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Abstract
Strong coupling between the resonant modes can give rise to many resonant states, enabling the manipulation of light-matter interactions with more flexibility. Here, we theoretically propose a coupled resonant system where an anisotropic borophene localized plasmonic (BLP) and Bloch surface wave (BSW) can be simultaneously excited. This allows us to manipulate the spectral response of the strong BLP-BSW coupling with exceptional flexibility in the near infrared region. Specifically, the strong longitudinal BLP-BSW coupling occurs when the system is driven into the strong coupling regime, which produces two hybrid modes with a large Rabi splitting up to 124 meV for borophene along both x- and y-directions. A coupled oscillator model is employed to quantitatively describe the observed BSW-BLP coupling by calculating the dispersion of the hybrid modes, which shows excellent agreement with the simulation results. Furthermore, benefited from the angle-dependent BSW mode, the BSW-BLP coupling can be flexibly tuned by actively adjusting the incident angle. Such active tunable BLP-SBW coupling with extreme flexibility offered by this simple layered system makes it promising for the development of diverse borophene-based active photonic and optoelectronic devices in the near infrared region.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Surface plasmon resonance (SPR) [1] is a type of electromagnetic surface wave that can confine the electromagnetic energy within a subwavelength scale, enabling tremendous promising applications in nanophotonics [2]. Especially, SPR observed in the atomic thin two-dimensional (2D) layered materials (for instance graphene [3], and black phosphorus [4]) exhibit a rich variety of distinctive advantages that are refreshingly different from plasmons in noble metals since they can be actively tuned by external gate voltage [5–8]. Moreover, by further coupling these SPR modes with the other kinds of resonant modes (phono polaritons [9], magnetic polaritons [10,11], molecular vibrational modes [12,13]), various exciting optical phenomena have been realized, which allow us to exploit novel device concepts. Despite of the exciting potential, the plasmonic resonant frequencies of these devices are limited within the terahertz or infrared region [14–16] due to their low carrier densities (∼1017 m-2). As a new member of 2D material family, monolayer boron sheet or borophene [17] is intrinsically metallic/semimetallic with extremely high density of Dirac electrons (∼1019 m-2) even though it is a semiconductor in bulk form, which enables it to support highly tunable confined plasmons in the visible and near-infrared region [18–21]. Additionally, borophene also exhibits an anisotropic plasmonic response due to the different in-plane effective electron masses along the two crystal axes [22,23]. These impressive features have triggered numerous theoretical research interests [18–23]since the successful synthesis of borophene on metallic (Au [24], Ag [25] etc.) surface.
As another type of electromagnetic surface wave, Bloch surface wave (BSW) [26,27] propagates at the interface between a truncated periodic dielectric multilayer (one-dimensional photonic crystal, 1D-PC) and an adjacent dielectric medium, which is considered to be a dielectric analog to SPR. Compared to SPR, BSW possesses superior properties owing to its metal-free structure, such as ultrawide resonant wavelength ranges (from UV [28] to visible [29] and mid-infrared [30]region), long propagation lengths (up to the order of millimeters [31] and even up to centimeters [32] in the visible), and high-quality resonances (with quality factor up to the order of 1000 [33]). Additionally, because of the simple geometry, easy fabrication and great freedom in the choice of dielectric materials (including CMOS-compatible [26]), BSW is a promising all-dielectric photonic platform for applications of various chip-integrated optoelectronic devices [34]. Recently, the applications of BSW in sensors [35], exciton-polariton excitation [36], surface-enhanced spectroscopies [37] as well as the BSW-induced effects such as nonlinear optical effects [38], giant Goos-Hanchen shift [39] and enhanced magneto-optical effect [40] have been investigated extensively. Whereas the studies on the coupling of a BSW mode to a SPR mode are limited, especially the coupling of a BSW mode to an active tunable SPR mode in 2D materials have yet to be demonstrated, which could further expand the application of BSW platform for highly integrated optical technologies.
In this paper, we theoretically proposed a coupled resonant system where the transverse magnetic (TM) polarized BSW mode and the anisotropic borophene localized plasmonic (BLP) mode can be simultaneously excited. The longitudinal strong coupling between the BSW mode and the anisotropic BLP mode results in two absorption peaks, corresponding to the two hybrid polariton modes, of which the dispersion can be quantitatively described by a coupled oscillator model. Moreover, such strong BSW-BLP coupling can be further actively modulated by tuning the borophene electron density and the incident angle of light. Our results may inspire related researches about the hybrid light-matter interactions and should be useful for constructing the active hybrid resonant system in the near-infrared region.
