High-Quality Graphene-Based Tunable Absorber Based on Double-Side Coupled-Cavity Effect

Graphene-based devices have important applications attributed to their superior performance and flexible tunability in practice. In this paper, a new kind of absorber with monolayer graphene sandwiched between two layers of dielectric rings is proposed. Two peaks with almost complete absorption are realized. The mechanism is that the double-layer dielectric rings added to both sides of the graphene layer are equivalent to resonators, whose double-side coupled-cavity effect can make the incident electromagnetic wave highly localized in the upper and lower surfaces of graphene layer simultaneously, leading to significant enhancement in the absorption of graphene. Furthermore, the influence of geometrical parameters on absorption performance is investigated in detail. Also, the device can be actively manipulated after fabrication through varying the chemical potential of graphene. As a result, the frequency shifts of the two absorption peaks can reach as large as 2.82 THz/eV and 3.83 THz/eV, respectively. Such a device could be used as tunable absorbers and other functional devices, such as multichannel filters, chemical/biochemical modulators and sensors.

Recently, great attention has been devoted to graphene-based absorbers. However, the absorption of monolayer graphene is very low. In order to solve this problem, some strategies have been proposed. For instance, high-efficiency absorbers have been realized by designing graphene patterns with different shapes, such as split ring, fishing-net and E-shaped models [14,15]. Another method is to use periodic metal units to excite surface plasmon polaritons to enhance the absorption of graphene [16][17][18][19][20][21]. In addition, periodic multi-layer dielectric materials have also been reported to enhance the absorption of graphene based on optical resonance effect [22][23][24][25][26]. These achievements open up a new way for designing graphene-based absorbers. However, in these approaches, no matter if adding periodic metal units or multi-layer dielectric materials to monolayer graphene, it needs a complicated fabrication process or it may give rise to a big size of device. Moreover, in these studies, only single-side coupled-cavity effect has been applied. These would bring limitations to applications. Therefore, graphene-based absorbers with high-efficiency and ultra-compact structure are necessary to be further explored.
In this paper, in order to improve the absorption of monolayer graphene and simultaneously keep the size of device compact, a new kind of absorber with monolayer graphene sandwiched between two layers of dielectric rings is proposed. The double-layer dielectric rings equivalent to resonators that are symmetrically added to both sides of monolayer graphene. Due to the double-side coupled-cavity effect, almost complete absorption is obtained. The influence of geometrical parameters on absorption performance is further discussed in detail. Also, the result demonstrates that a wide frequency range of absorption spectrum could be tuned by varying the chemical potential of graphene. Such a device could be used as tunable absorbers and other functional devices, such as multichannel filters, chemical/biochemical modulators and sensors.
Here, there are two points need to be emphasized. Firstly, different from some research in which only single-side coupled-cavity structures have been investigated [27][28][29], the double-layer dielectric rings we proposed to add to both sides of monolayer graphene could be more efficient to enhance the electric-field intensities on the upper and lower surfaces of graphene, whose double-side coupled-cavity effect could increase the absorption of graphene. Secondly, the double-layer dielectric rings do not require fabrication of metallic nanostructures [30] and could make the size of device more compact, compared with the periodic metal units or multi-layer dielectric materials reported in the other articles. The use of dielectric rings shows advantages in terms of fabrication cost since they have a lower price.

Structure Design and Calculation Method
The schematic of graphene-based absorber we proposed is shown in Figure 1a. On an insulator substrate with refractive index n sub = 1.45, monolayer graphene is sandwiched between two layers of identical dielectric-ring arrays. The dielectric is chosen as silicon with refractive index n silicon = 3.4. The upper and lower layers of dielectric-ring arrays are symmetric with respect to the graphene plane. The lower layers of dielectric-ring arrays are embedded in polymethyl methacrylate (PMMA) layer with refractive index n PMMA = 1.48. The thickness of the insulator substrate is set as D sub = 12 µm. A gold layer is deposited at the back side of the insulator substrate to present any transmission through the structure. In this way, perfect absorption can be obtained when the reflection is close to zero. The structure is periodic in the x-axis and the y-axis directions with periodic constant P = 8.5 µm. Figure 1b shows the simulation model of a pair of dielectric rings. Figure 1c shows the x-y cross section of a single dielectric ring. The inner and outer radii of dielectric ring are assumed as R in = 1.4 µm and R out = 2.4 µm, respectively. The height is set as H ring = 3 µm. The PMMA layer has the same thickness H PMMA = H ring = 3 µm. For convenience in the following analysis, a new parameter W ring = R out − R in standing for the width of the dielectric rings is defined.
