1. Introduction

The rapid development of terahertz technology has been aided by the emergence of metamaterials, which played an important role in promoting them [1–6]. However, several technical problems are yet to be solved [7]. The resonant frequency of traditional metamaterials absorbers remains unchanged owing to their fixed structural parameters and resonant structures and limits their scope for application [8]. Recently, graphene has attracted considerable attention owing to its unique electromagnetic properties [9], which can be changed by external factors such as the external voltage; thereby achieving control over the absorption frequency [7,10]. As a two-dimensional material consisting of carbon atoms arranged in a honeycomb lattice [11–16], graphene has extensive applications in filters [17], modulators [18], switches, photodetectors [19,20], sensors, solar cells, and so on [21] owing to its electromagnetic properties.

Circuit theory can explain the fact that when a graphene layer is periodically formed on the substrate with an appropriate design and the equivalent input impedance of the absorber structure is matched to the medium [22–24], the absorber has no reflection. In this work, we propose a tunable terahertz perfect absorber with a graphene-based double split-ring structure. In this absorber, gold, silicon dioxide, and graphene are placed layer by layer in sequence, with silicon as the base. We have not only discussed the impact of different Fermi energies and the disparate relaxation times on the regulation of absorption, but also explored the optimal structure parameters required for achieving perfect absorption. Compared to other metamaterial absorbers, our design has the advantages of absorption of a wide range of tunable dual-bands, thin, wide operation, simple configuration and so on. We believe that this design will enrich the THz absorber family and inspire practical applications such as sensing and imaging.

2. Structure design

A tunable terahertz perfect absorber with a graphene-based double split-ring structure is shown in Fig. 1. Figure 1(a) depicts the top view of the graphene-based absorber. Here, we set the outside and interior lengths of the square split ring at 1800nm and 1600 nm, respectively. The inner and outer radii of the interior round split ring are 500 nm and 600 nm, respectively. In order to achieve better resonance, we set the split width (a) to 100 nm. The absorber elements have a period of 2450 nm. Silicon forms the base on which gold, silicon dioxide, and graphene are placed layer by layer in order, as shown in Fig. 1(b). The silicon dioxide layer sandwiched between the metals and graphene can capture more light energy to produce ohmic loss and achieve perfect absorption. This layer is 4200 nm thick, and its permittivity is selected as 3.9. Transmission can be eliminated by ensuring that the thickness of the gold layer exceeds the skin depth of the electromagnetic wave [25–27]. In this work, we determined the thickness of the gold layer and its electrical conductivity to be 450 nm and $\sigma = \textrm{4}\textrm{.56} \times \textrm{1}{\textrm{0}^7}$S/m, respectively. In order to achieve better absorption, the thickness and relative permittivity of the silicon layer were set at 2000nm and 11.9, respectively.

 

Fig. 1. (a) Top view of a unit cell of the absorber. (b)Schematic representation of a unit cell of graphene-based perfect absorber with the relevant geometric parameters.

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3. Simulation and discussions

The permittivity of graphene, ${\varepsilon _g},$ can be expressed as [28,29]

Here, ${\sigma _{\textrm{intra}}}$ and ${\sigma _{\textrm{inter}}}$, in the Kubo formula represent the intraband and interband conductivities, defined in the following [30]:

Here $\tau ,$ the relaxation time, depends on the Fermi energy, Fermi velocity ${V_F} = \textrm{1}{\textrm{0}^6}\; m/s$, and carrier mobility $\mu = 1{m^2}/Vs$ as $\tau = {E_f}\mu /({eV_f^2} )$ [29,30]. In this work, the Fermi energy and relaxation time were assumed to be 0.9 eV and 1.1 ps, respectively. Under these circumstances, owing the nearly perfect impedance matching condition, the absorption can reach a maximum.

The absorption of the double split-ring absorber was simulated using CST Microwave Studio. From Fig. 2(a), it is obvious that the absorber reaches 99.7% and 99.4% at 10.96 THz and 12.71 THz, respectively, i.e., nearly perfect absorption. To better illustrate the physical mechanism of perfect absorption, we simulated the distribution of the electric fields at 10.96 THz and 12.71 THz. As shown in Fig. 2(b), the electric field is mainly concentrated in the outer rectangular split ring at 10.96 THz. As the frequency of the electromagnetic waves increases, the electric field is concentrated on the middle circular split ring in Fig. 2(c) at 12.71 THz. In Fig. 2(d), we can get that the structure we designed has a good absorption effect, clearly. Owing to the plasma resonance of the local surface in the graphene array, there are two electric dipole resonances at 10.96 THz and 12.71 THz. The electric dipole resonance and the lower metal film will have a strong coupling effect. The lower metallic layer produces a current opposite to the surface graphene, leading to a magnetic dipole resonance in the structure. Therefore, this strong electromagnetic resonance can produce nearly 100% absorption [30].

