IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 53, NO. 1, FEBRUARY

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1 IEEE JOURNAL OF QUANTUM ELECTRONIC, VOL. 53, NO. 1, FEBRUARY Deriving urface Impedance for 2-D Arrays of Graphene Patches Using a Variational Method aeedeh Barzegar-Parizi, Mohammad Reza Tavakol, and Amin Khavasi Abstract In this paper, we extract the fundamental resonant mode of a graphene patch using a variational method. We use 2-D eigenvalue problem obtained from the integral equation governing the surface current on graphene patterns under quasistatic approximation. To compute the eigenvalues, we propose three trial eigenfunctions, which meet the boundary conditions. We investigate the accuracy of these eigenfunctions with comparing to the results obtained by full wave simulations. Finally, we analyze square-lattice arrangements of graphene patches using the most accurate proposed eigenfunction and derive a very accurate surface impedance for it. The proposed surface impedance is much more precise than the surface impedance already reported by Padooru et al. for this structure. Index Terms Graphene patch, surface impedance, fundamental mode, variational method. I. INTRODUCTION GRAPHENE, a single layer of carbon atoms arranged in a two-dimensional honeycomb lattice, has been introduced as an exclusive material due to its extraordinary properties of electrical and thermal conductivity [1], high density and mobility of charge carriers, optical conductivity [2] and controllable plasmonic properties [3] [5]. These properties make the graphene structures as potential candidates in many applications. For example, periodically patterned graphene surfaces appeared in one-dimensional (graphene ribbons [6]) and twodimensional (graphene patches [7] and disks [8]) arrangements possess dual inductive-capacitive nature. This property leads to many interesting applications such as tunable planar filters, metasurface conformal cloaks [9] and absorbers [10] [12]. Moreover, multilayered structures composed of these patterned arrays separated by conventional dielectric slabs are interesting due to the existence of band gap. Hyperbolic dispersion could be seen in band gap region where the effective permittivity is appeared as a negative value [13], [14]. In consequence, an accurate and fast analysis of the electromagnetic behavior of these graphene structures is very critical for better understanding of these properties and designing new devices based on them. Recently, accurate and fast analytical solutions have been presented to extract the resonant modes of a single graphene Manuscript received August 24, 2016; revised November 2, 2016; accepted December 6, Date of publication December 21, 2016; date of current version January 12, Barzegar-Parizi is with the Electrical and Computer Engineering Department, irjan University of Technology, Kerman , Iran ( barzegarparizi@sirjantech.ac.ir). M. R. Tavakol and A. Khavasi are with the Electrical Engineering Department, harif University of Technology, Tehran , Iran ( khavasi@sharif.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /JQE ribbon, a graphene disk and the arrays of them in [6] and [8]. In these papers, by using a quasi-static approximation, the integral equation governing the surface current on graphene patterns is described as an eigenvalue problem. Then, the eigenvalue problem is solved by the expansion of the eigenfunctions as the series of sinusoidal (for graphene ribbons)/bessel functions (for graphene disks). Another important structure is a graphene patch that can be used in designing graphene-based antennas in Terahertz band for wireless communication systems [15]. Moreover, in its array form, it can be used in designing absorbers [10] [12]. It is not an easy task to find an analytical solution for a single graphene patch or its array form. One may employ numerical methods such as method of moments to solve the integral equation governing the surface currents. However, these methods are usually time-consuming and memory hungry. In this paper, we present a simple analysis to extract the fundamental mode of a graphene patch based on a variational method. We use the two dimensional integral equation governing the surface current on the graphene under quasistatic approximation, in the form of an eigenvalue problem [8]. Using the method of separation of variables, we write the two dimensional eigenfunction as a product of two one dimensional functions. In the variational method, trial approximate eigenfunctions that meet boundary conditions and any other physical constraints, are selected. In this paper, we propose three forms of trial functions and investigate their accuracy in extracting the fundamental resonant mode of graphene patch. We assume that the current vector is along the x-direction. As the first trial function, we consider a uniform distribution for the y-direction, and a sinusoidal distribution for the x-direction. The sinusoidal function is appropriately selected to meet the boundary conditions. This eigenfunction is the simplest one, considered here to extract the eigenvalues corresponding to the resonant frequencies of graphene patch with an acceptable accuracy. However, one can consider the graphene patch as a truncated graphene ribbon. Thus, the second trial function can be defined based on the eigenfunction of a single graphene ribbon [6] in x-direction, instead of the sinusoidal function. This trial function is more complicated however leading to more accurate results. The third trial function is similar to the second one but it has a non-uniform (hyperbolic cosine) distribution along the y-direction. This trial function is the most accurate although the most complicated one. Finally, we present an equivalent surface impedance for square lattice arrays of graphene patches by using the third trial function and including the effect of coupling between the patches. It is demonstrated that the proposed impedance is more accurate than the surface impedance already reported IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. ee for more information.