For the structure, monolayer graphene is modeled as a conductive surface with thickness D g = 0.35 nm in the simulations. The surface conductivity σ(ω, µ c , Γ, T) in THz range can be expressed by the Drude model according to the Kubo formula, as given by [15], where ω is the radian frequency, µ c is the chemical potential of graphene, Γ = (2τ) −1 is the phenomenological scattering rate with τ standing for the relaxation time due to charge carrier scattering, T is the temperature,h is the reduced Plank constant, κ B is the Boltzmann constant, e is the electron charge. The relaxation time satisfies the relationship τ = µµ c /eν F 2 . Here, ν F = 10 6 m/s represents the Fermi velocity and µ is the carrier mobility of graphene. According to reported result, µ can be changed and the value as high as 4 m 2 /Vs can be obtained at room temperature [31]. Different µ c is obtained by changing the dc voltage bias. and y-axis directions to simulate the periodic structure. Perfectly matched layers (PMLs) are used at the outside of the top and bottom planes of calculation space to absorb the electromagnetic wave. In x-z plane, θ0 represents incident angle. When the incident electromagnetic wave reaches the gold layer, it is completely reflected if the thickness of gold substrate is set large enough. Therefore, the transmission is zero. The absorption of structure is calculated by A = 1 − R with R = Pre/Pin, in which Pin and Pre stand for the input and reflection energy flows, respectively. Pre is detected at the back of incident wave plane.

Results and Discussions
In Figure 2a, it gives the absorption spectrum of the proposed graphene-based structure, as shown by the black line. For the parameters, T = 300 K, τ = 1 ps, μc = 1.0 eV and μ In the calculation, the finite difference time domain (FDTD) method is employed to calculate the reflection properties and simulate the electromagnetic field distributions of structure. As shown in Figure 1b, periodic boundary conditions are applied in both x-axis and y-axis directions to simulate the periodic structure. Perfectly matched layers (PMLs) are used at the outside of the top and bottom planes of calculation space to absorb the electromagnetic wave. In x-z plane, θ 0 represents incident angle. When the incident electromagnetic wave reaches the gold layer, it is completely reflected if the thickness of gold substrate is set large enough. Therefore, the transmission is zero. The absorption of structure is calculated by A = 1 − R with R = P re /P in , in which P in and P re stand for the input and reflection energy flows, respectively. P re is detected at the back of incident wave plane.

