**4. Conclusion**

overloaded resources (sources) and the underloaded ones (receivers), depending on their load information by using the threshold values. Grid Agent and Cluster Agent

> **Location policy**

receiver resource is the most under loaded and it has the highest communication load with the migrated Agent

receiver resource is the least loaded host

The destination node is the node with the least LC (Location Credit) value

having desired configuration

available worker agent in **Decision making**

Migration decision is taken by Cluster Agent

Migration decision is taken by migration management agent

Migration decision is taken locally by the LBC Agent

Migration decision is taken by Load agent

Migration decision is taken by

**Migration condition**

Cluster state is unbalanced;

host is overloaded,

Local load value is greater than the load threshold value

fitness value of a VM becomes less than or equal to threshold value

network traffic analysis

**Implementation**

Jade [6] + Alea2 [7] simulator

AST-RTI [11] version 2.0 + C+ +

Java + Jade

Java

Jade

interfaces are shown in **Figures 5** and **6**.

**3.3 Model comparison with some works**

**Load information gathering policy**

Event-based information gathering

periodicbased information gathering

Event-based information gathering

periodicbased information gathering

periodicbased information gathering

**Selection policy**

*Artificial Intelligence - Latest Advances, New Paradigms and Novel Applications*

Credit -based concept

computation and communication cost

Credit -based concept

Not cited receiver VM

task cost receiver is

**configuration**

contains a set of computing resources hierarchy of control, with six types of agents

The system contains entity, federate, VM, and host with migration management agent

The system contains a set of nodes decentralized in control, with seven agents

contains a set of VMs centralized in control, with three agents

The system contains a set of machines centralized in

A2LB [12] The system

N LB WITH STRONG MIGR ATION IN AN

**68**

**Model System**

**Figure 6.**

*Cluster agent interface.*

ABLBM The system

VM, dynamic balancing [10]

LB in distributed MAS [4]

Recognizing key factors to prove the convergence of grid and MAS and models is not a simple task. We note that the current state of GRID and MAS research activities are necessarily developed to enable justifying the study of the path towards an integration of the two fields.

We have presented a theoretical comparison between some related works and the proposed model. The proposed model has some unique features. It is hierarchical, which facilitates the circulation of information through the tree and defines the flow of messages between agents. Also, the proposed Agent-based load balancing model uses an event-driven information gathering policy, the latter being especially beneficial in terms of economy of usage of network resources. Furthermore, it can achieve excellent performance with significantly less computational load and system instability than a periodic information gathering policy. To select the migrating agent, we use the credit-based concept, accordingly, some factors are considered to calculate the credit value. Moreover, in the selection of receiver resources, we take into consideration the resource loads and the communication between the receiver resources and the migrating agent for avoiding the migration for external resources and reducing the communication cost. The migration decision is taken locally by Cluster Agent, where each cluster agent to balance its load among its associated resources. If it fails, the Cluster Agent migrates worker agents to underloaded clusters based on the load information received by other clusters. Finally, it supports flexibility and expandability, thus, various intelligent agents have been deployed to decrease system complexity by modularization. Moreover, it is easy to modify its components, and add more features and functions to it.

In theory, the multi-agent architecture of load balancing systems introduces important improvements, such as better average performance when one computer is not working and a lower system-error probability. In terms of the development process, fault-tolerance, and scalability, the agent approach offered the expected improvements, both in objective real-world measurements and in the subjective observations of designers, developers, and users.

On another hand, we could not overcome several well-known problems when designing distributed systems. For example, handling failed entities, synchronization problems, and query-response-related issues turned out to be the same as in any distributed programming. It is important to be aware of the advantages and disadvantages of the agent and non-agent approaches, but the most important point is whether the advantages prevail. For load balancing systems, our theoretical analysis and practical experiences both indicate that the advantages of agent-based load balancing systems clearly be more than the observed disadvantages.

The system performance was not studied yet. Thus, there is a need to analyze execution efficiency and compare it to available Agent-based load balancing platform evaluations. Further research is going to concentrate on execution performance.

*Artificial Intelligence - Latest Advances, New Paradigms and Novel Applications*

**References**

ADCAIJ201984518

MGS-200326.

