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Abstract
In this paper, we propose a graphene-dielectric metasurface to enhance the light-matter interactions in graphene. The dielectric metasurface consists of periodically arranged silicon split rings placed on the silica substrate, which supports a symmetry-protected bound state in the continuum (BIC). When perturbation is introduced into the system to break symmetry, the BIC will transform into the quasi-BIC with high quality (Q)-factor. As the graphene layer is integrated with the dielectric metasurface, the absorption of graphene can be enhanced by the quasi-BIC resonance and a bandwidth-tunable absorber can be achieved by optimizing the Fermi energy of graphene and the asymmetry parameter of the metasurface to satisfy the critical coupling condition. By varying the Fermi energy of graphene, the quasi-BIC resonances can be effectively modulated and the max transmission intensity difference is up to 81% and a smaller asymmetry parameter will lead to better modulation performance. Our results may provide theoretical support for the design of absorber and modulator based on the quasi-BIC.
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1. Introduction
The so-called bound state in the continuum (BIC), its frequency lies within the continuum spectrum, while it is a completely confined state without any radiation [1,2]. Initially, this conception was reported in quantum mechanics which can occur due to the direct and via-the-continuum interaction between quasi-stationary states [3]. Since then, many mechanisms for the formation of BICs in different systems have been found, which have been observed in the experiments of acoustic waves [4,5] and water waves [6], and so on. In recent years, photonic structures have become an attractive platform for the study of BICs because of their strong maneuverability to materials and structures, which is difficult to achieve in quantum systems. In 2008, D. C. Marinica et al. [7] revealed a BIC in the photonic nanostructure where the electromagnetic wave with a specific frequency can be trapped by the structure with infinite lifetime. In fact, a series of studies around the BICs based on photonic structures have been carried out, including its topological characteristics [8], the influence of substrate and structural roughness on BIC [9], anisotropic-induced BIC [10], supercavity modes [11], and embedded photon eigenvalues [12]. A photonic quasi-BIC usually possesses ultra-high Q-factor and strong electromagnetic field enhancement [13], which exhibits many potential applications, including on-chip vortex beam control [14], on-chip optical communication [15], laser [16], sensor [17], filter [18], and the enhancement of the Goos-Hänchen shift [19,20]. In fact, ideal BICs are usually transformed to high Q-factor quasi-BICs, which can be achieved by introducing perturbations into these systems. The research groups of Yi Xu [21] and Yuri Kivshar [22] successively introduced BICs into the metasurface, which may help us gain more degrees of freedom to manipulate the modal properties of this system by changing the symmetry of the structures or excitation. Furthermore, results demonstrated that a large variation of the Q-factors can be realized by modifying the unit-cell asymmetry.
Given that quasi-BICs normally possess strong electromagnetic field enhancement, it may have great potential for enhancing light-matter interactions in 2D materials. As we know, the veil of 2D materials world in the research community has been lifted since graphene was successfully mechanical exfoliated in 2004 [23,24]. 2D materials of atomic thickness, owing to their electron confinement within two dimensions, do exhibit their uniqueness in electronic, optical, chemical, mechanical properties, and so on, which has become the center of attention both in industries and academia [25,26]. Nevertheless, light-matter interactions in 2D materials are usually weak due to their atomic-scale thickness. For example, the absorption coefficient of monolayer graphene is only 2.3%, which is a constant independent of the wavelength of incident light. [27]. It has limited the performance of graphene-based devices to some extent, and many efforts have been made to overcome this deficiency. One of the most obvious approaches is to excite the graphene plasmon polariton. In fact, graphene is often regarded as an excellent plasmonic material because doped graphene supports plasmon polaritons across the terahertz (THz) and mid-infrared (mid-IR) spectral ranges. In 2013, Xiaolong Zhu et al. introduced a hybrid graphene-metal system where the light-matter interactions in the graphene were significantly enhanced and the absolute light absorption was increased by 30% with the aid of large electromagnetic field enhancement generated by the plasmon resonances [28]. While at higher frequencies, such as the communication wavelength, graphene behaves like a lossless medium and cannot support plasmons. Recently, the research groups of Jian Zi [29] and Shuyuan Xiao [30,31] presented a general method successfully for tailoring the absorption bandwidth of graphene via critical coupling in the near-infrared regime. On the other hand, some studies have shown that the amplitude of the light can be modified by tuning Fermi energy of graphene or Dirac-semimetals [32–35]. In general, the modulation performance will be better if electromagnetic field enhancement is stronger [36]. To sum up, quasi-BIC resonance will be a good candidate for enhancing light-matter interactions in 2D materials thanks to its tunability of radiation loss and the possession of strong electromagnetic field enhancement.
