Multi-band terahertz metamaterial absorber composed of concentric square patch and ring resonator array

Somayyeh Asgari; Tapio Fabritius

doi:10.1364/OPTCON.506061

1. Introduction

Materials that can absorb light are fundamental elements in optical engineering, and they are utilized in various fields where wave absorption is crucial. These domains encompass wireless communication, sensing, and imaging. Over time, significant efforts have been dedicated to developing versatile tools in the terahertz (THz) range. This is primarily because of their possibilities in creating optical devices such as modulators, sensors, and absorbers [1–3]. The graphene-based resonator arrays play an indispensable role in constructing absorbers that achieve high performance [4–8].

Numerous studies have used arrays of graphene resonators within a bias scheme. This scheme emphasizes the significance of the Fermi energy level in controlling the device behavior across a range of frequencies [4–8]. Extensive research has been undertaken to realize various graphene resonator-based absorbers operating in the THz range [4,6,7,9,10].

Typically, comprehensive simulations like Finite Element Methods (FEM) [9–11] are employed to model such devices, which demand substantial computational power and memory resources. However, the adoption of the Equivalent Circuit Model (ECM) and the concept of equivalent conductivity (EC) [10,12–15] to represent graphene configurations offers a simplified mathematical framework model these metamaterials. In our earlier paper [10], the ECM is based on EC and transfer matrix, and the geometry is contained of two finite parallel graphene ribbons per unit cell. In our earlier paper [15], the ECM is based on EC and admittance-based transmission line theory, and the geometry is contained of a double-sided comb resonator array. In our earlier paper [16], the ECM is based on ABCD matrix and S-parameters, and the geometry is contained of U-shaped resonator array. Additionally, the Coupled Mode Theory (CMT) approach has been applied to metamaterials featuring multiple resonators per unit cell, referred to as supercells [17,18].

Among the published configurations, the graphene square patch arrays [19–21] have already well-established ECM representations. In addition to them, different kind of THz absorber designs have been introduced, including wideband, narrowband, and multi-band absorbers, all based on graphene patterns and some including theoretical approaches of ECM or CMT methodologies [22–24].

In this paper, we present a tunable metamaterial absorber operating in the terahertz (THz) region, utilizing a graphene-based resonator array. This absorber comprises an array of concentric square patch and ring resonators, arranged to exhibit multi-band absorption characteristics. Recently, similar kind of absorber design was reported but it was based on aluminum resonator array with graphene ribbons with more complicated geometry acting as a THz dual-band filter with transmission over 80%. That paper is also lacking any theoretical approach to model its characteristics. They used graphene ribbons to make the transmission spectrum of the filter tunable [25]. To enhance the functional characteristics of metamaterial, we replaced a metal resonator array with graphene and modelled its behavior exploiting ECM. The ECM is based on EC, impedance-based transmission line theory, and variational method (eigenvalue and eigenfunction problem). The impedance of the overall pattern is modelled by EC. The impedance of inner square patch array is modelled by variational method [20]. The impedance of the outer square patch is then obtained which is the combination of EC (overall pattern) and variational method (inner square patch array). Since the metamaterial design is made of supercells, the CMT approach is proposed as well. The designed metamaterial was evaluated as a multi-band absorber, but it can also be used in other applications like filtering and sensing in THz devices and systems.

2. Materials and methods

Fig. 1 displays the metamaterial absorber design from different angles: periodic, unit cell, front, and side views. The layers making the metamaterial from the top to the bottom are respectively ion gel, graphene, Teflon, and gold (Fig. 1(d)). A layer of ion gel with a permittivity value of 2.0164 and a thickness of 150 nm is positioned atop the metamaterial. This ion gel layer serves the purpose of biasing the underlying graphene resonator layer [26].

Fig. 1. (a) Periodic, (b) unit cell, and (c) front, and (d) side views of the designed graphene-based THz metamaterial absorber composed of concentric square patch and ring resonator array. The layers from the top to the bottom are respectively ion gel, graphene, Teflon, and gold. A layer of ion gel is used for biasing the resonator array and a gold layer is used beneath the metamaterial to avoid the transmission of the electromagnetic waves.

