1 Introduction

With the increasing development of communication technology and the use of frequency in the terahertz range, the need for tools and equipment in this frequency range is inevitable. This need drew the attention of all terahertz range researchers to metamaterials.

A metamaterial is any material engineered to have a property that is not found in naturally occurring materials (Kshetrimayum 2004). Metamaterials have special properties such as negative refraction (Valentine et al. 2008; Veselago 1968), cloak (Shin 2012; Cai 2007) superlunary (Smith et al. 2004; Pendry 2000) and backward wave (Song et al. 2012; Simovski et al. 2003), and for this reason, they have been highly regarded by researchers in the field of electromagnetism. In addition, metamaterials are very useful in the construction and design of solar energy absorbers (Zhao et al. 2022; Zhao et al. 2021), switchable terahertz devices (Zheng et al. 2022), refractive index sensors (Wu et al. 2021), waveguide-Cavities (Deng et al. 2018, 2015), and etc.

One of the most important and practical metamaterials is graphene. Graphene is a metamaterial made of graphite, and graphite itself is made up of layers of carbon atoms arranged hexagons. There is a Van der Waals bond (a rather weak chemical bond) between these layers. By breaking these bonds according to Fig. 1, we can reach the graphene metamaterial, which is a single layer of carbon atoms, and because of this, it is called a two-dimensional metamaterial (Katsnelson 2007).

Fig. 1
figure 1

Exfoliation of graphite to create graphene

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Graphene has received a great deal of attention in the construction of antennas and alternating structures, including absorbers, electromagnetic shields, and electromagnetic sensors, due to its small size, two-dimensionality, high conductivity and its adjustable conductivity (Ghods and Rezaei 2018a, b; Pop 2012). In recent decades, extensive research and developments have been done in the field of design and improvement of tools and devices that have been made and designed using graphene. Due to the use of graphene in antennas has brought us significant benefit such as extreme miniaturization, monolithic integration with graphene RF Nano- electronics, efficient dynamic tuning, and even transparency and mechanical flexibility (Perruisseau-Carrier 2012), the idea of designing graphene antennas has been a step to ahead.

Since we can change the chemical potential of graphene by external bias voltage or chemical doping, graphene is an adjustable metamaterial (Zamzam et al. 2021). We used this graphene capability to design the antennas with performance in far-field and terahertz range central frequency due to control its polarization just by changing the Fermi energy level of the graphene (Jafari-Chashmi et al. 2020a, b; Jafari-Chashmi et al. 2020a, b; Kiani et al. 2021a, b; Jafari-Chashmi et al. 2019).

Another application of graphene is the design and construction of absorbers that operate in the microwave (Kiani et al. 2021a, b; Ghods and Rezaei 2018a, b) and terahertz (Gong et al. 2021; Chahkoutahi et al. 2021) range. Tunable absorbers can be obtained by changing the Fermi level of graphene (µc) (Xiao et al. 2018). Electromagnetic absorbers are specifically selected or designed materials that can inhibit the reflection or transmission of electromagnetic radiation. Features that are important in absorbers are full absorption (close to 100% adsorption), insensitivity to polarization of radiation and light incident angle.

Absorbers that use metamaterials in their design and have absorption close to 100% are called metamaterial perfect absorbers (MPA) (Ni et al. 2017; Zamzam and Rezaei 2021; Ke et al. 2020; Yao et al. 2016). There are two types of absorbers: resonant (narrow band) absorbers (Fu et al. 2021; Li et al. 2017; Barzegar-Parizi 2019) and wideband absorbers (Deljoo et al. 2021; Elrashidi and Tharwat 2021; Bağmancı et al. 2017). Terahertz band absorbers have many applications such as sensing, imaging, modulating, detection, filtering, etc. (Bai et al. 2021; Song et al. 2018; Meng et al. 2018).

The mechanism of absorption of multilayer absorbers is such that the underlying conductive layer does not allow radiation to pass through the structure. In addition, the intermediate dielectric layer and the metamaterial placed on the dielectric must also be designed so that the waves reflected from inside the structure have a phase difference aiming to neutralize each other, in order to the structure reach to the perfect absorption.

