A tunable broadband graphene-based metamaterial absorber in the far-infrared region

Abstract

This paper reports a new design of a broadband absorber composed of graphene, dielectric, and gold layers. The designed absorber has four absorbent modes close to each other, which results in the formation of broadband absorption. The relative bandwidth, a key parameter to assess the bandwidth improvement, shows a significant increase in the proposed design compared to similar structures published in recent years. The numerical results also reveal this metamaterial absorber can be used for applications in the far-infrared frequency range due to choosing optimized dimensions and the graphene Fermi level. Unlike other graphene-based metamaterials, which require complicated structures to be able to attain broadband absorption, the physical structure of the proposed design has a relatively simple fabrication process. For further investigations, the effect of split geometry on the absorption spectrum is studied. Also, the use of graphene in this metamaterial absorber provides dynamic adjustability through electrostatic doping in order to tune the amount of absorption. This characteristic has been studied by changing the graphene Fermi level. This feature can be widely used in electro-absorption switches and modulators.

Introduction

Due to widespread applications of electromagnetic waves in physics and engineering, the interaction of such waves with matter is of great importance. The unique capabilities of metamaterials in increasing interactions between waves and materials have caused these structures to gain widespread attention in the literature. Incident waves enhance the electric or magnetic fields and thus increase the absorption by stimulating the resonant modes of the structure. Due to the nature of these resonators, the absorption is usually sharp with narrow spectral bandwidth. These resonances with high-quality factors have applications in the design and fabrication of filters (Butt et al. 2010; Hosseinzadeh Sani et al. 2020a; Mccrindle et al. 2014; Naghizade and Saghaei 2020a; Tavakoli et al. 2019), modulators (Shrekenhamer et al. 2013; Watts et al. 2014), sensors (Hosseinzadeh Sani et al. 2020b; Wang et al. 2015; Yahiaoui et al. 2015), logic gates (Mehdizadeh et al. 2017c; Moniem 2016; Naghizade and Saghaei 2020b; Sani et al. 2020), encoders (Haddadan et al. 2020; Mehdizadeh et al. 2017b; Naghizade and Khoshsima 2018), decoders (Alipour-Banaei et al. 2015; Parandin et al. 2018; Salimzadeh and Alipour-Banaei 2018), optical fibers (Aliee et al. 2020; Diouf et al. 2017; Ghanbari et al. 2018; Ghanbari et al. 2017; Saghaei et al. 2016a; Saghaei et al. 2016b; Saghaei et al. 2015), demultiplexers (Mehdizadeh and Soroosh 2016; Saghaei et al. 2011; Saghaei and Seyfe 2008; Talebzadeh et al. 2017; Wen et al. 2012), PhC fibers (Ebnali-Heidari et al. 2014; Raei et al. 2018; Saghaei 2018; Saghaei and Van 2019), switches (Alipour-Banaei et al. 2015; Chen et al. 2006; Mehdizadeh et al. 2017a), interferometers (Gu et al. 2007; Saghaei et al. 2019). However, broadband absorbers would be more suitable for other applications such as electromagnetic energy harvesting and efficient signal processing. Therefore, one of the interesting practical cases corresponds to structures known as perfect metamaterial absorbers (PMAs), which are designed based on close resonance modes in the resonator. PMAs typically consist of three layers (Peng et al. 2012), namely: a metal array acting as the resonator in the upper layer, the middle dielectric layer that can help the absorption process, and a metal substrate that acts as a reflector at the end of the structure. Various designs of these absorbers have been proposed in different wavelength regions based on vertical stacking (He et al. 2015; Pan et al. 2016; Zhang et al. 2015), super-lattice co-planar-based structures (Cui et al. 2011; Ming and Tan 2017; Naghizade and Saghaei 2020a), as well as multi branches metamaterials (Wang et al. 2019a, b).

