Equivalent circuit model of graphene chiral multi-band metadevice absorber composed of U-shaped resonator array

Somayyeh Asgari; Tapio Fabritius

doi:10.1364/OE.412107

1. Introduction

Chiral metadevices are in two categories of three dimensional (3D) [1] and two dimensional (2D) [2]. Chiral metadevices have attracted great attention due to their properties such as circular dichroism (CD) [3], circular conversion dichroism (CCD) [4], and/or linear dichroism (LD) [5] in terahertz (THz) frequency range. Many chiral metadevices, including graphene-based devices, have been proposed recently to achieve chirality responses such as CD, CCD, and/or LD which could be controlled [1–10].

Graphene, a 2D layer of carbon atoms arranged in a honeycomb lattice, has fascinated remarkable heads in photonics and optoelectronics [11] because it can support electromagnetic waves in THz region and its possible to control the electromagnetic waves actively by just altering the applied electric bias voltage without need to refabricate the metadevices [4,6–8]. Nowadays, many kinds of graphene-based devices such as filters [12,13], logic gates [14,15], demultiplexers [16,17], and absorbers [18–21] are proposed and designed in the THz and mid-infrared (MIR) regions. In the case of graphene, the chirality responses such as CD, CCD, and/or LD become actively tunable and controllable by its Fermi energy which is not possible by other commonly used materials such as dielectrics or metals. Basically, this means that by varying the bias voltage of graphene, its spectral properties can be tuned without need to refabricate the device.

2D graphene chiral metadevices composing of single- and double-layered graphene resonator array with tunable chirality responses of CD and/or CCD have recently been designed and proposed in THz region [4,6–8,22–24]. These reported devices are made of graphene, having CCD and CD respectively up to 20% and 60%. However, chiral tunable devices with strong chirality responses of CD, CCD, and/or LD are more desirable.

Recently, some chiral absorbers are designed and proposed [25–30] but none of them are tunable and controllable with single or dual band absorptions. There is a lack of simple multi-band absorber concepts with excellent absorption properties. That is because simplified geometry tends to limit the number of absorption bands. Conventional multi-band absorbers are based either on the combination of multiple resonators with different sizes into one super unit cell [31,32] or stacking the resonators as a multilayer structure with resonators with different geometrical parameters and separate them with dielectric spacers [32]. The graphene-based multi-band simple or chiral metamaterial absorber based on the above frequency superposition methods has a complicated structure requiring of high precision preparation technique. Thus, it is necessary to find a new method to realize graphene-based multi-band metamaterial absorber with simple geometry and few layers. So, implementation of tunable multi-band perfect metamaterial absorber with single-layered simple geometry is very beneficial to save material, cost, and time.

The future THz communication systems, where it is necessary to obtain more information through multi-band and multi-beam absorbers to promote multi-task systems, are potential applications for multi-band THz absorbers [33–36]. Similar kind of needs might be found in the sensing applications as well [37–39].

Theoretical and numerical simulations are not convenient to be used for engineering driven metamaterial design because of their complexity. That is why a simple, fast, and efficient equivalent circuit model (ECM) approach is proposed for a tunable graphene chiral multi-band metadevice absorber composed of tunable single-layered graphene U-shaped resonator array which has very strong LD response, working in THz region. Our proposed multi-band metadevice absorber with ECM model is expected to accelerate the development of such metadevices for some other potential applications such as polarization filter, converter, and biological sensor in the THz region.

2. Device, material, and equivalent circuit model

The 3D schematic view of the proposed tunable graphene-based chiral multi-band metadevice absorber consisted of periodically patterned graphene-based U-shaped resonator array is shown in Fig. 1(a). The substrate is made of quartz with refractive index of 1.96 (ɛ_d₂=3.84) [6]. The conductivity of gold layer is 4.56×10⁷ S/m [40]. Based on the penetration depth of the THz electromagnetic waves, the thickness of 0.5 µm for the gold reflector is chosen to ensure that the incident electromagnetic wave cannot transmit through it [40]. The ion gel layer is described by a nondispersive permittivity of ɛ_d₁ = 1.82 [41], which is overlaid on the graphene U-shaped resonator array, with a thickness of 0.5 µm for electrostatic bias of the graphene resonator array. Then, a gold electrode is manufactured on the ion gel layer. Simulations are done in CST Microwave Studio Software [6,7,42,43]. The chiral metadevice absorber (Figs. 1(a) and 1(b)) is composed of U-shaped graphene resonator array (USGRA). The USGRA is given in Fig. 1(c).

