1. Introduction

Metamaterial is an artificial resonant structure that is designed to obtain non-natural features, for example, a metamaterial absorber (MMA), which offers advantages over traditional absorbers, is one of the uses [1]. MMA is a structure that attenuates electromagnetic wave energy, converts incident energy into heat, and reduces reflected energy back to the source. It is widely employed in a variety of light bands, including microwave [2], millimeter wave [3], terahertz (THz) [4], near-infrared, and visible frequencies [5,6] etc. Because of their prospective uses [7] in communication, imaging, detection, sensing, and stealth, MMAs in the THz region have recently garnered a significant amount of interest and become one of the most popular research topics. For instance, Tao et al. presented a THz MMA via a resonant ring of bilayer metal [8], and by individually modifying the permeability and permittivity, the structure achieved a maximum absorption of more than 90%. In addition, four-band [9], five-band [10], six-band [11], hepta-band [12], and octa-band [13] absorbers were also proposed while an infrared ultra-broadband MMA was achieved experimentally [14]. Many studies have shown that most MMAs are caused by surface plasmon resonance (SPR), which happens when light waves and free electrons interact at the point where dielectric-metal merged [15]. Despite this, the absorbance of MMAs based on metal would be fixed because of the metal’s steady conductivity and the immutable geometrical structure after manufacturing, which severely hampers their practical applicability. MMAs with changeable conductivity or functionalities that can be dynamically configured are therefore in high demand. In the past few years, numerous effectively adjustable materials, including graphene [16], vanadium dioxide ($VO_{2}$) [17], and molybdenum disulfide $(MoS_{2})$ [18] have been used to tune MMAs.

Among the previously mentioned flexible materials, graphene (formed of a monolayer of honeycomb-shaped carbon, developed through highly sophisticated manufacturing, and possesses excellent carrier mobility at ambient temperature) based THz MMAs have drawn the attention of an increasing amount of researchers. Since one graphene sheet has an absorbance of approximately 2.3% [19], several graphene designs are devised to stimulate electromagnetic resonances on the basis of surface plasmon polariton (SPP) in order to increase graphene’s absorption. For instance, a single-band perfect MMA using graphene micro-ribbons [16] as well as dual-band absorption [20] MMA based on graphene strip arrays were achieved. Besides that, a triple-band MMA was obtained by Wu et al. which is extremely sensitive to the polarization angle and required graphene widths variation [21]. Moreover, many forms of perfect multi-band MMAs consisting of various graphene shapes have also been suggested [22,23], but these multi-band MMAs have peak absorption at some particular frequencies, not over a continuous range of frequencies. In addition to these single-band and multi-band MMAs based on graphene, several broadband MMAs based on graphene have also been reported. For instance, asymmetric graphene-based broadband MMA was proposed by Liu and colleagues [24], but the multilayer structure and use of different chemical potentials make the MMA very complex. For broadband absorption, a graphene design with repeated sinusoidal shapes was utilized [25], limited to one range of frequencies. Moreover, Kim et al. suggested a broadband MMA that achieved broadband characteristics in dual frequency bands; however, the structure is extremely complex [26]. Although many tunable broadband MMAs have been designed over the years, the vast majority of them have several limitations: lower bandwidth, significant dependency on incident angle and polarization. Some works showed much higher bandwidth, but they had complex multi-layer structures or multiple resonators, which made them difficult to fabricate. The existing multi-band MMAs, ranging from dual-band to octa-band absorption, could not function as broadband absorbers due to their narrowband characteristics. On the other hand, some of the reported MMAs claim broadband characteristics in continuous band but their bandwidth is very limited and can operate in a maximum of two bands; however, no broadband MMAs that can simultaneously function in three distinct wide frequency bands have been reported.

