2. Conductivity model of graphene

Due to its mono-atomic thickness, graphene can be considered as an infinitesimally thin surface. With graphene's gapless electronic band structure, conductivity is the most appropriate parameter to characterize its electromagnetic properties. Hence the graphene sheet is modeled by surface conductivity, which relates the surface current to the tangential electric field in the graphene plane.

As shown in Figure 1, consider a graphene sheet in the presence of an electric field E ⇀ <sup>¼</sup> <sup>b</sup>xEx and a static magnetic field B ⇀ <sup>0</sup> <sup>¼</sup> <sup>b</sup>zB0. The electron in graphene is accelerated along �x direction by the electric field force F ⇀ <sup>e</sup> ¼ �bxeEx. As the electron moves with a speed <sup>v</sup> ⇀¼ �bxv, the magnetic

Figure 1. A graphene sheet biased with a static magnetic field.

field force F ⇀ <sup>m</sup> ¼ �e v⇀ � <sup>B</sup> ⇀ ¼ �byeB0<sup>v</sup> is generated to deflect the electron toward �y direction. Therefore, the induced current has two components, i.e., J ⇀ <sup>¼</sup> <sup>b</sup>xJx <sup>þ</sup> <sup>b</sup>yJy <sup>¼</sup> <sup>b</sup>xσxxEx <sup>þ</sup> <sup>b</sup>yσyxEx. Similarly, in the case of an electric field E ⇀ <sup>¼</sup> <sup>b</sup>yEy and a static magnetic field <sup>B</sup> ⇀ <sup>0</sup> <sup>¼</sup> <sup>b</sup>zB0, the induced current becomes J ⇀ <sup>¼</sup> <sup>b</sup>xJx <sup>þ</sup> <sup>b</sup>yJy ¼ �bxσyxEy <sup>þ</sup> <sup>b</sup>yσxxEy. Hence, an interesting property of graphene is magnetically induced gyrotropy which can be stated as

J ⇀ ¼ σ� E ⇀ , (1)

in which

[13–22]. For example, the well-known double negative materials, which was proposed by Veselago [23], can support the backward propagating waves [24], and near-zero material can

In recent years, metasurfaces have caused huge research interest [26–30]. Metasurfaces may be considered as the two-dimensional counterparts of metamaterials. Due to its subwavelength thickness, metasurfaces are easier to fabricate by using planar fabrication technology. Different from the effective medium characterization of the volume metamaterial, metasurfaces modulate the behaviors of electromagnetic waves through specific boundary conditions. By designing subwavelength-scaled patterns in horizontal dimensions, characteristics of electromagnetic waves including phase, magnitude, and polarization can be flexibly manipulated. One of the most important applications by means of the metasurface is to control the wavefront of the electromagnetic waves by imparting local, gradient phase shift to the incoming waves, which

Graphene [31, 32] consisting of a single layer of carbon atoms arranged in a hexagonal lattice has attracted increasingly attention of the research community due to its extraordinary mechanical, electric, optical, and heating properties. Intrinsic graphene is a zero band-gap semiconductor which is very promising for nanoelectronics applications, because its conduction and valence bands meet at Dirac points. Graphene's transport characteristic and conductivity can be tuned by either electrostatic or magnetostatic gating or via chemical doping [31, 33]. The Fermi level of intrinsic graphene can be engineered to support surface plasmons polariton (SPP) [34, 35]. These fascinating characteristics promise graphene a nature candidate in the metamaterial/metasurface-based devices including absorbers [36–39], cloaks [40], filters [41], antennas [42, 43], nonlinear optical devices [44], etc. In this chapter, we first review the conductivity model and equivalent circuit model of graphene, respectively. Next, a graphenebased metamaterial absorber with good performance including tunable absorbing bandwidth, wide angle, and polarization insensitive characteristic is developed at mid-infrared frequencies. Finally, a wideband tunable graphene-based metamaterial reflectarray is proposed to

Due to its mono-atomic thickness, graphene can be considered as an infinitesimally thin surface. With graphene's gapless electronic band structure, conductivity is the most appropriate parameter to characterize its electromagnetic properties. Hence the graphene sheet is modeled by surface conductivity, which relates the surface current to the tangential electric