 

Fig. 2. (a) Absorption in the terahertz region as a function of the frequency. (b) Electric field distribution when the resonance frequency is 10.96 THz. (c) Electric field distribution when the resonance frequency is 12.71 THz. (d) out-of-plane field distribution respectively at 10.96 THz and 12.71 THz

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It is known that the application of a gate voltage and chemical dropping can effectively control the surface conductivity of graphene. The relation between the Fermi energy and gate voltage can be expressed as [31,32]

In the above formula, ${V_f}$ is the Fermi velocity, V is the gate voltage, ${t_s}$ is the thickness of the plasma, ${\varepsilon _0}$ is the dielectric constant, and ${\varepsilon _r}$ is the relative dielectric constant., We set the Fermi velocity as ${V_f} = {10^6}$m/s to simplify the calculation. Figure 3(a) clearly illustrates the relationship between the Fermi energy and absorption. It is seen that when the Fermi energy increases from 0.7 eV to 1.0 eV, the absorption peak shifts until ${E_f}$ = 0.9 eV where the absorption approaches 100%. Figure 3(b) shows the schematic representation of the gate voltage for controlling the Fermi energy of graphene. The current flows through the brain and source. From Eq. (6), it is obvious that the Fermi energy increases as the gate voltage V is increased. Thus, tuning can be achieved easily by controlling the gate voltage.

 

Fig. 3. (a) Absorption spectra for different Fermi energies. (b) Schematic representation of the gate voltage for controlling the Fermi energy of graphene.

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Since the placement of organic molecules on graphene can effectively enhance the carrier mobility and hence, the relaxation time [31,32], we studied the impact of the relaxation time on the absorption. As shown in Fig. 4, when $\tau $ = 1.1 ps and ${E_f}$ = 0.9 eV, the charge carriers are conductive to plasma oscillation absorption, which leads to the peak attaining a maximum value. If the relaxation time is continuously increasing because most of the energy of the microwave will be reflected, which leads to a reduction in absorption instead of consistent growth, and the absorption peak gradually declines [30–32]. In addition, it is significant that as the relaxation time increases, the absorption peak is no longer nearly perfect.

 

Fig. 4. Absorption spectra for different relaxation times with other parameters set up

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As for the polarization performance of designed structure, we stimulated the different angles θ from 0° to 50° and ϕ from 0° to 45°. And θ is an angle between incident of electromagnetic wave and normal incident, and ϕ is express as the angle between H and Y directions [33]. With the increase of the θ, the absorption rate of absorber become lower in Fig. 5(a). Moreover, the reduction is not significant, which means the absorber has a very great wide-angel incoming characteristic. As shown in Fig. 5(b), the impact of the change of θ on the absorption performance of the structure is relatively large. So, the absorber is polarization-sensitive, and it has an application on filtering.

 

Fig. 5. Simulate absorptivity with different (a)θ, (b) ϕ (TM polarization)

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As the outer length of the square split ring increases, as shown in Fig. 6(a), the absorption peak shifts towards shorter outer length, it will disturb the impedance matching condition. This is the reason why the absorption peak declines as we increase the length. As shown in Fig. 6(b), the absorption peak blueshifts as the interior length of the square (L₂) increases from 1400 nm to 1600 nm. When L₂ increases to 1600 nm, the absorption peak reaches a maximum owing to impedance matching. The diminished absorption peak observed when L₂ = 1700 nm arises from mismatched impedances. To further investigate the relationship between the structural parameters and the absorption, we considered different radii of the interior split ring, as shown in Fig. 6(c) and (d). In Fig. 6(c), as the outer radius of the split ring increases, double absorption peaks, located at 10.96 THz and 12.71 THz, emerge gradually. Then, owing to the resonance condition transformed, as the outer radius of the split ring is continuously increased, the distance between the double absorption peaks gradually widens. As depicted in Fig. 6(d), as the radius (r₂) gradually increases, double absorption peaks emerge at r₂ = 500 nm. If the radius is increased further, the two peaks will disappear. Under these conditions, impedance matching is no longer satisfied.

 

Fig. 6. Absorption rates at (a) different outer lengths of the square split ring (1700 nm - 1850 nm), (b) different interior lengths of the square split ring (1400 nm - 1700 nm), (c) different radii of the outer round split ring (550 nm - 750 nm), (d) different radii of the interior round split ring (400 nm - 550 nm) keeping other parameters unchanged.

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4. Conclusion

In conclusion, we have discussed a tunable terahertz perfect absorber consisting of a graphene-based double split-ring structure and presented the results obtained using CST Microwave Studio. The simulated results illustrate that the double absorption peaks reach 99.7% and 99.4% at 10.96 THz and 12.71 THz, respectively. By controlling the gate voltage, we can indirectly adjust the Fermi energy of graphene to tune the absorption peak. Furthermore, the relaxation time may be used to manipulate the absorption peak owing to the small impact on the resonance frequency. When the parameters of the proposed absorber structure are properly adjusted, the impedance may be mismatched so that it cannot reach perfect absorption.