Fig. 1. (a) Three dimensional schematic view of the tunable graphene chiral multi-band metadevice absorber array composed of graphene U-shaped resonators (b) View of the unit cell of the metadevice absorber. (c) Front view of the U-shaped graphene resonator array (USGRA). The substrate is made of quartz dielectric with refractive index of 1.96. Gold reflector with conductivity of 4.56×10⁷ S/m is used beneath the structure to prevent the wave transmission. For uniform electrostatic biasing of graphene U-shaped resonators, ion gel with refractive index of 1.35 is used.

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The substrate thickness d₂ has considered to be 7 µm in all simulations. The length of the U-shaped resonator is L = 20 µm. The length and width of the gap are respectively considered l = 7.5 µm and w = 1.5 µm. The considered values for the parameters of the proposed metadevice absorber are summarized in Table 1.

Table 1. Considered values for the parameters of the chiral multi-band metadevice absorber of Fig. 1.

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The thickness of the monolayer graphene is considered as Δ = 0.335 nm [44]. The relative permittivity of graphene is [42,45]:

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

in which σ, ω, and ɛ₀ are respectively the surface conductivity of the graphene, angular frequency, and vacuum permittivity. σ consists of the inter- and intra-band electron transition contributions by assumption of the incident electromagnetic wave as e^jωt; as follows [46]:

(2a)$$\sigma {\; } = {\sigma _{intra}}(\omega )+ {\sigma _{inter}}(\omega ), $$

(2b)$${\sigma _{intra}}(\omega )= \frac{{2{k_B}{e^2}T}}{{\pi {\hbar ^2}}}ln\left[ {2cosh\left( {\frac{{{E_f}}}{{2{k_B}T}}} \right)} \right]\frac{j}{{j{\tau ^{ - 1}} - \omega }}, $$

(2c)$${\sigma _{inter}}(\omega )= \frac{{{e^2}}}{{4\hbar }}\left[ {H\left( {\frac{\omega }{2}} \right) - \frac{{4j\omega }}{\pi }\mathop \smallint \limits_0^\infty \frac{{H(\zeta )- H\left( {\frac{\omega }{2}} \right)}}{{{\omega^2} - 4{\zeta^2}}}d\zeta } \right], $$

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

in which ℏ is the reduced Plank’s constant, k_B = 1.38×10⁻²³ J/K is the Boltzmann’s constant, e = 1.6×10⁻¹⁹ C is the electron charge, T is the temperature equals to 300 K, and ζ is the integral variable. τ is the relaxation time as [42,47]:

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

in which v_f = 10⁶ m/s is the Fermi velocity and µ=1.25 m²/(V.s) is the carrier mobility. In the THz band, we have: $\hbar \omega \ll {E_f}$. So, the inter-band transition is assumed to be negligible and the intra-band transition dominates [42].

The propagation constant of the electromagnetic wave on graphene-vacuum configuration can be calculated by [48]:

(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. The spectra of the real part of the propagation constant β, of the graphene layer for three different values of E_f are calculated and given in [42]. As the E_f increases, the real part of the β decreases. External bias voltage should be applied between the graphene U-shaped resonator array and the bottom gold reflector for manipulating of the E_f . The relationship between E_f and bias voltage (V>0) is approximately considered as [49]:

(5)$${E_f} = \hbar {v_F}\sqrt {\pi {\varepsilon _0}{\varepsilon _{d2}}\frac{V}{{e{d_2}}}} , $$

Transverse magnetic (TM) and transverse electric (TE) polarized electromagnetic waves are launched separately to the metadevice absorber composed of U-shaped resonator array in THz region. The ECM of the metadevice absorber differs for TM polarized wave and TE polarized wave which proves the chirality nature (lack of mirror symmetry) of the proposed metadevice.