One of the characteristics that makes the absorber structure unique for different applications is the insensitivity of the structure to polarization (Shalini 2021; Bendelala et al. 2018) and different light incident angles (Pan et al. 2021).

In this paper, we present a novel design of a multi-band perfect metamaterial absorber that has one graphene disk at the center and four graphene solid triangles with semicircular cuts on them that surround the central disk. This new design is based on tuning and optimization of various design parameters such as structural dimensions, dielectric constant, the conductivity of the conductive layer, chemical potential, and relaxation time of graphene. The dielectric of the middle layer and the conductive of underlying layer of the structure are SiO2 and Cu, respectively. One of the advantages of this structure is in its construction, because in this structure, copper conductor is used in the lower layer to prevent radiation transmission, which is more economical than structures in which gold is used in the lower layer (Feng et al. 2020; Yi et al. 2019; Wu 2019; Zhang et al. 2018). The proposed absorber is a resonant absorber that has four perfect absorption peaks, which are 99.5%, 99.7%, 94.71% and 97.06% at frequencies of 4.24 THz, 5.89 THz, 9.66 THz, and 10.62 THz, respectively.

2 Graphene conductivity and theoretical discussion

Graphene is a two-dimensional metamaterial that has semi-metallic properties in the terahertz range. Graphene conductivity can be calculated using Kubo formulas (Eqs. (1–3)) (Zamzam et al. 2021). The conductivity of graphene according to Eq. 1 consists of two parts, inter-band and intra-band. The inter-band part can be calculated by Eq. (2) and the intra-band part can be calculated by Eq. (3).

$$ \sigma \left( {\omega ,\tau , \mu_{C} , T} \right) = \sigma_{{{\text{inter}}}} \left( {\omega ,\tau , \mu_{C} , T} \right) + \sigma_{{{\text{intra}}}} \left( {\omega ,\tau , \mu_{C} , T} \right) $$
(1)
$$ {\upsigma }_{{{\text{inter}}}} \left( {\omega ,{\uptau }, \mu_{C} , T} \right) = - j\frac{{e^{2} }}{4\pi \hbar }ln\left( {\frac{{2\mu_{C} + \left( {\omega - j{\uptau }^{ - 1} } \right)\hbar }}{{2\mu_{C} - \left( {\omega - j{\uptau }^{ - 1} } \right)\hbar }}} \right) $$
(2)
$$ {\upsigma }_{{{\text{intra}}}} \left( {\omega ,{\uptau }, \mu_{C} , T} \right) = - j\frac{{e^{2} k_{B} T}}{{\pi \hbar^{2} \left( {\omega - j{\uptau }^{ - 1} } \right)}}\left[ {\frac{{\mu_{C} }}{{k_{B} T}} + 2ln\left( {\exp \left( { - \frac{{\mu_{C} }}{{k_{B} T}}} \right) + 1} \right)} \right] $$
(3)

where ω, τ and μc represent the angular frequency, the relaxation time and chemical potential of graphene, respectively, T, e, kB, and ℏ show the ambient temperature (in this paper T = 300 K), the electron charge (here e = 1.6 × 109 c), the Boltzmann’s constant, and the reduced Plank’s constant, respectively.

According to the Pauli exclusion principle and since we have μc ≫ kBT in the terahertz and infrared bands, the inter-band part of graphene conductivity can be ignored in Eq. (1) and this equation can be simplified to the following equation (like a Drude model):

$$ {\upsigma }\left( {\omega ,{\uptau }, \mu_{C} } \right) = \frac{{je^{2} \mu_{C} }}{{\pi \hbar^{2} \left( {\omega + j\tau^{ - 1} } \right)}} $$
(4)

The relaxation time of graphene in Eq. (4) is \(\tau = v\mu_{C} /eV_{F}^{2}\),Where v and \(V_{F}\) represents the electron mobility and Fermi velocity (approximately 106 m/s).