<sup>e</sup> ¼ �bxeEx. As the electron moves with a speed <sup>v</sup>

<sup>0</sup> <sup>¼</sup> <sup>b</sup>zB0. The electron in graphene is accelerated along �x direction by

⇀

⇀¼ �bxv, the magnetic

<sup>¼</sup> <sup>b</sup>xEx and

As shown in Figure 1, consider a graphene sheet in the presence of an electric field E

tunnel electromagnetic waves through very narrow channels [25].

172 Metamaterials and Metasurfaces

results in generalized Snell's transmission and reflection laws [26].

generate an orbital angular momentum (OAM) vortex wave in terahertz.

2. Conductivity model of graphene

⇀

⇀

field in the graphene plane.

a static magnetic field B

the electric field force F

$$
\overline{\sigma} = \begin{bmatrix}
\sigma\_{\text{xx}} & -\sigma\_{\text{yx}} \\
\sigma\_{\text{yx}} & \sigma\_{\text{xx}}
\end{bmatrix}.
\tag{2}
$$

The conductivity tensor in Eq. (2) can be determined from Kubo formalism, and the explicit expressions of σxx and σxy for zero energy gap are [45, 46]:

$$
\sigma\_{xx} = \frac{c^2 \nu\_r^{\prime} \left[ \mathrm{e} B\_0 (\mathrm{a} - \mu \mathrm{\mathcal{T}}) \sum\_{n=0}^{\infty} \left[ \frac{n\_r (M\_s) - n\_r (M\_{s+1}) + n\_f (-M\_{s+1}) - n\_r (-M\_s)}{\left( (M\_{s+1} - M\_s) \left[ (M\_{s+1} - M\_s)^2 - \hbar^2 (\alpha - j \mathrm{\mathcal{T}} \mathrm{\mathcal{T}})^2 \right] \right)} + \right.} \right. \tag{3}
$$

$$
\frac{n\_F (-M\_s) - n\_r (M\_{s+1}) + n\_F (-M\_{s+1}) - n\_r (M\_s)}{\left( M\_{s+1} + M\_s \right) \left[ \left( M\_{s+1} + M\_s \right)^2 - \hbar^2 (\alpha - j \mathrm{\mathcal{T}} \mathrm{\mathcal{T}}^2) \right]} \tag{4}
$$

$$
\sigma\_{yx} = -\frac{c^2 \nu\_F^2 \sigma B\_0}{\pi} \sum\_{n=0}^{\infty} \left\{ \frac{n\_F (M\_n) - n\_F (M\_{n+1}) - n\_F (-M\_{n+1}) + n\_F (-M\_n)}{\left[ \left( M\_{n+1} - M\_n \right)^2 - \hbar^2 \left( \alpha - j \mathrm{\mathcal{T}} \mathrm{\mathcal{T}} \right)^2 \right]} \tag{4}
$$

$$
+ \frac{n\_F (M\_n) - n\_F (M\_{n+1}) - n\_F (-M\_{n+1}) + n\_F (-M\_n)}{\left[ \left( M\_{n+1} + M\_n \right)^2 - \hbar^2 \left( \alpha - j \mathrm{\mathcal{T}} \mathrm{\mathcal{T}} \right)^2 \right]},
$$

in which vF ≈ 106 m=s is the Fermi velocity, nFð Þ¼ y 1= 1 þ exp y � μ<sup>c</sup> � �=ð Þ kBT � � � � is Fermi-Dirac distribution, Mn ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2nℏv<sup>2</sup> <sup>F</sup>j j eB<sup>0</sup> q is the energy of the nth Landau level, μ<sup>c</sup> is chemical potential, τ ¼ 1=ð Þ 2Γ is the scattering time, �e is the charge of an electron, ℏ is the reduced Planck's constant, T is temperature, and kB is Boltzmann constant.