For the TE polarized incident wave, the U-shaped graphene resonator array (USGRA) of Fig. 1(c), is modeled with RLC circuit and the gap of the resonator is modeled by a parallel capacitance. In this case, the incident electric field is normal to the gap, the ECM is depicted in Fig. 2(a). The gap of the resonator is modeled by a capacitance which is calculated by:

(6)$${C_{gap}} = {\varepsilon _{eff}}\frac{l}{w}, $$

which is 158 pF based on the parameters given in Table 1, where

(7)$${\varepsilon _{eff}} = {\varepsilon _0}\frac{{{\varepsilon _{d1}} + {\varepsilon _{d2}}}}{2}, $$

Fig. 2. The equivalent circuit models (ECMs) of the U-shaped graphene resonator array (USGRA) of Fig. 1(c), when (a) TE and (b) TM electromagnetic waves are launched to the multi-band metadevice absorber.

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For the TM polarized incident wave, the U-shaped resonator array could only be modeled with RLC circuit. However, C_gap = 0 because the incident electric field is parallel to the gap. The ECM of this case is shown in Fig. 2(b).

The ECM approach is based on the ABCD matrix and the S-parameters [50]. The ABCD matrices of ion gel/substrate (IG/S) and U-shaped graphene resonator array (USGRA) respectively are obtained by [50]:

(8)$$[{{T_{IG/S}}} ]= \left[ {\begin{array}{cc} {\cos ({{\phi_{IG/S}}} )}&{j{Z_{IG/S}}\sin ({{\phi_{IG/S}}} )}\\ {\frac{j}{{{Z_{IG/S}}}}\sin ({{\phi_{IG/S}}} )}&{\cos ({{\phi_{IG/S}}} )} \end{array}} \right], $$

(9)$$[{{T_{USGRA,TM/TE}}} ]= \left[ {\begin{array}{cc} 1&0\\ {\frac{1}{{{Z_{USGRA,TM/TE}}}}}&1 \end{array}} \right], $$

in which the electrical length and the impedances of IG/S are as follows:

(10)$${\phi _{IG/S}} = \frac{\omega }{c}\sqrt {{\varepsilon _{r,IG/S}}} , $$

(11)$${Z_{IG/S}} = \frac{{{Z_0}}}{{\sqrt {{\varepsilon _{r,IG/S}}} }}, $$

The ABCD matrix of the graphene chiral metadevice absorber without considering ion gel and gold reflector can be obtained by [50]:

(12)$$[{{T_{USGRA\& S,TM/TE}}} ]= [{{T_{USGRA,TM/TE}}} ].[{{T_S}} ]= \left[ {\begin{array}{cc} {{A_{USGRA\& S,TM/TE}}}&{{B_{USGRA\& S,TM/TE}}}\\ {{C_{USGRA\& S,TM/TE}}}&{{D_{USGRA\& S,TM/TE}}} \end{array}} \right], $$

where ω, c, and Z₀ are respectively the angular frequency, speed of light, and impedance of vacuum.

In order to calculate Z_{USGRA, TM/TE}, we use the field reflection coefficients of TM/TE modes for the structure containing USGRA and substrate, r_{USGRA&S,TM/TE}, which are calculated by finite element method (FEM) simulations [50]:

(13)$${r_{USGRA\& S,TM/TE}} = \frac{{{A_{USGRA\& S,TM/TE}} + \frac{{{B_{USGRA\& S,TM/TE}}}}{{{Z_0}}} - {Z_0}.{C_{USGRA\& S,TM/TE}} - {D_{USGRA\& S,TM/TE}}}}{{{A_{USGRA\& S,TM/TE}} + \frac{{{B_{USGRA\& S,TM/TE}}}}{{{Z_0}}} + {Z_0}.{C_{USGRA\& S,TM/TE}} + {D_{USGRA\& S,TM/TE}}}}, $$

So, Z_{USGRA, TM/TE} is calculated by:

