Frequency-tunable terahertz absorber with wire-based metamaterial and graphene

Recently, extensive research has been carried out on the design and fabrication of cloaks from microwave to optical frequency due to their potential applications in stealth technology [1–5]. Metamaterial absorbers (MAs), as a kind of cloaking device, have provoked extensive interest, and a variety of metamaterial structures have been designed [6–9]. With these kinds of novel absorbers, theoretical unity absorptivity can be achieved by matching the impedance of MA to the free space [10, 11]. However, once the structure of MA is fabricated, the position and magnitude of the absorption peak is fixed. Therefore, more and more attention has been focused on the design of tunable MAs. Graphene, a single atom carbon layer arranged in a honeycomb lattice, has attracted a great deal of attention due to its intriguing properties in photonics and optoelectronics [12]. Meanwhile, because of its high electron mobility, unique doping capabilities, and tunable conductivity by bias voltage, the absorbers based on graphene can be tunable in the THz and optical regime [13, 14], which are important properties in some applications. However, the general principles of selecting structure parameters for the desired MGAs have not been discussed. For engineers, it is more important to understand the relationship between the chemical potential, structure parameters, and the absorption properties. More recently, an ultra-broadband multilayered graphene absorber has been studied in [15]. However, there is a lack of general strategy to analyze the absorption frequency of the ultra-broadband from its transmission line model.

Wire-based metamaterial or graphene ribbon absorber, as simple but versatile artificial structures, have been widely investigated in previous literature (e.g. [16–21]). At the same time, some theoretical approaches, such as effective surface conductivities through a sheet retrieval method [17], transmission line method [20, 22], accurate analytical model [23], effective medium theory [24] and simulation analysis method [25, 26] have been exploited to explain the physics mechanism. When the metamaterial is in close proximity to a graphene sheet, however, to the best of the authors' knowledge, little attention has been paid on discussing the equivalent impedance of this whole in previous literature.

In this paper, a wire-based metamaterial and graphene absorber is proposed in which a periodic array of metal wire is placed on a graphene sheet. The absorption spectra can be shifted dynamically by modulating the chemical potential of graphene. We use the retrieved equivalent impedance of wire-based metamaterial and graphene to analyze the mechanism. Meanwhile, an equivalent circuit model is applied to further reveal the influences of the constitutive parameters on the absorption properties for this proposed MGA.

The schematic of the unit cell of the studied MGA is illustrated in figure 1. The metal wire is separated from the bottom gold plate by an ultrathin graphene and insulator dielectric substrate. The gold is made of lossy metal with an electric conductivity of σ  =  4.56  ×  10⁷ S m⁻¹ and the thickness is 0.1 µm. The real part of the dielectric constant of the dielectric spacer is 11.9, and the thickness is t  =  3 µm. The other dimension parameters are fixed to be p  =  10 µm, l  =  9 µm and w  =  1.2 µm. All of these geometrical sizes are carefully designed to acquire the optimal effect.

Zoom In Zoom Out Reset image size

For a graphene sheet, the surface conductivity of graphene σ_g can be given by the Kubo formula [27]. When there is no external magnetic field present, the graphene exhibits isotropy. In this case, the surface conductivity can be approximated as follows for k_BT $\ll$ μ_c, $\hbar \left( \omega \right)$:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\sigma }_{{\rm g}}}=&\,\frac{j{{e}^{2}}{{k}_{{\rm B}}}T}{\pi {{\hbar }^{2}}\left( \omega +j/\tau \right)}\left[ \frac{{{\mu }_{{\rm c}}}}{{{k}_{{\rm B}}}T}+2{\rm ln}\left( {{{\rm e}}^{-{{\mu }_{{\rm c}}}/{{k}_{{\rm B}}}T}}+1 \right) \right]\nonumber \\ &+\frac{j{{e}^{2}}}{4\pi \hbar }{\rm ln}\left[ \frac{2\left| {{\mu }_{{\rm c}}} \right|-\left( \omega +j/\tau \right)\hbar }{2\left| {{\mu }_{{\rm c}}} \right|+\left( \omega +j/\tau \right)\hbar } \right],\nonumber \end{align} \tag{ 1 }$$

where k_B is the Boltzmann constant, e is the electron charge, ℏ is the reduced Planck constant, T is the temperature, μ_c denotes the chemical potential, ω is the angular frequency, and τ is the relaxation time, respectively. All simulations are performed in T  =  300 K, the thickness of graphene sheet is 1 nm and τ  =  0.1 ps throughout this study. The surface impedance of a graphene sheet can be expressed by Z_g  =  1/σ_g.

Performance simulations of this MGA were demonstrated using the full-wave electromagnetic simulation software CST Microwave Studio. The incident electric field vector is parallel to the metal wire, and the magnetic field vector is perpendicular to the metal wire. Because the bottom gold plate prevents any transmission of terahertz radiation, the absorption properties of this MGA are characterized by reflection coefficients in this work.