Since the majority of applied metamaterials work in the microwave frequency region, there is an increasing need to develop absorbers in higher frequencies such as THz, infrared, and visible regions. The natural materials produce a weak response to far-infrared and THz radiations (Williams 2006); the development of metamaterials in this frequency range seems to be an important issue. On the other hand, most of the existing metamaterials are applicable in specific frequency ranges, and there exists no adjustability both in the central frequency and the absorption intensity for these structures after their fabrication process. Therefore, to overcome these issues, graphene has received particular attention as an effective material for use in plasmonic structures due to its unique electromagnetic characteristics (Mostaan and Rasooli Saghai 2018; Tabrizi et al. 2021). The fabrication processes of such graphene-based absorbers have been presented by several research groups (Lee et al. 2017; Shu et al. 2018; Yang et al. 2017). Yang et al. (Yang et al. 2017) proposed an efficient liquid-phase exfoliation technique for graphene nanosheets (GNSs) and they fabricated NiCo2/GNS nanohybrids using the single-mode microwave-assisted hydrothermal method. The capability of using graphene in THz, and far-infrared frequency ranges, as well as its dynamical adjustability, have provided new opportunities for producing new devices. Extensive research has recently been conducted on graphene-based metamaterial absorbers. Most of these metamaterials have two-dimensional structures known as metasurfaces, which are further explained in (Cen et al. 2020; Chen et al. 2017; Tavakoli et al. 2019; Wang et al. 2018; Yi et al. 2019). The most critical problems of such structures are their limited number of frequency modes and their relatively low absorption that they reduce the absorption bandwidth. In order to overcome this issue and increase the absorption bandwidth, metamaterials based on more complex structures have been presented and studied by various research groups (Cai and Xu 2018; Su et al. 2015; Xu et al. 2018; Zhang et al. 2020; Zhou et al. 2018). The two main issues of the proposed designs are their lack of attention to the far-infrared region and the complexity of the designed structures. These problems further enhance the motivation for designing a graphene-based PMA with a simple design for far-infrared absorption applications.

In order to overcome the abovementioned issues, this paper presents a graphene-based broadband absorbent metamaterial. The main feature of the proposed structure is its simplicity that by creating a split in a rectangular graphene sheet, four resonant modes close together can be achieved in the far-infrared frequency range. The proposed structure has a spectral bandwidth of 10 THz for absorptions higher than 50%. Moreover, the relative bandwidth of this structure has been improved significantly compared to the previous designs in the literature. Since many modes are affected by the split in the structure, changing the length and width of this split can result in changes in the absorption spectrum under study. Another critical issue is the use of three-layer graphene in this structure in order to increase the interaction of the incident light with the matter.

Due to the simple structure design compared to the super-lattice and co-planar structures, applying the gate voltage to change the Fermi level will be easier in this design. The absorption can be significantly controlled by changing the Fermi level. This characteristic has the potential of being used in electro-absorption-based modulators. It should be noted that by using the initial idea of this structure and changing its parameters, other frequency ranges can also be covered.

Physical structure and simulation method

Figure 1a shows the 3D view of the proposed metamaterial absorber. This structure has three layers consisting of an array of graphene rectangles with splits (top layer), dielectric glass (middle layer), and rectangular metal in the reflective layer made of gold. The top layer is made of three-layered graphene so that it can increase the interaction of electromagnetic waves with graphene and dielectric glass and thus increase the absorption in a wide bandwidth. The middle dielectric layer has a thickness of toxid = 3 µm and a refractive index n = 2.1 + 0.05i. The gold substrate is used as an electrode layer, and applying a voltage changes the graphene Fermi level. Other metals such as silver, aluminum, platinum were studied as substrate material, but the results were almost the same in all cases. The reason for such an effect is raised from the fact that for low frequencies, metals act as a perfect electric conductor. The permittivity of the gold film is according to the Drude model (i.e., \(\varepsilon_{m} = 1 - \omega_{p}^{2} /(\omega^{2} + i\omega \gamma_{0} )\)) (Etchegoin et al. 2006; Vial and Laroche 2008; Walther et al. 2007).

Fig. 1
figure1

a Three-dimensional view of the proposed broadband metamaterial absorber, and b two-dimensional view of the unit cell (top layer)

Figure 1b demonstrates a unit cell of the metamaterial absorber, and its optimized parameters, which were calculated through the trial and error method. The lattice constants along the x and y axes are supposed to be Px = 2.3 µm and Py = 2.7 µm, respectively. The width and length of the graphene plate are equal to Lx = 1.8 µm and Ly = 2.7 µm, respectively. Also, the width and length of the split are w = 0.25 µm and L = 1.3 µm, respectively. In the simulations, the graphene plate is considered as an ultra-thin film, without any thickness, on the dielectric bed. The Kubo equation consisting of two interband and intraband terms is used to calculate the surface conductivity (Chen et al. 2016; Lao et al. 2014):

$$\sigma = \sigma_{{{\text{interband}}}} + \sigma_{{{\text{interaband}}}}$$
(1)