This paper presents the design of a broadband MMA which shows near-unity absorption in three different wide bands and consists of a hybrid graphene-gold patch embedded into $SiO_{2}$, where a bottom gold plane is used to neglect transmission and increase absorption. The device could provide wide-angle, polarization-insensitive, triple-broadband absorption with flexible active tuning by combining the advantages of metal and graphene. This is, as far as we know, the first paper to propose a MMA with broadband characteristics in three distinct wide THz bands. Simulation results indicate that the designed MMA has three broadband absorptions of more than 90% absorbance with a total bandwidth of more than 4.3 THz for TE as well as TM polarized waves. Finally, the investigation of polarization and incident angle sensitivity demonstrates that the proposed MMA is completely insensitive to polarization for both the TE and TM waves and retains a high absorption performance for both the TE and TM waves when the angle of incidence is less than 40°.

2. Structural design and basic theory

2.1 Suggested design

Figure 1(a) depicts the structure’s fundamental unit cell, whereas Fig. 1(b) depicts the top view, in which the metallic layer is embedded in the $SiO_{2}$ layer. The embedded metallic layer serves to form the metasurface layer, which is comprised of a hybrid gold-graphene rectangular patch. The dielectric layer in which the hybrid patch is embedded is used to achieve the analogy of a transmission line [27]. The gold layer at the bottom is enough thick to avoid transmission and achieve perfect reflection. The period of the unit cell is represented by p, whereas the side of the inner patch is represented by b. The width and height of each graphene patch are a1 and b1, respectively. In addition, the thickness of the bottom $SiO_{2}$ layer (between the ground metallic layer and the patch) is $t_{s}$, and the top $SiO_{2}$ layer is $t_{s2}$, whereas the hybrid resonator’s thickness is denoted by $t_m$. The optimal geometric parameters are tabulated in Table 1.

 

Fig. 1. (a) The structure’s fundamental unit cell consisted of a hybrid structure of graphene and gold patch embedded in $SiO_{2}$ substrate and a gold ground plane is used as a bottom layer, (b) prospective top view, and (c) equivalent transmission line model for the structure.

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Table 1. Optimized Parameters of the Proposed Structure

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It is considerd that $SiO_{2}$ has a relative permittivity of 3.9 [12], while gold’s conductivity is $4.09 \times 10^{7} S/m$ [28]. The expected range of frequencies is between 0.1 THz and 12 THz. Within the analyzed spectrum, the thickness of the bottom gold layer is significantly greater than its skin depth, prohibiting transmission [29]. Notably, we chose gold both for the bottom layer and for the resonator because, in the THz range, gold has a higher electrical conductivity with a lower rate of scattering for free electrons and also has a lower Ohmic and non-radiative loss. Moreover, compared to other metals, gold is chemically stable in a wide range of conditions and does not oxidize easily [30]. Consequently, the gold’s inclusion enables the resonant structure to preserve a higher Q factor as well as attain a wider range of applications. The thickness of the gold in the patch is 1 nm, whereas a medium with an effective thickness of 1 nm is assumed to substitute the three layers of graphene sheets since a single layer of graphene is 0.34 nm thick [31].

2.2 Basic theory

COMSOL Multiphysics, a simulation program based on the full-vector finite element method, is used to characterize the performance of the suggested MMA structure. To provide illumination for the unit’s infinite array, periodic boundary conditions are set up along the x and y axes. A floquet port has been added on the uppermost layer of our designed structure that emits electromagnetic waves in the reverse direction of the z axis on the surface of the structure. From the aspect of energy dissipation, the absorber’s absorption characteristics can be analyzed. When the incoming wave hits the absorber’s surface, the energy of the incident wave is split into three parts: the amount of energy that reflected, transmitted, and absorbed by the absorber. Thus, the absorbance of the proposed MMA, $A (\omega )$ is represented as [32,33]

Moreover, the analogy of a transmission line is employed to analytically characterize the properties of absorption for the suggested MMA. The equivalent circuit model of the suggested structure is depicted in Fig. 1(c). In the $SiO_2$ dielectric, a transmission line with perfect magnetic and electric boundary conditions can be used to characterize the plane wave’s propagation. The thin conducting layer of composite gold and graphene at the flat edge between two dielectrics is like a load at the point where two transmission lines meet. The ground gold plane acts as a short circuit, as depicted in Fig. 1(c).