Electromagnetic absorbers are mainly composed of three layers: two perfect conductor layers and a dielectric layer, which is located between these two conductive layers and is also known as the spacer layer. The general formula for calculating the absorption of electromagnetic absorbers can be written as follows:

$$ A = 1 - T - R $$
(5)

where A is the absorption coefficient of the structure, T is the transmission coefficient of electromagnetic waves from the structure and R shows the reflection coefficient of electromagnetic waves from the absorber. In many structures, such as the proposed structure, metamaterials such as graphene can be used in the top layer instead of the perfect conductors. In addition, it should be noted that the thickness of the bottom conductor layer should be greater than the penetration depth of terahertz waves in order to reduce the terahertz wave transmission coefficient from the structure to zero (T = 0) and simplify Eq. (5) to the following equation:

$$ A = 1 - R $$
(6)

Figure 2 shows the equivalent circuit model of the proposed absorber. A short transmission line with ZCu = 0 impedance is used to model the conductor of the substrate, which is made of copper in the proposed structure. SiO2 dielectric modeling is also performed using a transmission line with impedance of ZTL and electric length of \(E_{TL} = 2 \times \beta_{TL} \times T_{S}\). For graphene modeling, an R, L, and C branch is used. If the number of graphene layers in the structure of an absorber is more than one, for each additional layer, another R, L, and C branch will be added in parallel to the other branches. Considering the free space impedance equal to 377 Ω = 120 π, the input surface impedances of the presented model in Fig. 2 can be calculated as follows (Norouzi-Razani and Rezaei 2022):

$$ Z_{1} = jZ_{TL} {\text{tan}}\left( {\beta_{TL} T_{s} } \right) $$
(7)
$$ \beta_{TL} = 2\pi f\sqrt {\varepsilon_{0} \varepsilon_{r} \mu_{0} } $$
(8)
$$ Z_{TL} = \sqrt {\mu_{0} /\varepsilon_{0} \varepsilon_{r} } $$
(9)
$$ Z_{G} = R + jX_{G} = R + j\left( {2\pi fL - \frac{1}{2\pi fC}} \right) $$
(10)
$$ Z_{in} = \frac{{Z_{1} .Z_{g} }}{{Z_{1} + Z_{g} }} $$
(11)

where βTL and εr show the wavenumber and relative permittivity of the dielectric, respectively; in addition, ε0 and μ0 represent the relative permittivity and magnetic permeability of the free space, respectively. The values of R, L, and C can be calculated using the following equations (Rezagholizadeh et al. 2020):

$$ R = \frac{{KL^{2} }}{{\pi S^{2} }}\frac{{\hbar^{2} }}{{e^{3} \mu_{C} \tau }}\quad \quad L = \frac{{KL^{2} }}{{\pi S^{2} }}\frac{{\hbar^{2} }}{{e^{3} \mu_{C} }} \quad \quad C = \frac{{\pi^{2} S^{2} }}{{L^{2} k}}\frac{\varepsilon }{{q_{1} }} $$
(12)

where ε, q1, L, and ℏ are the permittivity of the dielectric, the first eigenvalue of the equation governing the surface current on the graphene layer, the period of unit cells, and the reduced Plank’s constant, respectively. K and S are the integral of the internal multiplication of an eigenfunction with the corresponding eigenvalue in the convolution of this eigenfunction and the integral of the eigenfunctions of the equation governing the graphene surface current, respectively, and can be calculated by referring to Ref. (Barzegar-Parizi et al. 2015). According to these equations, the reflection efficiency (Γ) can be calculated using the following equation:

$$ {\Gamma } = \frac{{Re\left( {Z_{in} } \right) - Z_{0} }}{{Re\left( {Z_{in} } \right) + Z_{0} }} $$
(13)
Fig. 2
figure 2

The equivalent circuit model of the proposed perfect absorber

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On the other hand, since R = Γ2, Eq. (1) can also be defined as follows:

$$ A = 1 - {\Gamma }^{2} $$
(14)

3 Structure and design

As shown in Fig. 3a, we present a three-layer absorber. The bottom layer of the absorber is a cube with dimensions L × L (L = 3 µm) and a thickness of 0.5 µm. The material of this substrate is copper with conductivity of σCu = 5.813 × \(10^{7} \) S/m (Tahseen and Kishk 2016). This conductor has the task of reducing the radiation transmission from the structure to zero. Our middle layer is a cube with the same dimensions L × L and a thickness of 4.9 µm. In addition, its dielectric material is SiO2 with a relative permittivity of ϵr = 3.9 (Deng et al. 2014; Chen and Alù 2011).