Results and Discussions
In Figure 2a, it gives the absorption spectrum of the proposed graphene-based structure, as shown by the black line. For the parameters, T = 300 K, τ = 1 ps, µ c = 1.0 eV and µ = 1 m 2 /Vs are chosen. The period unit for calculating is given in the inset. Here, Under TE-polarized normal incident wave (TE wave, the electric field along x-axis direction), it can be seen that there are two obvious absorption peaks at f 1 = 5.18 THz and f 2 = 7.02 THz. For convenience, they are denoted as modes M 1 and M 2 , respectively. The two peaks exhibit ultra-high absorption, reaching to 98.6% and 99.8%, respectively. In order to explore this absorption phenomenon, the electric-field amplitude distributions corresponding to the two absorption peaks are simulated and shown in Figure 2b-e. Figure 2b,c represent the electric-field amplitude distributions of mode M 1 at the x-z plane (y = 0) and x-y plane (z = 0), respectively, and Figure 2d,e represent that of mode M 2 . It can be observed that the electric-field amplitude distributions of modes M 1 and M 2 are highly localized at the interfaces of the upper-layer dielectric rings and graphene layer, and the lower-layer dielectric rings and graphene layer. Not only that, there are also strong electric-field amplitude distributions localized in the inner space of the double-layer dielectric rings. As shown in Figure 2f, the double-layer dielectric rings added on both sides of the graphene layer are equivalent to double-side coupled-cavity model. For modes M 1 and M 2 , the electric-field intensities of the upper and lower graphene interfaces are simultaneously enhanced by the resonance of double-side coupled cavities, which reinforces the graphene-wave interaction and enhances the absorption of graphene layer. On the other hand, referring to the literature [32], when the relaxation time of graphene is reduced to τ = 0.05 ps (µ c = 0.5 eV), the absorption spectrum is also calculated, as shown by the green line in Figure 2a. We can see that the absorption is reduced due to the decrease in the corresponding conductivity of graphene. = 1 m /Vs are chosen. The period unit for calculating is given in the inset. Here, Under TEpolarized normal incident wave (TE wave, the electric field along x-axis direction), it can be seen that there are two obvious absorption peaks at f1 = 5.18 THz and f2 = 7.02 THz. For convenience, they are denoted as modes M1 and M2, respectively. The two peaks exhibit ultra-high absorption, reaching to 98.6% and 99.8%, respectively. In order to explore this absorption phenomenon, the electric-field amplitude distributions corresponding to the two absorption peaks are simulated and shown in Figure 2b-e. Figure 2b,c represent the electric-field amplitude distributions of mode M1 at the x-z plane (y = 0) and x-y plane (z = 0), respectively, and Figure 2d,e represent that of mode M2. It can be observed that the electric-field amplitude distributions of modes M1 and M2 are highly localized at the interfaces of the upper-layer dielectric rings and graphene layer, and the lower-layer dielectric rings and graphene layer. Not only that, there are also strong electric-field amplitude distributions localized in the inner space of the double-layer dielectric rings. As shown in Figure 2f, the double-layer dielectric rings added on both sides of the graphene layer are equivalent to double-side coupled-cavity model. For modes M1 and M2, the electric-field intensities of the upper and lower graphene interfaces are simultaneously enhanced by the resonance of double-side coupled cavities, which reinforces the graphene-wave interaction and enhances the absorption of graphene layer. On the other hand, referring to the literature [32], when the relaxation time of graphene is reduced to τ = 0.05 ps (μc = 0.5 eV), the absorption spectrum is also calculated, as shown by the green line in Figure 2a. We can see that the absorption is reduced due to the decrease in the corresponding conductivity of graphene. In order to further explain the numerical results above, the absorption performance based on the double-side coupled-cavity effect is also investigated by coupled mode theory (CMT) [23]. According to the simulation model of a single-period of the proposed graphene-based structure in Figure 1b, the upper-layer and lower-layer dielectric rings are respectively denoted as cavity C 1 and cavity C 2 , and the unit cell of graphene layer is denoted as plane-I. Cavity C 1 and cavity C 2 are symmetric related to plane-I. The model of plane-I coupled with cavity C 1 and cavity C 2 can be considered as a coupled-cavity system, as shown in Figure 2f, which also can be described by the following formula based on the CMT analysis [33,34].
