3. Equivalent circuit model of multilayer graphene

The isolate graphene sheet can be characterized by its complex conductivity. In some designs of graphene-based metamaterial/metasurface devices, we need to consider electromagnetic wave interaction with the stacked periodic graphene sheets. Figure 6 shows a stack of graphene sheets separated by material slabs. In order to study plane wave reflection and transmission by a graphene-material stack, an equivalent circuit model [43] is developed in this section. Consider a uniform transverse electromagnetic (TEM) wave normally incident on the multilayer structure. Assume no higher-order modes are excited. Therefore, each graphene sheet is equivalent to a shunt admittance Ygi (i = 1, 2, …, N), and each material slab is regarded as a transmission line segment with a characteristic admittance Ymi and an electric length θ<sup>i</sup> = βidi, in which β<sup>i</sup> is the phase constant in each material slab and di is the thickness of the material slab. The corresponding equivalent circuit model is depicted in Figure 7.

According to the transfer matrix approach, the ABCD matrix can be written as:

$$
\begin{bmatrix} A & B \\ C & D \end{bmatrix} = \left( \prod\_{i=1}^{N} \begin{bmatrix} 1 & 0 \\ Y\_{\mathcal{S}^i} & 1 \end{bmatrix} \cdot \begin{bmatrix} \cos\theta\_i & j\frac{1}{Y\_{mi}}\sin\theta\_i \\ jY\_{mi}\sin\theta\_i & \cos\theta\_i \end{bmatrix} \right) \begin{bmatrix} 1 & 0 \\ Y\_{m\mathcal{N}+1} & 1 \end{bmatrix}. \tag{12}
$$

Figure 7. The equivalent circuit model of the multilayer graphene-based structure.

ns <sup>¼</sup> <sup>2</sup> πℏ<sup>2</sup>v<sup>2</sup> f

Figure 5. Total conductivity of the graphene as a function of the chemical potential.

Figure 4. The graphene sheet with an external gate voltage.

176 Metamaterials and Metasurfaces

tunable graphene-based devices.

ð ∞

ε fð Þ� ε f ε þ 2μ<sup>c</sup>

A single-layer graphene is grown on a substrate, for example, oxidized Si, and a gate voltage Vg can be applied, as shown in Figure 4. The gate voltage modifies the graphene carrier density as [32]

in which ε<sup>0</sup> and ε are permittivities of free space and the substrate, respectively, and t is the thickness of the substrate. Solving Eqs. (9) and (10), a relation between μ<sup>c</sup> and Vg can be

r

Figure 5 shows the conductivity variation of isotropic graphene with chemical potential. Here Γ ¼ 0:3291 meV and T ¼ 300 K. By changing the chemical potential, the conductivity of graphene can be flexibly adjusted, which provides us large degrees of freedom to design

ffiffiffiffiffiffiffiffiffiffiffiffiffi ε0ε et Vg

obtained. An approximate close-form expression to relate μ<sup>c</sup> to Vg can be given as [33]

μ<sup>c</sup> ¼ ℏvf

� � � � dε: (9)

ns ¼ ε0εVg=te, (10)

: (11)

0

Furthermore, the S parameters can be obtained as:

$$S\_{11} = \frac{A + B/Z\_0 - CZ\_0 - D}{A + B/Z\_0 + CZ\_0 + D}, \\ S\_{12} = \frac{2(AD - BC)}{A + B/Z\_0 + CZ\_0 + D} \tag{13}$$

average surface current density J

in which

expression:

to Zg<sup>i</sup> = 1/ σi.

patch array can be written as [49–51]

Zgi <sup>¼</sup> <sup>1</sup> Ygi

circuit and full-wave simulation, in good agreement.

⇀ind

<sup>¼</sup> Di Di � gi σ<sup>i</sup>

Etot

tan ¼ Zgi � J

Here, σ<sup>i</sup> is the conductivity of graphene and εeq is equivalent permittivity with the following

It is shown that when the gap tends to be zero, the corresponding graphene patch array becomes a whole graphene sheet and thus the surface impedance given by Eq. (16) is reduced

Consider a graphene-based metasurface structure, as shown in Figure 9. Here T = 300 K and Г = 3.2914 meV are used for each graphene sheet. The chemical potential of the top graphene layer is denoted as μc1 and the bottom one is μc2. Relative permittivity of the silicon material is 11.9. To validate the effectiveness of the equivalent circuit model, we consider three cases. The first case is L1 = 20 μm and μc1 = 0.1 meV and μc2 = 0.2 meV. The corresponding resonant frequency of the metasurface structure obtained by the equivalent circuit is 0.95 THz, which is same as the result obtained by the full-wave simulation. In the second case of L1 = 15 μm and μc1 = μc2 = 0.15 meV, the resonant frequencies solved by the equivalent circuit and the fullwave simulation are 1.18 THz and 1.2 THz, respectively. For the third case, i.e., L1 = 12 μm and μc1 = 0.2 meV and μc2 = 0.1 meV, the resonant frequencies solved by two methods are 1.39 THz and 1.4 THz, respectively. Figure 10 demonstrates the absorption obtained by equivalent

Figure 9. A graphene-based metasurface structure. All dimensions are in micrometer: L1 = 5, D = 20, and g = 2.