where

$$\sigma_{{{\text{interband}}}} = \frac{{e^{2} }}{4h }\left[ {\frac{1}{2} + \frac{1}{\pi }\arctan \left( {\frac{{h\omega - 2E_{f} }}{{2k_{B} T}}} \right) - \frac{i}{2\pi }{\text{ln}}\left( {\frac{{\left( {h\omega + 2E_{f} } \right)^{2} }}{{\left( {h\omega - 2E_{f} } \right)^{2} + \left( {2k_{B} T} \right)^{2} }}} \right)} \right]$$
(2)
$$\sigma_{{{\text{intarband}}}} = \frac{{2ie^{2} 2k_{B} T}}{{\pi h^{2} \left( {\omega + i\tau^{ - 1} } \right)}}\ln \left( {2\cosh \left( {\frac{{E_{f} }}{{2k_{B} T}}} \right)} \right)$$
(3)

wherein h represents the reduced Planck constant, T denotes the temperature (300 °C), kB is the Boltzmann constant, ω denotes the optical angular frequency, τ is the carrier relaxation time, and finally, Ef denotes the graphene Fermi level. In the abovementioned equation, the carrier relaxation time is calculated by τ = h/2Г where Г is the scattering rate that is assumed to be 0.002 eV. The Kubo equation shows that the graphene surface conductivity and absorption rate depend on the Fermi level. Now using the Amper law in the stationary regime and the Ohm law, one can calculate the permittivity of graphene (Ma et al. 2019) as follows:

$$\varepsilon_{g} = 1 + i\frac{N\sigma }{{t_{g} \varepsilon_{0} \omega }}$$
(4)

where N is the number of graphene layers, tg represents the thickness of the graphene plate, and ɛ0 denotes the permittivity of vacuum. Equations (1)–(4) show that graphene’s permittivity depends on the number of layers and the Fermi level.

The structure proposed in this article was examined by solving Maxwell equations using the finite-difference time-domain algorithm (Sullivan 2013). The incident light is assumed to be a plane wave. It is emitted from the top of the periodic structure along the z-axis, where its electric field is parallel to the y-axis unless another form is expressed (0°polarization). The boundary conditions are assumed to be periodic along the x and y axes, and perfectly matched layer (PML) along the z-axis. Mesh cell size is 10 nm along with the x and y directions and 100 nm along the z-direction.

Simulation results and analysis

Figure 2a shows the absorption in terms of wavelength for two structures of metamaterial absorber with a complete rectangular graphene sheet (MA-1) and a rectangle graphene sheet with a split shown in Fig. 1a (MA-2). We assumed the graphene layer and Fermi level to be 3, and 0.6 eV, respectively. As can be observed in the figure, MA-1 has three resonance modes at frequencies p1 = 6.47 THz, p2 = 14.5 THz, and p3 = 17.57 THz.

Fig. 2
figure2

a Absorption spectra for perfect graphene (MA-1) (the blue curve) and graphene with split (MA-2) (the red curve). b Absorption spectra of MA-2 for 0°, and 90° polarization angles

The formation of these three resonance modes is due to the different effect of enclosing the electromagnetic field inside the graphene structure. The red curve in Fig. 2a illustrates that a split inside the rectangular graphene sheet (MA-2) creates an absorption band with four close resonance modes at frequencies f1 = 8.82 THz, f2 = 11 THz, f3 = 14.44 THz, and f4 = 16 THz. In fact, the split in the graphene sheet is the main factor in the formation of an almost perfect absorption band. The absorption in resonance modes of f1 = 8.82 THz, f2 = 11 THz, f3 = 14.44 THz, and f4 = 16 THz are equal to 95%, 96%, 98%, and 93%, respectively. This structure has a continuous spectrum for absorptions higher than 80%, with a bandwidth of 8.48 THz. Another key parameter about wideband electromagnetic absorbers is the relative bandwidth, which provides a benchmark for comparison of our structure with other structures. For the structure under study with the central frequency of 12.43 THz, the relative bandwidth is 68.22%. This value shows a significant improvement compared to the similar graphene absorbers as well as the metal counterparts. Due to the asymmetric nature of the structure. The absorption spectrum is sensitive to the polarization of the incident wave. Figure 2b demonstrates that only two resonance modes are formed as we rotate the polarization by 90°which is very similar to the perfect absorber (MA-1). In fact, with 90°rotation in polarization, the split of the structure is ignored and the incident wave sees the graphene as a perfect sheet.

In order to quickly understand the concept of resonance frequencies, we compare them between MA-1 and MA-2. In this comparison, the length and width of the graphene sheet, the number of graphene layers, Fermi level are assumed to be the same for both absorbers. Thus, we first examine the excited modes in perfect graphene. It should be noted that the electric field is monitored at a distance 50 nm above the graphene sheet. Distributions of |E| and EZ fields for the three main modes are represented in Fig. 3.