where a i (i = 1, 2) is the mode amplitude of cavity C i , ω i (i = 1, 2) is the resonance frequency of cavity C i , S +i and S -i are the amplitudes of the incoming and the outgoing waves for cavity C i , respectively, and κ is the mutual coupling coefficient between the two cavities C 1 and C 2 , β i is the coupling coefficient between cavity C i and free space. We have ω 1 = ω 2 and β 1 = β 2 if considering the symmetry in structure, so new symbols ω 0 = ω 1 = ω 2 and β = β 1 = β 2 are defined. In our case, it has S +2 = 0. Assuming that S ±i and a i (i = 1, 2) have time dependence exp(jωt), we can obtain the following relation from Equation (3): From Equations (2), (4) and (5), the absorption of structure can be calculated as: As can be observed from Equation (6), at resonance frequency (ω = ω 0 ), if β is equal to κ, perfect absorption occurs. This critical coupling condition suggests that the impedance matching between cavities C 1 and C 2 must be equal to that between cavity C 1 (or cavity C 2 ) and free space. When the incident wave is chosen at the resonance frequency (ω = ω 0 ) and the system satisfies critical coupling condition (β = κ), resonant modes are excited at the interface of the upper-layer dielectric ring and graphene layer (called upper-layer resonant mode), and simultaneously excited at the interface of the lower-layer dielectric ring and graphene layer (called lower-layer resonant mode), as can be shown by the electric-field amplitude distributions of modes M 1 and M 2 in Figure 2b,d. The upper-layer and lower-layer resonant mode distributions are symmetric related to the graphene layer due to the high symmetry in structure. Based on the coupled-cavity effect, electric field intensity around the monolayer graphene is enhanced, which reinforces the graphene-wave interaction. As a result, the reflection wave of structure nearly vanishes and almost all the incident energies are absorbed. Perfect absorption occurs.
For comparison, two relatively simple structures are considered, one is the structure only with the upper-layer dielectric rings (called simple structure 1), and the other is the structure only with the lower-layer dielectric rings (called simple structure 2). Their absorption spectra are plotted in Figure 3a that there are strong field distributions at the lower-layer dielectric rings and graphene layer. The lower-layer dielectric rings are equivalent to resonance cavities added below the graphene layer, as shown in Figure 3g. Comparing Figure 2 with Figure 3, we conclude that the absorption spectrum (black line) in Figure 2a is approximately a combination of the coupled effect of simple structures 1 (only with the upper-layer dielectric rings) and simple structures 2 (only with the lower-layer dielectric rings). The double-layer dielectric rings added on both sides of graphene layer can bring higher absorption efficiency than that of only adding a single layer of dielectric rings on one side of the graphene layer, which is because a double-side coupled-cavity model can yield stronger resonance effect than a single-side-cavity model.   In addition, the influence of the geometrical parameters (W ring , H ring in Figure 1) on the absorption performance are investigated. These analyses provide some guidance to structural design and fabrication.
First, we investigate the influence of the width W ring of the double-layer dielectric rings on the absorption performance. In Figure 4, it shows the change of absorption spectrum with the increase of W ring . We observe that when W ring is relatively small, such as W ring = 0.3 µm, there is only one absorption peak at 8.91 THz with absorption 87.7%. As W ring increases, two absorption peaks appear. It shows that when W ring = 1.0 µm, two absorption peaks appear at 5.18 THz with absorption 98.6%, and 7.02 THz with absorption 99.8%, respectively. When W ring further increases, the number of absorption peaks continues to increase. For instance, when W ring = 2.6 µm, five high-absorption peaks appear at 3.56 THz with absorption 95.5%, 4.42 THz with absorption 99.8%, 5.49 THz with absorption 95.5%, 6.24 THz with absorption 96.6%, and 7.15 THz with absorption 83.3%, respectively. For this case, the frequency range of absorption above 75% is from 3.35 THz to 7.22 THz, while that for the case of W ring = 0.3 µm is only from 8.69 THz to 9.14 THz. It means that the absorption band is greatly widened when W ring increases. This is because the relatively wide dielectric rings could bring an increase in contact area of the dielectric rings and graphene layer, then more resonance mode can be supported. The result also can be illustrated from the electric-field amplitude distributions of the absorption peaks for different W ring , as shown in Figure 4g-l. The characteristic provides a significant design guide for developing graphene-based absorbers with multiple resonance peaks.   