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The metamaterial consists of a lower gold layer characterized by high conductivity (4.56 × 10⁷ S/m) and a relatively large thickness (0.5 µm) relative to the wavelengths in the THz range. This layer acts as a reflective mirror against THz radiation. On this gold layer, a dielectric layer made of Teflon is placed, upon which graphene patterns are arranged. In this setup, graphene is represented as an ultra-thin layer with a thickness of 0.335 nm and is placed over the Teflon layer with a thickness of 8 µm, also known as polytetrafluoroethylene (PTFE). Teflon possesses a low permittivity value (ɛ_d = 2.1) and an exceptionally low loss tangent (tanδ = 0.0003), making it a suitable substrate for THz absorbers. The design also features a graphene-patterned layer consisting of periodic arrays of concentric square patch and ring resonators, placed on the Teflon dielectric layer.

To fabricate the metamaterial, the following steps can be followed: Initially, deposit an 8 µm thick Teflon dielectric with gold coating on one side using chemical vapor deposition (CVD) [27]. Next, generate the graphene patterns through a standard lithography process on the Teflon dielectric [28]. The ion gel dielectric is then put on top of the graphene resonator array via thermal evaporation [29].

The optimized dimensional values are listed in Table 1. Numerical simulations were performed based on the finite element method (FEM) in the frequency domain solver of CST microwave studio 2018. Periodic boundary conditions were applied in the x and y directions, while an absorbing boundary condition was implemented in the z direction. The tetrahedral mesh was utilized for meshing the metamaterial [10,15,30]. The metamaterial was optimized using the genetic algorithm optimization technique within the CST software [10,15,31]. The unit cell dimensions were chosen as P_x = P_y = 16 µm, which is smaller than λ_min = 46.15 µm, the wavelength corresponding to f_max = 6.5 THz, the upper limit of the simulated frequency range. This sizing choice aimed to prevent the propagation of higher-order Floquet modes [32,33].

Table 1. Structural dimensions and their optimized values for the metamaterial absorber of Fig. 1

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The overall thickness of the metamaterial, comprising ion gel, graphene, Teflon, and gold layers, was 8.65 µm, approximately 0.187 ×λ_min. This indicated that the metamaterial’s thickness remained relatively thin within the simulated frequency range.

The relative permittivity of graphene is modeled by [10,15]:

(1)$${\varepsilon _g} = 1 - j\frac{{{\sigma _g}}}{{\omega {\varepsilon _0}\Delta }}$$

where σ_g represents the surface conductivity of graphene, ω stands for angular frequency, ε₀ is the permittivity of a vacuum, and Δ corresponds to the thickness of graphene (assumed as 0.335 nm). The value of σ_g encompasses inter- and intra-band electron transition contributions as determined by the Kubo formula [34]:

(2a)$${\sigma _g}(\omega )= {\sigma _{{\mathop{\rm int}} er}}(\omega )+ {\sigma _{{\mathop{\rm int}} ra}}(\omega )$$

(2b)$${\sigma _{{\mathop{\rm int}} er}}(\omega )= \frac{{{e^2}}}{{4\hbar }}\left[ {H\left( {\frac{\omega }{2}} \right) - \frac{{4j\omega }}{\pi }\int_0^\infty {\frac{{H(\xi )- H\left( {\frac{\omega }{2}} \right)}}{{{\omega^2} - 4{\xi^2}}}d\xi } } \right]$$

(2c)$${\sigma _{{\mathop{\rm int}} ra}}(\omega )= \frac{{2{k_B}{e^2}T}}{{\pi {\hbar ^2}}}\ln \left[ {2\cosh \left( {\frac{{{E_f}}}{{2{k_B}T}}} \right)} \right]\frac{j}{{j{\tau ^{ - 1}} - \omega }}$$

(2d)$$H(\xi )= \frac{{\sinh \left( {\frac{{\hbar \xi }}{{{k_B}T}}} \right)}}{{\cosh \left( {\frac{{{E_f}}}{{{k_B}T}}} \right) + \cosh \left( {\frac{{\hbar \xi }}{{{k_B}T}}} \right)}}$$

Here, ${\hbar}$ represents the reduced Planck constant, k_B which equals 1.38 × 10⁻²³ J/K is Boltzmann's constant, e denoting the electron charge is 1.6 × 10⁻¹⁹ C, T stands for temperature (300 K), and ξ is used as the variable of integration. The parameter τ signifies the relaxation time, obtained by [35]:

(3)$$\tau = \frac{{\mu {E_f}}}{{ev_f^2}}$$

where v_f = 10⁶ m/s is the Fermi velocity of graphene and µ denotes the carrier mobility of graphene.