In the low magnetic field limit, Eqs. (3) and (4) can be rewritten as [47]:

$$
\sigma\_{xx} = \frac{-j\varepsilon^2(\omega - 2j\Gamma)}{\pi\hbar^2} \left[ \frac{1}{\left(\omega - 2j\Gamma\right)^2} \right]\_0^\infty \left( \frac{\partial u\_F(\varepsilon)}{\partial \varepsilon} - \frac{\partial u\_F(-\varepsilon)}{\partial \varepsilon} \right) d\varepsilon - \tag{5}
$$

$$
\left[ \frac{\kappa\_F(-\varepsilon) - n\_F(\varepsilon)}{\left(\omega - j2\Gamma\right)^2 - 4\left(\varepsilon/\hbar\right)^2} d\varepsilon \right],
$$

$$
\sigma\_{yx} = -\frac{\varepsilon^2 v\_F^2 e B\_0}{\pi\hbar^2} \left[ \frac{1}{\left(\omega + 2j\Gamma\right)^2} \right]\_0^\infty \left( \frac{\partial u\_F(\varepsilon)}{\partial \varepsilon} + \frac{\partial u\_F(-\varepsilon)}{\partial \varepsilon} \right) d\varepsilon + \tag{6}
$$

$$
\left[ \frac{1}{\left(\omega + j2\Gamma\right)^2 - 4\left(\varepsilon/\hbar\right)^2} d\varepsilon \right],
$$

Figure 2 shows real and imaginary parts of the intra-band and intra-band terms of an isotropic graphene surface at 100 GHz and 400 THz for Γ ¼ 0:3291 meV and T ¼ 300 K, respectively. According to Figure 2, we can see that at low-terahertz frequencies the conductivity of graphene is mainly governed by the intra-band contribution, while at infrared frequencies, both the intraband and inter-band conductivities are dominant. In addition, we can know that the intra-band conductivity has a negative imaginary part. Figure 3 gives the corresponding total conductivities at 100 GHz and 400 THz. It can be observed that the imaginary part of the total conductivity

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175

One of the main advantages of graphene is that its chemical potential can be tuned by the implementation of a gate voltage or chemical doping. For an isolate graphene sheet, the carrier

can be positive or negative depending on the operation conditions of the graphene.

Figure 2. Intraband and interband conductivities of an isotropic graphene sheet at different frequencies.

Figure 3. Total conductivities of an isotropic graphene sheet at different frequencies.

density ns is related with μ<sup>c</sup> via the following expression [48]

in which ω is radian frequency. Note that Eqs. (5) and (6) are valid under the condition ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>ℏ</sup>j j eB<sup>0</sup> <sup>v</sup><sup>2</sup> <sup>F</sup>=c q ≤ Γ. Specially, in the absence of biased magnetic field, we have σyx ¼ 0 and thus graphene is reduced to be isotropic. It is worthwhile pointing out that in Eqs. (5) and (6), there are two terms: the first one is related to intra-band contribution and the second one corresponds to inter-band contribution. The intra-band term in Eq. (5) can be analytically obtained as [48]:

$$\sigma\_{\rm xx,intra} = -j \frac{e^2 k\_B T}{\pi \hbar^2 (\omega - j2\Gamma)} \left[ \frac{\mu\_c}{k\_B T} + 2 \ln \left( e^{-\frac{\mu\_c}{k\_B T}} + 1 \right) \right]. \tag{7}$$

According to Eq. (7), the intra-band term follows the Drude model form. The real part of the intra-band term is greater than zero and its imaginary part is less than zero.

Generally, the inter-band term in Eq. (5) cannot be analytically obtained. However, when kBT ≤ μ<sup>c</sup> � � � � and kBT ≤ ℏω, the inter-band term in Eq. (5) can be approximately expressed as [46]:

$$\sigma\_{\rm xx,inter} = -j \frac{e^2}{4\pi\hbar} \log \left[ \frac{2|\mu\_c| - (\omega - j2\Gamma)\hbar}{2|\mu\_c| + (\omega - j2\Gamma)\hbar} \right]. \tag{8}$$