Fig. 3
figure 3

a Unit cells structure from three-dimensional view. b Dimensions of the absorber structure from a two-dimensional view

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As mentioned, graphene is a two-dimensional material that is placed on our SiO2 layer. According to Fig. 3b, the central graphene disk has a radius of R = 0.55 µm. The four triangles that surround the central disk are equal in size. The height of each triangle is w = 0.95 µm and the base of each of them is equal to L = 3 µm. We also cut each triangle into a semicircle so that all the semicircles are equal and have a radius R = 0.55 µm.

4 Simulation results

We performed the simulations at room temperature (T = 300 K) using the CST Studio Suite software. As shown in Fig. 4, by setting the parameters of graphene to µc = 0.8 eV and τ = 1.6 ps, the structure has four perfect absorption peaks, which are 99.5%, 99.7%, 94.71% and 97.06% at 4.24 THz, 5.89 THz, 9.66 THz and 10.62 THz, respectively.

Fig. 4
figure 4

Absorption spectra of structure with µc = 0.8 eV and τ = 1.6 ps

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This structure is tunable; it means that we can shift the frequency of absorption to a required frequency for the intended application or increase and decrease the amount of absorption at the desired frequency by changing Fermi level or relaxation time of graphene. We changed the Fermi level of graphene to different values, and you can see the frequency changes of the absorption peaks in Fig. 5. We also changed the relaxation time of graphene in Fig. 6, where you can see the changes in the absorption peaks of the structure.

Fig. 5
figure 5

Absorption spectra of structure for different Fermi level (µc) at τ = 1.6 ps

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Fig. 6
figure 6

Absorption spectra of structure at µc = 0.8 eV for a τ = 0.8 ps, b τ = 1.6 ps, c τ = 3.5 ps, and d τ = 7 ps

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As you can see in Fig. 7, one of the good advantages of this structure is that we can increase the number of absorption peaks by changing the Fermi level of graphene (for example changing it to 0.5 eV) and changing the relaxation time of graphene (for example changing it to 3 ps). By this setting our structure has six absorption peaks, four of those are above 90%, one of those is above 85% and the other one is above 75%. This setting is used for applications where the number of absorption peaks takes precedence over the absorption rate.

Fig. 7
figure 7

Increasing the number of absorption peaks by setting µc = 0.5 eV and τ = 3 ps

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One feature that is very important in absorbers is the insensitivity of the structure to polarization. As you can see in Fig. 8a, the absorber presented in this paper is polarization insensitive because the absorption peaks remain unchanged at different angles of polarization. This feature makes the proposed absorber very suitable for applications such as imaging, detecting, filtering, sensing, etc.

Fig. 8
figure 8

Absorption spectra of structure for a different polarization angles and b different light incident angles

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As shown in Fig. 8b, it is obtained that the absorption peaks at frequencies 5.89 THz and 10.62 THz with different light incident angles (From 0 to 45 degrees) remain almost unchanged and the absorption peaks at frequencies 4.22 THz and 9.66 THz have a high tolerance (with a little frequency shift) against different light incident angles (From 0 to 45 degrees). Especially at the 10.62 THz peak, the absorption rate increases slightly with increasing light incident angle. In order to investigate the non-change of absorption peaks at frequencies 5.89 THz and 10.62 THz with increasing light incident angle up to 45 degrees, in Fig. 9, the plots of the real part and the imaginary part of the normalized impedance of the proposed structure are presented for the light incident angles of 0 and 45 degrees. As you know, when the real part of the normalized impedance gets closer to 1 and the imaginary part gets closer to 0, the structure gets closer to the impedance matching conditions. Figure 9a and b also show that under a 45-degree light incident angle at absorption peak frequencies, the amount of real parts is close to 1 and the amount of imaginary parts is close to 0 (just same as the 0-degrees light incident angle). Therefore, by increasing the impact angle up to 45 degrees, the structure still has perfect absorption in 4 bands. In addition, the reason for the increase in the absorption rate at the peak of 10.62 THz at the light incident angle of 45-degrees compared to the 0-degrees is that at this frequency, under an oblique light incident angle, the structure becomes closer to the impedance matching conditions. The frequency shift of the absorption peaks is also due to the parasitic resonance generated at the large light incident angles. All of these make the proposed absorber more widely used in various applications.