� <sup>j</sup> <sup>π</sup>

2ωεeqDi ln csc πgi

induced on it. The average boundary condition for the

Design of Graphene-Based Metamaterial Absorber and Antenna

=2Di

εeqr ¼ ð Þ ε<sup>i</sup> þ ε<sup>i</sup>þ<sup>1</sup> =2: (17)

ind, (15)

, (16)

http://dx.doi.org/10.5772/intechopen.78608

179

$$\mathbf{S}\_{21} = \frac{2}{A + \mathbf{B}/\mathbf{Z}\_0 + \mathbf{C}\mathbf{Z}\_0 + \mathbf{D}},\\\mathbf{S}\_{22} = \frac{-A + \mathbf{B}/\mathbf{Z}\_0 - \mathbf{C}\mathbf{Z}\_0 + \mathbf{D}}{A + \mathbf{B}/\mathbf{Z}\_0 + \mathbf{C}\mathbf{Z}\_0 + \mathbf{D}}.\tag{14}$$

To determine the shunt admittance, Ygi of ith graphene sheet given in Figure 6, assume ith graphene sheet as an array of graphene patches with a period of Di, as shown in Figure 8. The gap between the graphene patches is g<sup>i</sup> (g<sup>i</sup> < Di), and permittivities of the material slabs at the top and bottom part of the graphene array are ε<sup>i</sup> and ε<sup>i</sup> + 1, respectively. In the case of the normally incident wave, the graphene patch array can be characterized by a surface impedance Zgi, which relates the average tangential components of the total electric field E ⇀tot tan and the

Figure 8. Periodic graphene patch array.

average surface current density J ⇀ind induced on it. The average boundary condition for the patch array can be written as [49–51]

$$E\_{\rm tan}^{\rm tot} = Z\_{\rm gi} \cdot \mathbf{J}^{\rm ind},\tag{15}$$

in which

Furthermore, the S parameters can be obtained as:

178 Metamaterials and Metasurfaces

<sup>S</sup><sup>21</sup> <sup>¼</sup> <sup>2</sup>

Figure 8. Periodic graphene patch array.

<sup>S</sup><sup>11</sup> <sup>¼</sup> <sup>A</sup> <sup>þ</sup> <sup>B</sup>=Z<sup>0</sup> � CZ<sup>0</sup> � <sup>D</sup>

<sup>A</sup> <sup>þ</sup> <sup>B</sup>=Z<sup>0</sup> <sup>þ</sup> CZ<sup>0</sup> <sup>þ</sup> <sup>D</sup> , S<sup>12</sup> <sup>¼</sup> <sup>2</sup>ð Þ AD � BC

To determine the shunt admittance, Ygi of ith graphene sheet given in Figure 6, assume ith graphene sheet as an array of graphene patches with a period of Di, as shown in Figure 8. The gap between the graphene patches is g<sup>i</sup> (g<sup>i</sup> < Di), and permittivities of the material slabs at the top and bottom part of the graphene array are ε<sup>i</sup> and ε<sup>i</sup> + 1, respectively. In the case of the normally incident wave, the graphene patch array can be characterized by a surface imped-

ance Zgi, which relates the average tangential components of the total electric field E

<sup>A</sup> <sup>þ</sup> <sup>B</sup>=Z<sup>0</sup> <sup>þ</sup> CZ<sup>0</sup> <sup>þ</sup> <sup>D</sup> , S<sup>22</sup> <sup>¼</sup> �<sup>A</sup> <sup>þ</sup> <sup>B</sup>=Z<sup>0</sup> � CZ<sup>0</sup> <sup>þ</sup> <sup>D</sup>

<sup>A</sup> <sup>þ</sup> <sup>B</sup>=Z<sup>0</sup> <sup>þ</sup> CZ<sup>0</sup> <sup>þ</sup> <sup>D</sup> , (13)

<sup>A</sup> <sup>þ</sup> <sup>B</sup>=Z<sup>0</sup> <sup>þ</sup> CZ<sup>0</sup> <sup>þ</sup> <sup>D</sup> : (14)

⇀tot

tan and the

$$Z\_{\rm gi} = \frac{1}{Y\_{\rm gj}} = \frac{D\_i}{\left(D\_i - g\_i\right)\sigma\_i} - j\frac{\pi}{2\omega\varepsilon\_{eq}D\_i\ln\left[\csc\left(\pi g\_i/2D\_i\right)\right]}\tag{16}$$

Here, σ<sup>i</sup> is the conductivity of graphene and εeq is equivalent permittivity with the following expression:

$$
\varepsilon\_{eqr} = (\varepsilon\_i + \varepsilon\_{i+1})/2. \tag{17}
$$

It is shown that when the gap tends to be zero, the corresponding graphene patch array becomes a whole graphene sheet and thus the surface impedance given by Eq. (16) is reduced to Zg<sup>i</sup> = 1/ σi.

Consider a graphene-based metasurface structure, as shown in Figure 9. Here T = 300 K and Г = 3.2914 meV are used for each graphene sheet. The chemical potential of the top graphene layer is denoted as μc1 and the bottom one is μc2. Relative permittivity of the silicon material is 11.9. To validate the effectiveness of the equivalent circuit model, we consider three cases. The first case is L1 = 20 μm and μc1 = 0.1 meV and μc2 = 0.2 meV. The corresponding resonant frequency of the metasurface structure obtained by the equivalent circuit is 0.95 THz, which is same as the result obtained by the full-wave simulation. In the second case of L1 = 15 μm and μc1 = μc2 = 0.15 meV, the resonant frequencies solved by the equivalent circuit and the fullwave simulation are 1.18 THz and 1.2 THz, respectively. For the third case, i.e., L1 = 12 μm and μc1 = 0.2 meV and μc2 = 0.1 meV, the resonant frequencies solved by two methods are 1.39 THz and 1.4 THz, respectively. Figure 10 demonstrates the absorption obtained by equivalent circuit and full-wave simulation, in good agreement.

Figure 9. A graphene-based metasurface structure. All dimensions are in micrometer: L1 = 5, D = 20, and g = 2.

Figure 10. Absorption of the graphene-based metasurface structure.