Fig. 3
figure3

The distribution of ac absolute electric field |E|, and df real electric field along the z-axis (Ez) monitored at a distance 50 nm above the perfect graphene plane of MA-1 corresponding to the resonance modes: p1( a and d), p2 (b and e), and p3 (c and f)

It is seen from Fig. 3a, that the electric field distribution of mode p1 is mainly accumulated at the upper and lower edge of the resonator. Also, Fig. 3d demonstrate that EZ is distributed in the same resonator areas. It is clearly seen that the opposite modes are distributed on two sides of the rectangle, which indicates the excitation of dipolar mode in the designed structure. As can be observed in Fig. 3b, the intensity of |E| for the second mode has its highest values, mainly about the left and right edges. Moreover, a small concentration of electrical field exists in the middle of the upper and lower edges. According to Fig. 3e, the opposite nodes are distributed at the upper and lower edges along with the polarization of the electric field. The distribution of EZ, almost similar to the first mode, has its dominant nodes at the top and bottom edges. Thus, the mechanism of the second mode is due to the resonator dipolar response to incident waves occurring at this frequency. However, the distribution of the real part of EZ in this mode is different from the first mode. It is obvious from Fig. 3e that in addition to the formation of opposite nodes in the corners of the upper and lower edges, a distribution of opposite nodes is also observed in the middle areas of the upper and lower edges.

The formation of opposite nodes in the middle of the upper and lower edges reduces the effective length of resonance in the structure compared to the first mode, which in turn would increase the frequency of the second mode. In the third mode, a large field resonance is observed about the left and right edges, which are shown in Fig. 3c. The distribution of EZ for this mode shows two nodes along the y-axis, which indicate higher-order mode resonance in the simulated resonator.

In order to analyze the proposed MA-2, it is necessary to investigate the field distributions based on the resonance modes. It is seen from Fig. 4a that |E| has enhanced mainly in the two right corners and the two edges of the split on the left side. Normal field distribution in Fig. 4e shows concentrations at the same points. The concentration of opposite nodes at the right and left sides along the y-axis imply a dipole resonance on two sides of the structure. However, the distribution of Ez along the x-axis on the two upper and lower edges are also different.

Fig. 4
figure4

The distribution of ad absolute electric field |E|, and eh real electric field along the z-axis (Ez) monitored 50 nm above the perfect graphene plane of MA-2 corresponding to the resonance modes: f1 (a and e), f2 (b and f), f3 (c and g), and f4 (d and h)

Therefore, the hybrid mode, f1, is the structure’s dipole mode, which naturally reduces the resonance frequency relative to the first mode of the perfect structure. Mode f2 in the MA-2 is inherently quite different from the other modes in the MA-1. The electric field distribution indicates the field accumulation on the split side. As can be observed in Fig. 4f, one can realize from normal field distribution, the two modes of the upper and lower arm dipoles are coupled on the left side of the resonator. According to Fig. 4c, the distribution of the electric field in mode f3 experiences severe resonance in the corners and part of the split. It is clear from Ez in Fig. 4g that the resonance mode is a hybrid mode consisting of the modes p2 and f2. Since f3 is very close to the mode p2 in terms of frequency, the dominant mode in MA-2 is p2, and in fact, this mode is the main source of the mode f3. Finally, the distribution of Ez, shown in Fig. 3h represents a quadrupole mode. This distribution is similar to the normal field in the third mode of MA-1 with four dominant nodes, with the difference that the effect of split results in the coupling of dipole modes created in the upper and lower parts of the resonator. Moreover, the impact of split results in the shift of the f4 mode to lower frequencies compared to the original structure.

Based on these analyses, the role of the split in the graphene sheet is significant, and changing the split geometry leads to changing all four resonance modes. The change in geometry has a more significant impact on the f2 and f4 modes formed by the split. Figure 5a shows that due to the increase in split width w, the second and fourth modes shift to higher frequencies while fewer changes occur in the first and third modes. In fact, increasing w, the length of the upper and lower arms are reduced, which in turn reduces the resonance length of the modes in the upper and lower dipoles, resulting in an increase in the frequency. Moreover, the impact of changing the split length, L, is illustrated in Fig. 5b. Since changing L does not affect the length of upper and lower parts of the graphene, it will have a little effect on the second and fourth modes of the design structure, and does not affect the first and second modes.