[1] A. Wided, K. Okba, and B. Fatima. Load balancing with Job Migration Algorithm for improving performance on grid computing: Experimental Results. Advances in Distributed Computing and Artificial Intelligence Journal, 2019; 8(4): 5–18. DOI:10.14201/

*Agent Based Load Balancing in Grid Computing DOI: http://dx.doi.org/10.5772/intechopen.94219*

(CCPE),2002, 14:13–15. DOI: 10.1002/

[9] N. Li, X.-Y. Peng, M.-H. Zhang, M. Wang, and G.-H. Gong, Multimedia communication over HLA/RTI, SimulationModelling Practice and Theory,2006,14(2): 161–176. https://doi. org/10.1016/j.simpat.2005.03.003

[10] A.Singh,D. Juneja, M.Malhotra. Autonomous agent based load balancing

[11] X. Song, Y. Ma, D.Teng. A load balancing scheme using federate

cloud simulations, Mathematical Problems in Engineering, 2015,pp.1–11. https://doi.org/10.1155/2015/506432

an agent based grid using CSP

Computing & Applications (IJGCA),2012, 3(4),43–53. DOI:

10.5121/ijgca.2012.3404

workload/l\_metacentrum

migration based on virtual machines for

[12] Z. A .Sayar and N.Erdogan. Network load balancing with strong migration in

Approach. International Journal of Grid

[13] http://www.cs.huji.ac.il/labs/parallel/

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algorithm in cloud computing.

Computing Technologies and Applications, 2015. DOI: 10.1016/j.

procs.2015.03.168

cpe.710

[2] A. Wided, K. Okba . A new agent based load balancing model for improving the grid performance. Multiagent and Grid Systems Journal, 2020; 16(2):153–170. DOI: 10.3233/

[3] Chow K-P, Kwok Y-K. On load balancing for distributed multiagent computing. IEEE Trans Parallel Distrib

[4] A.M. Metawei, G. Salma,M.H. Sahar ,

Syst ,2002;13(8):1153–61.

M.N.Salwa .'Load balancing in distributed multi-agent computing systems', Ain Shams Engineering Journal,2012, 3(3):237–249.

[5] R. Gupta . Coding categorical variables in regression models: dummy and effect coding.Cornell Statistical Consulting Unit,2008,4(2):202–210.

[6] F.L. Bellifemine, G.Caire, and D. Greenwood, Developing Multi-Agent Systems with JADE. John Wi-ley & Sons,2007, NJ. ISBN: 978–0–470-05747-6

[7] D. Klusáček and H.Rudová. Alea 2 job scheduling simulator. In Proceedings of the 3rd International Conference on Simulation Tools and Techniques (SIMUTools 2010), Torremolinos, MalaMA, Spain. DOI: 10.4108/ICST.

[8] R.Buyya, and M.Murshed. Gridsim: a toolkit for the modeling and simulation of distributed resource management and scheduling for grid computing, The

Computation: Practice and Experience

SIMUTOOLS2010.8722

Journal of Concurrency and

**71**


**2.5 Pharyngeal microbiota**

In the nasal, oral and pharyngeal human cavities live hundreds of microbial species, including between 25 and 40 families archeas, bacteria, amoebae and fungi, as evidenced in a wide range of cultures. The number of newly discovered species has increased considerably due to the discovery of noncultivable species [88]. A data published in the Human Microbiome Project (HMP), 5 main bacterial phyla have been identified in the pharynx: Firmicutes, Proteobacteria, Bacteroidetes, Fusobacteria and Actinobacteria. Interestingly, the pharyngeal microbiome is distinguished from other parts of the body (intestines, skin and vagina) by having more Bacteroidetes. The proportion of bacterial phyla in the pharyngeal microbiome comprises 27% of Bacteroidetes and only 10% of Proteobacteria, compared to the salivary microbiome, whose proportion corresponds to 9% of Bacteroidetes and 51% of Proteobacteria. These two phyla are mentioned because they are the main pathogens in human infections, for example, periodontitis is caused by Bacteroidetes and the most common Gram-negative pathogens (*Acinetobacter*, *Moraxella*, *Pseudomonas*, *Haemophilus*, *Klebsiella*, and *Legionella* spp) [89]. However, the two genera that dominate the micro-ecosystem of the pharynx are *Streptococcus* and *Prevotella* [90].