The graphene sheet is intended to be 1 nm thick. Interband and intraband elements are used to characterize the conductivity of graphene [27,34]

In the THz frequencies, where the energy of photon $\hbar \omega \ll K_{B}T$, the interband part can be neglected from Eq. (5). So, Drude-like model is used to describe graphene’s conductivity in the THz regime. The conductivity is linearly proportional to the Fermi energy when $E_F \gg K_{B}T$

Graphene is typically modelled in numerical simulations as a material having small thickness with an effective permittivity in the in-plane direction

The effective permittivity’s normal component, $\varepsilon _{eff,n}=1$, implies that graphene can be compared to a metal of small thickness whose plasma frequency depends on the Fermi energy

By varying the $E_{F}$ of graphene and the $\mathrm{\tau}$, it is obvious that the conductivity of graphene can be changed dynamically to tune the optical properties of specified metamaterial functional devices.

The entire cell structure was meshed using the physics-controlled mesh available in COMSOL Multiphysics in order to numerically solve the model using the finite element method, and an extremely fine mesh setting was used to obtain the highest level of accuracy. As both the graphene and gold layers are extremely thin (1 nm), the transition boundary condition was applied to these interior layers, thereby reducing computational time without sacrificing the precision.

3. Results and discussion

After optimization of different parameters (as illustrated in Table 1) for the proposed model, the final result is achieved with the normal angle of incidence, $\mathrm{\tau}$ = 0.1 ps, and $E_F$ = 0.9 eV, as illustrated in Fig. 2. The solid black line depicts the absorption spectrum for a TE-polarized wave, while the solid red line depicts the absorption spectrum for a TM-polarized wave. Simulation findings show that if the graphene’s electrochemical potential (Fermi energy) is adjusted to 0.9 eV, the absorber has three broad-band absorptions greater than 90% in (1.1-2.72) THz, (5.39-6.47) THz, and (9.25-10.97) THz, with a total bandwidth of 4.42 THz for the TE waves and in the frequency spectrum of (1.14–2.64) THz, (5.5–6.55) THz, and (9.18–10.98) THz, with an absorbance greater than 90% and a total bandwidth of 4.29 THz for TM waves.

 

Fig. 2. Simulated absorption spectrum for the designed broadband MMA, where the black and red solid lines represent the absorption spectra for TE and TM polarized waves, respectively.

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3.1 Optimization of the absorption spectrum

The analysis started with only gold in the dumbbell shape within the rectangular patch (with gold ground plane), i.e., there is no graphene layer. All the following analysis were done with the normal angle of incidence and with the TE wave (polarized electric field along the direction of the y axis). In Fig. 3, the blue line represents the absorption spectrum for only the gold layer with TE polarization, from which it is clear that having only one gold layer can achieve three absorption bands, however the maximum absorption is low for the first two bands. The maximum absorbance for the first band is less than 60%, having a resonance centre frequency at 1.9 THz whereas the absorption for the second band is less than 82%, having a resonance centre frequency of 5.7 THz. The maximum absorption for both these two bands is much lower than unity, and the bandwidth is quite low as well. On the other hand, the maximum absorption for the third band, which has a resonance peak at 10 THz, is more than 90% for frequencies between 9.41 THz and 10.6 THz. As indicated in Fig. 1, the graphene patches are utilized to improve the bandwidth and maximum absorbance in the two lowest bands as it is widely known that graphene’s surface conductivity increases in the lower frequencies [35]. Because the addition of graphene sheets can result in the excitation of graphene SPPs, there is an increase in the interaction of light-matter as well as the dissipative loss rate, which helps to confine a portion of the incoming electromagnetic field and cause it to be absorbed within the structure. The red line in Fig. 3 depicts the absorption spectrum produced by graphene in four rectangular patches containing gold in dumbbell configuration. It is evident from the results that the maximum absorption in the two lowest bands has increased, and bandwidth has also increased, which justifies our design assumption. Both the graphene layer and gold are modelled as 1 nm-thick effective mediums.

