1 Introduction

Terahertz (THz) multiband absorbers have garnered significant consideration in latest years caused by their pivotal role in various industries such as medical, chemical and electrical engineering branch (Norouzi and Rezaei 2022a; Asl and Khajenoori 2021), ranging from telecommunications (Ye et al. 2022) and medical imaging (D’Arco et al. 2020) to security screening (Tribe et al. 2004) and spectroscopy (Yang et al. 2021). The terahertz region of the electromagnetics frequency range spanning approximately 100 GHz–10 THz (Kim et al. 2016), offers unique opportunities for material interaction and characterization. However, harnessing these capabilities requires efficient control over THz radiation, which is where multiband absorbers come into play (Khani and Hayati 2021).

At first a Russian physicist, Victor Veselago, presented metamaterials in 1968. These materials are artificial characterized by engineered structures and exhibit electromagnetic specifications that are not exist in natural material for example negative permittivity’s imaginary part, negative refractive index and so on (Varshney and Giri 2021; Khani and Hayati 2022; Kiani et al. 2021). Metamaterials can be divided into several categories, graphene is one of these engineered structures which is two-dimensional with unique properties (Abohmra et al. 2022), a metamaterial that is consists of breaking graphite bonds which are demonstrated in Fig. 1. The unique properties of graphene such as small size and it is adjustable chemical potential that affects the graphene conductivity (Wang et al. 2016), cause use them in manufacture antenna (Failed 2023) to reduce the structure size and increase the antennas gain with controllable polarization (Chashmi et al. 2020), the design of resonators, electromagnetic shields, cloaking (Zhang et al. 2023; Zhimin et al. 2023), sensors (Norouzi Razani et al. 2022; Alipour et al. 2023; Kiani et al. 2020) and absorbers to inhibit the reflection or transmission.

Fig. 1
figure 1

Graphite exfoliation in order to create graphene

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Terahertz metamaterial absorbers, by controlling the graphene chemical potential with the bias voltage (external) specifically designed to achieve high absorption efficiency (near to 100%) and absorption bandwidth over specific frequency ranges, have attracted attention for their ability to surpass the limitations of conventional materials (Ghods and Rezaei 2018; Zhang et al. 2022). Conventional absorbers designed for single-frequency operation fall short when confronted with the complexity and diversity of modern THz applications. This ability to manipulate and absorb distinct frequencies opens up avenues for enhanced signal processing, improved imaging resolution and enhanced security protocols (Zamzam and Rezaei 2021; Khodadadi et al. 2022; Zamzam et al. 2023a; Kim and Hopwood 2019).

In absorbers, high absorption and polarization insensitivity are two important parameters (Zamzam et al. 2021). These features can cause absorber flexibility to be used in different applications such as Modulators (Emami-Nejad and Mir 2017), solar cells (Failed 2022; Sahin and Kabacelik 2018; Tang et al. 2016) and sensors for example Li et al(2021) designed early-stage cervical cancer identification with dual band terahertz metamaterial biosensor (Li et al. 2021).

Since graphene offers several advantages compared to aluminum when used in terahertz (THz) absorbers such as tunable absorption (Norouzi and Rezaei 2022b), ultra-thin material, compatibility with nanofabrication and environmental stability (Yong Zhang and He 2021) so, in this article, a multiband perfect metamaterial absorber (PMA) (Feng et al. 2020) is suggested by using Dongxia et al.’s article design and substitute graphene as a structure material with aluminum. At first simulations outcomes demonstrate the improvement of the suggested model to achieve the absorptivity rate of more than 90% in four picks of 0.56, 0.79, 1.02 and 1.25 THz. The multiband presented absorber is made of only one graphene layer, in comparison with multi-layer metamaterial, this feature of our suggested structure makes it easier to manufacture and inexpensive. This absorber consists of a square-shaped graphene surface layer, a silicon as an insulated spacer, along with the golden ground layer. From the simulation results, it can be obtained that, the suggested structure is very proper for sensing, imaging and other applications (Khodadadi et al. 2023a; Al-Ibadi 2022; Zamzam et al. 2023b; Nickpay et al. 2022).