The propagation constant of the electromagnetic wave on graphene-vacuum configuration can be calculated by [16]:

(4)$$\beta = {k_0}\sqrt {1 - {{\left( {\frac{2}{{{\eta_0}\mathrm{\sigma }}}} \right)}^2}} $$

where β, k₀, and η₀ are respectively the propagation constant of electromagnetic wave on graphene-vacuum configuration, the wave vector of incident light wave, and the vacuum impedance. External bias voltage is applied between the graphene resonator array and the bottom gold layer to change the E_f . The relationship between E_f and bias voltage (V > 0) is approximately considered as [16]:

(5)$${E_f} = \hbar {v_F}\sqrt {\pi {\varepsilon _0}{\varepsilon _d}\frac{V}{{ed}}} $$

The metamaterial is illuminated by TE mode. The ECMs of the graphene resonator layer and the whole metamaterial (shown in Fig. 1) are given in Fig. 2. The inner patch resonator is modeled by Z₁ impedance. As the metamaterial is illuminated by TE mode, the incident electric field which is along the y-axis is normal to the upper and lower gaps between the resonators. So, these gaps are modeled by the c capacitance in the ECM. The whole impedance of the inner square patch resonator and the distances between the inner patch and outer ring is modeled by Z₃. The impedance of the outer ring resonator is modeled by Z₂. The impedance of the whole graphene resonator layer is modeled by Z_g. The whole metamaterial is modeled by the transmission lines. The input impedances for each section of the metamaterial absorber are shown in Fig. 2(b). The thickness of the graphene layer is much smaller than the minimum wavelength in the considered simulated wavelength range, so the graphene resonator layer is modeled as a point load [36,37].

Fig. 2. Equivalent circuit model (ECM) of (a) the graphene-based resonator array and (b) the whole metamaterial absorber.

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In order to calculate the absorption by use of the ECM approach, the formulas are calculated by a MATLAB code. The parameters inside the formula are considered with their SI unit in the code.

The equivalent conductivity of the graphene resonator layer σ_g is calculated by the Fresnel equation [13]:

(6)$${\sigma _g} = \frac{{\sec ({{\theta_{in}}} )- \sqrt {{\varepsilon _d}} \sec ({{\theta_{out}}} )- r\left( {\sec ({{\theta_{in}}} )+ \sqrt {{\varepsilon_d}} \sec ({{\theta_{out}}} )} \right)}}{{{Z_0}({1 + r} )}}$$

where θ_in stands for the angle of the incident illuminated wave, ε_d represents the relative permittivity of the dielectric substrate (Teflon), θ_out represents the angle of the transmitted wave, r shows the reflection of the graphene resonator array obtained in CST. To calculate r and minimize the effect of the Teflon substrate on r since graphene should always be located on a substrate for further investigation, the graphene resonator array is located on the Teflon dielectric half-space. The obtained r from CST is substituted in the MATLAB code for Eq. 6. and Z₀ corresponds to the impedance of vacuum (377 Ω). The relation between θ_in and θ_out is:

(7)$$\sin ({{\theta_{out}}} )= \sqrt {\frac{1}{{{\varepsilon _d}}}} \sin ({{\theta_{in}}} )$$

Conductivity is equal to admittance. Impedance is the inverse of admittance, so we have:

(8)$${Z_g} = \frac{1}{{{Y_g}}} = \frac{1}{{{\sigma _g}}}$$

(9)$${Z_g} = \frac{{{Z_0}({1 + r} )}}{{\sec ({{\theta_{in}}} )- \sqrt {{\varepsilon _d}} \sec ({{\theta_{out}}} )- r\left( {\sec ({{\theta_{in}}} )+ \sqrt {{\varepsilon_d}} \sec ({{\theta_{out}}} )} \right)}}$$

Extraction of the fundamental mode of the graphene-based inner square patch array could be done by a variational approach. The equivalent impedance of the inner square patch array is calculated by [20]:

(10)$${Z_1} = {P_x}{P_y}\left( {\sigma_g^{ - 1} + \frac{q}{{j\omega {\varepsilon_{eff}}}}} \right)\frac{{{k_1}}}{{s_1^2}}$$

where P_x and P_y are the unit cell dimensions in the x and y directions, ε_eff, q, k₁, and s₁ are respectively the effective permittivity, the eigenvalues, and the coefficients of the array of the inner square patches calculated by the proposed trial eigenfunctions and the geometrical constant PxPy.