The frequency tenability of graphene absorber is an important characteristic in practical application. In order to explore the tunable effects of this proposed MGA, we first merely vary the chemical potential of graphene sheet. Reflection coefficient spectra for different chemical potential are represented in figure 2. At normal incidence. At figure 2(a), one can observe a progressive reduction and blue shift of the reflection peaks with increasing μ_c from 0 to 0.4 ev. The peak positions shift from 6.91–8.1 THz, and the reflection peaks continuously decrease from  −39.7 dB to  −7.24 dB. To clearly display these relationships, figure 2(b) shows the dependence of the relationship among chemical potential, reflection coefficient and absorption frequency. The reason for this behavior will be elucidated in the next section. From these results, we can see that graphene plays an important role in achieving the tunable function for this MGA. Meanwhile, the properties of the proposed MGA structure can be modulated conveniently by controlling the graphene's chemical potential.

Zoom In Zoom Out Reset image size

We first use a transmission line model to investigate this proposed absorber. As show in figure 3(a), Z_t is the characteristic impedance of the dielectric substrate. Z_mg presents the effective impedance of the metamaterial-graphene (MG) film. The gold ground plane is considered as a short circuit in the transmission line model analysis due to its zero transmission. (Although it is exact only for perfect electric conductors, it is accurate enough in the terahertz regime as long as the metal backplane is sufficiently thick.) The impedance of the grounded substrate can be expressed as [28–30]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{Z}_{1}}=j\frac{{{Z}_{0}}}{\sqrt{{{\varepsilon }^{\prime }}-j{{\varepsilon }^{\prime\prime }}}}{\rm tan}\left( {{k}_{0}}\sqrt{{{\varepsilon }^{\prime }}-j{{\varepsilon }^{\prime\prime }}}t \right),\nonumber \end{align} \tag{ 2 }$$

where Z₀ is the wave impedance in the free-space and is equal to 376.7 Ω, ${{k}_{0}}=\omega \sqrt{{{\mu }_{0}}{{\varepsilon }_{0}}}$ is the free space propagation constant, t is the thickness of the dielectric, ${{\varepsilon }^{\prime }}$ and ${{\varepsilon }^{\prime\prime }}$ are real and imaginary parts of the relative dielectric constant of the dielectric material. The effective surface impedance of the ground substrate is depicted in figure 3(b). It can be seen that at 7.22 THz the imaginary part of Z₁ is zero, indicating resonance occurring at near this frequency when there only exists gold-backed silicon dielectric substrate.

Zoom In Zoom Out Reset image size

If Z_mg is calculated, the input impedance of the absorbing structure Z_in is equal to the parallel connection between the MG impedance Z_mg and the complex surface impedance of the grounded dielectric slab Z₁ [31],

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{Z}_{{\rm in}}}=\frac{{{Z}_{{\rm mg}}}{{Z}_{1}}}{{{Z}_{{\rm mg}}}+{{Z}_{1}}}.\nonumber \end{align} \tag{ 3 }$$

Finally, the reflection coefficient can be expressed as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle R=\frac{{{Z}_{{\rm in}}}-{{Z}_{0}}}{{{Z}_{{\rm in}}}+{{Z}_{0}}}.\nonumber \end{align} \tag{ 4 }$$

So in the following section, we should first calculate the value of Z_mg.

To better understand the physics mechanism of the reflection and explain quantitatively the relationship between the frequency shift and chemical potential, the effective impendence of the MG sheet must be established first. Theoretically calculating the impedance of MG sheet is much more difficult since the mutual coupling between the metal wire and graphene sheet. There are several methods to determine the effective surface resistance of the thin layer [24, 32, 33]. Here, we retrieved the effective impendence of the MG sheet Z_mg from the simulated reflection coefficient S₁₁ [32]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{1}{{{Z}_{{\rm mg}}}}={{Y}_{0}}\frac{1-{{S}_{11}}}{1+{{S}_{11}}}-{{Y}_{1}}\frac{{{{\rm e}}^{-j{{k}_{0}}nt}}+{{{\rm e}}^{\,j{{k}_{0}}nt}}}{{{{\rm e}}^{-j{{k}_{0}}nt}}-{{{\rm e}}^{{{k}_{0}}nt}}},\nonumber \end{align} \tag{ 5 }$$

where ${{Y}_{0}}=1/{{Z}_{0}}~$ the admittance of free space, Y₁ is the intrinsic admittance of the dielectric spacer, and n is the refractive index of the dielectric spacer.

