e-jωt with frequency ω, showing time dependence as dai/dt = -jωai. The relationships between the amplitudes of the input and output waves can be obtained by
When the light is only inputted along + z axis, the transmission and reflection coefficient of a single-ribbon-resonator can be written as
Here, ti = A(i)out,f/A(i)in,f is the wave transmission coefficient through the single ribbon and ri equals to A(i)out,b/A(i)in,f corresponding to the reflection coefficient. For simplicity, the relationships of amplitudes between four channels can be described by matrices
Equation (8) effectively reveals that proposed multilayer borophene ribbons structure has similar properties to the F-P oscillator model. In addition, the phase difference between two adjacent ribbons should be considered when traveling wave propagates steadily
Here φ stands for the phase difference between two ribbons (φ = βh). Considering the dielectric is non-dispersive, the propagation constant β can be expressed as
where c is the speed of light in a vacuum, and nsilica stands for the refractive of silica (nsilica = 1.45). While in a direct coupling scheme, the coupling strength is sensitive to slight change of separation distance between two ribbons which is too small to hold a single wave (h << λ). A previous work of a framework of theory of complex waves gave a analysis of dispersion features of a graphene-coated waveguides [35]. It points that the waveguide mode will be trapped on surface when the propagation constant meets certain condition. This enlightening idea almost distinguish the coupling schemes well (shown in Fig. 1(b), dotted circle region indicates the ‘trapped surface’). For simplicity, the traveling wave can be characterized by introducing an imaginary coefficient jκi(i+1) into the phase term of Eqs. (9),(10),Note that the modulation to amplitude is not only from the periodic variation of real part of phase difference φi, but also related to imaginary part κi(i+1). Substitute with followed matrices
After that, transfer properties of dual-layer borophene-coupled system can be well describe by the transfer matrix method (TMM)
Also neglecting the bottom input, the transmissivity formula of proposed borophene metamaterials can be simply written as
Followed the aforementioned theoretical analysis, numerical investigations are performed by using the finite-difference time-domain (FDTD) method. The periodic boundary condition is used in the x-axis direction of the two-dimension (2D) simulation region, and the perfectly matched absorption boundary (PML) is applied in the z-axis direction to ensure the numerical convergence of the results. Also the uniform mesh which is set as 1 nm in the x-direction and 10 nm in the z-direction is used to divide the calculation area. The maximum mesh step inside the borophene ribbon is set as Δx = Δz = 0.3 nm and Δy = 10 nm. The period of proposed borophene meatamaterials P is set to 60 nm and the width of ribbon W is set to 30 nm. As for the materials dispersion relation, α phase borophene is preferred used in the upcoming discussion.
Considering the intraband transitions largely determine the plasmon response in borophene, the diagonalized conductivity can be written by the Drude model [36]:
where ξ represents the orientation of optical axes of 2D borophene pattern, which is selected as x or y axes in our work. mξ is the effective electron mass and Dξ is the Drude weight. ω, e and τ stand for respectively the incident frequency, electron charge, and relaxation time of electron. n represents the carrier density of borophene which can be tuned by gate voltage [37,4]. Fig. 1(c) shows the transmission response of proposed system under different plasmonic excited orientation. The 2D borophene has a higher plasmonic energy when electrons vibrate in the x-direction as well as a sharper resonance dip, demonstrating the anisotropy of the borophene ribbon [38]. The different electron effective mass that mx is large than my is the result of such orientation-dependent optical properties. Based on this consideration, the x-direction has been selected for the following study.3. Results and discussions
Note that from Eqs. (16)-(18), the transmission hinge strongly on the incident frequency ω, separation h, and carrier density n of borophene ribbons. At first, the influence of carrier density on plasmon response of the borophene F-P oscillator system is discussed, with the fixed separation h = 538 nm, and ribbon widths W1 = W2 = 30 nm. As shown in Fig. 2(a), the system can be modulated by the difference of carrier density between two ribbons with a center value n = 4×1019 m−2. The plasmon resonant of the top ribbon has obvious blueshifts as Δn increases and crosses with that of the bottom ribbon plasmon mode when carrier density difference is about zero. We can see that the phase coupling between the two borophene plasmon modes with a moderate density difference Δn = 0.6×1019 m−2 (n1 = 3.7×1019 m−2 and n2 = 4.3×1019 m−2) gives rise to a PIT peak with narrow bandwidth and high transmission in Fig. 2(e). When Δn is far above 0.6×1019 m−2, two modes behave independently with a large bandwidth of PIT peak. Each of the simulation results based on FDTD are well consistent with CMT theoretical solutions. Figs. 2(b)–2(f) also indicate that the borophene plasmon possess great flexibility in near-infrared region, confirming the advantage of borophene operating band compared with other 2D materials. Based on above, we employ the carrier densities of two ribbons as n1 = 3.7×1019 m−2 and n2 = 4.3×1019 m−2, respectively, in followed works.