ε_eff are calculated by:

(11)$${\varepsilon _{eff}} = \frac{{{\varepsilon _0}({{\varepsilon_{ig}} + {\varepsilon_d}} )}}{2}$$

k₁ and s₁ are calculated by [20]:

(12)$${k_1} = {\int\limits_S {|{{\xi_x}({x,y} )} |} ^2}dS$$

(13)$${s_1} = \int\limits_S {{\xi _x}({x,y} )dS}$$

In the context of a periodic arrangement of square graphene patches, the interaction between these patches leads to a modification in the eigenvalues of an individual graphene patch. By disregarding the influence of the disturbance on the eigenfunctions, the eigenvalues for the ensemble of graphene square patches can be expressed as [20]:

(14)$$q = \lambda + \frac{1}{{\int\limits_S {{{|{\xi ({x,y} )} |}^2}dS} }} \times \sum\limits_{({p,q} )\ne `0} {\int\limits_S {\int\limits_{S^{\prime}} {\frac{{{{\nabla ^{\prime}}_T}.\xi ({x^{\prime},y^{\prime}} ){\nabla _T}.\xi ({x,y} )}}{{4\pi \sqrt {{{({x - x^{\prime} - p{p_1}} )}^2} + {{({y - y^{\prime} - q{p_1}} )}^2}} }}} } } dS^{\prime}dS$$

where q is the shifted eigenvalue. The third trial eigenfunction is reported to be the most accurate and complicated one and it can be obtained as [20]:

(15)$$\lambda = 0.262\frac{\pi }{l}$$

In this method, suitable trial eigenfunctions must adhere to the boundary conditions. Considering the polarization of the incident wave, the current vector is presumed to align with the x-direction. The normal component of the surface current on the edges of graphene should be zero. As a result, the boundary condition that needs to be met is as follows [20]:

(16)$${\xi _x}({x,y} )= {\left. { - \frac{{\partial \psi ({x,y} )}}{{\partial x}}} \right|_{x ={\pm} {\raise0.7ex\hbox{$l$} \!\mathord{/ {\vphantom {l 2}}}\!\lower0.7ex\hbox{$2$}}}} = 0$$

By employing the technique of variable separation, we express the eigenfunction of the fundamental mode in two dimensions as a multiplication of two distinct one-dimensional functions, as follows [20]:

(17)$$\psi ({x,y} )= \Psi (x ).\Phi (y )$$

The approximates of three trial eigenvalues and eigenfunctions for the fundamental mode of the graphene-based inner square patch array satisfying Eq. 17 for the ψ (x, y) are extracted and given in Table 1 of [20]. The second and third eigenfunctions are as follows [20]:

(18)$${\Psi _2}(x )= 1.2\left( {\frac{l}{2}} \right)\left[ {\frac{1}{2}\left[ {\left( {\frac{{2x}}{l}} \right)\sqrt {1 - {{\left( {\frac{{2x}}{l}} \right)}^2}} + {{\sin }^{ - 1}}\left( {\frac{{2x}}{l}} \right)} \right]} \right] - 0.106\left( {\frac{l}{2}} \right)\left[ { - \frac{{2x}}{l}{{\left( {1 - {{\left( {\frac{{2x}}{l}} \right)}^2}} \right)}^{\frac{3}{2}}}} \right]$$

(19)$${\Phi _3}(y )= \cosh \left( {\alpha \frac{{2y}}{l}} \right)$$

The fitting parameter α ≈ 0.67 is based on the best fit to the current distribution obtained from full-wave simulation results in [20].

Since the metamaterial is illuminated by TE wave, the incident electric field is along the y-axis and normal to the upper and lower gaps between the inner and outer resonators. These gaps are modeled by capacitances. The capacitances are calculated by:

(20)$$c = {\varepsilon _{eff}}\frac{l}{g}$$

The impedance of the capacitances is obtained by:

(21)$${Z_c} = \frac{{2g}}{{j\omega {\varepsilon _0}({{\varepsilon_{ig}} + {\varepsilon_d}} )l}}$$

By having Z₁ and c, Z₃ (shown in Fig. 2(a)) can be obtained by:

(22)$${Z_3} = {Z_1} + 2{Z_c}$$

So:

(23)$${Z_3} = {P_x}{P_y}\left( {\sigma_g^{ - 1} + \frac{q}{{j\omega {\varepsilon_{eff}}}}} \right)\frac{{{k_1}}}{{s_1^2}} + \frac{{4g}}{{j\omega {\varepsilon _0}({{\varepsilon_{ig}} + {\varepsilon_d}} )l}}$$

Then, following the circuit of Fig. 2(a) which is containing of three parallel impedances, Z_g is obtained by:

(24)$${Z_g} = \frac{{{Z_2}.{Z_3}}}{{{Z_2} + {Z_3}}}$$

By substituting Eqs. 9 and 2323 in Eq. 24, the impedance of the outer square ring array Z₂ can be calculated by:

{Z_2} = \frac{{{a_1}}}{{{a_2}}}

{a_1} = {Z_0}({1 + r} )[{l{P_x}{P_y}{k_1}({j\omega {\varepsilon_{eff}} + q{\sigma_g}} )+ 2g{\sigma_g}s_1^2} ]

(25)$$\begin{array}{l} {a_2} = [{l{P_x}{P_y}{k_1}({j\omega {\varepsilon_{eff}} + q{\sigma_g}} )+ 2g{\sigma_g}s_1^2} ].\left[ \begin{array}{l} \cos ({{\theta_{in}}} )- \sqrt {{\varepsilon_d}} \cos ({{\theta_{out}}} )- \\ r\left( {\cos ({{\theta_{in}}} )+ \sqrt {{\varepsilon_d}} \cos ({{\theta_{out}}} )} \right) \end{array} \right] - \\ {Z_0}({1 + r} )j\omega {\varepsilon _{eff}}l{\sigma _g}s_1^2 \end{array}$$

The gold layer impedance Z_gold is zero as it acts as a perfect reflector in the considered simulation ranges [15]. Z_in₁ in Fig. 2(b) is calculated by [36]:

(26)$${Z_{in1}} = {Z_d}\frac{{{Z_{gold}} + j{Z_d}\tan ({{\beta_d}d} )}}{{{Z_d} + j{Z_{gold}}\tan ({{\beta_d}d} )}}$$

where Z_d and β_d are respectively the impedance of the Teflon dielectric substrate and the propagation constant of the THz electromagnetic waves in the Teflon substrate. Since Z_gold is zero, Eq. 26 simplifies to:

(27)$${Z_{in1}} = j{Z_d}\tan ({{\beta_d}d} )$$

As shown in Fig. 2(b), Z_in₂ is made of Z_in₁ and Z_g which are parallel. So:

(28)$${Z_{in2}} = \frac{{j{Z_d}\tan ({{\beta_d}d} ){Z_0}({1 + r} )}}{{{Z_0}({1 + r} )+ j{Z_d}\tan ({{\beta_d}d} )\left[ {\cos ({{\theta_{in}}} )- \sqrt {{\varepsilon_d}} \cos ({{\theta_{out}}} )- r\left( \begin{array}{l} \cos ({{\theta_{in}}} )\\ + \sqrt {{\varepsilon_d}} \cos ({{\theta_{out}}} )\end{array} \right)} \right]}}$$

The input impedance of the whole metamaterial absorber Z_in can be obtained by [36]:

(29)$${Z_{in}} = {Z_{ig}}\frac{{{Z_{in2}} + j{Z_{ig}}\tan ({{\beta_{ig}}{d_{ig}}} )}}{{{Z_{ig}} + j{Z_{in2}}\tan ({{\beta_{ig}}{d_{ig}}} )}}$$

where Z_ig and β_ig are respectively the impedance of the ion gel and the propagation constant of the THz electromagnetic waves in the ion gel. The impedance of ion gel/Teflon Z_ig/d can be obtained by [36]:

(30)$${Z_{ig/d}} = \frac{{{Z_0}}}{{\sqrt {{\varepsilon _{ig/d}}} }}\cos ({{\theta_{in/d}}} )$$

where θ_in and θ_d are respectively the angle of the incident illuminated wave and the electrical length of the substrate. θ_d is calculated by:

(31)$${\theta _d} = \frac{{d\omega \sqrt {{\varepsilon _d}} }}{c}$$

β_ig/d are calculated by:

(32)$${\beta _{ig/d}} = \frac{\omega }{c}\sqrt {{\varepsilon _{ig/d}}}$$

The scattering parameter S₁₁ is calculated by:

(33)$${S_{11}} = \frac{{{Z_{in}} - {Z_0}}}{{{Z_{in}} + {Z_0}}}$$

The absorption is calculated by:

(34)$$A = 1 - {|{{S_{11}}} |^2}$$

The summarized procedure of the ECM approach is as follows:

Step 1: reflection of the graphene resonator array layer r in CST

Step 2: Z_g obtained by Eq. 9

Step 3: Z₁ obtained by Eq. 10

Step 4: c obtained by Eq. 20

Step 5: Z₃ obtained by Eqs. 22 and 23

Step 6: Z₂ obtained by Eq. 25

Moreover, the absorption characteristics of metamaterial absorbers can be elucidated using the coupled-mode theory (CMT). CMT falls under the category of parametric theoretical models, providing a framework to unveil the underlying physical principles governing the interaction between artificial atoms of the super cells (unit cells containing more than one resonator) of the metamaterials. According to the principles of CMT, the absorption can be determined by the expression outlined as [17,18]:

(35)$$A = \sum\limits_{i = 1}^n {\left( {\frac{{4{\gamma_i}{\delta_i}}}{{{{({\omega - {\omega_i}} )}^2} + {{({{\gamma_i} + {\delta_i}} )}^2}}}} \right)}$$

Since our designed metamaterial has three absorption resonances, n which shows the number of the resonances of the metamaterial is equal to 3. ${\omega_{i}}$ represents the resonance frequency, γ_i and δ_i denote the time rate of the amplitude change and the dissipative losses in the guided resonance of the metamaterial, respectively.

3. Results and discussion

Absorption spectrum of the designed metamaterial of Fig. 1 obtained by numerical simulation in CST software is given in Fig. 3(a). As its shown, the metamaterial has three resonances in the considered frequency range of 4-6.5 THz. The average of the absorption peaks is 99%. Moreover, the electric field distributions of the metamaterial absorber in the three resonance frequencies are given in Fig. 3(b-d). As it is clear from the field distributions, the whole pattern combination of the inner and outer resonators creates the resonances. Some parts of the pattern have more influence and some parts of the pattern have less influence in formation of the resonances. The inner square patch array or the outer square ring array cannot produce the resonances itself.

Fig. 3. (a) Absorption spectrum of the metamaterial absorber of Fig. 1. Electric field distributions of the metamaterial absorber of Fig. 1 in (b) 4.66 THz, (c) 5.70 THz, and (d) 6.22 THz.

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Fabrication of perfect graphene patterns is challenging, and the presence of defects in graphene leads to a reduction in its mobility, which subsequently affects its relaxation time [16,38]. According to Eq. 3, change of τ will change µ while the other parameters in Eq. 3 kept unchanged. Considering this condition, to assess the performance of the absorber under varying relaxation time of graphene, the absorption spectra of the device were obtained for three relaxation time values: τ = 1.5, 1.6, and 1.7 ps, all while keeping E_f unchanged and equal to 0.7 eV. Consequently, based on Eq. 3, we have different values for µ as well. The results are presented in Fig. 4. Remarkably, the change of the τ (consequently µ) has no influence on the resonance frequencies of the absorption. However, an increase in τ (consequently µ) corresponds to increased absorption peaks within the absorption spectra. This phenomenon is attributed to the fact that an elongated τ results in an augmented contribution of carriers to plasma oscillation. Therefore, the absorption of the metamaterial increases due to the increase of τ (consequently µ) [16,39,40].

Fig. 4. Absorption spectra of the absorber of Fig. 1 for three different values of τ (consequently µ) while keeping E_f unchanged and equal to 0.7 eV,

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The spectra of the real part of the β (Eq. 4), of the graphene layer for three different values of E_f are calculated and plotted in Fig. 5(a). As it is shown, the real part of the β decreases as the E_f increases. Thus the resonance frequencies of the absorption peaks increase with the increase of E_f, leading to a blueshift effect [41,42]. By increasing E_f, the imaginary part of the β (Eq. 4) for the graphene layer (Fig. 5(b)) which shows the losses of the electromagnetic waves in graphene decrease. Thus, the absorption increases with the E_f increase. Absorption spectra of the metamaterial absorber of Fig. 1 for three different values of E_f are given in Fig. 5(c). External bias voltage which is applied between the graphene resonator array and the bottom gold reflector should be changed for manipulating the E_f. The relationship between E_f and the external bias voltage (V > 0) is given in Eq. 5 [16,39]. Overall, the three absorption resonances show high average absorption for this absorber in the E_f of 0.7-0.8 eV. However, based on Eq. 3, the change of E_f will change τ. As shown in Fig. 4, the change of τ will slightly affect the absorption spectrum which can be neglected.