 

Fig. 3. The absorption spectrum of the designed broadband MMA with TE waves, where the blue and red lines illustrate the absorption spectrum with only gold in the dumbbell shape within the patch and after adding graphene in the four rectangular patches, respectively.

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Figures 4(a)–4(c) illustrate the distribution of electric field at three resonant centre frequencies of 0.85 THz, 5.6 THz, and 9.85 THz, respectively. The way the electric field is spread out can be used to figure out how the proposed structure absorbs energy. Figure 4(a) demonstrates that the first broad absorption band with a centre frequency of 0.85 THz is achieved as a result of the strong localization of opposite charges at the top and bottom ends of the patch. The subsequent two broadband absorptions, on the other hand, are achieved as a result of the strong localization of opposite charges at the graphene-gold junction’s inner edges and at the edges of the gold patches, respectively. The buildup of electric charges at the two opposing ends causes them to function as two capacitors, each of which is able to store and release a portion of the energy contained in the electric field. The production of absorption in the resonator can be attributed to the impacts of dipolar resonances as well as the dissipation of capacitors in the absorber.

 

Fig. 4. Distribution of electric field of the suggested structure for TE waves at resonant centre frequencies (a) 0.85 THz, (b) 5.6 THz, and (c) 9.84 THz in hybrid graphene-gold patch.

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3.2 Optimization of the geometric parameters

To maximize the absorption bandwidth, the geometric parameters are optimized. First of all, the side length (b) of the rectangular patch is changed from 9.5 µm to 12.5 µm with 0.5 µm intervals. Figure 5(a) illustrates the absorption spectrum at various values of b with TE waves (all the geometric parameters are optimized for TE waves), while Fig. 5(b) depicts the absorbance as a function of b and frequency. In the 2-D color plot, as depicted in Fig. 5(b), the red regions indicate the highest absorption, which decreases as it moves towards the blue regions. Both of these figures demonstrate that an increase in b results in a corresponding increase in bandwidth. This is due to the fact that elevating b not only extends the overall resonant length of the outer capacitor as well as the inner capacitor, but it also enhances the amount of interaction between the two capacitors. As a result, the peak absorption in the low-frequency region will be red-shifted, while the peak absorption in the high-frequency region will be blue-shifted. Finally, the total absorption bandwidth will become more broad. However, the variation of the peaks (specially the third peak) in the absorption bands is due to the mismatch of the impedance caused by the variation of parameter b. When b is between 9.5 µm and 10 µm, the maximum absorption for the middle band is less than 90%, and when b is between 11 µm and 12.5 µm, the maximum absorption is greater for the first and second bands but decreases for the third band, and the peak absorption is not constant. For b = 10.5 µm, however, both the peak absorption and the bandwidth of peak absorption are greater than 90% in all the three bands, therefore b is set to 10.5 µm.

 

Fig. 5. For TE waves, absorption spectrum at different values of (a) b, (c) a1, and (e) b1, and absorbance as a function of (b) b, (d) a1, and (f) b1 and frequency.

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The width of the graphene patches (a1) is now varied with a 1 µm interval from 1 µm to 5 µm. The absorption graph for various values of a1 is depicted in Fig. 5(c), whereas Fig. 5(d) illustrates the absorbance as a function of a1 and frequency. Because a1 increases the width of the graphene layer, the graphs clearly demonstrate that as a1 rises, so does the maximum absorption for the first broad absorption band. As it is widely known, graphene’s surface conductivity increases more rapidly at lower frequencies [35], resulting in a higher absorption peak for the first band; however, the variation of a1 has a negligible effect on the subsequent two bands as those bands are mostly determined by the gold resonator. The slight variation of the peaks of the second and third bands is due to a mismatch of the impedance of the absorber with the free space. But if a1 exceeds 4 µm, however, the peak absorption as well as the bandwidth of the peak absorption begin to decrease because a further increase in a1 can make the graphene layer in the resonator more conductive and cause a mismatch of impedance between the proposed MMA and the free space, hence a1 is set at 4 µm.