2. Principle and modeling
The schematic of the proposed hybrid system is sketched in Fig. 1(a), which consists of a patterned borophene nanoribbons array and a dielectric multilayer structure (SiO2/Si/SiO2) on a Si prism. The corresponding side view is shown in Fig. 1(b). In practice, such structure can be fabricated by the following steps. Firstly, SiO2/Si/SiO2 multilayer are subsequently deposited on Si prism by magnetron sputtering. Then, monolayer borophene is synthesised on metallic substrate under ultrahigh-vacuum conditions and transferred onto the prism by a PMMA-assisted method. Finally, the borophene nanoribbons can be fabricated by e-beam lithography followed by oxygen plasma etching.
Fig. 1. (a) Schematic of the proposed hybrid system consisting of patterned borophene nanoribbons and dielectric multilayer structure on a Si prism. (b) Side view of the structure.
To investigate the strong longitudinal coupling between the anisotropic BLP mode and the BSW mode in such configuration, simulations are performed using finite element method software. In the simulations, the period and width of the borophene nanoribbons array are set as P=120 nm and W=60 nm, respectively, while the thicknesses of the dielectric multilayer are optimized as T1=50 nm, T2=130 nm and T3=2000nm throughout the work, as illustrated in Fig. 1. The refractive index of SiO2 and Si are set as 1.47 + 0.001*i and 3.48. Monolayer borophene is modeled as the surface current density boundary condition without considering its thickness using anisotropic conductivity σjj described by the semiclassical Drude formula as [21–23]
To simultaneously excite the BLP mode and the BSW mode the proposed hybrid system, s-polarized light is considered as the probing light. As the light irradiates upon the Si prism with incident angle θ larger than the critical angle of the total internal reflection (TIR), it can couple into the BSW mode if the parallel component of its propagation constant matches that of BSW mode and propagates along the surface. While the rest of light will be reflected by total internal reflection at the top interface, resulting in the reduced reflected light at a specific wavelength and a resonant peak in the absorption spectrum. In the meantime, anisotropic plasmons can be excited in borophene by the external incident light once the period of the borophene nanoribbon and the dispersion relationship fulfill the following phase matching condition
where k0=ω/c is vacuum wavevector, c is the speed of light, m is the order of the plasmonic mode, and q(ω) is the wavevector of borophene plasmonic wave that can be expressed as3. Results and discussion
The active tuning of anisotropic plasmons behavior in borophene nanoribbons without the perturbation of the BSW mode is first investigated by setting the incident angle θ = 0 deg since BSW cannot be excited under this condition. The absorption spectra with varying electron density for borophene nanoribbons along x- and y-directions are shown in Fig. 2(a) and 2(b), respectively. It can be seen that the resonant wavelength blue shifts as the borophene electron density increases for both x- and y-directions. Whereas the resonant wavelength of the borophene nanoribbons along x-direction is smaller than that along y-direction for the same electron density. This is due to the smaller electron mass of borophene along x-direction than that along y-direction. Note that the observed absorption lines at the lower left corner is the second order borophene plasmonic resonant mode. One important feature is that the absorption of the BLP is significantly enhanced at some specific wavelengths for both x- and y-directions, as indicated by the vertical dashed lines. Such phenomenon can be ascribed to the Fabry Perot (FP) resonant effect induced by the multilayer structure, which can be confirmed by the electric field distribution in Fig. 2(c) where a standing wave is formed within the SiO2 layer.
Fig. 2. Simulated absorption spectra with varying electron density for borophene nanoribbons along (a) x- and (b) y-direction when θ=0 deg. (c) Electric field |E| intensity distribution of a FP mode. (d) Simulated absorption spectra with varying incident angle for borophene nanoribbons along x-direction without the top borophene nanoribbons. (e) Simulated absorption spectra with varying incident angle for borophene nanoribbons along x-direction when n=4.75×1019 m-2. (f) Electric field |E| intensity distribution of a BSW mode.
The active tuning of the BSW mode of the multilayer structure is then investigated without and with the top borophene nanoribbons. The calculated absorption spectra with varying incident angles without the top borophene nanoribbons are shown in Fig. 2(d). It can be observed that the BSW can be only excited at specific angle-wavelength combinations when the incident angle is larger than the TIR critical angle. For the proposed configuration, the considered transverse magnetic BSW mode is forbidden when the incident angle is larger than 27.3 deg. It can be effectively excited within an angle range from 24.4 to 26.2 deg, with BSW resonant wavelength varied from 1.34 to 2.28 µm and resonant absorption in a range between 0.2 and 1. A typical electric field intensity distribution of a BSW propagating at the surface of a multilayer structure is shown in Fig. 2(f). It can be seen that a large portion of the electric field is localized at the SiO2/Si as well as SiO2/Air interface, which is expected to have a large mode overlap when another resonant mode is excited at this interface and induce the strong mode coupling. As expected, when borophene nanoribbons are placed at the SiO2/Air interface, the spectral response of the hybrid structure is significantly changed, as can be seen from the absorption spectra with varying incident angles displayed in Fig. 2(e). It can be seen that a BLP mode is excited and enhanced when the incident angle is larger than the TIR critical angle. Additionally, the resonant wavelength of BSW mode is nearly independent of the incident angles. Such angle-independent BLP mode will couple with the angle-dependent BSW mode, and alter the spectral response of the BSW mode. As can be seen from Fig. 2(e), a Rabi splitting phenomenon occurred when the incident angle is larger than 24 deg, where the BLP mode and BSW mode are simultaneously excited. More details about the observed Rabi splitting phenomenon will be discussed later.
