Graphene is a monolayer of carbon atoms arranged in a honeycomb structure which exhibits remarkable properties including high electron mobility, mechanical flexibility, and saturable absorption. In this chapter, the conductivity model of the graphene is first reviewed. Based on the conductivity model of graphene, the equivalent circuit model of graphene is discussed. By varying graphene’s chemical potential via external biasing voltage, graphene conductivity can be flexibly tuned in the terahertz and infrared frequencies. With the tunable characteristic, graphene-based metamaterial absorber and reflectarray have been designed. Good performance in these examples illustrates that graphene promises sufficient flexibility in the design of metamaterial devices.
- conductivity model
- equivalent circuit model
Metamaterial [1, 2, 3, 4, 5, 6] has attracted much attention in the scientific communities over the past 20 years. The metamaterial is a macroscopic composite of periodic or non-periodic structures whose scale is smaller than the wavelength. Metamaterials derive their properties not from the properties of base materials but from their newly designed structures. The property of the subwavelength structure in the metamaterial can be described by effective medium parameters [7, 8, 9, 10, 11, 12] including electric permittivity and magnetic permeability. The design of the structure allows effective medium parameters to be tailored to special parameters, for example, near-zero values and negative values, and the resultant metamaterials could flexibly manipulate the behavior of electromagnetic waves in ways that have not been conventionally possible [13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. For example, the well-known double negative materials, which was proposed by Veselago , can support the backward propagating waves , and near-zero material can tunnel electromagnetic waves through very narrow channels .
In recent years, metasurfaces have caused huge research interest [26, 27, 28, 29, 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 results in generalized Snell’s transmission and reflection laws .
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, 37, 38, 39], cloaks , filters , antennas [42, 43], nonlinear optical devices , etc. In this chapter, we first review the conductivity model and equivalent circuit model of graphene, respectively. Next, a graphene-based 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 generate an orbital angular momentum (OAM) vortex wave in terahertz.
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 and a static magnetic field . The electron in graphene is accelerated along −x direction by the electric field force . As the electron moves with a speed , the magnetic field force is generated to deflect the electron toward −y direction. Therefore, the induced current has two components, i.e., . Similarly, in the case of an electric field and a static magnetic field , the induced current becomes . Hence, an interesting property of graphene is magnetically induced gyrotropy which can be stated as
in which is the Fermi velocity, is Fermi-Dirac distribution, is the energy of the nth Landau level, is chemical potential, is the scattering time, is the charge of an electron, is the reduced Planck’s constant, is temperature, and is Boltzmann constant.
in which is radian frequency. Note that Eqs. (5) and (6) are valid under the condition . Specially, in the absence of biased magnetic field, we have 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 :
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.
It can be seen from Eq. (8) that for and , becomes a purely imaginary number with a positive imaginary part. In the case of and , is a complex number, whose imaginary part is still positive and real part is for . 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 meV and 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 intra-band 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.
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 is related with via the following expression 
in which and 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 and Vg can be obtained. An approximate close-form expression to relate to Vg can be given as 
Figure 5 shows the conductivity variation of isotropic graphene with chemical potential. Here meV and 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.
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  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 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:
Furthermore, the S parameters can be obtained as:
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 gi (gi < Di), and permittivities of the material slabs at the top and bottom part of the graphene array are εi and εi + 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 and the average surface current density induced on it. The average boundary condition for the patch array can be written as [49, 50, 51]
Here, σi is the conductivity of graphene and εeq is equivalent permittivity with the following expression:
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 Zgi = 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 full-wave 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.
4. Graphene-based metamaterial absorber
With extraordinary electronic and optical properties, graphene has caused enormous research interest in recent years. The conductivity or carrier density of graphene can be tuned by the chemical potential via an external gate voltage. Various intriguing applications such as tunable cloaks , reflectarray [42, 43], nonlinear optical devices , etc., have been proposed and experimentally demonstrated. On the other hand, since Landy et al. proposed a thin and near-perfect metamaterial absorber in 2008 , various metamaterial absorbers have been demonstrated from microwave to optical frequencies [53, 54]. The fascinating property of the tunable conductivity promises graphene a good candidate for the design of the tunable metamaterial absorber. In this section, a broadband tunable, wide-angle, and polarization-insensitive graphene-based metamaterial absorber is designed.
Figure 11 shows the designed metamaterial absorber. The unit cell of metamaterial absorber with a periodicity of 2 μm consists of four layers from the top to the bottom: a dual ELC unit composed of Au with a thickness of 0.1 μm, a graphene sheet, a BaF2 material with a thickness of 0.24 μm, and an Au material with a thickness of 0.5 μm. Figure 12 shows the absorbing spectra of the proposed absorber for the chemical potential of μc = 0.5 eV when TE- and TM-polarized plane waves are normally incident on the proposed absorber, respectively. It can be observed that for TE polarization, a wide absorption characteristic of 90% with a 41.12% fractional bandwidth from 27.78 to 42.16 THz is obtained. Similarly, the absorbing band for the TM polarization covers from 26.78 to 40.06 THz with a 39.74% fractional bandwidth.
