1. Introduction

EM absorbers are structures and substances of great importance that are utilized in many applications such as electromagnetic compatibility (EMC), stealth technology, reducing electromagnetic (EM) pollution, medical treatment, and several more [1–7]. In plenty of practical applications, we need high absorption at the desired frequency band [8].

Graphene is a flat monolayer of carbon atoms arranged in a honeycomb [9]. One of graphene’s extraordinary properties is the tunability of surface conductivity and comfortable adjustment [10]. Graphene can be tuned by chemical doping [11] and electrical or magnetic bias [12,13]. Moreover, the amount of graphene conductivity in the microwave band is almost constant for different frequencies. We could use these properties to match graphene’s impedance in the desired frequency band [14]. Therefore, graphene could be one of the best substance choices for designing a planar absorber in the millimeter-wave band.

In this work, an ultra-broadband absorber with polarization-independent property is proposed, in which multilayered graphene is placed a quarter wavelength of the reflector plane (copper has been used as the ground) and other layers. Furthermore, due to the symmetrical design of the unit cell, this structure is a polarization-independent absorber.

The simplicity of implementation and bias in the design of the graphene pattern is considered. The relative bandwidth for one, two, and three-layer graphene are obtained close to 178.8%, 189.9%, and 190.1%, respectively, with above 90% absorption. By comparing the relative bandwidth obtained in this work with the other works such as [15–17], it can be seen that the bandwidth has been significantly improved. The first graphene sheet is placed a quarter wavelength distance from the reflector plane. The space between those is filled with a dielectric which constructs the absorber structure. The admittance of the structure is matched to free space at the central frequency. Full-wave software, CST Studio Suite, which uses the finite integration technique (FIT), is applied for simulation. Moreover, periodic boundaries for the designed unit cell are considered.

We have utilized the destructive interference and resonance approach to design this absorber [18]. This paper is organized as follows: In Section 2, the design of the proposed structure is presented and described. In Part 3, we investigate the effect of a few parameters on the absorption coefficient, and in Section 4, the results are shown.

2. Structure and circuit model

In this section, at first, the one-layer absorber is designed and then extended to a multilayered absorber. The schematic of the proposed structure is shown in Fig. 1(a). The first layer of the graphene sheet is placed at one-fourth wavelength from the reflector plane. Between the reflector plane and the graphene layer is filled with a dielectric that forms the absorber structure. By adjusting the chemical potential of graphene, we could match the admittance of the structure to free space in a central frequency. The reflector plane is a good conductor that we used copper, which prevents wave transmission. For the two and three-layer absorbers, the same graphene and dielectric pattern as the one-layer absorber have been used, with the difference that the conductivity of each graphene layer is different from each other, which is done to achieve impedance matching of the absorber structure with the free space. By changing the chemical potential of graphene, its conductivity can be adjusted. Increasing the number of absorbent layers to achieve a higher absorption coefficient is done in a wide frequency band, although we are facing a limitation in increasing these layers. Considering homogenous region and normal incident wave, the schematic of the transmission line model is shown in Fig. 1(b). In this model, the copper is approximately a short circuit that will be shaped approximately an open circuit in a quarter wavelength. Moreover, the input admittance of the structure is approximately equal to graphene surface admittance at the central frequency (${f_0}$). To match with free space, the surface admittance of the graphene is adjusted at the central frequency. The thickness of the graphene sheet is about 0.34 nm [19]. Although in the millimeter frequency band, we can consider the more amount in simulation in software due to the wavelength of incident wave in comparison with the thickness of graphene is so much greater. The thin layer (sheet) of graphene with surface conductivity ${\sigma _G}$ by the well-known Kubo’s formula is modeled. Because of our frequency band in this paper, in Kubo’s formula, Intraband distribution is dominated which written [20,21]

 

Fig. 1. (a) Schematic of the proposed absorber, with period 5.7 mm and width of graphene w = 3 mm. (b) Transmission line model of the absorber.

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In attention to Fig. 1(b), in homogenous regions, obtain ${\beta _s} = \omega {n_s}/c$, ${y_s} = n/{\eta _0}$, where c is the speed of light in the vacuum and ${n_s}$ is the refractive index of substrates that we consider PMI (Polymethacrylimide) with ${n_s} = \sqrt {{\varepsilon _s}} \simeq 1.05$ [22], ${\varepsilon _s}$ is the permittivity of PMI. Also, the conductivity of copper (reflector plane) is regarded as $5.8 \times {10^7}\; S/m$. We can write as

For perfect absorption, the real part of input admittance must equal $1/{\eta _0}$, and the imaginary part equal zero. In central frequency, according to the disruptive interference approach, we have $2{\beta _d} = \pi $ that causes reflected waves from the reflector plane (copper) and reflecting waves from the surface of the structure (graphene sheet layer) to be summed with a 180° phase difference that cancels each other. The aim is to design an absorber with ultra-broadband thus must utilize the below condition [23]:

Due to the symmetrical graphene pattern in the unit cell design, our structure will be polarization independent.

For the design multilayer absorber, as seen in Fig. 1(a), one layer of dielectric and graphene with the same dimension is added to the previous layer but with a difference in graphene sheet conductivity. For the matching absorber to the free space, the chemical potential should be determined in such a way that an appropriate surface conductivity is obtained. Employing Eq. (1), for different amounts of ${\mu _c}$, we have drawn the conductivity variation in the desired frequency band as shown in Fig. 2. In Table 1 appropriate values for each graphene layer have been presented. For the simulation in full-wave software, each of the graphene layers is modeled with a surface impedance which by inversing of the conductivity achieves (See Table 1).