7000106 IEEE JOURNAL OF QUANTUM ELECTRONIC, VOL. 53, NO. 1, FEBRUARY 2017 Fig. 1. A graphene patch with dimensions w x and w y which is subject to a normally incident plane-wave whose electric field is in the x-direction.

2 IEEE JOURNAL OF QUANTUM ELECTRONIC, VOL. 53, NO. 1, FEBRUARY 2017 Fig. 1. A graphene patch with dimensions w x and w y which is subject to a normally incident plane-wave whose electric field is in the x-direction. in [7]. To validate our approach, the results are compared with those obtained by full-wave simulations carried out by a commercial software (HF). This paper is organized as follows. In section II, the formulation of the problem and the proposed trial eigenfunctions corresponding to the fundamental mode are presented. The eigenvalues are then computed for these trial functions. In section III, we present an equivalent surface impedance for array of graphene patches and compare the results with those obtained by full-wave simulations carried out by the HF in section IV. Finally, the paper will be concluded in section V. II. THE FUNDAMENTAL MODE OF A GRAPHENE PATCH Consider a single graphene patch with dimensions w x and w y, subject to a normally incident plane wave whose electric field is in the x-direction (Fig. 1). The patch is surrounded by free space. The surface conductivity of the graphene patterns (σ s ) can be derived using the well-known Kubo formula [16] as: σ s = 2e2 k B T π ħ 2 j ω + jτ 1 ln [2 cosh (E F /2k B T )] [ 2EF ħ(ω jτ 1 ] ) 2E F + ħ(ω jτ 1 ) je2 4πħ ln (1) where e is the electron charge, E F is the Fermi energy, h is the Plank constant, ω is the frequency, T = 300K is the temperature and τ is the relaxation time. According to [8], the eigenvalue problem of a two-dimensional graphene pattern under quasi-static approximation can be written as: T T.ξ(ρ ) 4π ρ ρ d = λξ(ρ) (2) where ρ = x ˆx + yŷ, T =ˆx / x +ŷ / y and ξ(ρ) is the eigenfunction corresponding to eigenvalue λ which describes the current distribution on the graphene pattern. In (2), ξ can be written as the gradient of a scalar function [8]: ξ = T ψ. o, the scalar form of the eigenvalue problem appears as: 2 T ψ(ρ ) 4π ρ ρ d = λψ(ρ) (3) Based on the operator defined in (3), the eigenvalues can be computed as: <ψ H ψ > λ = (4) <ψ ψ > with 2 H [ ] = T [ ] dx dy 4π (5) (x x ) 2 + (y y ) 2 Thus, the eigenvalues can be extracted by calculating the following integral relation: λ = ψ(x,y) 2 T [ψ(x,y )] 4π (x x ) 2 +(y y ) 2 dx dy dxdy [ψ(x, y)] 2 dxdy Given the eigenvalues, the resonant wavelengths can be obtained by the relation 2λ = Re[k p (ω)] [8] in which k p (ω) = 2jωε ef f /σ s is the wavenumber of the plasmonic wave propagating on an infinite sheet of graphene. The aim of this work is the extraction of the fundamental mode of the graphene patch using a variational approach. In the variational method, appropriate trial eigenfunctions should meet the boundary conditions. According to the polarization of the incident wave, the current vector is assumed to be along the x-direction. The normal component of the surface current on the edges of graphene has to be zero. Therefore, the boundary condition which should be satisfied is: ψ(x, y) ξ x (x, y) = x x=±wx /2 = 0 (7) Using the method of separation of variables, we write the two dimensional eigenfunction of the fundamental mode as a product of two one-dimensional functions as ψ(x, y) = (x) (y). Three forms of trial eigenfunctions, all satisfying equation (7), are investigated for this mode as tabulated in Table.I. As the first trial eigenfunction, we consider a simple trial eigenfunction leading to simpler calculations. A uniform distribution is assumed for the y-direction ( 1 (y) = 1) while for the x-direction, a sinusoidal distribution ( 1 (x) = sin(π x/w x )) is appropriately selected to meet the boundary condition given in (7). The eigenvalue computed by substituting the trial function in (6) is compared with the one extracted by a full-wave simulation. As given in Table. I the error is 4.2%. For the second trial eigenfunction, again a uniform distribution for the y-direction ( 2 (y) = 1) is assumed. However, we use the integral of distribution of the first mode of a single graphene ribbon [6] for the x-direction, because the graphene patch considered here can be assumed as a truncated graphene ribbon. Note that, according to (6), the derivative of (x) should be similar to the eigenfunction of graphene ribbon. Therefore, we have ( wx ) 2 (x) = ( ) ( ) 2x 2x sin 1 ( 2x ) 2 2 w x w x w x ( wx ) x ( ( ) ) 2x 2 3/2 1 2 w x w x (6)

