An Analytical Method to Investigate Propagation Properties of Magnetostatic Biased Graphene Layers

The transmission and reflection properties of magnetostatic biased graphene layers are investigated based on analytical method, where graphene is treated as a bulk material. Considering the anisotropic nature of the magnetostatic biased graphene (magnetized graphene), the generalized reflection and transmission coefficients of a plane wave incidence on anisotropic multilayer structures are first derived. The accuracy of this method is validated by a numerical example of a plane wave incidence on a magnetized monolayer graphene in free space. Based on this model, the transmission coefficient decreases with increasing the incident angle for the TE (transverse electric) polarized wave incidence, while an opposite phenomenon occurs for the TM (transverse magnetic) polarized wave incidence. The transmission properties can be also modulated by changing the external static magnetic field. The electromagnetic propagation properties of composites consisting of magnetized graphene and dielectric layers are also analyzed based on this method. The transmission properties degrade with increasing the number of magnetized graphene layers.


I. INTRODUCTION
Graphene has attracted much attention due to its superior electrical, thermal, optical and mechanical properties [1]. Surface conductivity is a key parameter usually used for describing the electromagnetic properties of graphene. Generally, the expression of surface conductivity can be categorized into three forms [2]: (i) in the case of spatial dispersion but with neither electrostatic nor magnetostatic bias, the surface conductivity can be expressed by an operator; (ii) when graphene is biased by a static electric field with neither static magnetic field nor spatial dispersion, the surface conductivity is a scalar and the corresponding graphene can be regarded as an isotropic material; (iii) when graphene is biased by a static magnetic field and also possibly a static electric field (but with no spatial dispersion), the surface conductivity is a tensor. Particularly, in the case of (iii), The associate editor coordinating the review of this manuscript and approving it for publication was Chan Hwang See . graphene is an anisotropic material with gyrotropic and electromagnetic nonreciprocal properties, and thus has a great potential in applications of nonreciprocal and gyrotropic microwave/terahertz devices [3], [4].
So far, two main methods have been reported to establish the electromagnetic model for investigating the interaction between the electromagnetic field and the graphene. In one method, graphene is regarded as a two-dimensional (2-D) conductive sheet [3]- [7]; in the other method, graphene is treated as a bulk material [8]- [14]. Based on the first method with graphene as a 2-D sheet, Sounas and Caloz successfully obtained the analytic expressions of the reflection and transmission coefficients when a plane wave obliquely incidents on a magnetostatic biased graphene sheet in free space [3]. This original work has opened up a new strategy in revealing the fundamentals of gyrotropy and nonreciprocity of graphene. Therefore, it has received much attention. Recently, considering that periodic arrays of magneticallybiased graphene ribbons are one of the most popular graphene-based structures, Memarian et al. proposed an analytical method for the analysis of this structure. Based on integral equations governing the field and the surface current density on magnetically biased graphene ribbons, this new method is precise, reliable, and affordable [7]. It is noted that it is not easy to experimentally prepare monolayer graphene and alternatively graphene used in practice usually consists of several layers. The above facts suggest that it is more suitable for graphene to be modeled as a bulk material than a 2-D sheet under this circumstance.
On the other hand, besides the interaction between the electromagnetic field and the surface of graphene, the propagation of the electromagnetic wave inside graphene also needs to be considered when graphene is regarded as a bulk material. This significantly enhances the difficulty of analytic modeling, and thus leads to a fact that the numerical algorithms (e.g. the finite-difference time-domain (FDTD) algorithm) are mainly adopted when graphene is regarded as a bulk material [8]- [11]. However, the ultra-thin nature of the graphene would inevitably increases the numbers of grids (and thus degrading the simulation efficiency) for the conventional FDTD algorithm. To address this issue, many improved FDTD algorithms (e.g. unconditionally stable and weakly conditionally stable FDTD algorithms) have been developed, which can improve the simulation efficiency without sacrificing the simulation precision [8]- [10]. However, the numerical algorithms reported in the literatures only solved the simple problem of normal plane wave incidence on a monolayer graphene, and the problems associated with oblique plane wave incidence on multilayer graphene are rarely solved due to computational efficiency.
To the best of our knowledge, little effort has been made to provide an effective method to study the electromagnetic transmission and reflection properties of magnetized graphene layers when the graphene is modeled as a bulk material. Therefore, this paper offers the analytical formulations of the generalized transmission and reflection coefficients of a plane wave incidence on anisotropic multilayer structures based on the transfer matrix method (TMM). It is noted that, this work mainly focuses on the applications of graphene in the fields of electromagnetic shielding (EMC) and absorbers, therefore, the linearly polarized wave incidence, rather than the circularly polarized wave incidence is utilized in this work [12]- [14]. Then the electromagnetic propagation properties of both magnetized monolayer graphene and composites consisting of magnetized graphene layers and dielectrics can be analyzed.
The paper is organized as follows. Firstly, the problem of an oblique plane wave incidence on an anisotropic multilayer structure is solved to infer the generalized reflection and transmission coefficients. Secondly, a numerical example of a plane wave incidence on a magnetized monolayer graphene in free space is calculated to validate the accuracy of this modeling method by comparison with the work in the literature [3]. The influences of incident angle and external magnetic field on the propagation properties of this graphene layer are discussed. Finally, the paper provides numerical examples of oblique plane wave incidence on composites consisting of magnetized graphene layers and dielectrics, and analyzes the reflection and transmission properties of these structures.