In this paper, a graphene-dielectric metasurface is proposed to enhance the light-matter interactions in graphene, and its potential applications in absorber and modulator are discussed. Using the finite-difference time-domain (FDTD) method for numerical calculations, we have examined the band structure of the structure, determined the resonance frequency of the BIC point of the dielectric metasurface, and discussed the influence of asymmetry on the BIC resonance. The results showed that the BIC will transform into the quasi-BIC with high Q-factor when perturbation is introduced into the system to break symmetry, and Q-factor can be adjusted by altering the asymmetry parameter of the metasurface. We have further proved that the quasi-BIC resonance with great field enhancement can enhance the absorption of graphene, and a bandwidth-tunable absorber can be achieved by optimizing the Fermi energy of graphene and the asymmetry parameter of the metasurface to satisfy the critical coupling condition. By varying the Fermi energy of graphene, the quasi-BIC resonances could be effectively modulated and the max transmission intensity difference was up to 81% and a smaller asymmetry parameter will lead to better modulation performance. Our results may also provide potential applications in optical filter and bio-chemical sensing.
2. Design and simulations
The schematic of the proposed dielectric metasurface is illustrated in Fig. 1(a). The Si rings are split along their diameter into two equal parts, and they have splits of length corresponding to θ1 and θ2, which are periodically arranged with SiO2 as the substrate. The Si split rings have thickness h1 = 200 nm, inner radius r1 = 230 nm, outer radius r2 = 400 nm, the periods Px and Py of the metasurface are 1100 nm, and the thickness h2 of the substrate is 500 nm, respectively. The refractive indices of Si and SiO2 are equal to 3.5 and 1.44, respectively. We note that, unless otherwise stated, throughout our study the proposed metasurface is excited by plane waves polarized along x axis and incidents along z axis, and |E0| is set to 1. When θ1 = θ2, the Si split ring in a unit cell is of mirror symmetry about yz and xz plane, and it theoretically supports symmetry-protected BIC. The asymmetry parameter β is defined as sin α and α is calculated as |θ1 - θ2|/2. At the beginning, θ1 = θ2 = 80°, that is, α = 0° and asymmetry parameter β = 0. As the symmetry of the metasurface decreases, i.e., the right and left parts of the split ring rotate clockwise and counterclockwise by the same degree, respectively, asymmetry parameter β is no longer equal to 0, it will introduce a perturbation to open the radiation channel and the BIC will transform into the quasi-BIC. The graphene-dielectric metasurface is shown in Fig. 1(b). The graphene layer is placed on top of the Si spilt rings and modeled as a 2D flat plane, which can be approximated as an ultra-thin lossy film. The optical conductivity of graphene can be obtained through the random phase approximation (RPA) in the local limit, and the surface conductivity is represented by the sum of intraband and interband contributions [37,38],
Fig. 1. (a) Schematics of the dielectric metasurface and the asymmetry parameter β = sin α (α = |θ1 - θ2|/2). (b) Schematic of the graphene-dielectric metasurface. Parameters: Px = Py = 1100 nm, r1 = 230 nm, r2 = 400 nm, h1 = 200 nm, and h2 = 500 nm.
Here, T = 300 K, τ = μEF /ev2 F is the relaxation time, where vF ≈ c/300 is the Fermi velocity, the carrier mobility μ = 10000 cm2 / (V·s) and EF is the Fermi energy. In the proposed hybrid graphene-dielectric metasurface, the Fermi energy of the graphene layer can be modulated by using transparent electrodes which are placed between either the silica substrate or the silicon spilt-rings and the graphene layer [39]. The numerical simulations are conducted using the finite-difference time-domain (FDTD) method. Periodic boundary conditions are used in the x and y directions, and a perfectly matched layer is used in the z direction.