{Z_{USGRA,\; TM/TE = }}

(14)$$\frac{{{Z_0}.({1 + {r_{USGRA,\; TM/TE}}} )({{Z_0}.\cos ({{\phi_s}} )+ j{Z_s}.\sin ({{\phi_s}} )} )}}{{({1 - {r_{USGRA,\; TM/TE}}} ).({{Z_0}.\cos ({{\phi_s}} )+ j{Z_s}.\sin ({{\phi_s}} )} )- {Z_0}.({1 + {r_{USGRA,\; TM/TE}}} ).\left( {\cos ({{\phi_s}} )+ j\left( {\frac{{{Z_0}}}{{{Z_s}}}} \right).\sin ({{\phi_s}} )} \right)}}, $$

For calculation of Z_g_{, TE}, we have:

(15)$${Z_{USGRA,TE}} = {Z_{g,TE}}{Z_{gap}} \Rightarrow {Z_{g,TE}} = \frac{{{Z_{gap}}.{Z_{USGRA,TE}}}}{{{Z_{gap}} - {Z_{USGRA,TE}}}}, $$

For the state of TM mode, we have:

(16)$${Z_{USGRA,TM}} = {Z_{g,TM}},$$

After calculation of Z_{USGRA, TM/TE}, we calculate the ABCD matrix of the total metastructure T_{tot, TM/TE} which is composed of ion gel (IG), U-shaped graphene resonator array (USGRA), and substrate (S):

(17)$$[{{T_{tot,TM/TE}}} ]= [{{T_{IG}}} ].[{{T_{USGRA,TM/TE}}} ].[{{T_S}} ]= \left[ {\begin{array}{cc} {{A_{tot,TM/TE}}}&{{B_{tot,TM/TE}}}\\ {{C_{tot,TM/TE}}}&{{D_{tot,TM/TE}}} \end{array}} \right], $$

The ECMs of the proposed chiral multi-band metadevice absorber for TE and TM modes are respectively given in Figs. 3(a) and 3(b).

Fig. 3. The ECMs of the graphene chiral multi-band metadevice absorber when (a) TE and (b) TM electromagnetic waves are launched to it.

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The input impedances of TM and TE modes of the chiral metadevice absorber Z_in,TM/TE (shown in Fig. 3) is calculated by [50]:

(18)$${Z_{in,TM/TE}} = \frac{{{A_{tot,TM/TE}}{Z_{Gold}} + {B_{tot,TM/TE}}}}{{{C_{tot,TM/TE}}{Z_{Gold}} + {D_{tot,TM/TE}}}}, $$

where Z_GOLD is the impedance of the ground plate. The ground plate acts as a short circuit. So, we set Z_Gold to zero. Then, Eq. (18) is simplified to [50]:

(19)$${Z_{in,TM/TE}} = \frac{{{B_{tot,TM/TE}}}}{{{D_{tot,TM/TE}}}}, $$

which is equal to:

(20)$${Z_{in,\; TM/TE}} = \frac{{\left( {j{Z_S}\sin ({{\phi_S}} ).\left( {\cos ({{\phi_{IG}}} )+ \left( {j{Z_{IG}}\sin ({{\phi_{IG}}} ).\frac{1}{{{Z_{USGRA,TM/TE}}}}} \right)} \right)} \right) + ({j{Z_{IG}}\sin ({{\phi_{IG}}} ).\cos ({{\phi_S}} )} )}}{{\left( {j{Z_S}\sin ({{\phi_S}} ).\left( {\frac{j}{{{Z_{IG}}}}\sin ({{\phi_{IG}}} )+ \left( {\cos ({{\phi_{IG}}} ).\frac{1}{{{Z_{USGRA,TM/TE}}}}} \right)} \right)} \right) + ({\cos ({{\phi_{IG}}} ).\cos ({{\phi_S}} )} )}}, $$

Then, the reflection of the total metastructure r_tot,TM/TE is calculated by [50]:

(21)$${r_{tot,TM/TE}} = \frac{{{Z_{in,TM/TE}} - {Z_0}}}{{{Z_{in,TM/TE}} + {Z_0}}}, $$

Finally, the total absorption of the chiral metadevice absorber is calculated by [50]:

(22)$${A_{TM/TE}} = 1 - {|{{r_{tot,TM/TE}}} |^2}, $$

We can promote our ECM for the oblique incident angle (angle respect to -z axis in Fig. 1(b) which is not zero.) of ϕ_inc ≠ 0, as well. In this regard, Z_IG/S (Eq. (11)) for TE and TM modes are not equal, and they should respectively be replaced with [51]:

(23)$$Z_{IG/S}^{TE} = \frac{{{Z_{IG/S}}}}{{cos({{\phi_{inc}}} )}}, $$

(24)$$Z_{IG/S}^{TM} = {Z_{IG/S}}cos({{\phi_{inc}}} ), $$

Also, the electrical length (Eq. (10)) for the oblique incident angle should be modified as [51]:

(25)$${\phi _{IG/S}} = {\beta _ \bot }{d_2}, $$

in which:

(26)$${\beta _ \bot } = {\beta _0}\sqrt {{\varepsilon _{d2}} - si{n^2}({{\phi_{inc}}} )} , $$

where β_⊥ and β₀ are respectively the normal component of the wave number and the propagation constant of vacuum.

3. Results and discussion

Numerical simulations were performed in CST Microwave Studio Commercial Software by use of finite element method (FEM) with a frequency domain solver [52,53]. We have applied periodic boundary conditions in x and y directions, and open boundary condition in z direction in our simulations. In addition, tetrahedral mesh type is used for meshing of the structure. Our proposed structure is excited by the z-polarized transverse electric (TE) and transverse magnetic (TM) incident electromagnetic lightwaves.

In order to obtain the absorption spectra of TM/TE modes for the whole absorber structure of Fig. 1 by the ECM approach, the impedance of the U-shaped graphene resonator array Z_{USGRA, TM/TE} should be calculated. For calculation of Z_{USGRA, TM/TE} (as given in Eq. (14)), the field reflection coefficients of TM/TE modes for the structure containing of USGRA on substrate, r_{USGRA&S,TM/TE}, are needed. The structure containing USGRA on substrate (Fig. 4) with simulation parameters given in Table 1 is simulated and r_{USGRA&S,TM/TE} are obtained. The obtained r_{USGRA&S,TM/TE} which are given in Figs. 5(a) and 5(b) are put in Eq. (14) and Z_{USGRA, TM/TE} are determined. Z_{USGRA, TM} (Fig. 3(b)) is containing of RLC elements. The gap is not modeled by a capacitance in TM mode because the incident electric field is parallel to the gap. Z_{USGRA, TE} (Fig. 3(a)) is containing of RLC elements parallel with C_gap element. The gap is modeled by a capacitance in TE mode because the incident electric field is normal to the gap. Those RLC elements in Figs. 3(a) and 3(b) for TE and TM modes are respectively shown by Z_{g, TE} and Z_{g, TM}.

Fig. 4. View of the unit cell of the structure containing USGRA on substrate.

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Fig. 5. Reflection spectra of the structure of Fig. 4 for (a) TE and (b) TM modes. Real parts of (c) Z_{g, TE} (Fig. 3(a)) and (d) Z_{g, TM} (Fig. 3(b)) of the structure of Fig. 4._.

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The real parts of Z_{g, TE} (Eq. (15) and Fig. 3(a)) and Z_{g, TM} (Eq. (16) and Fig. 3(b)) of the USGRA are respectively plotted in Figs. 5(c) and 5(d). In addition, the imaginary parts of Z_{g, TE} (Eq. (15) and Fig. 3(a)) and Z_{g, TM} (Eq. 16) and Fig. 3(b)) of the USGRA are respectively given in Figs. 6(a) and 6(b). By increasing of E_f, the resonance frequencies of the absorber increase, which tends to exhibit a blueshift. It is because the real part of the β in Eq. (4) decreases as the µ_c increases (shown in [42]) [54]. Therefore the resonance values of the real and the imaginary parts of the impedances increase by the increase of E_f. At the same time, the losses of the TM and TE waves supported by the graphene decrease with the E_f increase. So, the absorption of the absorber increases correspondingly with the E_f increase (shown in [42]) [54]. Therefore the values of the real and the imaginary parts of the impedances increase by the increase of E_f.

Fig. 6. Imaginary parts of (a) Z_{g, TE} (Fig. 3(a)) and (b) Z_{g, TM} (Fig. 3(b)) of the structure of Fig. 4._.