Fig. 2. (a) Transmission spectra of borophene F-P oscillator structure with varying carrier density difference Δn between two ribbons. Geometric parameters set as h = 538 nm, W1 = W2 = 30 nm. (b)-(f) Transmission spectra calculated by FDTD (blue spheres) and theoretically fitted by CMT (red curves) for different Δn, namely n1 = 2.8×1019 m−2, 3.1×1019 m−2, 3.4×1019 m−2, 3.7×1019 m−2 and n2 = 5.2×1019 m−2, 4.9×1019 m−2, 4.6×1019 m−2, 4.3×1019 m−2.
Phase difference plays a vital role in the optical response of F-P oscillator system, consequently the separation distance between two ribbons should be the major study variables. Fig. 3(a) shows that the transparency properties can be optimized by varying the separation h. While the system is operating in the large separation regime about 100 to 1200 nm, the frequency of the transparency window is periodically tuned by the spacing distance, accompanied with slight blue/red-shifting of PIT peaks. This phenomenon exhibits the analogous features of the F-P oscillator, that the difference of phase between the output lights depends solely on the difference of optical path between two mirrors. When separation h is taken as 269, 403, 538, 673 nm (meet the conditions of constructive and destructive interferences), the results show obvious different transmission bands at the PIT peaks. For a given h = 538 nm, the PIT peak arrives at the highest transmission value. In addition, the result of h = 269 nm, which satisfies the condition of destructive interference, shows a lowest value.
Fig. 3. (a) Transmission spectra of borophene F-P oscillator structure varying as separation distance h from 25 to 1200 nm. Other parameters set as n1 = 3.7×1019 m−2 and n2 = 4.3×1019 m−2, W1 = W2 = 30 nm. (b)-(f) Transmission spectra calculated by FDTD (blue spheres) and theoretically fitted by CMT (red curves) for different h.
To elucidate the underlying mechanism of the transmission band, the field distributions at the three wavelengths indicated in Fig. 3(e) are depicted in Figs. 4(a)–4(c) respectively. And 1D Ez distribution obtained by the a linear monitor placed 8 nm from the center along z-axis are plotted in Figs. 4(d)-(f). The dual-layer borophene ribbons could serve as two plasmonic resonators with frequency detunings. Highly localized electric fields on the surface of borophene ribbons can be observed when the incident light excites the resonances of individual ribbons (λ1 = 1.484 µm and λ3 = 1.604 µm), that result in the strong reflectivity of this system corresponding to the transmission dips in the spectrum. As for the distribution at center wavelength (transparency peak at λ2 = 1.545 µm), Fig. 4(b) shows that two plasmonic resonances are both excited simultaneously and there are minimal power exists in the system. The corresponding ‘side views’ also give the clear localize distributions of Ez in Figs. 4(a)–4(c). Provided ribbon can be seen as a lossless mirror, the inner light reflects repeatedly and finally outputs with enhancement, which is similar to the mechanism of the F-P oscillator model.
Fig. 4. (a)-(c) Numerical field distributions (Ez) in a unit cell with the excitation wavelength at λ1 = 1.484 µm (left dip), λ2 = 1.545 µm (PIT peak), and λ3 = 1.604 µm (right dip), respectively. (d)-(f) The one dimensional electric field distributions Ez indicated in (a) (calculating along the dash line in Fig. 4(a)) corresponding to (a)-(c).
The PIT spectral features in small separation situation demonstrate more information. The origin of this near-field interaction is attributed to the excitation of plasmons upon the ribbons which work as a dipole source, and they effect each other through radiation, giving rise to a shift of plasmonic resonances wavelength. In Fig. 5(f), for h is set as 55 nm, the resonant wavelength of each ribbon has similar situation with indirectly coupling scheme. The results of our model are matched well with the simulation results, confirming that the shifting of the transparency window is caused by the coupled distances between two borophene plasmonic modes. When the separation decreases, there are growing differences between two resonance dips. The dipole-dipole coupling start once the separation is less than 55 nm (evanescent field of borophene plasmons). Moreover, as noted from Fig. 5(c), the bandwidth of each resonance dip is smaller than that of the transparency peak. In other words, when the system is operated in a strong coupling regime, the majority of energy is exchanged between two ribbons.