A. ELECTROMAGNETIC PARAMETERS OF THE MAGNETIZED GRAPHENE
When a graphene layer is biased with a perpendicular static magnetic field B 0 , it exhibits anisotropy and its surface conductivityσ is a tensor as discussed earlier and can be described by the following equation In the above equation, the diagonal conductivity σ xx , σ yy and the off-diagonal conductivity σ xy , σ yx can be expressed in the Drude model form according to [3], [15] where e is the single electron charge, τ is the scattering time, h is the reduced Planck's constant, k B is the Boltzmann constant, T is the operating temperature. µ c is the chemical potential, ω c is the cyclotron frequency and v F is the Fermi velocity. Formulas (2)-(5) are valid under the conditions of µ C k B T and µ C h ω [16], [17]. The permittivity and permeability of the magnetostatic biased graphene can be expressed bȳ where where ε 0 is the vacuum permittivity and µ 0 is the vacuum permeability.

B. PLANE WAVE IN MAGNETIZED GRAPHENE
The time harmonic form of the Maxwell's equations in a passive, lossless and anisotropic material is described as For an anisotropic material, the TE and TM polarized waves would couple with each other, that is, when the material is illuminated by a TE (or TM) polarized wave, both TE and TM modes exist in the reflected and transmitted waves simultaneously. Consequently, the wave equations in anisotropic materials are vector equations, which can be easily solved by using the state variable method [18]. In order to facilitate using the boundary conditions, we deduce the state equations of the transverse components of the electric and magnetic fields (suppose that the propagation direction of the plane wave is parallel to the z axis in the Cartesian coordinate system).
By decomposing the two curl equations of the Maxwell's equations into the transverse and vertical components respectively and eliminating the vertical components, we get the following state equation where V = [E x , E y , H x , H y ] T is the state variable, and H is a 4 × 4 matrix related to the parameters of frequency, permittivity, permeability and transverse wave number. H is also called the coupling matrix, which can be described as In the above equation, k x = ksinθ cosϕ and k y = ksinθ sinϕ are the transverse wave numbers, where k is the amplitude of wave vector, θ is the pitch angle and ϕ is the azimuth angle.
According to the boundary conditions, transverse components of electric and magnetic fields are continuous at the interface, so equation (14) is a first-order ordinary differential equation. Its general solution is inserting (16) into (14) yields which can be ascribed to an eigenvalue problem. Since H is a 4 × 4 square matrix, (17) has four eigenvalues and four eigenvectors. So the general solution described by (16) has the following form where A i is amplitude, β i is eigenvalue and a i is the eigenvector for the eigenvalue β i . The four components represent the TE and TM polarized wave propagating along the z and -z directions respectively. Suppose an infinite magnetized graphene layer with its two surfaces perpendicular to the z axis, when a plane wave obliquely incident on the graphene in the xoz plane, the coupling matrix H becomes By substituting (19) into (17), the eigenvalues and eigenvectors are where By substituting the eigenvalues and eigenvectors into (18), the electromagnetic field in magnetized graphene can be VOLUME 8, 2020 This arrangement means that the first two diagonal elements represent electromagnetic wave propagating along the -z direction, while the latter two diagonal elements represent electromagnetic wave propagating along the z direction.