We then investigate the dynamical longitudinal coupling between BLP mode and BSW mode by gate tuning the borophene electron density. The absorption spectra of the configuration are shown in Fig. 3(a) and 3(b) for borophene nanoribbons along x- and y-direction, respectively, with consecutively shifted borophene electron density from 2×1019 to 1.5×1020 m-2. Obviously, two absorption bands can be observed for both cases, which are formed by the resonant absorption peak of the resulting two hybrid modes. Additionally, the two absorption bands blue shift to smaller wavelengths as the borophene electron density increases. An important feature is that these two absorption bands do not cross to each other, but exhibit a spectral splitting with an energy gap. This is a typical Rabi splitting phenomenon that results from the longitudinal coupling between the BLP mode and the BSW mode. To further explain such coupling process, a standard coupled oscillator model is introduced, where the BLP mode and BSW mode are considered as the classical oscillators. Then the eigenvalues corresponding to energies of the hybrid modes can be characterized by [41,42]
Fig. 3. Simulated absorption spectra with varying electron density for borophene nanoribbons along (a) x- and (b) y-direction when θ=25 deg. Analytical dispersion relations of the BLP, BSW, and hybrid modes, and the comparison between the analytical and numerical hybrid dispersion relations for borophene nanoribbon along (c) x- and (d) y-direction.
According to Eq. (4) and Eq. (5), the dispersion curves of two hybrid modes (blue lines), BLP mode (red line) and BSW mode (black line) are plotted in Fig. 3(c) and 3(d) for x- and y-direction, respectively. Here, the BSW resonant wavelength is derived by solving the eigenmode equation via a transfer matrix method. The simulated absorption peaks (blue hollow stars) of two hybrid modes are also extracted for comparison. For the uncoupled GSP mode, its resonant wavelength blue shifts as the borophene electron density increases, and crosses with that of the BSW mode when the electron density is about 4.75×1019 m-2 and 1.14×1020 m-2 for x- and y-directions. Similarly, the dispersion curves of two hybrid modes also blue shift as the borophene electron density increases. However, they do not cross with each other, instead, they are separated by an energy gap. The energy gap of the mode splitting, i.e., the coupling strength is estimated to be 124 meV for both x- and y-directions. We further shown that the coupling strength between BLP and BSW modes can be modulated by changing the thickness of the SiO2 or the number of Si/SiO2 layers (not shown here), such that the weak coupling phenomenon also can be realized. Additionally, the analytical results match well with the numerical results, as compared in Fig. 3(c) and Fig. 3(d), confirming the observed Rabi splitting phenomenon is induced by the strong coupling between BLP and BSW modes.
To gain a deeper understanding of the coupling process between BLP and BSW modes, the energy exchange between the hybrid modes is further analyzed. Taking x-direction as an example, three representative absorption spectra when borophene electron density is 2×1019 m-2, 4.75×1019 m-2 and 7.5×1019 m-2 are extracted from Fig. 3(a) and shown in Fig. 4(a). While the corresponding electric field |Ey| distributions of the left peaks in Fig. 4(a) are displayed in Fig. 4(c). An energy evolution scheme is introduced to visually describe the modes coupling process in Fig. 4(b), which shows that the coupling between a BLP mode and a BSW mode with moderate energy levels can give rise to a hybrid mode with even higher energy (shorter wavelength) and a hybrid mode with even lower energy (longer wavelength). Based on such a model, the energy diagrams of the hybrid modes corresponding the cases in Fig. 4(a) are provided in Fig. 4(d). When the borophene electron density is 2×1019 m-2, the energy of the BLP mode (457 meV) is far lower than that of the BSW mode (700 meV), and thus the two modes behave independently. As shown in the left panel of Fig. 4(c), the |Ey| distribution indicates that the mode is a typical BSW-like hybrid mode. As the borophene electron density increases to 4.75×1019 m-2, the energy of the BLP mode (704 meV) increases and comes closed to that of the BSW mode. In this case, light energy exchanged intensively between the BLP and BSW modes via a hybrid mode, which can be confirmed by the |Ey| distribution in the middle panel of Fig. 4(c) that the hybrid mode possesses the features of both the BLP and BSW modes. Moreover, noted from the green line in Fig. 4(c) that the frequency splitting is larger than the linewidth of each mode. This indicates that the energy exchange rate between two modes is faster than the damping rate of each mode, meaning that the device is operating in the strong coupling regime in this case. As the borophene electron density further increases to 7.5×1019 m-2, the energy of the BLP mode (885 meV) is far larger than that of the BSW mode, and the hybrid mode gradually changes its nature with the BLP-like features (|Ey| distribution in the right panel of Fig. 4(c)). These results indicate that the coupling process are actually the indicative of the energy conversion process between the BLP-like and BSW-like hybrid modes.