It can be seen from Eq. (8) that for Γ ¼ 0 and 2 μ<sup>c</sup> � � � � > ℏω, σxx,inter ¼ jσ<sup>00</sup> xx,inter becomes a purely imaginary number with a positive imaginary part. In the case of Γ ¼ 0 and 2 μ<sup>c</sup> � � � � < ℏω, σxx,inter ¼ σ<sup>0</sup> xx,inter þ jσ<sup>00</sup> xx,inter is a complex number, whose imaginary part is still positive and real part is σ<sup>0</sup> xx,inter <sup>¼</sup> <sup>σ</sup>min <sup>¼</sup> <sup>π</sup>e<sup>2</sup>=2<sup>h</sup> for <sup>μ</sup><sup>c</sup> 6¼ 0. The intra-band conductivity mainly accounts for low-frequency electrical transport and the inter-band conductivity is for the optical excitations. Figure 2 shows real and imaginary parts of the intra-band and intra-band terms of an isotropic graphene surface at 100 GHz and 400 THz for Γ ¼ 0:3291 meV and T ¼ 300 K, respectively. According to Figure 2, we can see that at low-terahertz frequencies the conductivity of graphene is mainly governed by the intra-band contribution, while at infrared frequencies, both the intraband and inter-band conductivities are dominant. In addition, we can know that the intra-band conductivity has a negative imaginary part. Figure 3 gives the corresponding total conductivities at 100 GHz and 400 THz. It can be observed that the imaginary part of the total conductivity can be positive or negative depending on the operation conditions of the graphene.

in which vF ≈ 106

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>ℏ</sup>j j eB<sup>0</sup> <sup>v</sup><sup>2</sup> <sup>F</sup>=c

q

kBT ≤ μ<sup>c</sup> � � �

σxx,inter ¼ σ<sup>0</sup>

real part is σ<sup>0</sup>

distribution, Mn ¼

174 Metamaterials and Metasurfaces

m=s is the Fermi velocity, nFð Þ¼ y 1= 1 þ exp y � μ<sup>c</sup>

1 ð Þ <sup>ω</sup> � <sup>2</sup>j<sup>Γ</sup> <sup>2</sup>

> ð ∞

> > 0

1 ð Þ <sup>ω</sup> <sup>þ</sup> <sup>2</sup>j<sup>Γ</sup> <sup>2</sup>

> ð ∞

> > 0

τ ¼ 1=ð Þ 2Γ is the scattering time, �e is the charge of an electron, ℏ is the reduced Planck's

ð ∞

∂nFð Þε

nFð Þ� �ε nFð Þε ð Þ <sup>ω</sup> � <sup>j</sup>2<sup>Γ</sup> <sup>2</sup> � <sup>4</sup>ð Þ <sup>ε</sup>=<sup>ℏ</sup> <sup>2</sup> <sup>d</sup><sup>ε</sup>

> ∂nFð Þε ∂ε þ

1 ð Þ <sup>ω</sup> <sup>þ</sup> <sup>j</sup>2<sup>Γ</sup> <sup>2</sup> � <sup>4</sup>ð Þ <sup>ε</sup>=<sup>ℏ</sup> <sup>2</sup> <sup>d</sup><sup>ε</sup>

≤ Γ. Specially, in the absence of biased magnetic field, we have σyx ¼ 0 and thus

μc

� and kBT ≤ ℏω, the inter-band term in Eq. (5) can be approximately expressed as [46]:

2 μ<sup>c</sup> � � �

2 μ<sup>c</sup> � � �

� � �

low-frequency electrical transport and the inter-band conductivity is for the optical excitations.

kBT <sup>þ</sup> 2 ln <sup>e</sup>

� <sup>μ</sup><sup>c</sup> kBT þ 1

� � � �

� � ð Þ ω � j2Γ ℏ

" #

� þ ð Þ ω � j2Γ ℏ

� > ℏω, σxx,inter ¼ jσ<sup>00</sup>

xx,inter is a complex number, whose imaginary part is still positive and

xx,inter <sup>¼</sup> <sup>σ</sup>min <sup>¼</sup> <sup>π</sup>e<sup>2</sup>=2<sup>h</sup> for <sup>μ</sup><sup>c</sup> 6¼ 0. The intra-band conductivity mainly accounts for