Figures 13 and 14 illustrate the polarization and angular dependences of the proposed absorber with μc = 0.5 eV, respectively. As shown in Figure 13, when the incident angle varies from 0 to 60° for both TE and TM polarizations, the absorptions remain above 80% in the whole operating band. Especially, for incident angle below 50°, the absorption over 90% can be achieved. Due to the approximate symmetry of the designed absorber, the absorption is nearly independent of polarization, as shown in Figure 14.
Figure 15 demonstrates absorption variation of the proposed absorber with the chemical potential of graphene. For the TE polarization, the fractional absorbing band of the absorption of 90% with μc = 0.2 eV is 44.8% from 25.08 to 39.56 THz. With the increase of the chemical potential, the absorption curve has a blue shift accompanied by an approximately unchanged fractional band. By adjusting the chemical potential from 0.2 to 0.8 eV, the absorbing band continuously covers from 25.08 to 44.81 THz. By comparison, for the TM polarization, the fractional absorbing bandwidth firstly increases and then decreases as the chemical potential increases. With the variation of the chemical potential from 0.2 to 0.8 eV, the absorbing band of the absorption of 90% continuously covers from 25.74 to 40.06 THz.
5. Graphene-based metamaterial reflectarray for orbital angular momentum (OAM) vortex wave
The electromagnetic waves carry both linear and angular momentums. Angular momentum comprises spin angular momentum (SAM) and orbital angular momentum (OAM). The SAM is associated with the circular polarization states of electromagnetic beams. The OAM arises from spatial variations of amplitude and phase that render the beam asymmetric around its propagation axis [55, 56]. In 1992, Allen et al. found that light beam with an azimuthal phase dependence of exp.(ilϕ) carries an OAM, in which ϕ represents the azimuthal angle and l is the topological charge. For any given l, the OAM vortex wave has l interwinded helical phase fronts and a phase singularity with zero intensity on the beam axis. With a theoretically unlimited range of orthogonal eigenstates, OAM offers new degrees of freedom in communication in addition to traditional linear momentum and polarization degrees of freedom [57, 58, 59, 60].
An attractive feature of graphene is that its conductivity is changeable by controlling voltage applied to graphene via an external gate. With this characteristic, a graphene-based metamaterial reflectarray is designed for the generation of the wideband OAM vortex waves with tunable modes in this section. As shown in Figure 16, the designed reflectarray with a size of 10× 10λ comprises 12 regions, each of which has the same azimuthal angle. Here, λ is wavelength in free space at the frequency of 2.3 THz. In each region, a same graphene-based metamaterial structure is designed. By tuning the conductivities of the graphene sheets in the jth region, the reflection phase of πlj/6 (j = 1, …,12) is achieved such that the whole reflectarray can generate a helical profile of exp.(ilϕ). To guarantee independently adjustable conductivities of the graphene sheets in each region, a small height difference ∆h is introduced to ensure the insulation between two adjacent regions, as shown in Figure 16(b).
To obtain a desirable reflection phase in each region, a metamaterial unit cell composed of a three-layer sandwich structure has been designed, as shown in Figure 17. Each sandwich structure comprises graphene/Al2O3/SiO2 materials from the top to the bottom. An insulating layer of Al2O3 material is inserted between two adjacent sandwich structures. A ground consisting of an Au material is placed at the bottom of the unit cell. The designed unit cell periodicity is P = 20 μm, and the thicknesses of SiO2, Al2O3, and Au materials are 12 μm, 10 nm, and 5 μm, respectively. In each sandwich structure, an external DC voltage is applied between the graphene layer and the SiO2 layer to control the conductivity of graphene. For convenience, the chemical potentials of the graphene layers from the top to the bottom are denoted as “μc1,” “μc2,” and “μc3,” respectively. With the equivalent circuit model given in Figure 17(b), the maximum reflection phase range of the proposed unit cell can be obtained. As shown in Figure 18, the reflection phase range of the proposed unit cell can cover 360o in a wide frequency band from 1.8 to 2.8 THz, when three chemical potentials independently vary from −1 to 1 eV.
A wideband horn antenna as the excitation is used to generate a wave incident on the reflectarray. Figure 19 shows the OAM vortex waves with different modes generated by the reflectarray at 2.3 THz. We can clearly observe the spiral phase distributions of the OAM vortex waves with l = ±1, l = ±2, and l = ±3 modes and the doughnut-shaped intensity distributions. Note that the radiation patterns with singularity in the center greatly reduce the coupling between the reflectarray and the horn antenna. With the reflection phase range of 360° in a wide frequency band from 1.8 to 2.8 THz, as shown in Figure 18, the OAM vortex waves can be generated by the proposed reflectarray in the wide frequency band, as shown in Figure 20. It is observed that the desirable spiral phase distributions can be obtained in the whole frequency band.
In sum, several characteristics of graphene including the conductivity model and equivalent circuit model have been presented. Two graphene-based devices, i.e., metamaterial absorber and metamaterial reflectarray, have been designed. By varying graphene’s chemical potential, the wideband tunable absorption for the designed absorber and the broadband tunable OAM modes for the developed reflectarray have been demonstrated, respectively. Graphene provides more degrees of freedom for the design of the tunable metamaterial systems.
This work is supported by National Natural Science Foundation of China under Contract 61771359 Fundamental Research Funds for the Central Universities (No. JBF180202), and Technology Innovation Research Project of the CETC.
Conflict of interest
No conflict of interest was reported by the authors.