Fig. 5. (a) Transmission spectra of borophene F-P oscillator structure varying as separation distance h from 25 to 55 nm. Other parameters set as n1 = 3.7×1019 m−2 and n2 = 4.3×1019 m−2, W1 = W2 = 30 nm. (b)-(f) Transmission spectra calculated by FDTD (blue spheres) and theoretically fitted by CMT (red curves) for different h.
The component electric field distribution Ez is given in Figs. 6(a)–6(c). It can be seen that the hybridization between two modes results in a symmetric (λ4 at 1.470 µm) and an anti-symmetric (λ6 at 1.622 µm) charge oscillations distribution correspond to the hybrid mode of in phase and out of phase. Noticeably, the transparency window at λ5 = 1.577 µm can be attributed to the overlapping of the two hybrid modes. In other words, the electric field distribution shown in Fig. 6(a), displaying a in-phase mode, is resulted by the simultaneous excitation of two super resonances that one of it possess forbidden phase and another possess stronger enhanced phase. As depicted in Figs. 6(e) and 6(f), one can find that the decay length of electric distribution of the ribbons are overlapped due to the small distance, which is distinguished from the case of indirect coupling. More details, in Figs. 6(d) and (e), the positive Ez is distributed on both sides and negative Ez is squeezed in the middle. This distribution confirms the in-phase distribution between two ribbons when excited at λ5 and λ6. By the same token, Fig. 6(d) reveals a superposition of reverse amplitudes between two ribbons.
Fig. 6. (a)-(c) Numerical field distributions (Ez) in a unit cell with the excitation wavelength at λ4 = 1.470 µm (left dip), λ5 = 1.577 µm (PIT peak), and λ6 = 1.622 µm (right dip), respectively. (d)-(f) The one dimensional electric field distributions Ez indicated in (a) (calculating along the dash line in Fig. 6(a)) corresponding to (a)-(c).
To understand the discrimination condition of two coupling configurations, the fitting parameters of coupling strength κ12, intrinsic decay rate κo, and radiative decay rate κe as a function of the separation, which are based on our proposed formulations are plotted in Fig. 7. The radiative decay rate κe is decreased or increased when h varies from 25 nm to 55 nm and then remains substantially with the further grow of h. Which also implied in Fig. 5(b)–5(f) that the change trend of the bandwidth of two dips are contrary with decrease of separation. This happens because the small separation produces direct coupling between two plasmonic resonators. The field strong interacts each via the surface evanescent wave of the plasmon ribbons, that cause the frequency splits in spectra. It is also worthy of note that the intrinsic decay rate keeps roughly constant, indicating that it is dominantly modulated by electric density in our system, and the radiative damping is approximately four times larger than the non-radiative damping in our system. Figure 7 indicates a rapid decreasing trend of the coupling strength κ12 with separation h from 25 nm to 55 nm. Meanwhile, the energy superposing between two eigenmodes results in spectral shifting and a wider line width. This implies it plays an important role in the dipole-dipole coupling that the modulation of the non-zero imaginary component of the phase which is mentioned in Eq. (17). A positive κ12 would lead a wave propagates with exponential damped energy in the metamaterials. On the other hand, the transmission of the indirectly coupling system is merely tuned by the real part of the phase when κ12 equals zero, which is analogous to the phase-coupled PIT in a detuned cavity-coupled MIM waveguide [39].
Fig. 7. The parameters κo1, κo2, κe1 and κe2 used in fitting and the coupling coefficient κ12 of theoretical solution as function of separation distance h. Other parameters set as n1 = 3.7×1019 m−2 and n2 = 4.3×1019 m−2, W1 = W2 = 30 nm.
4. Conclusions
In summary, we have verified a model which enables us to uniformly describe the direct and indirect coupling in the borophene-integrated PIT system in communication band. With the modulation of carrier density of borophene ribbons, the remarkable PIT effect can be observed at 1.55 µm. By combining the CMT and TMM, as well as employing a coupling coefficient iκ12, the optical response of two coupling schemes can be clearly identified. One is the indirect coupling which through varying the separation distances between two borophene ribbons in a large range, achieving a periodical evolution in terms of tensity and symmetry for PIT peak. In addition, as the separation distance is reduced below 55 nm, the system is driven to a direct coupling regime, where the spectral bandwidth of transparency window is varied with the separation distance, because the coupling strength is mainly depends on radiation properties of ribbons. The theoretical fittings of proposed model show good consistency with FDTD simulations. This method of uniformly describing the PIT effect in borophene-based metamaterial can find potential applications in the development of compact elements such as tunable sensors, switchers and slow light devices in the communication band.
Funding
National Natural Science Foundation of China (11947062); Natural Science Foundation of Hunan Province (2020JJ5551, 2021JJ40523).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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