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The absorption spectra of the proposed tunable graphene-based multi-band metadevice absorber for TE and TM modes by considering of E_f= 0.8 eV are given in Fig. 7. As depicted, there are four resonances in 1.18, 2.21, 3.1, and 4.13 THz for TE mode and two resonances in 1.78 and 3.74 THz for TM mode. In fact, there are three absorption bands for TE mode and one absorption band for TM mode with absorption >50% in 0.5-4.5 THz. The absorption spectrum of TE is not coincided with that of TM which shows the chiral (asymmetrical) nature of the introduced metadevice absorber. In addition, the chirality nature of the device could be sensed by the ECMs of the USGRA in both TE and TM modes which are respectively given in Figs. 2(a) and 2(b). The gap is modeled by a capacitance in TE mode because the incident electric field is normal to the gap.

Fig. 7. Absorption spectra of the proposed multi-band metadevice absorber of Fig. 1 considering E_f= 0.8 eV, for TE and TM modes.

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The observed chiral asymmetry nature of the proposed metadevice absorber can also be demonstrated by its electrical field distributions. The electric field distributions of the proposed graphene chiral metadevice absorber of Fig. 1 under TM and TE incident lightwaves at the two different resonance frequencies of 1.78 and 2.21 THz are displayed in Fig. 8. The distributions show a clear dependence on the polarization of the incident light, resulting in a difference in the absorption of TM and TE lightwaves. The field distributions of the absorber for TM and TE normal incident illuminations (illuminations are in -z direction in Fig. 1(b)) are given in Figs. 8(a) and 8(b). They are not equal. The maximum absorption occurs for TM mode in 1.78 THz resonance frequency (shown in Fig. 7). So, the electric field is stronger in this resonance frequency for TM mode comparing with TE mode (shown in Figs. 8(a) and 8(b)). The same as for the electric field distributions for 2.21 THz resonance frequency in Figs. 8(c) and 8(d). Maximum absorption occurs for TE mode in 2.21 THz (shown in Fig. 7). So, the electric field is stronger for TE mode compared with TM mode in 2.21 THz (shown in Figs. 8(c) and 8(d)).

Fig. 8. Electric field distributions of the proposed graphene chiral metadevice absorber of Fig. 1 for (a) TM in 1.78 THz, (b) TE in 1.78 THz, (c) TM in 2.21 THz, and (d) TE in 2.21 THz.

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The absorption obtained by simulation (FEM method) and ECM (Eq. (22)) methods, for the proposed graphene chiral metadevice absorber when the structure is illuminated normally (illumination in -z direction) by TM and TE polarized incident waves for three different values of E_f = 0.6, 0.7, and 0.8 eV are compared and depicted in Figs. 9(a) and 9(b), respectively.

Fig. 9. Comparison of FEM and ECM results of the proposed graphene chiral multi-band metadevice absorber composed of U-shaped resonator array for normal incident wave (illumination in -z direction) for different values of E_f for (a) TE and (b) TM modes.

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The results of the both ways for the different values of E_f are in good agreement. As it can be seen, by increasing E_f, the resonance frequency increases with a tendency to exhibit a blueshift. Also, the absorption increases. The reason for this is the same as given for Figs. 5 and 6. The input impedances of the absorber are matched with the normalized vacuum impedance at the resonance frequencies which results in maximum absorption.

Fabrication of perfect graphene patterns is challenging, and defects decrease the mobility of the graphene [55]. In order to predict the device performance in the case of different mobilities of graphene, absorption characteristics of the device is obtained by FEM and ECM (Eq. (22)) methods when the structure is illuminated normally by TM and TE polarized incident waves with three different mobility values of µ = 0.85, 1.15, and 1.45 m²/(V.s) when µ_c=0.7 eV in Figs. 10(a) and 10(b), respectively. As depicted, the carrier mobility does not affect the resonance frequencies of the TE and TM absorption spectra. Increase in carrier mobility results in an increase in the absorption peaks of the TE and TM absorption spectra. The reason of enhanced absorption by increase of carrier mobility µ (increase of µ results in increase of the relaxation time τ; Eq. (3)) is that the contribution of carrier to plasma oscillation increases with the increase of µ (or τ), so that the absorption increases [56].