3 BARZEGAR-PARIZI et al.: DERIVING URFACE IMPEDANCE FOR 2-D ARRAY OF GRAPHENE PATCHE TABLE I THREE TRIAL APPROXIMATE EIGENFUNCTION FOR FUNDAMENTAL MODE OF GRAPHENE PATCH AND THE EIGENVALUE EXTRACTED.THE EXACT EIGNEVALUE I π/w x As observed in Table I, the second trial function leads to a much more accurate result (an error of 0.73%). The third trial function is inspired from the current distribution calculated by full-wave simulations (HF). Fig. 2 (a) shows the surface current distribution of a graphene patch with dimensions w x = w y = 0.8 μ m obtained by a fullwave simulation at fundamental mode (free space wavelength of 36.3μ m corresponding to eigenvalue π/w x ).The charge relaxation time and Fermi energy are assumed to be given by τ = sec, and E F = 0.2eV, respectively. As it is obvious, the surface current density has a non-uniform distribution in the y-direction. Fig. 2 (b) and 2 (c) show the current distribution on the x and y- axes, respectively, obtained by HF (circles). It is obvious from this figure that the current distribution in the y-direction is similar to a hyperbolic cosine function. This can be also understood based on the edge guided modes in graphene ribbons where the current exponentially increase in edges [17], so two coupled edge modes should have hyperbolic cosine distribution. On the other hand, Fig. 2(b) demonstrates that the current distribution in the x-direction is very similar to the one obtained from the second eigenfunction. Therefore, we choose the third trial function similar to the second one in the x-direction but assume a hyperbolic cosine distribution along the y-direction as: ( 3 (y) = cosh α 2y ) w y The fitting parameter α 0.67 is calculated by the best fit to the current distribution obtained from full wave simulation results. This trial function is the most accurate (see Fig. 2(b) and (c)) although the most complicated one. For a single patch this parameter is fixed, but it can be slightly altered due to coupling effects in an array of patches. Nevertheless, we will use first-order perturbation for considering the coupling effects and thus we neglect the change in the current distribution. As already stated the fundamental mode is only considered in this paper. For higher order modes we can use the higher order eigenfunctions of graphene ribbons in the x- direction. We also note that the second order mode in the y-direction may be approximated by a hyperbolic sine function. Moreover, for higher order modes (in the y-direction) cosine-like functions may be well fitted to the sought-after eigenfunctions. However, accurate estimation of higher order modes requires further investigations which are out of the scope of this work. In the next section, we analyze an array of graphene patches based on this eigenfunction. III. URFACE IMPEDANCE FOR ARRAY OF GRAPHENE PATCHE In this section, we investigate a periodic array of graphene patches with periods L x and L y in the x- andy-directions, respectively (Fig. 3). Thanks to the approximate surface current obtained in the previous section, an equivalent surface impedance is extracted for this array. We show that this impedance is more accurate than the one reported in [7]. As described in [6] and [8], for the periodic array of graphene patches, the interaction between the patches appears as a shift in the eigenvalues of a single graphene patch. Therefore, neglecting the effect of the perturbation on the eigenfunctions, the eigenvalues of the array of graphene patches are written as [8]: 1 q = λ + ξ(x, y) 2 d T.ξ(x, y ) T.ξ(x, y) d (p,q) =0 4π (x d x pl x ) 2 +(y y ql y ) 2 (8) with ξ defined in (7) and q is the shifted eigenvalue. uppose the array is surrounded by two semi-infinite homogenous media with permittivities of ε 1 and ε 2 placed above and below the array, respectively. One may extract the amplitude of the diffracted orders in the semi-infinite homogenous media by using the Rayleigh expansion of the electromagnetic fields inside homogenous media and applying the proper boundary conditions (the same procedure has been followed in [6] and [8]). Assuming sub-wavelength regime, only the zeroth order mode is propagating. Thus, the problem can be modeled as two transmission lines with the characteristic impedances of Z i = μ/ε i (i = 1, 2 ) equivalent with the semi-infinite homogenous media where the interface between the two transmission lines (array of graphene patches) has been

7000106 IEEE JOURNAL OF QUANTUM ELECTRONIC, VOL. 53, NO. 1, FEBRUARY 2017 Fig. 3. An array of graphene patches with periods L x and L y in the x- and y-direction, respectively.