Fig. 4. Three representative absorption spectra when the electron densities of the borophene are 2×1019 m-2, 4.75×1019 m-2, and 7.5×1019 m-2 for borophene nanoribbons along x-direction. (b) Energy diagram describing the longitudinal strong coupling between anisotropic BLP modes and BSW modes. (c) Electric field |Ey| intensity distribution at the left resonant peaks indicated in (a). (d) Energy diagrams corresponding to the cases in (a).
We finally investigate the active tuning of the strong BLP-BSW coupling employing the angle-dependent spectral response of BSW mode (as indicated in Fig. 2(d)). The simulated absorption spectra of the configuration with varying borophene electron density for x-direction when the incident angle is 24.7 and 25.3 deg are shown in Fig. 5(a) and (b), respectively. Clear Rabbi splitting phenomenon can be observed for both cases, excepting that the strong BLP-BSW coupling occurred at a higher borophene electron density and a shorter wavelength for the larger incident angle. Such phenomenon is easy to understand that the BSW resonant wavelength bule shifts as the incident angle increases, such that the BLP resonant wavelength also needs to shift to the smaller wavelength region to ensure the generation of the strong BLP-BSW coupling by increasing the borophene electron density. Moreover, the coupling strength of the BLP-BSW coupling becomes stronger as the incident angle increases. Quantitatively, the coupling strengths are calculated to be 95 and 157 meV when the incident angle is 24.7 and 25.3 deg, respectively. For y-direction, variation of the BLP-BSW coupling nearly follows the same trend as x-direction, excepting that the strong BLP-BSW coupling occurred at an even higher borophene electron density for the same incident angle. This can be attributed to the larger electron mass of borophene along y-direction than that along x-direction, thus it needs a higher electron density to excite the BLP mode at the same wavelength. Additionally, as compared in the figures, the analytical dispersion curves calculated by Eq. (5) shown good agreement with the simulated results for both x- and y-directions. These results indicate that the coupling strength of the BLP-BSW coupling can be actively modulated by tuning the incident angle, offering another route to manipulate the spectral response of configuration.
Fig. 5. Simulated absorption spectra varying electron density for borophene nanoribbons along x-direction when θ is (a) 24.7 deg and (b) 25.3 deg. Simulated absorption spectra varying electron density for borophene nanoribbons along y-direction when θ is (c) 24.7 deg and (d) 25.3 deg. Black and violet lines are the calculated dispersion curves of the BSW and BLP modes, and the blue star line is the analytical hybrid dispersion curves.
4. Conclusion
In conclusion, we theoretically investigate the active tuning of strong longitudinal coupling between the anisotropic BLP and BSW modes in a coupled resonant system. By varying the borophene electron density to drive the system into the strong coupling regime, Rabi splitting phenomenon arises, with a large splitting energy up to 124 meV for borophene along both x- and y-directions. A coupled oscillator model was employed to quantitatively describe the observed BSW-BLP coupling, which shows excellent agreement with the simulation results. Our investigations further reveal that the BSW-BLP coupling can be flexibly tuned by actively adjusting the incident angle benefited from the angle-dependent BSW mode. Specifically, the coupling strengths can be from 95 to 157 meV as the incident angle increases from 24.7 to 25.3 deg, allowing for the angle-dependent strong light-matter interaction. Such coupled resonant system with tunable optical responses may open up an avenue for the design of many borophene-based active photonics and optoelectronic devices in near infrared region.
Funding
China Postdoctoral Science Foundation (2021M692175); National Natural Science Foundation of China (61905147, 61935013, 91750205, U1701661); Leading Talents Program of Guangdong Province (00201505, 2019JC01Y178); Natural Science Foundation of Guangdong Province (2016A030312010, 2019TQ05X750, 2020A1515010598); Science, Technology and Innovation Commission of Shenzhen Municipality (JCYJ20180507182035270, KQTD2017033011044403, KQTD20180412181324255); Shenzhen University Starting Funding (2019073).
Acknowledgments
The authors would like to acknowledge the Photonics Center of Shenzhen University for technical support.
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that supports the findings of this study are available within the article.
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