## *2.5.1 The pharyngeal microecosystem*

The most common bacterial genera in the human pharynx are *Prevotella*, *Capnocytophaga*, *Campylobacter*, *Veillonella*, *Streptococcus*, *Neisseria*, *Haemophilus*, which represent 9.72% to 1.26% of the bacteria in the normal microbiome, according to HMP data. [89, 90]. As there are few studies of the pharyngeal microbiome, many interactions between the components of the microecosystem are not clear, however, the interactions of microorganisms with factors of the local environment are characteristic of the microbiome. From the above, it can be assumed that the pharyngeal microbiome may share common characteristics of other human microbiomes [89].


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 ⇀ ¼ bxEx and a static magnetic field B ⇀ 0 ¼ bzB0. The electron in graphene is accelerated along �x direction by the electric field force F ⇀ e ¼ �bxeEx. As the electron moves with a speed v ⇀¼ �bxv, the magnetic

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

field force F ⇀ m ¼ �e v⇀ � B ⇀ ¼ �byeB0v is generated to deflect the electron toward �y direction. Therefore, the induced current has two components, i.e., J ⇀ ¼ bxJx þ byJy ¼ bxσxxEx þ byσyxEx. Similarly, in the case of an electric field E ⇀ ¼ byEy and a static magnetic field B ⇀ 0 ¼ bzB0, the induced current becomes J ⇀ ¼ bxJx þ byJy ¼ �bxσyxEy þ byσxxEy. 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

e ¼ �bxeEx. As the electron moves with a speed v

0 ¼ bzB0. The electron in graphene is accelerated along �x direction by

⇀

⇀¼ �bxv, the magnetic

¼ bxEx 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 � μc � �=ð Þ kBT � � � � is Fermi-Dirac distribution, Mn ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2nℏv2 Fj j eB0 q is the energy of the nth Landau level, μc 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 μc via the following expression [48]

in which ω is radian frequency. Note that Eqs. (5) and (6) are valid under the condition ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ℏj j eB0 v2 F=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 ≤ μc � � � � 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 μc � � � � > ℏω, σxx,inter ¼ jσ00 xx,inter becomes a purely imaginary number with a positive imaginary part. In the case of Γ ¼ 0 and 2 μc � � � � < ℏω, σxx,inter ¼ σ0 xx,inter þ jσ00 xx,inter is a complex number, whose imaginary part is still positive and real part is σ0 xx,inter ¼ σmin ¼ πe2=2h for μc 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 ℏj j eB0 v2 F=c

q

kBT ≤ μc � � �

σxx,inter ¼ σ0

real part is σ0

distribution, Mn ¼

174 Metamaterials and Metasurfaces

m=s is the Fermi velocity, nFð Þ¼ y 1= 1 þ exp y � μc

1 ð Þ ω � 2jΓ 2

> ð ∞

> > 0

1 ð Þ ω þ 2jΓ 2

> 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 ω � j2Γ 2 � 4ð Þ ε=ℏ 2 dε

> ∂nFð Þε ∂ε þ

1 ð Þ ω þ j2Γ 2 � 4ð Þ ε=ℏ 2 dε

≤ 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 μc � � �

2 μc � � �

� � �

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

kBT þ 2 ln e

� μc kBT þ 1

� � � �

� � ð Þ ω � j2Γ ℏ

" #

� þ ð Þ ω � j2Γ ℏ

� > ℏω, σxx,inter ¼ jσ00

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

xx,inter ¼ σmin ¼ πe2=2h for μc 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

4πℏ log

imaginary number with a positive imaginary part. In the case of Γ ¼ 0 and 2 μc

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

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2nℏv2 Fj j eB0

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 πℏ2

> FeB0 πℏ2

σxx,intra ¼ �j e2kBT πℏ2

σxx,inter ¼ �j e2

It can be seen from Eq. (8) that for Γ ¼ 0 and 2 μc

xx,inter þ jσ00

2 4

q

σxx ¼ �je2

σyx ¼ � e2v2

� �=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, μc is chemical potential,

∂ε � ∂nFð Þ �ε ∂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 μc 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 θi = βidi, in which βi is the phase constant in each material slab and di is the thickness of the

� cos θi j 1

jYmi sin θi cos θi

Ymi

sin θi

3 5

1 A

Design of Graphene-Based Metamaterial Absorber and Antenna

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177

1 0 YmNþ1 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 � � ¼ Y

Figure 6. Planar multilayer graphene.

N

0 @

i¼1

in which ε0 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 μc and Vg can be obtained. An approximate close-form expression to relate μc 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.