However, there are some important differences: the eigenvalue q, the coefficients 1 and K 1 which are calculated by the proposed trial eigenfunctions and the geometrical constant L x L y.

4 IEEE JOURNAL OF QUANTUM ELECTRONIC, VOL. 53, NO. 1, FEBRUARY 2017 Fig. 3. An array of graphene patches with periods L x and L y in the x- and y-direction, respectively. where ε ef f = (ε 1 +ε 2 )/2. The expression obtained here is very similar to that already reported for graphene ribbons [6]. However, there are some important differences: the eigenvalue q, the coefficients 1 and K 1 which are calculated by the proposed trial eigenfunctions and the geometrical constant L x L y. In next section, we present some numerical examples to examine the accuracy of the proposed surface impedance. The validation is done by comparing the results with those obtained from full-wave electromagnetic simulations carried out by the commercial software HF. It should be noted that we employ the third eigenfunction in (10). IV. NUMERICAL REULT In this section, some numerical examples are presented for investigation of the accuracy of the proposed surface impedance for the analysis of periodic arrays of graphene patches. First, we plot the surface impedance of the array and then as second example, the scattering problem is studied. Consider a square lattice array of square graphene patches with period of L x = L y = L = 8μm and width of w x = w y = w = 7.2μm in free space. The charge relaxation time and Fermi energy are assumed to be given by τ = sec, and E F = 0.2eV, respectively. Fig. 4 shows the proposed surface impedance and the surface impedance reported in [7] as: Z s = L π j { } (11) wσ s 2ωε ef f L ln csc( π(l w) 2L ) Fig. 2. urface current distribution for the fundamental mode of the graphene patch on (a) patch surface obtained by HF (b) the x-axis (c) the y-axis. modeled by a shunt surface impedance of Z s. This equivalent surface impedance is computed as: ( Z s = L x L y σs 1 + q ) K1 (9) jωε ef f with 1 = ξ x (x, y) d, K 1 = s s 2 1 ξ x (x, y) 2 d (10) As it is observed, for low terahertz frequencies, the array of graphene patches is capacitive while it is inductive for higher frequencies where the transition point occurs at resonant frequency. To examine which of the surface impedances plotted in Fig. 4 is more accurate, we compare the results obtained by these surface impedances with those obtained by HF in Fig. 5. We plot the absorption spectra for the array presented in Fig. 4. The dots show the HF result. The results show that the proposed surface impedance is in an excellent agreement with the full-wave simulations, while the results produced by the surface impedance of [7] have considerable discrepancies. The reason behind this considerable discrepancy is that the surface impedance presented in [7] is obtained by a heuristic approach. In this approach, the graphene patch is assumed to behave like a metallic patch but with a finite conductivity. Therefore, the expression already available for subwavelength array of metallic patches is used with slight modification which takes into account the conductivity of the graphene. In our paper, however, the problem is treated with a mathematically more accurate approach although with some