Figure 5, when Hring is relatively small, such as Hring = 0.8 μm, the absorptions are 88.6% at 6.37 THz for the left mode M1, and 97.4% at 6.99 THz for the right mode M2. We can see that the absorptions are not high enough, which is because the dielectric rings with small height exhibit weak ability in z-direction spatial localization of electromagnetic wave (concluded from the comparison of Figure 5b-e), resulting in rel-  Figure 5, when H ring is relatively small, such as H ring = 0.8 µm, the absorptions are 88.6% at 6.37 THz for the left mode M 1 , and 97.4% at 6.99 THz for the right mode M 2 . We can see that the absorptions are not high enough, which is because the dielectric rings with small height exhibit weak ability in z-direction spatial localization of electromagnetic wave (concluded from the comparison of Figure 5b-e), resulting in relatively weak resonance effect and unable to obtain enough strong absorption of graphene. When H ring further increases, the absorption peaks M 1 has a red shift and the absorption peaks M 2 has no obvious change in frequency. Their absorptions are gradually enhanced. When H ring ≥ 3.0 µm, the two absorption peaks tend to be stable not only in frequency position, but also in absorption amplitude. The positions are basically fixed at about 5.18 THz for M 1 , and about 7.02 THz for M 2 . Moreover, the absorptions are above 98.5% for M 1 , and above 99.7% for M 2 . Therefore, 3.0 µm ≤ H ring ≤ 5.0 µm is the best range since H ring being too small (H ring < 3.0 µm) is not of avail to the localization of electromagnetic wave, and an H ring too large (H ring > 5.0 µm) is not necessary for fabrication in practice. On the other hand, one remarkable advantage of graphene is that its chemical potential μc can be manipulated over a wide range through adjusting direct-current bias voltage Vg that is added between the graphene layer and gold substrate. Due to this characteristic, graphene-based devices can be actively manipulated after final design and fabrication. As On the other hand, one remarkable advantage of graphene is that its chemical potential µ c can be manipulated over a wide range through adjusting direct-current bias voltage V g that is added between the graphene layer and gold substrate. Due to this characteristic, graphene-based devices can be actively manipulated after final design and fabrication. As shown in Figure 6, it illustrates the influence of changing the chemical potential µ c of graphene on absorption performance. The parameters W ring = 1.0 µm, H ring = 3.0 µm are chosen, referring to Figures 4 and 5. In this case, two absorption peaks M 1 and M 2 can be observed. In Figure 6, it shows that when µ c increases from 0.1 eV to 1.5 eV (µ = 1 m 2 /Vs), both of the two absorption peaks M 1 and M 2 exhibit obvious blue shifts and the frequency shifts can reach as large as 2.82 THz/eV for the left peak M 1 and 3.83 THz/eV for the right peak M 2 , which demonstrates the remarkable tunability of structure. Meanwhile, when µ c increases from 0.1 eV to 1.5 eV, the absorption of the left peak M 1 increases from 12.6% to 98.8%, and that of the right peak M 2 also increases from 10.7% to 99.8%. It indicates that the absorptions of the two peaks are enhanced with the increase of µ c . Such a characteristic could be used as tunable absorbers and other functional devices, such as multichannel filters, chemical/biochemical modulators and sensors.
Nanomaterials 2021, 11, x FOR PEER REVIEW 10 of 13 be used as tunable absorbers and other functional devices, such as multichannel filters, chemical/biochemical modulators and sensors. The frequency spectra of the absorption are represented in Figure 6 for different values of chemical potential obtained by increasing the dc voltage bias. The dielectric breakdown strength of the used materials is relatively high (large than 20 V/μm) in theory, which can satisfy the need of design. In practice, a layer of ionic gel would be applied over graphene layer. It can significantly reduce the voltage bias and avoid dielectric break down. This method has been demonstrated in the reference [35]. In the discussions above, absorption spectra have been investigated for normal incidence (θ0 = 0°). Finally, the absorption spectra at different θ0 are illustrated for TE and TM polarizations, as shown in Figure 7a,b, respectively. When the plane of incidence is set in the x-z plane, the incident angle θ0 is defined as the angle between the incident plane wave and the positive z-direction, as shown in Figure 1b. According to refraction theory, we have, where n0 is the refractive index of air, n1 is the effective refractive index of absorption domain and θ1 is the refraction angle. Here, n1 is complex due to the absorption. We can define n1 = n1r + i·n1i, where n1r and n1i are the real and imaginary components of n1. Therefore, the refraction angle θ1 is also complex. It can be expressed as θ1 = θ1r + i·θ1i, where θ1r The frequency spectra of the absorption are represented in Figure 6 for different values of chemical potential obtained by increasing the dc voltage bias. The dielectric breakdown strength of the used materials is relatively high (large than 20 V/µm) in theory, which can satisfy the need of design. In practice, a layer of ionic gel would be applied over graphene layer. It can significantly reduce the voltage bias and avoid dielectric break down. This method has been demonstrated in the reference [35].