Fig. 5. (a) The real part of β (Eq. (4)) of the graphene layer, (b) the imaginary part of β (Eq. (4)) of the graphene layer, and (c) Absorption spectra of the absorber of Fig. 1 for three different values of E_f.

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We have simulated the metamaterial absorber of Fig. 1 for different values of the Teflon substrate thickness d with the goal of achieving the highest average absorption. Absorption spectra of the absorber for three different values of d are given in Fig. 6(a). As it is shown, the highest average absorption of the absorber reaches 99% when d is equal to 8 µm. So 8 µm is the optimized value of the substrate thickness.

Fig. 6. Absorption spectra of the absorber of Fig. 1 for three different values of (a) d (thickness of Teflon substrate), (b) g and l, and (c) g and L.

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The resonance frequency depends on the gap, g. g cannot change itself and its change will also affect the other parameters. There are two ways for studying the change of g: 1-Change of g and l while other parameters are kept unchanged. 2-Change of g and L while other parameters are kept unchanged. The results are shown in Figs. 6(b) and 6(c), respectively. Change of g and l, or g and L will affect the resonance frequencies and also the amplitudes of the absorptions. As depicted in Fig. 6(b), the highest average absorption is obtained for g = 1 and l = 9 µm. As shown in Fig. 6(c), the highest average absorption is obtained for g = 1 and L = 11 µm.

The real and imaginary parts of the impedance of the inner square patch resonator array (Eq. 10 and Z₁ in Fig. 2(a)) are given in Fig. 7. The real part of Z₁ is almost constant and positive with a small value showing almost small constant loss and the resistive nature of the inner square patch resonator array. The imaginary part contains both positive and negative sections showing the inductive and capacitive natures of the inner square patch resonator array.

Fig. 7. The real and imaginary parts of the impedance of the inner square patch resonator array (Z₁ in Fig. 2(a)).

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Gap is defined as the distance between the inner square patch and outer square ring. The incident electric field is along the y axis, and it is perpendicular to the gap in the up and lower sides. These distances are modeled by capacitors in Fig. 2(a). Since the incident electric field is parallel to the gap in the right and left sides, these distances cannot be modeled by capacitors. The gap capacitance is shown by c in Fig. 2(a). c is obtained by Eq. (20) and it is equal to 164 pF.

To obtain the reflection spectrum of the graphene resonator layer (r at Eq. (9)), the graphene resonator array is located on a half-space Teflon substrate. The configuration is simulated, and the reflection of the resonator array is obtained. The result is given in Fig. 8.

Fig. 8. Reflection spectrum of the graphene resonator array layer when it is located on a half-space Teflon dielectric substrate to minimize the substrate effects on the reflection spectrum of the graphene layer.

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The obtained reflection in Fig. 8 is exploited in Eq. (9) to calculate the real and imaginary parts of the impedance of the whole graphene resonator array Z_g. The results are shown in Fig. 9. The real part is positive showing the resistive nature and the imaginary part has both positive and negative sections showing the inductive and capacitive natures of the resonator array. As a consequence, from Figs. 7(a) and 9(a), loss is mainly produced by the outer square ring resonator array in this metamaterial absorber.

Fig. 9. The real and imaginary parts of the impedance of the whole graphene resonator array layer (Z_g in Fig. 2(a)).

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The real and imaginary parts of the impedance of the outer square ring array are given in Fig. 10. As it is clear from Fig. 10(a), the outer ring array produces a great loss for the metamaterial. The real part of 2Z₂ is positive and the imaginary part contains both positive and negative sections.

Fig. 10. The real and imaginary parts of the half impedance of the outer square ring resonator array (2Z₂ in Fig. 2(a)).

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Absorption spectra of the metamaterial absorber of Fig. 1 obtained by CST, ECM, and CMT are given in Fig. 11. The obtained spectra are in good agreement. The parameters of the CMT approach in Eq. (35) and their values are given in Table 2.

Fig. 11. Absorption spectrum of the metamaterial absorber of Fig. 1 obtained by CST simulation, equivalent circuit model (ECM), and coupled mode theory (CMT) approaches.