Next, the height (b1) of the graphene’s rectangular patch is varied from 1.5 µm to 3 µm with 0.5 µm intervals and the absorption spectrum for different values of b1 is illustrated in Fig. 5(e), whereas Fig. 5(f) depicts the absorbance as a function of b1 and frequency. It is clear from the figures that when the value of b1 is increased from 1.5 µm, the absorption peak increases gradually for the first band, whereas the subsequent two bands are hardly affected by the values of b1. It is seen that the effect of b1 is quite similar to a1, because b1 also increases the amount of graphene in the four rectangular patches, hence increasing the surface conductivity of graphene at lower frequencies. Similarly, when b1 is beyond 2.5 µm, both the bandwidth and maximum absorption decrease, as a further increase in b1 can make the graphene layer in the resonator more conductive and cause a mismatch of the impedance between the suggested MMA and the free space, hence b1 is set at 2.5 µm. Finally, we can claim that the latter two bands are hardly impacted by structural parameters, but the first band is affected by the width (a1) and height (b1) of the graphene sheet; therefore, these two parameters must be maintained appropriately during the fabrication process. After that, these parameters cannot have any impact on the absorber’s performance, as once the structure is fabricated, we cannot change the geometric parameters.

Now the bottom $SiO_{2}$ layer’s (between the ground metallic layer and the patch) thickness ($t_{s}$) is varied from 8.5 µm to 10 µm with 0.5 µm intervals. Figure 6(a) illustrates the absorption spectrum for various values of $t_{s}$ with TE waves, whereas the absorption as a function of $t_{s}$ and frequency is illustrated in Fig. 6(b). Both the figures clearly show that with the increase in thickness ($t_{s}$), the absorption spectrum shifts to the left (red shift). This is expected as there is an inverse relationship between the substrate’s thickness and the resonance frequency [12], hence the increase in $t_{s}$ leads the absorption spectrum to be red-shifted. Then, the top $SiO_{2}$ layer’s thickness is varied from 10 µm to 15 µm and the absorption spectrum for various values of $t_{s2}$ under TE waves is depicted in Fig. 6(c), whereas the absorbance as a function of $t_{s2}$ and frequency is illustrated in Fig. 6(d). Both figures show that the thickness $t_{s2}$ has no effect on the absorption bands. As the outer layer acts as an open circuit, as illustrated in Fig. 1(c), a slight variation of the outer layer will keep the absorption spectrum unchanged.

 

Fig. 6. For TE waves, (a) absorption spectrum under different values of the thickness of inner dielectric layer ($t_{s}$), (b) absorption as a function of $t_{s}$ and frequency, (c) absorption spectrum under various values of the thickness of outer dielectric layer ($t_{s2}$), and (d) absorption as a function of $t_{s2}$ and frequency.

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3.3 Effects of graphene’s $E_F$ on the performance of the device

It is widely known that the structural parameters of metal-based optical devices cannot be modified once the device has been built. This means that the performance of the metal-based MMAs cannot be changed in real time, limiting their applications. However, we can dynamically tune the absorption performance of the proposed MMA by adjusting the $E_F$ of graphene, as graphene’s surface conductivity is primarily related to its $E_F$ and frequency [27], that can be manipulated easily with the application of bias voltage. Figures 7(a) and 7(b) depict the absorption spectra with different $E_F$ varying from 0 eV to 1.1 eV and the real (solid lines) and imaginary (dotted lines) conductivity components of graphene at various $E_F$, respectively. As graphene’s conductivity increases with increasing $E_F$ [27], it is evident that the level of absorption increases gradually. However, the variation of $E_F$ has a greater effect on the first band; the second band is only slightly affected, and the final band is completely unaffected. This is happening because the conductivity of graphene changes rapidly at lower frequencies with an increase in $E_F$, whereas at higher frequencies it is nearly constant and lower in values [35]. It can also be seen in Fig. 7(b), where the conductivity of graphene increases more rapidly at lower frequencies than at higher frequencies with the increase in $E_F$.

 

Fig. 7. (a) Absorption spectrum under different Fermi energies ($E_F$) for TE waves (b) graphene’s conductivity at different Fermi energies ($E_F$).

