1 Introduction

Metamaterials (MMs) are engineered materials and their properties are often not found in nature, for example: negative refractive index, cloaking behavior, negative permittivity, invisibility, reverse doppler effects and perfect absorption. Today MMs have expanded into high frequency domains such as microwave (Smith et al. 2000), mm-wave (Gokkavas et al. 2006), THz (Yen et al. 2004), infrared and optics (Pereira and Shulika 2011; Choi 2015). These materials derive their properties from their structure and not just from their composition. For the first time, Veselago offered a theory about material with a negative refractive index (simultaneously \(\varepsilon\) and \(\mu\) are negative) (Vesalago 1968). All materials affect electromagnetic waves through two properties of permittivity coefficient and permeability coefficient. Metamaterials are also introduced by complex electric permittivity \(\left( {\varepsilon \left( \omega \right) = \varepsilon_{1} + j\varepsilon_{2} } \right)\) and complex magnetic permeability \(\left( {\mu \left( \omega \right) = \mu_{1} + j\mu_{2} } \right)\). Most researchers focus on the real part \(\varepsilon\) and \(\mu\) to achieve a negative refractive index and try to minimize the imaginary part \(\varepsilon\) and \(\mu\) to reduce losses, but these imaginary parts have a great effect on some strange and unique properties of MMs. one of which is the property of absorbing electromagnetic waves. Therefore, MM can be used in terahertz absorber structures. Absorbers are structures through which electromagnetic energy is not reflected and not transmitted. The use of MM with metal creates a very strong perfect absorption structure. So, metamaterial perfect absorber (MPA) is an attractive branch in the terahertz frequency.

The classic and traditional form of MPA structures is that three layers of metal-dielectric-metal are formed so that the middle layer, which is made of a dielectric, is sandwiched between two strong conductive layers such as gold.

The advantage of having a high conductive layer on the top of the structure is that by matching the impedance to air, reflection can be eliminated. If a percentage of the EM wave passes through the top layer, it is wasted in the middle layer due to the middle layer’s loss. The third or bottom layer, called the ground Prevents any waves from passing through the structure. Therefore, perfect absorption can be achieved. Many kinds of MPAs such as single-band (Faruk and Sabah 2019; Ebrahimi 2020; Luo and Cheng 2020), dual-band (Xiao et al. 2018; Li and Cheng 2020), multi-band (wang et al. 2014; Hu et al. 2016; Wang and Wang 2016; Meng et al. 2018) and broad-band (Zhu et al. 2014; Jindal and kumal 2015; Arik et al. 2017; Tang et al. 2017; Barzegar-Parizi 2018; Dong and Chen 2018; Soheilifar 2018; Patel et al. 2020a, b) and also MPA based on graphene metasurface in terahertz region (He et al. 2015; Xiao et al. 2017; Ye et al. 2017; Huang et al. 2018; Zhang et al. 2018; Barzegar-Parizi 2019; Daraei et al. 2020; Mishra and Panwar 2020) and microwave region (Ghods and Rezaei 2018) has been introduced.

In this paper, a new classical MPA model with three layers (metal-dielectric-metal) but with a change in the height of the middle layer is introduced as a unit cell. In the discussion of filters, we use the phrase of MIM (metal-insulation-metal) (Khani et al. 2019a, b; Khani et al. 2019a, b). Numerical results show that a single-band with an absorption of 99.99% and also a dual-band with an average absorption of 97.18% is obtained. So, we have a single-band perfect absorption and a dual-band nearly perfect absorption with only one structure. Another advantage of this structure is that it is adjustable, ie by changing and optimizing the effective parameters, a nearly narrow peak or a nearly wide peak is obtained. The great advantage of this absorber is that all of these results are achieved without re-manufacturing the structure but also by changing the effective parameters. Based on the finite-difference time domain (FDTD) method, the proposed absorber in this paper was calculated and optimized numerically. In all simulation periodic boundary conditions were set in the x-and y-axis and perfectly matched layer boundary condition was used along z-axis. Here it is assumed that the source was the plane wave, which was normally incident onto the structure.

2 Structure design and simulation

Figure 1 shows a schematic of the absorber unit cell suggested in this paper. As can be seen in the Fig. 1, it is obvious that the structure consists of 3 layers. A perfect conductor (gold) with a thickness of 0.2 µm and with a conductivity of \(\sigma = 2.2 \times 10^{5} {\text{S}}/{\text{m}}\) is used for the top and bottom of the structure. The middle layer SiO2 consists of a dielectric with a constant of \(\varepsilon = \left( {2.79 + j0.31} \right)\) and thicknesses of m and h1. The presence of a perfect conductor (gold) on the bottom of the structure is because no waves pass through the structure because the thickness of this bottom layer (0.2 µm) is bigger than typical skin depth in the THz range. The absorptivity of the presented metamaterial absorber can be defined as A = 1 − T − R where T is transmissivity and R is reflectivity (Wang et al. 2015). Due to the reason stated above, T = 0, then the absorption can be obtained by A = 1 − R. Hence the bottom layer is also called the mirror. So A (absorption) can get very close to 1 when the reflection is close to zero (R = 0, impedance (z = 1) matched to the free space).