Fig. 5
figure5

The dependency of the absorption spectrum to a width, and b length of the split in MA-2 with three-layer graphene and a Fermi level of 0.6 eV

Finally, the effects of graphene parameters, such as the number of layers and Fermi level, are investigated. In this study, five cases of one-, two-, three-, four-, and five-layer of graphene, were examined. Figure 6a shows a reduction in the absorption spectrum by reducing the number of graphene layers. As seen in the figure, increasing the number of layers to four and five shifts the two upper modes to higher frequencies more rapidly, which reduces the absorption to less than 80%. In other words, every resonance mode has a specific bandwidth and a quality factor, increasing the gap between two modes, for instance, modes 2 and 3 as Fig. 6a, leads to the creation of a deep in the absorption profile at f = 15 THz and amount of absorption fall below 60%. Thus, according to this figure, the optimal number of layers is equal to three. On the other hand, considering the issues mentioned above, dynamic adjustability is of great importance for changing the absorption spectrum. In order to achieve this objective, the effect of changing the Fermi level on the absorption spectrum of the structure is studied. Figure 6b demonstrates that the optimal absorption of the MA-2 is achieved at the Fermi level of 0.6 eV. According to the figure, the reduction of graphene Fermi level leads to a significant reduction in the absorption profile. It should be noted that reducing the Fermi level and some layers cause a minor shift in the absorption spectral bandwidth towards the lower frequencies. As stated in the Kubo equation, Eq. (4), both surface conductivity and, consequently, the electrical transmission coefficient of the graphene depend on the number of layers and Fermi level. Therefore, increasing the Fermi level as well as increasing the number of graphene layers lead to an increase in the surface conductivity of graphene, which results in a shift of resonance modes to higher frequencies. All this increases the absorption values in resonance modes. As a result, the effect of applied changes, are well in agreement with the values predicted from the equations.

Fig. 6
figure6

The dependency of the absorption spectrum to a the number of graphene layers at Fermi level of 0.6 ev, and b the Fermi level for three-layer of graphene in MA-2

Table 1 represents a comparison between the proposed absorber, MA-2, and recent graphene-based absorbers presented in the literature. It reveals, this absorber is distinct from other previous designs and covers a different range of frequencies. On the other hand, although the relative bandwidth calculated for this absorber is not the best among all research papers, it has a simple physical structure, which significantly simplifies the fabrication process. For more details, our design is distinctively better than absorbers in rows 2, 6, 9, and 11 of the table in terms of relative bandwidth. The absorbers presented in rows 1, 3, 4, 8, and 10 have better relative bandwidth, but their designed structures are difficult for fabrication. Row 5 in the table has larger bandwidth in comparison, but on the other hand under the graphene layer, there exist three layers of polysilicon, dielectric, and metal which require an additional layer deposition. Finally, row 7 of the table with a central frequency of 3.7 THz has wider relative bandwidth. Although some works demonstrate better bandwidth in comparison to our work, this work presents a new frequency band in the far-infrared regime.

Table 1 Comparison of the designed absorber metamaterial, MA-2, with various recent designs of similar wideband absorbers in different frequency ranges

Conclusion

In summary, we designed a graphene-based broadband metamaterial absorber consisting of a three-layer structure. The first layer of this absorber was created by making a split in the rectangular graphene sheet. This structure increases the absorption bandwidth by producing four resonance modes close together. Simulation results showed that the change in geometry has a more significant impact on the f2 and f4 resonance modes formed by the split. This absorber has a relative bandwidth equal to 68.22% which is unique compared to corresponding graphene absorbers. Being in the far-infrared range, and almost 100% absorption is among the other significant features of the proposed structure. Furthermore, the effects of changing the graphene surface conductivity function through changing the number of graphene layers and the Fermi level were numerically studied, which implied a significant effect on the absorption and a very small impact on the center frequency of the absorption band. The proposed design can be used in a new class of metamaterial absorbers, which possess large bandwidths and adjustable absorption while needs a simple fabrication process. Using the initial idea of this design, one can introduce new devices for microwave, THz, and visible frequency ranges with enormous applications in optical sensors, modulators, and filters.

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Alden Mostaan, S.M., Saghaei, H. A tunable broadband graphene-based metamaterial absorber in the far-infrared region. Opt Quant Electron 53, 96 (2021). https://doi.org/10.1007/s11082-021-02744-y

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Keywords

  • Graphene
  • Metamaterial
  • Tunable absorber
  • Broadband
  • Far-infrared region