5 BARZEGAR-PARIZI et al.: DERIVING URFACE IMPEDANCE FOR 2-D ARRAY OF GRAPHENE PATCHE As a final remark, it is worth mentioning that, the eigenvalue corresponding to this array is π/w, that shows a considerable shift due to the interaction between the patches in comparison with the eigenvalue of a single graphene patch ( π/w). V. CONCLUION In this paper, the fundamental resonant mode of a graphene patch using a variational method was extracted. Twodimensional eigenvalue problem, under quasi-static approximation, has been employed with three trial eigenfunctions which satisfy the boundary conditions. We investigated the accuracy of the eigenvalues obtained by these eigenfunctions. Finally, a periodic array of graphene patches was analyzed and an accurate equivalent surface impedance was derived by using the most accurate trial eigenfunction and including the effect of the coupling between the patches. We demonstrated that the obtained surface impedance is much more accurate than the one reported in [7]. REFERENCE Fig. 4. (a) real and (b) imaginary part of the surface impedance of a periodic array of graphene patches with period L = 8μm and width w = 7.2 μm as a function of frequency. Fig. 5. Absorption spectra of a periodic array of graphene patches with period L = 8μm and width w = 7.2μm as a function of frequency. reasonable approximations due to the subwavelength nature of the problem. It should be noted that although the surface impedance proposed in this work is much more accurate than the one presented in [7], the surface impedance of [7] is presented in a simple closed-form expression while the integrals in our proposed formula have to be carried out numerically. [1] Q. Bao and K. P. Loh, Graphene photonics, plasmonics, and broadband optoelectronic devices, AC Nano, vol. 6, no. 5, pp , Apr [2] F. Wang et al., Gate-variable optical transitions in graphene, cience, vol. 320, no. 5873, pp , Apr [3] F. H. L. Koppens, D. E. Chang, and F. J. G. de Abajo, Graphene plasmonics: A platform for strong light matter interactions, Nano Lett., vol. 11, no. 8, pp , [4] A. N. Grigorenko, M. Polini, and K.. Novoselov, Graphene plasmonics, Nature Photon., vol. 6, pp , Nov [5] T. Low and P. Avouris, Graphene plasmonics for terahertz to midinfrared applications, AC Nano, vol. 8, no. 2, pp , [6] A. Khavasi and B. Rejaei, Analytical modeling of graphene ribbons as optical circuit elements, IEEE J. Quantum Electron., vol. 50, no. 6, pp , Jun [7] Y. R. Padooru, A. B. Yakovlev, C.. R. Kaipa, G. W. Hanson, F. Medina, and F. Mesa, Dual capacitive-inductive nature of periodic graphene patches: Transmission characteristics at low-terahertz frequencies, Phys. Rev. B, vol. 87, no. 11, p , Mar [8]. Barzegar-Parizi, B. Rejaei, and A. Khavasi, Analytical circuit model for periodic arrays of graphene disks, IEEE J. Quantum Electronics. vol. 51, no. 9, p , ep [9] Y. R. Padooru, A. B. Yakovlev, P.-Y. Chen, and A. Alù, Line-source excitation of realistic conformal metasurface cloaks, J. Appl. Phys., vol. 112, no. 10, p , [10] A. Khavasi, Design of ultra-broadband graphene absorber using circuit theory, J. Opt. oc. Amer. B, Opt. Phys., vol. 32, no. 9, pp , [11] K. Arik,. A. Ramezani, and A. Khavasi, Polarization insensitive and broadband terahertz absorber using graphene disks, Plasmonics, vol. 11, pp. 1 6, Jun [12] M.. Jang et al., Tunable large resonant absorption in a midinfrared graphene alisbury screen, Phys.Rev.B, vol. 90, p , Oct [13] K. V. reekanth, A. De Luca, and G. trangi, Negative refraction in graphene-based hyperbolic metamaterials, Appl. Phys. Lett., vol. 103, no. 2, p , [14]. Barzegar-Parizi, tudy of backward waves in multilayered structures composed of graphene micro-ribbons, J. Appl. Phys., vol. 119, no. 19, p , [15] I. Llatser, C. Kremers, A. Cabellos-Aparicio, J. M. Jornet, E. Alarcón, and D. N. Chigrin, Graphene-based nano-patch antenna for terahertz radiation, Photon. Nanostruct.-Fundam. Appl., vol. 10, no. 4, pp , [16] G. W. Hanson, Dyadic Green s functions and guided surface waves for a surface conductivity model of graphene, J. Appl. Phys., vol. 103, no. 6, p , Mar [17] A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno, Edge and waveguide terahertz surface plasmon modes in graphene microribbons, Phys.Rev.B, vol. 84, no. 16, p , 2011.

6 IEEE JOURNAL OF QUANTUM ELECTRONIC, VOL. 53, NO. 1, FEBRUARY 2017 aeedeh Barzegar-Parizi received the B.c. degree from the Iran University of cience and Technology, Tehran, Iran, in 2008, and the M.c. and Ph.D. degrees from the harif University of Technology, Tehran, in 2010 and 2015, respectively, all in electrical engineering. he has been with the Department of Electrical Engineering, irjan University of Technology, where she is currently an Assistant Professor. Her research interests include the numerical solving of periodic structures, such as periodic rough surfaces and artificial structures. Amin Khavasi was born in Zanjan, Iran, in He received the B.c., M.c., and Ph.D. degrees from the harif University of Technology, Tehran, Iran, in 2006, 2008, and 2012, respectively, all in electrical engineering. He has been with the Department of Electrical Engineering, harif University of Technology, where he is currently an Assistant Professor. His research interests include photovoltaics, plasmonics, and circuit modeling of photonic structures. Mohammad Reza Tavakol received the B.c. degree in electrical engineering from the AmirKabir University of Technology, Tehran, Iran. He is currently pursuing the M.c. degree with the harif University of Technology. His research interests include photonics, plasmonics, and optics.

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