2 Theoretical discussions and structure design

2.1 Theoretical discussions

The absorption ability of a metamaterial absorber (A) is calculated by the general absorption equation which is mentioned below (Zhong et al. 2020):

$$ {\text{A}} = 1 - {\text{T}} - {\text{R}} $$
(1)

In this equation T is transmissivity and R is reflectivity. As expressed before, graphene is a two-dimensional controllable metamaterial, the graphenes’ surface conductivity controlled with external gating which has tolerance from − 1 to 1 eV and it can also be obtained from the Kubo formula which is indicated as follows (Khodadadi et al. 2023b; Hanson 2008):

$$ \sigma_{intra} ({{\varvec{\upomega}}},E_{F} ,{{\varvec{\uptau}}},T) = \sigma_{intra} ({{\varvec{\upomega}}},E_{F} ,{\uptau },{\text{ T}}) \, + \sigma_{inter} ({{\varvec{\upomega}}},E_{F} ,{{\varvec{\uptau}}},{\text{ T}})\sigma_{intra} $$
(2)
$$ \sigma_{{{\text{int}} ra}} ({{\varvec{\upomega}}},E_{F} ,{{\varvec{\uptau}}},T) = \frac{{2k_{B} e^{2} T}}{{\pi h^{2} }}\ln \left[ {2\cosh \, \left( {\frac{{E_{f} }}{{2k_{B} T}}} \right)} \right]\frac{i}{{\omega + i\tau^{ - 1} }} $$
(3)
$$ \sigma_{inter} ({{\varvec{\upomega}}},E_{F} ,{{\varvec{\uptau}}},{\text{ T}}) = \frac{{e^{2} }}{4\hbar }\left[ {{\text{H}}\left( {\frac{\omega }{2}} \right) + \frac{4iw}{\pi }\int\limits_{0}^{\infty } {\frac{{H\left( \varepsilon \right) - H\left( {\frac{\omega }{2}} \right)}}{{\left( {w^{2} - 4e^{2} } \right)}}} } \right]d\varepsilon $$
(4)
$$ {\text{H(w)}} = {\text{sinh }}[\hbar \varepsilon {/}(K_{b} T)]/\left\{ {{\text{cosh}}\left[ {\frac{{E_{f} }}{{K_{b} T}}} \right] + {\text{cosh}}[\hbar \varepsilon {/}(K_{b} T)]} \right\} $$
(5)

where ω, σ, τ, \({\mu }_{c}\), \({E}_{F}\) are the angular frequency, the graphene conductivity, the relaxation time of graphene, the graphene chemical potential and the Fermi energy value, respectively. In addition, e is the charge of electron with 1.6 × \({10}^{-19}\) c value, T = 300 K, h = the constant of reduced Plank’s and \({K}_{b} \) = the constant of Boltzmann. It is noticeable that according to the Pauli Exclusion Principle, in terahertz frequency band, we will have \({E}_{F}\gg {K}_{b}T\) so the inter-band part of the above formula becomes negligible. Consequently, the graphene conductivity equation will be simplified as the Drude model follow (Huang et al. 2018a):

$$ \sigma \left( {{{\varvec{\upomega}}},E_{F} ,{{\varvec{\uptau}}},{\text{T}}} \right) = \frac{{e^{2} E_{f} }}{{\pi h^{2} }} \frac{i}{{\omega + i\tau^{ - 1} }} $$
(6)

As mentioned before the graphene’s relaxation time can be obtained by Eq. (4)

$$ {\uptau } = \frac{{\mu \mu_{c} }}{{eV_{F}^{2} }} $$
(7)

where VF and μ are the Fermi velocity with an approximately constant of \({10}^{6} \)m/s and the electron mobility.