Fig. 9
figure 9

Normalized impedance of proposed structure a real part and b imaginary part

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5 Discussion and comparison

The absorption mechanism of this absorber is such that the bottom layer of copper does not allow radiation to pass through the structure. In addition, the middle layer of SiO2 and graphene placed on the layer of SiO2 is also designed in such a way that the waves reflected from inside the structure have a phase difference and neutralize each other, thus the structure reaches four full absorption peaks.

Plasmons formed at the interface of a metal and dielectric are called surface plasmons. Surface plasmons are excited by visible or ultraviolet photons, a phenomenon called “surface plasmon resonance” (SPR). Surface plasmons are surface-bound plasmons that react strongly with light from polaritons. They occur at the interface between a vacuum and a material with a dielectric constant with a small positive real part and a large negative imaginary part. To study surface plasmon, the behavior of metals against light electromagnetic fields must first be studied. The optical response of metals is known by their dielectric function. Plasmon plays an important role in the optical properties of metals, which according to the intended application, we coordinate the frequency of light that we want to reflect with the frequency of plasma to resonate. Nanoparticles have a large number of surface atoms compared to the atoms within their volume. This in itself increases the importance of surface effects compared to volumetric effects. In fact, nanoparticles in response to external fields and forces show effects that depend on the size and shape of the particle and the same ratio to the dielectric constant of the environment and metal, so the size of nanoparticles can be estimated in the diagram of the light spectrum. The dependence of the optical spectrum of large nanoparticles on their size is an external effect that is controlled only by the dimensions of the particle relative to the electromagnetic radiation. Slight changes in the dielectric around the nanoparticle affect the resonance of surface plasmons so that these changes are reflected in the amount of beam scattered, the absorbed beam, or the change in wavelength. When the oscillation frequency of the generated plasmon is equal to the frequency of incident electromagnetic waves, the phenomenon of “local surface plasmon resonance” (LSPR) occurs. By doing this, the electromagnetic field is concentrated in a very small space of about 100 cubic nanometers. Any object that enters this area, called a nano-focus, affects the LSPR. We presented the LSPRs of the proposed structure by calculating electric fields at absorption peak frequencies in Fig. 10.

Fig. 10
figure 10

Electric field [real (Ez)] distributions in the normal incidence of TE waves at frequencies of a 4.24 THz, b 5.89 THz, c 9.66 THz, and d 10.62 THz

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Figure 10 shows the electric field [real (Ez)] distributions in the normal incidence of TE waves at frequencies of absorption peaks. As you can see in Fig. 10a, the field distribution is mainly concentrated on the lower two corners and along the sides of the triangles located in the x direction. Therefore, we can conclude that the first resonance at frequency of 4.24 THz is due to resonance at these points. Figure 10b shows that the electric field is mainly concentrated on a part of the surface of graphene triangles located in the y direction along the semicircular cuts and the graphene disk, and this concentration extends to the edges of the graphene disk. So the second resonance at the frequency of 5.89 THz is attributed to the resonance of these points. A similar analysis of Fig. 10c shows that the electric field is concentrated on the surface of the graphene triangles located in the x direction and on the inner edges of the graphene disk. As a result, the third resonance at wavelength of λ = 31,048.9 nm can be considered due to the graphene resonance at these points. Finally, in Fig. 10d, it is clear that the electric field is concentrated on the intersection of the graphene disk and the graphene triangle located in the y direction, Perimeter of semicircular cuts located in the y direction, the bottom two corners and along the sides of each graphene triangle. Therefore, we can conclude that the fourth resonance at frequency of 10.62 THz is due to resonance at these points.