3. Results and discussion
First, we consider symmetry protected BIC mode in the all-dielectric metasurface with the asymmetry parameter β = 0 (α = 0°, θ1 = θ2 = 80°). To obtain band structure and distribution of the magnetic field for BIC mode, dipole cloud and Bloch boundary conditions are used in FDTD Solutions. As shown in Fig. 2(a), the band structure of the metasurface reveals that such a periodic metasurface supports a symmetry-protected BIC at point Γ of the first Brillouin zone. Distribution of the magnetic field for the eigenmode, shown in Fig. 2(b), reveals the magnetic dipole (MD) nature of the BIC with its dipole moment along the z axis. Such an ideal BIC is unstable and it will transition into quasi-BIC mode when perturbation is introduced into the system to break symmetry of the metasurface.
Fig. 2. (a) Band structure associated with the magnetic dipole BIC mode. (b) Distribution of the normalized magnetic field for BIC mode (λ = 1551.5 nm), presented for the xy plane bisecting the split rings 100 nm above the substrate surface. The white arrows indicate the in-plane electric field vectors.
We prove this claim by simulating the transmission spectra of the all-dielectric metasurface with different asymmetry parameters β, and the results are shown in Fig. 3(a). When α = 0°, i.e., the asymmetry parameter β = 0, the resonance bandwidth is successively vanishing, and the Q-factor is theoretically infinite. When α = 3°, the transmission spectrum exhibits an asymmetric line shape with a narrow dip at 1550.9 nm, which is well fitted by the classical Fano formula [22]. To investigate the physical mechanism behind quasi-BIC, distribution of the electric fields |E| of the xy and xz plane of the metasurface with α = 3° at the resonance wavelength of 1550.9 nm are plotted in Fig. 3(c) and Fig. 3(d), respectively. When perturbation is introduced into the system to break symmetry, two electric dipole moments supported by the two arcs of the Si split ring is not equal, i.e., the net dipole moment of the entire system is nonzero. As a result, the system can exchange energy with the continuous free-space radiation mode, and manifests a magnetic dipole quasi-BIC resonance with asymmetric Fano line shape. What’s more, the angle between two electric dipoles is positively correlated with α of the metasurface within a certain range, resulting in an increase both in the radiation leakage and the bandwidth of the resonance. The values of Q-factor as a function of the asymmetry parameter β are summarized in Fig. 3(b). The results show that for small values of β the behavior of Q-factor for the metasurface is clearly inverse quadratic, that is,
where β = sin (|θ1 - θ2|/2) and Q0 is a constant determined by the metasurface design, being independent on θ [22]. The adjustment range of Q-factor exceeds three orders of magnitude which indicates that we can actively tune the radiation coupling rate of the resonance mode to a large extent. It can be used to design a system that satisfies critical coupling conditions to achieve perfect absorption when materials with non-radiative loss are introduced into the system. The bandwidth of the absorber can even be designed to be adjustable, when the materials with tunable non-radiative loss are introduced. Furthermore, the magnetic dipole quasi-BIC resonance is accompanied by a giant local field enhancement, as shown in Fig. 3(c) and Fig. 3(d), and most of the energy is localized in the Si split ring. Therefore, it is believed that quasi-BIC can enhance the interaction between graphene and light, for reason that the optical power absorbed by graphene is highly correlated with the overlapping electric field between graphene and dielectric metasurface. It can be described as [28]Fig. 3. (a) Evolution of transmission spectra for the all-dielectric metasurface with different α. (b) Dependence of the Q-factor on the asymmetry parameter β (log-log scale). (c, d) When α = 3°, the distribution of |E| at the resonance (λ = 1550.9 nm) presented for the xy plane 5 nm above the split rings surface and the xz symmetry plane of the unit cell, respectively.