Fig. 1
figure 1

The schematics of a single unit cell. a Perspective view. b Side view. (w = 25 µm, m = 3.5 µm, h1 = 19.5 µm, px = py = 9 µm)

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Our results are obtained through the period structures are illuminated by a normally incident plane wave and perfectly matched layers (PML) are applied along the z direction and period boundary conditions in the x and y directions.

3 Results and discussion

As shown in Fig. 2, the effects of changes in the height and width of the middle layer column (w, h1) on the absorption intensity are very noticeable. The basis of our discussion in this article is a normal incidence wave. In Fig. 2a, when the height of the middle layer columns is h1 = 22.5 µm, the structure has two absorption peaks of 88.81% and 88.45% at 3.88 THz and 6.54 THz, respectively. In this case, with the reduction of h1 from 22.5 to 19.5 µm, the intensity of peak absorption will reach 88.03% and 99.99% at frequencies of 4.08 THz and 6.86 THz, respectively. So, a dual-band absorber with an average absorption of 94.01% is obtained.

Fig. 2
figure 2

Absorption spectra of the presented MPA; a with different column height (h1), b with different column width (w)

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Moreover, according to Fig. 2b, as the width of the columns increases from 25 to 30 µm, the low frequency absorption intensity increases and the high frequency absorption intensity decreases. Thus that, when w = 25 µm, one of the absorption peaks is 99.99% at the high frequency of 6.86 THz and the absorption intensity of the second peak is 88.03% at the low frequency of 4.08 THz and when w = 30 µm higher absorption intensity occur at a lower frequency and lower absorption intensity occur at a higher frequency, ie the structure has two absorption peaks of 99.79% and 88.99% at 4.93 THz and 6.85 THz frequencies, respectively. But the best possible case is when w = 28 µm. Because in this case, a dual-band nearly perfect absorber with an average absorption of 97.18% at frequencies of 4.24 THz and 6.86 THz is obtained.

We will now see why the effects of changing the above parameters affect the intensity of absorption. As seen and discussed in Fig. 2, it is obvious that changes in the height of h1 have a direct effect on the absorption intensity of the second peak (higher frequency). By increasing the h1 thickness from 19.5 to 22.5 µm, the first peak (lower frequency) remains almost constant, but the second peak reaches the best state, ie 99.99% absorption. The reason for this phenomenon is that when the thickness of the dielectric is small, the interaction between two metal layers is very strong (top and bottom layers). Therefore, as this thickness increases, the interaction between these two layers becomes weak. So the thickness of h1 has a significiant influence on absorption intensity of the second peak.

The effect of m height changes on the first peak is also investigated (when u = 2 µm and h = 19.5 µm). As can be seen in Fig. 3, when m = 5.5 µm the first peak has an absorption of 82.29% at a frequency of 4.07 THz and the second peak has an absorption of 99.99% at a frequency of 6.85 THz. Then, with a decrease of (m) from 5.5 to 3.5 µm, the first peak remains at the same frequency has an absorption of 88.03% (at the frequency of 4.07 THz) and the second peak with the same absorption (99.99%) at the same frequency (6.85 THz).

Fig. 3
figure 3

Absorption spectra of the presented MPA with various in (m1) heights

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4 Equivalent circuit modeling

We use the transmission line mode to analyze the equivalent circuit model of the reported PMA unit, as shown in Fig. 4. These absorption intensity changes can be understood according to the circuit model of the proposed structure. As you can see in Fig. 4, We divided the equivalent circuit into six parts for further investigation from A to F. According to Equation \({\text{A}}\left( \omega \right) = 1 - {\text{T}}\left( \omega \right) - {\text{R}}\left( \omega \right)\), due to the strong conductivity of the bottom layer, \(T\left( \omega \right)\) becomes zero. So part A is the metal Gold plane at the bottom, which acts as a short circuit. Part B is a dielectric layer transmission line model that depends on the height m. Part C consists of two parallel impedances of zm and zg. The impedance zm is shown with the gold_dielectric interference impedance. zg is also considered as the only gold impedance. Part D consists of two parallel parts zd1 and the spacing between two dielectric columns, which depends on the height h1. Part E consists of two RLC circuits connected parallel corresponding to the top (gold) structure. Finally, Part A is a free space transmission line with characteristic impedance Z0.