III. REFLECTION AND TRANSMISSION PROPERTIES OF ANISOTROPIC MULTILAYER STRUCTURES
Regarding a multilayer structure with N layers of anisotropic materials in Fig.1, the generalized reflection coefficient (the ratio of the reflected field with the incident field) and transmission coefficient (the ratio of the transmitted field with the incident field) can be solved by using the transfer matrix method [18].

A. THE GENERALIZED REFLECTION COEFFICIENT
For the vacuum adjacent to the left side of the 1st layer (denoted as layer 1) in Fig.1, the state variable of electromagnetic field is The relationship between the electromagnetic fields in the vacuum and the 1st layer material at the interface z = z 0 can be expresses as

B. THE GENERALIZED TRANSMISSION COEFFICIENT
For the vacuum adjacent to the right side of the N th layer in Fig. 1, the state variable of the electromagnetic field is The relationship between the electromagnetic fields in the N th layer material and the vacuum at the interface z = z N can be expresses as where T 0N is the generalized transmission coefficient matrix. For transmitted wave of the TE and TM modes propagating along the z direction, A 3(N +1) and A 4(N +1) represent their amplitude respectively. Considering that there is no reflection wave along the -z direction in the vacuum, the state variable at z = z N can be written as The state variable of (23) in layer m can be rewritten as whereP m (z, z ) =ā m · e jβ m (z−z ) ·ā −1 m is the transfer matrix from z to z in layer m.
For layer m, the electromagnetic field at interfaces z = z m−1 and z = z m can be linked through According to the boundary conditions that transverse components of the electromagnetic field are continuous at the interface, the following equation can be obtained thus, whereP(z N , z 0 ) = 1 m=NP m (z m , z m−1 ) is the transfer matrix from z = z 0 to z = z N .
Substituting (27) and (30) into (34) yeilds A schematic of a plane wave incidence on a magnetized monolayer graphene in free space.
define  Fig. 2 shows the schematic of an infinite monolayer graphene in free space with its two surfaces perpendicular to the z axis. The graphene is biased by a static magnetic field B 0 which is along the z-axis direction. A plane wave incident on the graphene in the xoz plane with an incident angle θ . In this case, the transfer matrix of the graphene layer can be written asP where z 1 − z 0 = 0.34nm is the thickness of a single graphene layer [19]- [21]. When T = 300K , µ c = 0.117eV and B 0 = 2T , we first calculate the transmission coefficient of the magnetized graphene layer with different incident angles for both TE and TM polarized wave incidences. To validate the accuracy of our method, we also calculate the transmission coefficient by using the following formulation [3]: where η 0 = √ µ 0 /ε 0 is the wave impedance of free space, Z e = k z /(ωε 0 ) and Z h = ωµ 0 /k z are the wave impedance of the E(TM) and H (TE) polarized waves respectively. Comparison between these two results is shown in Fig. 3.
Graphene is modeled as a 2-D conductive sheet in [3], while it is treated as a bulk material in our work. The following conclusions can be obtained according to Fig. 3: (1) firstly, the results in our work coincide well with those  reported in [3] for both normal incidence and oblique incidence, thus validating the accuracy of our method. (2) Secondly, when the plane wave is normally incident, the transmission coefficient is equal for both TE and TM wave incidence, and thus the four curves (denoted as TE_0 o _ref [3], TE_0 o _our, TM_0 o _ref [3], TM_0 o _our) overlap.
(3) Thirdly, for the TE polarized wave incidence, the transmission property decreases with increasing the incident angle θ , while for the TM polarized wave incidence, it shows an opposite phenomenon. This could be explained as following. For the TE polarized wave, the electric field is always parallel to the graphene surface for all incident angles, and so graphene always interacts with the wave. As θ increases, the interaction between the incident wave and graphene enhances and thus leading to the transmission decreases. For the TM polarized wave, the component of the electric field parallel to the graphene surface decreases with increasing the incident angle θ , and the field becomes completely perpendicular to the graphene surface as θ is 90 o . Therefore, the interaction between the incident wave and the graphene attenuates with increasing the incident angle θ , thus leading to the transmission increases. In addition, all the transmission spectra displays minimum points. This should be due to that the worst impedance matching between the graphene characteristic impedance and the air characteristic impedance occurs at this frequency. VOLUME 8, 2020 FIGURE 5. A schematic of a plane wave incidence on a multilayer structure of alternate graphene and dielectric layers in free space. The transmission coefficient is also influenced by the external static magnetic field B 0 , as shown in Fig. 4. When B 0 increases from 0.2T to 2.0T , the transmission coefficient increases obviously for the TE polarized wave incidence. For the TM polarized wave incidence, its transmission coefficient also increases with increasing B 0 .