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Table 2. Parameters of the CMT approach in Eq. (35) and their values

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Our designed multi-band graphene-based THz metamaterial absorber is compared with some previously published ones in Table 3. As can be seen, the overall features of our proposed metamaterial are improved compared to each of Refs. [43–46].

Table 3. Comparison of the multi-band graphene-based THz metamaterial absorbers

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4. Conclusion

In this paper, a tunable multi-band metamaterial absorber is constructed from graphene-based concentric square patch and ring resonator array in the terahertz (THz) region. Equivalent circuit modeling (ECM) approach by MATLAB code is presented for the absorber. Moreover, the absorption behavior of the metamaterial is studied through coupled mode theory (CMT) since it is made of super cells. The simulation outcomes, obtained via the finite element method (FEM) in CST Microwave Studio Software, exhibit consistency with the ECM and the CMT results. The proposed metamaterial absorber is notable for being tunable, single-layered, featuring two resonators within each unit cell, showing three absorption bands with absorption reaching 100%, and demonstrating an average absorption of 99% for three bands. Furthermore, the suggested device holds promise for potential applications in future controllable THz devices and systems.

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 may be obtained from the authors upon reasonable request.

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Parameter	Value	Parameter	Value
Thickness of the ion gel layer (d_ig)	150 nm	Thickness of the Teflon dielectric substrate (d)	8 µm
Thickness of the gold layer (d_gold)	0.5 µm	Unit cell dimension in x direction (P_x)	16 µm
Unit cell dimension in y direction (P_y)	16 µm	Inner length of the outer square ring (L)	11 µm
Length of square patch resonator (l)	9 µm	Distance between the outer square ring and inner square patch (g)	1 µm
Fermi energy level of graphene (E_f)	0.8 eV	graphene relaxation time (τ)	2 ps
carrier mobility of graphene (µ)	25,000 cm²/(V.s)

γ₁	δ₁	$ω_{1}$
2.7 × 10¹¹	1.17 × 10¹²	2 × π×2.88 × 10¹²
γ₂	δ₂	$ω_{2}$
2.2 × 10¹¹	4 × 10¹¹	2 × π×4.66 × 10¹²
γ₃	δ₃	$ω_{3}$
3.3 × 10¹¹	6 × 10¹¹	2 × π×5.71 × 10¹²

Ref.	Number of resonators per unit cell	Max. number of absorption bands (over 50%)	Max. absorption (%)	Average of absorption peaks (%)	ECM
[10]	two	two	100	91.5	yes
[15]^a	one	two (TE)	100 (TE)	78 (TE)	yes
[15]^a	one	three (TM)	99 (TM)	97.3 (TM)	yes
[16]^a	one	three (TE)	97 (TE)	66. 25 (TE)	yes
[16]^a	one	one (TM)	96 (TM)	96 (TM)	yes
[43]	one	two	99.8	98.8	no
[44]	three	three	99.8	94.9	yes
[4]	two	three	99.2	95.7	yes
[45]	five	two	over 95	over 95	no
[46]	one	one	∼50	∼50	no
This work	two	three	100	99	yes

Parameter	Value	Parameter	Value
Thickness of the ion gel layer (d_ig)	150 nm	Thickness of the Teflon dielectric substrate (d)	8 µm
Thickness of the gold layer (d_gold)	0.5 µm	Unit cell dimension in x direction (P_x)	16 µm
Unit cell dimension in y direction (P_y)	16 µm	Inner length of the outer square ring (L)	11 µm
Length of square patch resonator (l)	9 µm	Distance between the outer square ring and inner square patch (g)	1 µm
Fermi energy level of graphene (E_f)	0.8 eV	graphene relaxation time (τ)	2 ps
carrier mobility of graphene (µ)	25,000 cm²/(V.s)

γ₁	δ₁	$ω_{1}$
2.7 × 10¹¹	1.17 × 10¹²	2 × π×2.88 × 10¹²
γ₂	δ₂	$ω_{2}$
2.2 × 10¹¹	4 × 10¹¹	2 × π×4.66 × 10¹²
γ₃	δ₃	$ω_{3}$
3.3 × 10¹¹	6 × 10¹¹	2 × π×5.71 × 10¹²

Multi-band terahertz metamaterial absorber composed of concentric square patch and ring resonator array

Abstract

1. Introduction

2. Materials and methods

3. Results and discussion

4. Conclusion

Disclosures

Data availability

References

Data availability

Cited By

Figures (11)

Tables (3)

Equations (40)

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