2.2 Absorber structure design

The unit cell’s geometrical structure is presented in Fig. 2, like most absorber structures, the presented absorber in this paper contains 3 layers. From upper to lowest layer, the absorber consists of a 0.01-μm thick graphene array on a middle layer of 190-μm thick high-resistance silicon as a dielectric or spacer layer with a constant of ε = 11.9. To avoid transmission to the other side of the metamaterial both of these layers are located on the gold as the third layer with a thickness of 0.5 μm and conductivity of \({\sigma }_{gold}\) = 4.56 × \({10}^{7}\). The most effective dielectric with high capacitance is an ion-gel, it can be compatible with tunable graphene plasmonic devices on various substrates by its good mechanical flexibility and thermal stability, the structure diagram of the ion-gel top gate which can be used to control the Fermi energy of graphene is shown in Fig. 2d. After patterning an array of graphene we cover it with the ion-gel layer, which induces carrier concentration and allows the absorber to enter the terahertz band, we can give a conclusion ion-gel has little effect on the simulation results and we omit ion-gel to simplify the calculation.

Fig. 2
figure 2

Schematic of the suggested structure. a Side view of unit cell, b top view of unit cell, c absorber arrangement under terahertz beam radiation, d a layer of ion-gel which added to biasing the graphene

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One part of the incident light interacts with the graphene. The remained part passes through the graphene to reach the gold surface, which is reflected back by the gold and then lost in the silicon, to get a strong absorption mechanism, the imaginary parts of the relative permeability and/or permittivity can provide high magnetic and/or electrical losses for traveling waves absorption, so, by using a gold layer for the base layer of the structure, the transmission coefficient of the EM wave was decreased [T(ω) → 0].

The noticeable point is that to decrease the coefficient of the transmission (T) to zero, the base conductor layer thickness would be more than the THz range typical skin depth, so the absorption equation simplified to Eq. (8):

$$ {\text{A}} = 1 - {\text{R}} $$
(8)

In our simulation, \({\mu }_{c}\) and τ were considered as 0.9 eV and 0.1 ps respectively, to get the optimum value We checked the various values of parameters and selected the best of them in the below table.

3 Simulation result and comparison

3.1 Simulation result

At first glance a quad-band absorber was designed as demonstrated in Fig. 3a with the same geometrical as mentioned in Table 1 and a thickness of 190 μm. Then by increasing the thick value to 600 μm absorption peak growth to XIII- band, so in this way the absorption peaks number is controllable with dielectric thickness changing and the claim to reach XIII-band absorber become true. Figure 3b illustrates the absorbance spectrum of the suggested absorber and it is zoom figures. All the simulation was implemented in CST software based on the full-wave finite-integration-technique (FIT) at normal room temperature (300 K) and the accuracy of mesh in the graphene layer are set to 93.33 and 0.1 nm tetrahedrons mesh type, in addition the boundary condition considered as \({z}_{max}\) open (add space), \({z}_{min}\) electric (Et = 0).

Fig. 3
figure 3

Absorbance spectrum of the planned absorber for various Si layer thickness a 190 μm, b 600 μm

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Table 1 The obtained values of the planned unit-cell (per μm)
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Figure 4 simulation results, shows various value of (w), the graphene chemical potential and relaxation time to get the optimum result. As illustrated, only by substituting aluminum with graphene and a little change in the in-line width of the structure, both of absorption peak and average absorptivity increase. Four absorption peaks and more than 99.6% for average absorptivity. By using graphene, the claim that the proposed structure is tunable becomes true because changing the graphene fermi level (μc) can shift the absorption frequency without remanufacturing and even parameters changing. The maximum absorption peaks resulted by \(\upmu _{{\text{c}}} =\)0.9 eV in Figs. 4a and 8 for graphene line width in Fig. 4b also τ = 0.1 ps is considered as the optimum value of graphene relaxation time in Fig. 4c.