 

Fig. 2. The surface conductivity of graphene for different chemical potentials (In Eq. (1) (Kubo’s formula) the temperature and carrier scattering rate are chosen as T = 300° K and Γ=5 THz).

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Table 1. The values of chemical potential, surface impedance, and relative bandwidth (RB)

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To set the impedance of graphene equal to the value obtained in Eq. (3), the chemical potential can be adjusted according to what is shown in Fig. 2 in order that the conductivity of graphene reaches the desired value (see Table 1). The appropriate value for the thickness of the designed absorber was obtained from Eq. (4). In Fig. 3, are plotted the absorption diagrams for one, two, and three-layer absorbers. As shown in Fig. 3, with increasing the number of layers, amplitude and bandwidth also increased. For the absorption coefficient above 90%, the relative bandwidth (RB) is shown in Table 1. Due to our structure type, the Fabry_Perot resonances have existed, and the number of those increases with increases of layers. Although, It improves the amplitude and bandwidth but must pay attention that we have limitations to increasing layers. For example, as increasing the number of graphene layers, the thickness of the structure increases. Also, the resonances may interfere with each other and reduce the absorption amplitude at some desired frequencies.

 

Fig. 3. The absorption diagram for the absorber with one, two, and three-layer.

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3. Parametric study

In Fig. 4, for the incident angle, the effect of each one of angles ϕ (angle between x and y axes in Fig. 1(a)) and θ (the angle between the incident wave and the z-axis in Fig. 1(a)) on the absorption coefficient is studied. The effect of θ) angle variation, the angle between the incident wave and z-axis, is investigated. As is shown in Fig. 4(a) for one layer absorber with a growing theta angle, the amplitude and bandwidth decrease. Until $\theta = 50^\circ $, absorption approximately is above 80% but in less bandwidth. Also, it is probed in Fig. 4 for a three-layer absorber. In comparison to one layer, the absorption and bandwidth are so stable with rising the theta angle and for $\theta = 60^\circ $, amplitude still remains approximately above 85%. In Fig. 4(c), as it is shown, due to the symmetry of the unit cell (graphene pattern), variation of phi angle (ϕ) in the incident wave does not change the absorption and the bandwidth.

 

Fig. 4. (a) The absorption coefficient of the one-layer absorber that is illuminated by incident wave with considering the different values for the theta angle. (b) Such as section (a) but for the three-layer absorber. (c) The absorption coefficient of the one-layer absorber, which incident wave illuminated it with different phi angle.

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With the increase in the layer of the absorber, we raise the surface impedance of the graphene sheet (Table 1) for matching impedance and the outer layer has the closest amount to free space impedance. Because of this, the effect of impedance variation with the change of incident wave angle is decreased on matching and the amplitude of absorption and bandwidth are less sensitive on incident angles. Therefore, our structure is an independent polarization.

Let us probe the effect of a few parameters of our proposed structure on absorption and bandwidth. In this section, we consider a one-layer absorber. As shown in Fig. 5(a), with decreasing dielectric thickness (d = 0.5 mm, blue diagram, and d = 0.6 mm, red diagram), a blue shift with less bandwidth occurs. Moreover, a redshift with less bandwidth happens when dielectric thickness increases (d = 0.8 mm, cyan diagram, and d = 0.9 mm, black diagram). The absorber with dielectric thickness d = 0.7 mm (green diagram) has the best function because according to what has been said before at central frequency a disruptive interference occurs in a quarter wavelength from the reflector plane. According to Eq. (4), when dielectric thickness decrease, the resonances happen at a higher frequency, and when increase dielectric thickness resonances occur at a lower frequency.

As shown in Fig. 5(b), the absorption diagram moves to a lower frequency with increasing epsilon's parameter. Moreover, this parameter affects the amplitude and decreases bandwidth. Again, with attention to Eq. (4) can see with increasing epsilon, the resonances happen in lower frequency. Also in higher frequencies, the variation of the epsilon changes the amount of ${\beta _s}$ which could affect the structure’s impedance matching with free space and also the amplitude of absorption.

 

Fig. 5. (a) The absorption coefficient for the different dielectric thicknesses. (b) The absorption coefficient for the different relative permittivity of dielectric. (c) The influence of graphene width on absorption coefficient.

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In Fig. 5(c), the influence of the width of the cross graphene ribbon is shown. By changing the width of the graphene, the position of frequencies that resonances change. With the increasing width of graphene from $w = 2.2\; \textrm{mm}$ until $w = 3$ mm, the bandwidth is increasing but the amplitude decreases slightly. As it increases, the resonances get closer again and the bandwidth decreases.

4. Conclusion

In summary, we have proposed an ultra-broadband multilayer graphene absorber in the millimeter band by applying the graphene layers in a quarter wavelength above the reflector plane and other graphene sheets. Moreover, the conductivity appropriately has adjusted for each of the graphenes that were designed with the proper pattern. For just the one-layer absorber is obtained an absorption coefficient above 90% in relative bandwidth of 178.8%. Furthermore, with the increase in the number of layers, more resonances occur. For two and three-layer absorbers, extraordinary bandwidths of 189.9% and 190.1% have been obtained, respectively. On the other hand, the absorber does not depend on polarization due to the symmetry of the unit cell. Besides, the parameters’ analysis has been carried out on absorption and bandwidth.