The retrieved effective surface impedances Z_mg from S-parameters are plotted in figures 4(a) and (b). Referring to figure 4(a), it can be seen that the magnitude of the real part of Z_mg decreases at first and then increases with the augment of ${{\mu }_{{\rm c}}}$. Figure 4(c) shows this relationship intuitively. By comparing figures 2(b) and 4(c), one can find that the absorption frequency of MGA shows a similar exponential increase with the increase of μ_c. However, comparing the reflection coefficient to the impedance Z_mg of MG sheet, it can also find that when the real part of Z_mg decreases from 717.77 Ω to 213.4 Ω, the reflection coefficient increases from  −38 dB to  −7.29 dB. Therefore, we can confirm that the absorption frequency and absorptivity were determined by the effective impendence Z_mg, which was tunable by the chemical potential.

Zoom In Zoom Out Reset image size

In order to explain how the constitutive structure parameters affect the reflection properties, an equivalent circuit model (shown in figure 5(a)) was utilized, which is equivalent to figure 3(a). The inductance L is determined primarily by the parallel wire, capacitance C is created between the wire and the bottom gold plate. In addition, R_m, R_g and R_t were caused by the metal wire, graphene sheet and dielectric losses, respectively [16]. Compared with the thickness of the substrate, the thickness of graphene sheet is ultrathin. For simplicity, R_m and R_g can be considered as a whole, represented by R_mg. The inductance L and capacitance C are presented by ${{L}^{\prime }}$ and ${{C}^{\prime }}$ (shown in figure 5(b)). As a matter of fact, it is difficult to obtain the accurate analytical formulations of inductance ${{L}^{\prime }}$ and capacitance ${{C}^{\prime }}$.

Zoom In Zoom Out Reset image size

From this equivalent circuit model, the absorption frequency can be expressed as [16]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \omega _{0}^{2}{{L}^{\prime }}{{C}^{\prime }}=\frac{{{L}^{\prime }}-R_{{\rm mg}}^{2}{{C}^{\prime }}}{{{L}^{\prime }}-R_{t}^{2}{{C}^{\prime }}}.\nonumber \end{align} \tag{ 6 }$$

Due to $R_{t}^{2}{{C}^{\prime }}\ll {{L}^{\prime }}$, so the term of $R_{t}^{2}{{C}^{\prime }}$ can be neglected in equation (6). Furthermore, the inductance ${{C}^{\prime }}$ can be approximated by the formula of a parallel capacitor as ${{C}^{\prime }}\propto {{\varepsilon }_{r}}lw/t$, and the inductance L as $~L\propto {{\mu }_{r}}tl/w~$ [16, 34, 35], so the absorption frequency from equation (6) can be given by:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{f}_{0}}\propto \frac{\sqrt{1-R_{{\rm mg}}^{2}{{\varepsilon }_{r}}{{w}^{2}}/{{t}^{2}}}}{l\sqrt{{{\varepsilon }_{r}}}}.\nonumber \end{align} \tag{ 7 }$$

From relation (7), we can find that the absorption frequency is related to the length l, width w, permittivity ${{\varepsilon }_{r}}$, loss R_mg and thickness t. In this study, the influence of spacer permittivity ${{\varepsilon }_{r}}$, loss R_mg and thickness t were not discussed. We first investigated the relationship between length l and absorption frequency. Figure 6(a) plots the reflection coefficient for various wire lengths when chemical potential is set at 0 ev and w, t, ${{\varepsilon }_{r}}$ at the optimized value. It is clear that the absorption peak shows the trend of first decreasing and then increasing. Meanwhile, as the wire length l decreases, the absorption frequency experiences a blue shift, displaying an inversely proportional relationship as indicated by equation (7). Figure 6(b) shows the dependence of the absorption frequency on the length of wire clearly.

Zoom In Zoom Out Reset image size

Finally, the influence of the width on the absorption frequency was also investigated at normal incident case and the results are presented in figure 7(a). Here, we have varied the width w from 0.8 µm to 1.2 µm. It can be found that the absorption frequencies show continuously red shift with increasing the width w. These results agree well with the conclusion of equation (7). In order to clearly display this relationship, figure 7(b) shows this dependency.

Zoom In Zoom Out Reset image size

In conclusion, we have investigated a frequency-tunable terahertz absorber based on wire-based metamaterial and graphene. The influence of the MG surface resistance on the chemical potential and absorption frequency has been studied in detail. Meanwhile, the wire length and width were discussed by using an equivalent circuit model. With this transmission line model and equivalent circuit model, it can be easy to understand the physical mechanism. The most important thing is that this will provide valuable guidance on how to develop more advanced MGAs in the future.

It should be noted that only the proposed method was verified in this model. Therefore, in our future work, we will focus on the experimental verification of the acquired numerical results by measuring fabricated prototypes. In addition, the dependence of the absorption frequency on the structural parameters will also be explored in other absorbers.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61501067, 61661012), Foundation and Advanced Research Projects of Chongqing Municipal Science and Technology Commission (Grant Nos. Cstc2016jcyjA0377), and Opening Project of Guangxi College Key Laboratory of Microwave and Optical Wave Applications Technology (Grant Nos. MLLAB2016001).