Fig. 4
figure 4

The planned absorber absorption spectrum for various a graphene fermi-level, b graphene line width, c graphene relaxation time

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Most multiband absorbers are challenging to fabricate because they make use of over one layer of metamaterials in their form or have a complicated construct although they’re single layers. However, the presented structure uses only single layer of metamaterial in its construction and has an elementary plan. Therefore, they are completely lot simpler to fabricate than lookalike multiband structures.

To describe the proposed absorber absorption mechanism, we mention the gold bottom layer considered as a ground layer which does not permit the radiated wave to pass through the arrangement. Furthermore, a silicon mid layer and a designed graphene layer located on top, neutralize the reflected waves by their phase difference, as a result perfect absorber seems reasonable.

Figure 5 displays the electrical field distribution in a normal incidence for TE modes. Similarly, Fig. 6 displays the electrical field distributions of TM modes briefly.

Fig. 5
figure 5

Electrical field distribution of TE waves at a 0.49 THz, b 0.76 THz, c 1.05 THz, d 1.34 THz

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Fig. 6
figure 6

Electrical field distributions of TM waves at a 0.49 THz, b 0.76 THz, c 1.05 THz, d 1.34 THz

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Here to compare the plot in both mode of excitation we show the figure of absorption where TE and TM mode of excitation plotted in a figure, so as shown in Fig. 7 TE and TM mode of excitation shows negligible differences on absorption peak.

Fig. 7
figure 7

TE and TM mode of excitation

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3.1.1 Stability analysis

In this section, the angular and polarization stability of suggested absorber is considered by exposure to terahertz radiation along the ‘z’ axis. For evaluating the stability of polarization, TM and TE propagation modes are applied by increasing ϕ angle to 90° when the incidence angle (θ) is 0°. Even though, the suggested terahertz absorber is not in rotational form however Fig. 8a shows that the TM and TE modes are similar so, it’s obvious that the absorption is stabile for both TM- and TE-modes therefore the claim that the presented absorbers are polarization insensitive become true. Moreover, Fig. 8b illustrates the study of how electromagnetic waves with a different angle of incidence affect their ability to be absorbed.

Fig. 8
figure 8

The absorption spectrum for a insensitivity of polarization, b changes of incident angle

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We evaluate the level of angular stability by looking at the range of angles, called θ, where the built absorber appears to have similar properties. So, the suggested terahertz absorber works well at any angle up to 45°. The terahertz absorber we showed works the same for different polarizations, so the way it behaves when the waves come at an angle is similar for both types of waves. Therefore, the planned absorber is stable when it comes to changes in angles and polarization.

Table 2 tries to compare this paper with reported results of other research from the standpoint of the peak numbers, average absorption, incident angle and polarization insensitive. It is obvious that in addition to polarization insensitivity the average absorption of absorber reached 99.60% in XIII bands which in comparison with other results in the table, it is slightly better.

Table 2 The comparison of suggested absorber with similar reported work
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In Table 2 the multiband absorber presented in this paper is compared with some other similar structures that were reported recently. Our final structure has the benefits of simple design, polarization insensitivity and high incident angle tolerance with the desired absorption peak with average absorption of 99.60%. All the characteristics make the final structure very flexible and a good candidate for sensing, imaging, spectroscopy and some other absorbing application.

4 Conclusion

In this paper, at first, a dual-band THz absorber based on aluminum was designed for sensor application to diagnose early-stage cervical cancer. Because of graphene’s several advantages compared to aluminum by substituting the surface material of the sensor with graphene and adding a gold ground layer our proposed model formed. In comprehension between the new suggested absorber and the previous model, two absorption peaks were added and overall absorptivity was increased. We investigated an multi band tunable metamaterial structure that is polarization insensitive and tolerates incident angles up to 45°, in addition to all these features the flexibility and high absorbance characteristics, make it a candidate for our proposed model for filtering, imaging, sensing and other applications.