Considering the difference in the amplitude of the electric field |E| along the thickness of the Si split ring, the absorption spectra of graphene with EF = 0.4 eV at different positions of the metasurface are calculated using Eq. (3), the results are illustrated in Fig. 4(a) and Fig. 4(b). When the graphene layer is moved from a plane embedded halfway into the Si split ring (z = 100 nm plane) to the surface of the Si split ring (z = 200 nm plane), the resonance wavelength blueshifts and the absorption intensity of graphene is around 0.5. As z > 200 nm, the graphene layer is suspended on the metasurface, its absorption intensity significantly reduces as z increases, mainly due to the greatly decrease of the electric field intensity. It can be concluded that the absorption is positively correlated with the intensity of the electric field, so we can utilize the strong field enhancement characteristics of quasi-BIC to increase the absorption of graphene.
Fig. 4. (a) When α = 9° and the Fermi energy of graphene is 0.4 eV, the absorption spectra of graphene at different positions in the structure. (b) For (a), the absorption peaks and resonance wavelengths as a function of coordinate z. (c) The critical coupling results of the monolayer graphene at different positions in the structure. (d) For (c), the FWHMs and resonance wavelengths as a function of coordinate z.
Another benefit is that the dissipation loss of the graphene is strongly related to its permittivity and conductivity which can be tuned by adjusting the Fermi energy of the graphene. While the radiation loss of the dielectric metasurface can be modulated by triggering BICs transform into quasi-BICs, i.e., by tuning the asymmetry parameter. According to the temporal coupled-mode theory (TCMT), when only one single-mode optical resonator is considered, the absorption in the resonance system can be written as [40,41],
Once γ is equal to δ, the system is under the critical coupling condition. This single-mode two-port system has a theoretical maximum Amax = 0.5. That’s to say, in the graphene-dielectric metasurface, we can theoretically design and optimize the system to satisfy the critical coupling conditions to achieve perfect absorption at the resonance frequency ω = ω0. In addition, absorption bandwidth [the full width at half maximum (FWHM)], can be calculated as,
The absorption intensity depends on the ratio of γ and δ, while the absorption bandwidth is related to the total damping rate. Therefore, the absorption performance of the hybrid system can be effectively modulated by these two parameters.
When the graphene layer with EF = 0.4 eV is placed on different z-planes of the metasurface, the system can be optimized to satisfy the critical coupling condition by tuning α of the dielectric metasurface. Figure 4(c) and Fig. 4(d) show the critical coupling results of the graphene-dielectric metasurface while graphene layer is placed on different z planes. As the increase of z, the resonance wavelength redshifts, and the bandwidth narrows. The physical mechanism of bandwidth tunability is mainly because the δ of absorbing material is not only dependent on its inherent absorption coefficient, but also related to the field distribution of the structural resonance state and the position of the absorbing material in the structure. From Fig. 3(d), we can know that most of the energy is confined in the Si split rings, and the energy maximum is near the plane z = 100 nm. Obviously, the field intensity becomes lower as z increases from 100 nm, resulting in the less light coupled to the resonator being absorbed by the graphene, thereby effectively reducing δ. At z = 350 nm, the minimum bandwidth is 0.8 nm. It is known that the γ of the BIC metasurface can be as low as 0, which means that when the absorbing material is placed in the ultra-weak field region of the structure, the absorption bandwidth can be close to infinitely narrow.
We considered the situation where the graphene layer is placed on the metasurface (z = 200 nm plane). As shown in Fig. 5, the graphene-dielectric metasurface is carefully optimized to reach the critical coupling state, thus achieving maximum absorption of the system. As shown in Fig. 5(a), the maximum absorption is achieved at the wavelength of 1547.3 nm and the bandwidth is 11 nm when the Fermi energy of graphene is equal to 0.1 eV and α = 11°. As EF increased from 0.1 eV to 0.4 eV at a step of 0.1 eV, in order to satisfy the critical coupling condition, the asymmetry parameter should be correspondingly reduced, resulting in a tunable resonance bandwidth. When EF = 0.4 eV and α = 7°, the maximum absorption is achieved at the wavelength of 1551.5 nm and the bandwidth is 5 nm, as shown in Fig. 5(d). In conclusion, the absorption of graphene can be effectively enhanced by quasi-BIC resonances, and an absorber with tunable bandwidth can be designed based on these results.