Fig. 4
figure 4

Equivalent circuit model of proposed PMA

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The absorption of the proposed absorber can be obtained from:

$$A\left( w \right) = 1 - \left[ {\left| {\frac{{z_{in} \left( w \right) - z_{0} }}{{z_{in} \left( w \right) + z_{0} }}} \right|} \right]$$
(1)

Zin is the input impedance of the PMA, expressed as:

$$Z_{in} (\omega ) = \frac{1}{{z_{f} \left( \omega \right)}} + \frac{1}{{z_{d1} \left( \omega \right)||z_{0} }} + \frac{1}{{z_{g} \left( \omega \right)}} + \frac{1}{{z_{m} \left( \omega \right)}} + \frac{1}{{z_{d} \left( \omega \right)}}$$
(2)

Here in:

$$z_{f} \left( \omega \right) = \frac{1}{{R_{1} + j\omega L_{1} + \frac{1}{{j\omega C_{1} }}}} + \frac{1}{{R_{2} + j\omega L_{2} + \frac{1}{{j\omega C_{2} }}}}$$
(3)
$$z_{d1} \left( \omega \right) = j\sqrt {\frac{{\omega_{r} \omega_{0} }}{{\varepsilon_{r} \varepsilon_{0} }}} \cdot \tan \left( {kd_{1} } \right)$$
(4)
$$z_{g} = \frac{1}{{\sigma_{g} }}$$
(5)
$$z_{m} = \frac{{z_{{d_{1} }} \left( \omega \right)}}{{\frac{{2\sqrt {n_{1} n_{2} } }}{T} - \left( {n_{1} + n_{2} } \right)}}$$
(6)
$$z_{d} \left( \omega \right) = j\sqrt {\frac{{\omega_{r} \omega_{0} }}{{\varepsilon_{r} \varepsilon_{0} }}} \cdot \tan \left( {kd} \right)$$
(7)
$$k = \frac{{k_{0} }}{{\sqrt {\varepsilon_{r} k_{r} } }}$$
(8)

where k, ωr, εr, are wavenumber, permeability, and relative permittivity of the SiO2 dielectric layer, and k0, ωr, εr are wavenumber, permeability, and relative permittivity of the free space, respectively. Also, n1 and n2 are the refractive indexes of dielectric and gold material and Z0 is air impedance (~ 377 Ω). d and d1 also correspond to the metamaterial heights shown in Fig. 1. When \({\text{Z}}_{in} \left( \omega \right) = Z_{0}\), PMA can achieve perfect absorption (Patel et al. 2020a, b). Then by changing the geometric parameters of structure can vary the input impedance of PMA.

In order to better justify the absorption mechanism of the proposed structure, the distribution of the electric field in the two absorption frequencies is shown in Fig. 5. As shown in Fig. 5 (a), at the first peak (4.39 THz) most transient wave losses occur in the center strip of the middle layer columns. In fact, above the height of m, the two columns together with the air distance between the two columns are a combination of 2 dielectrics, which when we increase the width of these columns w due to the stronger and higher constant dielectric of this combination and smaller distance of air, the absorption rate of the first peak is much higher. In Fig. 5 (b) most of the electric field is concentrated in the center and sides of the two columns of the dielectric layer, and most of the losses occur in the air space between the two columns, which was observed that the absorption intensity of the second peak decreases significantly with increasing width w.

Fig. 5
figure 5

Simulated unit cell perspective view distributions of the electric field at two frequencies. a 4.39 THz (first peak); b 6.86 THz (second peak)

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Table 1 tries to compare the work of this article with several other articles in terms of average absorption and it is observed that in 2 frequencies of 4.24 THz and 6.86 THz, the average absorption reached 97.18% which is better than other works in the table.

Table 1 Comparison of the proposed absorber with the reported state-of-the-art
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As can be seen from Table 1, the proposed article has been compared with several other articles. The main advantage of the proposed paper over other papers is in the results. We obtained higher absorption peaks at different frequencies. The proposed article has an average absorption of 97.13%, which has a higher absorption percentage than other articles.

5 Conclusion

As a conclude, a novel nearly perfect and adjustable dual-band absorber is introduced. This structure is the same as the classical and traditional structure of adsorbents, ie metal-dielectric-metal, with the difference that the dielectric layer of the adsorbent proposed in this article has a change in height or in other words, multiple layers. A continuous dielectric layer with height m and then two dielectric columns of the same material is placed on the top of height m. The height m on the first peak or lower frequency and also the height of these columns h1 on the second peak or higher frequency is very effective. Then by optimizing this value, we will reach a dual-band nearly perfect absorber with absorption of 97.18% at 4.24 THz and 6.86 THz. Finally, the electric field distribution of this structure at two peak frequencies was investigated. This nearly perfect and tunable terahertz absorber has potential in filtering, detection and imaging.