In the discussions above, absorption spectra have been investigated for normal incidence (θ 0 = 0 • ). Finally, the absorption spectra at different θ 0 are illustrated for TE and TM polarizations, as shown in Figure 7a,b, respectively. When the plane of incidence is set in the x-z plane, the incident angle θ 0 is defined as the angle between the incident plane wave and the positive z-direction, as shown in Figure 1b. According to refraction theory, we have, n 0 sin θ 0 = n 1 sin θ 1 (7) where n 0 is the refractive index of air, n 1 is the effective refractive index of absorption domain and θ 1 is the refraction angle. Here, n 1 is complex due to the absorption. We can define n 1 = n 1r + i·n 1i , where n 1r and n 1i are the real and imaginary components of n 1 . Therefore, the refraction angle θ 1 is also complex. It can be expressed as θ 1 = θ 1r + i·θ 1i , where θ 1r and θ 1i are the real and imaginary components of θ 1 . Therefore, the complex refraction angle θ 1 is related to the parameters θ 0 , n 0 , n 1r and n 1i . With FDTD Solutions software, the simulation model can be calculated by setting the parameters θ 0 , n 0 , n 1r and n 1i , in which n 0 , n 1r and n 1i depend on the parameters of graphene and dielectric materials. It should be noted that the refracted plane wave into the absorber is nonuniform since it is closely related to the corresponding material in the refraction region. The absorption of structure can be indirectly calculated by collecting the total reflection energy flow at the detection plane, as shown in Figure 1b. In Figure 7, we can find that the absorption peaks M 1  and θ1i are the real and imaginary components of θ1. Therefore, the complex refraction angle θ1 is related to the parameters θ0, n0, n1r and n1i. With FDTD Solutions software, the simulation model can be calculated by setting the parameters θ0, n0, n1r and n1i, in which n0, n1r and n1i depend on the parameters of graphene and dielectric materials. It should be noted that the refracted plane wave into the absorber is nonuniform since it is closely related to the corresponding material in the refraction region. The absorption of structure can be indirectly calculated by collecting the total reflection energy flow at the detection plane, as shown in Figure 1b. In Figure 7, we can find that the absorption peaks M1 and M2 are almost unchanged at incident angles as large as 35° for TE polarization and 45° for TM polarization. This phenomenon is attributed to the high symmetry in the structure. The results would have important applications for wide-angle absorbers in practice.

Conclusions
In summary, we introduce a new kind of high-quality graphene-based absorber using a double-side coupled-cavity effect. In this structure, monolayer graphene is sandwiched between two layers of dielectric rings, which are equivalent to resonators coupled

Conclusions
In summary, we introduce a new kind of high-quality graphene-based absorber using a double-side coupled-cavity effect. In this structure, monolayer graphene is sandwiched between two layers of dielectric rings, which are equivalent to resonators coupled to both sides of graphene layer. The double-side coupled-cavity effect can greatly enhance the absorption of graphene. The result shows that two peaks with almost complete absorption are obtained. The influence of geometrical parameters on absorption performance are investigated in detail. Furthermore, through varying the chemical potential of graphene, the frequency shifts of the two absorption peaks can reach as large as 2.82 THz/eV and 3.83 THz/eV, respectively. The proposed structure also could be used for applications related to multichannel filters, chemical/biochemical modulators and sensors.