Fig. 10. Comparison of FEM and ECM results of the proposed graphene chiral multi-band metadevice absorber composed of U-shaped resonator array for normal incident wave (illumination in -z direction) for different values of µ when µ_c=0.7 eV for (a) TE and (b) TM modes.

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In addition, the TE and TM absorption spectra of the proposed absorber of Fig. 1 for oblique incident angle of 40° with both methods of FEM and ECM are respectively given in Figs. 11(a) and 11(b). Good agreement between the FEM and ECM results indicates that the proposed ECM can predict the absorption spectra, even at oblique incidences.

Fig. 11. Comparison of FEM and ECM results of the proposed graphene chiral metadevice absorber for oblique incident angle of 40° for (a) TE and (b) TM modes.

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To make a comparison between TE and TM absorption spectra of 0° (illuminated by the incident lightwave in -z direction) and 40° (the angle between the illuminated lightwave and -z direction) oblique incident waves, TE and TM simulated absorption spectra by FEM for the both mentioned states are respectively given in Figs. 12(a) and 12(b). For TE mode (Fig. 12(a)), by increasing of the incident lightwave angle to 40°, the resonance frequencies are unchanged and stable, but the absorption values are decreased. For TM mode (Fig. 12(b)), by increasing of the incident angle to 40°, the first and second resonance frequencies respectively increase and decrease slightly, and the absorption values for the first and second resonances are respectively unchanged and increased. As a consequence, the overall performance of the absorber structure for both TE and TM modes are maintained by change of the 0° incident angle to the oblique 0°-40° ones.

Fig. 12. Comparison of FEM results of the proposed graphene chiral metadevice absorber for normal and oblique 40° incident angles for (a) TE and (b) TM modes.

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The characteristics of our proposed tunable graphene chiral absorber are compared with other published chiral absorbers. The comparison is given in Table 2. All the reported works are tunable by change of geometrical parameters or incident angle. Graphene chiral absorber (our work) is also tunable by change of bias voltage which is a beneficial feature to serve material, cost, and time. Also, the characteristics of our proposed graphene chiral metadevice are compared with other graphene chiral metastructures in Table 3. Comparison of our multi-band graphene chiral absorber with other previously published multi-band graphene absorbers is given in Table 4.

Table 2. Comparison of the characteristics of the chiral metadevice absorber of Fig. 1 with other chiral absorbers.

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Table 3. Comparison of the characteristics of the graphene chiral metadevice absorber of Fig. 1 with other graphene chiral metastructures.

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Table 4. Comparison of the characteristics of the multi-band graphene chiral metadevice absorber of Fig. 1 with other published multi-band graphene absorbers.

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Even though the fabrication of the proposed metastructure is out of the scope of this paper, the implementation feasibility is necessary to be considered. Usually the quality of graphene in such devices is lower than expected in theoretical and numerical calculation due to the imperfections caused by fabrication process. The consequence of those defects is the lower carrier mobility of graphene [55]. Fabrication procedure of the proposed chiral metadevice absorber could be as follows: The quartz dielectric is transferred on gold metal reflector through thermal evaporation. The graphene layer is coated on the quartz dielectric by chemical vapor deposition (CVD). The graphene U-shaped resonator array is written by electron beam etching [61,62]. The ion gel dielectric is transferred on graphene resonator array through thermal evaporation. The proposed metamaterial model assumes that all the layers are fabricated without any defects or imperfections. The fabrication of the proposed device and its influence on metamaterial performance needs further investigation.

4. Conclusion

In this paper, equivalent circuit modeling (ECM) approach of a tunable graphene-based chiral metadevice absorber composed of U-shaped resonator array by use of a simple and fast MATLAB code is proposed, designed, and analyzed in terahertz (THz) region. Simulation results are done by the finite element method (FEM) in CST Microwave Studio Software which are in good agreement with the ECM ones. Our proposed ECM approach could also be used for modeling of other chiral metadevices. Our proposed absorber structure is tunable, single-layered, containing one resonator in each unit cell, having strong linear dichroism (LD) response of 94%, absorption of 99% for both of TE and TM electromagnetic waves, and having three absorption bands for TE mode and one absorption band for TM mode. Also, the proposed device could find some other potential applications in controllable polarization sensitive devices such as polarization filter and biological sensor in the future.