As mentioned in the previous sections, the chemical potential of graphene can be adjusted by electric gating and chemical doping, which is done with an external DC-bias voltage and an electrode layer, as shown in Fig. 11. The electrode layer is usually made of polysilicon and is placed at a very small distance from the graphene, and the thickness of this layer is considered as small as possible so as not to interfere with the performance of the absorber. According to Eq. (16), the VG voltage, shown in Fig. 2a, regulates the chemical potential of graphene (μc) (Hanson 2008).

$$ V_{G} = \frac{{n_{s} eT}}{{\varepsilon_{0} \varepsilon_{r} }} $$
(15)

where T is the thickness of the dielectric layer between the graphene layer and the electrode layer, εr is the relative permittivity of the dielectric, and ns can be calculated by Eq. (17):

$$ n_{s} = \frac{2}{{\pi h^{2} V_{F}^{2} }}\mathop \smallint \nolimits_{0}^{\infty } \varepsilon \left[ {f_{d} \left( {\varepsilon ,\mu_{c} ,T} \right) - f_{d} \left( {\varepsilon + 2\mu_{c} ,\mu_{c} ,T} \right)} \right]d\varepsilon $$
(16)

where VF = 9.5 × 105 m/s is the Fermi velocity, ε represents energy, \(f_{d} \left( {\varepsilon ,\mu_{c} ,T} \right) = \frac{1}{{\left( {e^{{\left( {\varepsilon - \mu_{c} } \right)/(K_{B} T)}} + 1} \right)}}\), and \(f_{d} \left( {\varepsilon + 2\mu_{c} ,\mu_{c} ,T} \right) = \frac{1}{{\left( {e^{{\left( {\varepsilon + \mu_{c} } \right)/(K_{B} T)}} + 1} \right)}}{ }\).

Fig. 11
figure 11

External DC-bias voltage of graphene

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It should be noted that when we want to physically implement the regulation of the chemical potential of graphene in such structures, the graphene components with a gap between them must be connected either in series or each must be connected to a separate external DC-bias voltage (With the same voltage VG).

As mentioned, more absorption peaks, high absorption rate, insensitivity to polarization and light incident angle are some of the features that distinguish the metamaterial absorber. For this purpose, in Table 1, we have compared the metamaterial absorber presented in this paper with some of the metamaterial absorbers presented in recent years in terms of having or not having these properties.

Table 1 Comparison between the features of the absorber structure of different articles with the features of the proposed absorber structure in this paper
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As it is known, the proposed absorber has more absorption peaks than some of the compared absorbers (Xing and Jian 2018; Meng et al. 2019; Wang et al. 2020; Li and Cheng 2020; Aksimsek 2020; Rahmanshahi et al. 2021) and has a higher absorption rate than some of them (Appasani et al. 2019; Wang et al. 2020, 2021; Lou et al. 2020; Jiang et al. 2021). In addition, the proposed absorber has the advantage of insensitivity to polarization compared to some of the absorbers (Wang 2017; Xing and Jian 2018; Appasani et al. 2019; Rahmanshahi et al. 2021) presented in Table 1 due to its symmetrical design, and it is also insensitive to the light incident angle at two absorption peaks. As a result, the proposed absorber is more flexible and suitable for many applications such as imaging, detecting, sensing, filtering, etc. due to multi-band absorption, complete absorption, insensitivity to polarization and insensitivity to the light incident angle at its two absorption peaks.

6 Conclusion

In recent years, many terahertz absorbers have been introduced. The absorber presented in this paper is a multi-band perfect metamaterial absorber that has one graphene disk at the center and four graphene solid triangle with semicircular cuts on them that surround the central disk. The proposed absorber is a resonant absorber that has four perfect absorption peaks, which are 99.5%, 99.7%, 94.71% and 97.06% at frequencies of 4.24 THz, 5.89 THz, 9.66 THz, and 10.62 THz, respectively. This structure has the ability to adjust and increase the absorption peaks only by changing the Fermi level or the graphene relaxation time, without the need for structural change. In addition, this absorber is not sensitive to polarization due to its symmetrical design and its two absorption peaks remain almost unchanged in various light incident angles (from 0 to 45 degrees). All these features make the absorber very suitable for applications such as imaging, detection, sensing and filtering.