0 ε

ð ∞

0

in which ω is radian frequency. Note that Eqs. (5) and (6) are valid under the condition

graphene is reduced to be isotropic. It is worthwhile pointing out that in Eqs. (5) and (6), there are two terms: the first one is related to intra-band contribution and the second one corresponds to inter-band contribution. The intra-band term in Eq. (5) can be analytically obtained as [48]:

According to Eq. (7), the intra-band term follows the Drude model form. The real part of the

Generally, the inter-band term in Eq. (5) cannot be analytically obtained. However, when

ð Þ ω � j2Γ

<sup>4</sup>π<sup>ℏ</sup> log

imaginary number with a positive imaginary part. In the case of Γ ¼ 0 and 2 μ<sup>c</sup>

intra-band term is greater than zero and its imaginary part is less than zero.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2nℏv<sup>2</sup> <sup>F</sup>j j eB<sup>0</sup>

constant, T is temperature, and kB is Boltzmann constant.

In the low magnetic field limit, Eqs. (3) and (4) can be rewritten as [47]:

2 4

ð Þ ω � 2jΓ πℏ<sup>2</sup>

> <sup>F</sup>eB<sup>0</sup> πℏ<sup>2</sup>

<sup>σ</sup>xx,intra ¼ �<sup>j</sup> <sup>e</sup><sup>2</sup>kBT πℏ<sup>2</sup>

<sup>σ</sup>xx,inter ¼ �<sup>j</sup> <sup>e</sup><sup>2</sup>

It can be seen from Eq. (8) that for Γ ¼ 0 and 2 μ<sup>c</sup>

xx,inter þ jσ<sup>00</sup>

2 4

q

<sup>σ</sup>xx <sup>¼</sup> �je<sup>2</sup>

<sup>σ</sup>yx ¼ � <sup>e</sup><sup>2</sup>v<sup>2</sup>

� �=ð Þ kBT � � � � is Fermi-Dirac

dε�

(5)

(6)

: (7)

: (8)

xx,inter becomes a purely

� � � � < ℏω,

3 5,

dεþ

is the energy of the nth Landau level, μ<sup>c</sup> is chemical potential,

<sup>∂</sup><sup>ε</sup> � <sup>∂</sup>nFð Þ �<sup>ε</sup> ∂ε

> ∂nFð Þ �ε ∂ε

> > 3 5,

� �

� �

One of the main advantages of graphene is that its chemical potential can be tuned by the implementation of a gate voltage or chemical doping. For an isolate graphene sheet, the carrier density ns is related with μ<sup>c</sup> via the following expression [48]

Figure 2. Intraband and interband conductivities of an isotropic graphene sheet at different frequencies.

Figure 3. Total conductivities of an isotropic graphene sheet at different frequencies.

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

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

$$m\_s = \frac{2}{\pi \hbar^2 v\_f^2} \Bigg[ \varepsilon \left[ f(\varepsilon) - f(\varepsilon + 2\mu\_c) \right] d\varepsilon. \tag{9}$$

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]

$$m\_s = \varepsilon\_0 \varepsilon V\_\lg / \text{te}\_\prime \tag{10}$$

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

� cos <sup>θ</sup><sup>i</sup> <sup>j</sup> <sup>1</sup>

jYmi sin θ<sup>i</sup> cos θ<sup>i</sup>

Ymi

sin θ<sup>i</sup>

3 5

1 A

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177

1 0 YmNþ<sup>1</sup> 1

� �: (12)

material slab. The corresponding equivalent circuit model is depicted in Figure 7.

2 4

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

1 0 Ygi 1 � �

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

A B C D � � <sup>¼</sup> <sup>Y</sup>

Figure 6. Planar multilayer graphene.

N

0 @

i¼1

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

$$
\mu\_c = \hbar v\_f \sqrt{\frac{\varepsilon\_0 \varepsilon}{et}} V\_\mathcal{g}.\tag{11}
$$

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 tunable graphene-based devices.