B. PROPAGATION PROPERTIES OF MULTILAYER STRUCTURES CONTAINING MAGNETIZED GRAPHENE LAYERS
Since the graphene/dielectric composites are widely used in applications, a typical multilayer structure of this composite is shown in Fig. 5, which consisting of alternate graphene and dielectric layers. The thicknesses of each graphene and dielectric layers are d 1 and d 2 respectively, and the relative dielectric constant of the dielectric layer is ε r2 .
The transmission and reflection coefficients of this multilayer structure with different number of layers (denoted as N )  under the TE polarized wave incidence are shown in Fig. 6. Here, B 0 is 2T , ε r2 is 2.25 [13], d 2 is 1µm, and the incident angle θ is π /6. From the result it can be concluded that the transmission properties degrade while the reflection properties improve with increasing the number of magnetized graphene layers.
When the number of layers is N = 6, the transmission and reflection coefficients versus different incidence angles at 1GHz and 1THz are shown in Fig. 7. It can be found that for the TE polarized wave incidence, the transmission decreases while the reflection increases when the incident angle θ increases, the reason is the same as that of TE and TM waves incidence on a monolayer graphene as mentioned before.
As it mentioned earlier, the graphene in practice is usually consisted of several layers due to the preparation constraint. For instance, the graphene prepared by the supercritical fluid stripping method is about 5 layers. So another composite structure (called the M + 1 structure below) is shown in Fig. 8. This structure is consisted of M continuous layers of magnetized graphene and one single layer of dielectric whose relative dielectric constant is ε r2 and thickness is d 2 . Fig. 9 shows the transmission coefficient and reflection coefficient of this structure for the TE polarized wave incidence with an incident angle θ = π /6 when M takes different numbers. The chemical potential µ c of the M + 1 structure is assumed to be the same as that of the monolayer graphene structure. It can be found that when M increases from 1 to 5, the transmission coefficient declines obviously while the reflection coefficient takes a remarkable increase.

V. CONCLUSION
The transmission and reflection properties of magnetized graphene layers with graphene treated as a bulk material are studied. The accuracy of the proposed analytical method is validated by the numerical example. The method can be also used to analyze electromagnetic propagation properties of structures containing other anisotropic materials thus has a great potential for applications.