Funding

Academy of Finland (320017).

Disclosures

The authors declare no conflicts of interest.

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Parameter	Value
Unit cell dimension in x direction (P_x)	20 µm
Unit cell dimension in y direction (P_y)	20 µm
Thickness of the ion gel layer (d₁)	0.5 µm
Thickness of the quartz dielectric spacer (d₂)	7 µm
Thickness of the gold reflector (d₃)	0.5 µm
The length of the U-shaped resonator (L)	15 µm
The length of the gap (l)	7.5 µm
The width of the gap (w)	1.5 µm
graphene Fermi energy (E_f)	0.8 eV

		[25]	[26]	[27]	[28]	[29]	[30]	This work
Tunability by change of	geometrical parameters & incident angle	yes	yes	yes	yes	yes	yes	yes
Tunability by change of	bias voltage	no	no	no	no	no	no	yes
Number of layers		two	one	one	two	one	one	one
Number of resonators in unit cell		eight	one	one	eight	one	six	one
Unit cell dimension		10 × 10 mm²	320 × 420 nm²	300 × 450 nm²	80 × 80 µm²	235 × 335 nm²	380 × 700 nm²	20 × 20 µm²
Theoretical approach		no	no	no	no	no	no	yes
Max. absorption (%)		93.2	80	50	97	not reported	85	99
Number of absorption bands		one	two	two	two	two	one	four
Max. CD (%) / based on		86 / transmission	50 / absorption	12 / absorption	70 / transmission	88 (0) / reflection (transmission)	50 / absorption	0 / absorption
Max. CCD (%) / based on		0 / transmission	N/A	N/A	N/A	0 (15) / reflection	N/A	0 / absorption
Max. LD (%) / based on		N/A	N/A	N/A	N/A	N/A	N/A	94 / absorption

	[4]	[6]	[7]	[8]	[22]	[23]	[24]	[42]	This work
Number of layers	one	one	one	one	one	one and two	three	two	one
Unit cell dimension	1.6 × 1 µm²	80 × 80 µm²	0.25 × 0.3 µm²	2 × 4 µm²	0.27×0.27 µm²	4.4 × 3.65 µm²	46 × 46 µm²	22 × 22 µm²	20 × 20 µm²
Theoretical approach	no	no	no	no	no	no	no	yes	yes
Max. absorption (%)	N/A	45	N/A	N/A	N/A	N/A	90	60	99
Max. CD (%) / based on	N/A	Not possible	0 / transmission	0 / transmission	0 / transmission	0 / transmission	60 (60) / absorption (reflection)	20 / absorption	0 / absorption
Max. CCD (%) / based on	15 / transmission	Not possible	6 / transmission	4 / transmission	20 / transmission	19 / transmission	0 (0) / absorption (reflection)	N/A	0 / absorption
Max. LD (%) / based on	N/A	Not possible	N/A	N/A	N/A	N/A	N/A	N/A	94 / absorption

	[57]	[58]	[59]	[60]	This work
Number of graphene layers	one	one	one	one	one
Number of resonators in unit cell	five	one	one	one	one
unit cell dimension (µm²)	2 × 2	4 × 4	43.8×43.8	1.8 × 1.8	20 × 20
Theoretical approach	yes	no	no	yes	yes
Max. absorption (%)	99	100	99	100	99
Number of absorption bands (>50%)	two	three	two	two	four (three for TE and one for TM)

Parameter	Value
Unit cell dimension in x direction (P_x)	20 µm
Unit cell dimension in y direction (P_y)	20 µm
Thickness of the ion gel layer (d₁)	0.5 µm
Thickness of the quartz dielectric spacer (d₂)	7 µm
Thickness of the gold reflector (d₃)	0.5 µm
The length of the U-shaped resonator (L)	15 µm
The length of the gap (l)	7.5 µm
The width of the gap (w)	1.5 µm
graphene Fermi energy (E_f)	0.8 eV

Equivalent circuit model of graphene chiral multi-band metadevice absorber composed of U-shaped resonator array

Abstract

1. Introduction

2. Device, material, and equivalent circuit model

3. Results and discussion

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (12)

Tables (4)

Equations (30)

Optics Express