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3.4 Polarization and angle dependencies

The insensitivity of the MMA’s absorption performance to the angle of polarization ($\phi$) as well as the angle of incidence ($\theta$) will have a significant impact on its potential application in some particular fields. These properties have a variety of possible uses, including detection, sensing, and optoelectronic applications [12]. As a result, we firstly investigated the absorption characteristics of the suggested MMA at different values of $\phi$ with normal incident. Figures 8(a) and 8(c) depict the absorption spectra at different values of $\phi$ ranging from 0$^{\circ }$ to 90$^{\circ }$ for the TE and TM waves, respectively, whereas the corresponding Figs. 8(b) and 8(d) show the absorbance as a function of $\phi$ and frequency for the TE and TM waves, respectively. The results of the simulation show that both TE and TM waves don’t affect absorber’s performance. This is happening because the transverse field of electromagnetic waves incident with any polarization can typically be split into two polarized components that are orthogonal to each other [36]. In our case, the structure has an orthogonal symmetry in the x-y plane, hence interacts with each field components equally, resulting in insensitive to the electromagnetic wave with varying polarizations.

 

Fig. 8. The simulated absorption response at various polarization angles for the (a) TE, (c) TM waves, and at different angles of incidence for the (e) TE waves, and (g) TM waves, absorbance as a function of frequency and polarization angle for the (b) TE, and (d) TM waves, and absorbance as a function of frequency and incident angles for the (f) TE waves, and (h) TM waves.

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In order to cover a broader range of application fields, it is also important to investigate the absorption characteristics of the suggested MMA under a variety of incident angles with the optimal design parameters indicated above. Figures 8(e) and 8(g) depict the absorption spectrum at various incident angles ranging from 0$^{\circ }$ to 50$^{\circ }$ for TE and TM polarized waves, respectively, while Figs. 8(f) and 8(h) illustrate the absorbance as a function of ($\theta$) and frequency for TE and TM waves, respectively. When the angle of incidence for the both waves is less than 40°, simulations indicate that the wide range of absorption can be kept at a high level, but it has a tendency to blue-shift toward higher frequencies. When the angle of incidence further increases, continuously, the magnetic field’s projection on the absorber’s surface will become very weak, which makes it hard to get electromagnetic resonance to work well. So, the maximum absorption will start to go down, and the broad band of absorption will start to blue-shift quickly, causing dual broadband absorption. Finally, from the results, we can say that the designed structure is completely insensitive to polarization for both the TE and TM waves and can retain its absorption characteristics for a wide angle of incidence, which can make it more practical in real-world applications.

3.5 Suggested fabrication process

We were not able to fabricate the MMA in the laboratory because of the lack of fabrication facilities; however, we have proposed a simple fabrication process for the designed MMA as shown in Fig. 9. It is a simple process that can be completed using the following steps [28]

 

Fig. 9. Possible steps in the fabrication process of the proposed MMA.

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4. Conclusion

In conclusion, we have designed a broadband MMA that shows broadband characteristics in three separate wide bands and is exceptionally different from the previously reported multi-band absorbers (sharp absorption peak, limited absorption band) as well as broadband absorbers (peak absorption in a maximum of two wide frequency bands). The use of the graphene layer enables us to actively tune the obtained absorption spectrum by an external biasing voltage. Simulation results show that when the electrochemical potential (Fermi energy) of graphene is adjusted to 0.9 eV, the absorber exhibits three broadband absorptions of more than 90% absorbance with a total bandwidth of 4.42 THz and 4.29 THz for the TE waves and TM waves, respectively. Parametric study is carried out to optimize the structure, while the absorption mechanism is analyzed using electric field distribution. Due to the orthogonal symmetry, the structure is completely insensitive to polarization for both TE and TM waves and maintains a high absorption performance for incident angles less than 40°. Lastly, we expect that the proposed structure will be applicable to a range of potential uses, such as electromagnetic wave absorption, signal detection, tunable filtering, modulators, etc.