Fig. 5. (a) - (d) Absorption spectra of graphene with Fermi energy EF = 0.1 - 0.4 eV under critical coupling. (Graphene is deposited on top of the Si split rings, at z = 200 nm plane.)
In turn, quasi-BIC resonances can be modulated by graphene and the transmission spectra of the metasurface with the graphene layer are shown in Fig. 6(a). We started with a moderate α = 4°, the EF of graphene is 0.1 eV, 0.4 eV and 0.6 eV, respectively. While the resonance is broadened and becomes flatter with the decrease of the EF, this can be attributed to the strong coupling between the in-plane electric field of dielectric metasurface and the interband absorption of the graphene. The resonance wavelength exhibits a red shift as the EF decreases from 0.6 eV to 0.4 eV, while a blue shift is observed when the EF continues to decrease to 0.1 eV. This leaves the resonance wavelength changed little comparing the EF equals to 0.1 eV and 0.6 eV, which provides a good basis for designing optical modulator based on quasi-BIC resonance. The underlying physics mechanism is that graphene features metallic behavior when the EF = 0.6 eV, while it gradually transforms into a lossy dielectric with the decrease of the EF. The performance of the optical modulator can be displayed by calculating the absolute value of the transmission difference ΔT = |T(EF = 0.6 eV) – T(EF = 0.1 eV)| as shown in Fig. 6(b). Results show that the max of ΔT can reach 81% at the resonance wavelength of 1548 nm. It means that an efficient and tunable amplitude modulator can be designed in the near-infrared wavelength range, and it also has broad prospects in biosensing and telecommunications applications.
Fig. 6. (a) Transmission spectra of the graphene-dielectric metasurface under different Fermi energies of the monolayer graphene (α = 4°). (b) The absolute value of the transmission difference between graphene with EF = 0.1 eV and EF = 0.6 eV (α = 4°).
To further investigate the modulation effect of graphene on quasi-BIC resonances, the influence of element α is also considered. As shown in Fig. 7, when the graphene layer with EF = 0.1 eV is introduced, the transmission spectra of the graphene-dielectric metasurface with different α is calculated. The cases of the metasurface without the graphene layer are also added to the figures as a reference. When α = 0°, that is, in the absence of perturbation, the transmission spectra changes little after the graphene layer with EF = 0.1 eV is introduced. While the transmission spectra changes dramatically in the case of α = 2°, the max difference of transmission is up to 0.94 at the wavelength of 1552 nm. As the α is further increased from 2° to 6° at a step of 2°, we can see from the Figs. 7(b) – 7(d) that the contrast ratio of the quasi-BIC resonances gradually increases, demonstrating the decrease of the modulating ability of graphene. The graphene-dielectric metasurface with minimal asymmetry supporting high Q-factor quasi-BIC has great potential and excellent performance in optical modulator applications.
Fig. 7. Comparison of the transmission between the metasurface without and with graphene (EF = 0.1 eV).
4. Conclusion
In summary, we have proposed a graphene-dielectric metasurface to enhance the light-matter interactions in graphene. The dielectric metasurface consisted of periodically arranged silicon split rings placed on the silica substrate, which supported symmetry-protected bound states in the continuum (BIC). When perturbation was introduced into the system to break symmetry, the BIC transformed into the quasi-BIC with high Q-factor. When the graphene layer was integrated with the dielectric metasurface, the absorption of graphene could be enhanced by the quasi-BIC resonance and a bandwidth-tunable absorber could be achieved by optimizing the Fermi energy of graphene and the asymmetry parameter of the metasurface to satisfy the critical coupling condition. In addition, the quasi-BIC resonances could be effectively modulated and the max transmission intensity difference was up to 81% by varying the Fermi energy of graphene. Finally, we considered the influence of the asymmetry of the metasurface on the transmission modulation, and the results showed that smaller asymmetry parameter had a better modulation performance. Our results may provide theoretical support for the design of absorber and modulator based on the quasi-BIC.
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 maybe obtained from the authors upon reasonable request.
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