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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 13 — Jul. 1, 2013
  • pp: 15490–15504
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Graphene-based plasmonic switches at near infrared frequencies

J. S. Gómez-Díaz and J. Perruisseau-Carrier  »View Author Affiliations


Optics Express, Vol. 21, Issue 13, pp. 15490-15504 (2013)
http://dx.doi.org/10.1364/OE.21.015490


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Abstract

The concept, analysis, and design of series switches for graphene-strip plasmonic waveguides at near infrared frequencies are presented. Switching is achieved by using graphene’s field effect to selectively enable or forbid propagation on a section of the graphene strip waveguide, thereby allowing good transmission or high isolation, respectively. The electromagnetic modeling of the proposed structure is performed using full-wave simulations and a transmission line model combined with a matrix-transfer approach, which takes into account the characteristics of the plasmons supported by the different graphene-strip waveguide sections of the device. The performance of the switch is evaluated versus different parameters of the structure, including surrounding dielectric media, electrostatic gating and waveguide dimensions.

© 2013 OSA

1. Introduction

The field of plasmonics represents a new exciting area for the control of electromagnetic waves at scales much smaller than the wavelength. It is based on the propagation of surface plasmon polaritons (SPPs) [1

1. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004) [CrossRef] [PubMed] .

3

3. Y. Wang, E. W. Plummer, and K. Kempa, “Foundations of plasmonics,” Adv. Phys. 60, 799–898 (2011) [CrossRef] .

], which are electromagnetic waves propagating along the surface interface between a metal (or a semiconductor) and a dielectric. SPPs are typically obtained in the visible range by using noble metal such as gold or silver [2

2. J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007) [CrossRef] .

], but they are also supported at lower frequencies by composite materials [1

1. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004) [CrossRef] [PubMed] .

, 4

4. J. Elser, A. A. Govyadinov, I. Avrutsky, I. Salakhutdinov, and V. A. Podolskiy, “Plasmonic nanolayer composites: Coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation,” J. Nanomaterials 2007, 79469 (2007) [CrossRef] .

]. Surface plasmons have served as a basis for the development of nanophotonic devices [5

5. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003) [CrossRef] [PubMed] .

], merging the fields of photonics and electronics at the nanoscale [2

2. J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007) [CrossRef] .

] and finding application in different areas such as imaging [6

6. S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Sensor Actuat. B. 3, 388–394 (2009).

] or sensing [7

7. J. Homola, S. S. Yeea, and G. Gauglitzb, “Surface plasmon resonance sensors: review,” Sensor Actuat. B. 54, 3–15 (1999) [CrossRef] .

].

Graphene [8

8. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Mater. 6, 183–91 (2007) [CrossRef] .

] provides excellent possibilities to dynamically manipulate electromagnetic waves [9

9. I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant faraday rotation in single and multilayer graphene,” Nature Phys. 7, 48–51 (2010) [CrossRef] .

11

11. M. Tamagnone, J. S. Gómez-Díaz, J. R. Mosig, and J. Perruisseau-Carrier, “Reconfigurable thz plasmonic antenna concept using a graphene stack,” Appl. Phys. Lett. 101, 214102 (2012) [CrossRef] .

]. Its unique electric properties, which can be controlled by simply applying an external magnetostatic or electrostatic field, allows the propagation of surface plasmons in terahertz and infra-red frequency bands [12

12. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nature Photon. 6, 749–758 (2012) [CrossRef] .

]. Compared to conventional materials, such as silver or gold, SPPs on graphene present important advantages [13

13. F. H. Koppens, D. E. Chang, and F. J. G. de Abajo, “Graphene plasmonics: A plaftform for strong light-matter interactions,” Nano Lett. 11, 3370–3377 (2011) [CrossRef] [PubMed] .

] including tunability, low-losses, and extreme mode confinement. Several authors have studied the characteristics of plasmons propagating along 2D graphene sheets [14

14. G. W. Hanson, “Dyadic green’s functions and guided surface waves for a surface conductivity of graphene,” J. Appl. Phys. 103, 064302 (2008) [CrossRef] .

17

17. J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Propagation of hybrid transverse magnetic-transverse electric plasmons on magnetically-biased graphene sheets,” J. Appl. Phys. 112, 124906 (2012) [CrossRef] .

] and ribbons/strips [18

18. 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 84, 161407 (2011) [CrossRef] .

20

20. J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. G. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6, 431–440 (2012) [CrossRef] .

], and different configurations have already been proposed to enhance their guiding properties [21

21. E. H. Hwang and J. D. Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75, 205418 (2007) [CrossRef]

].

The ability to allow or to forbid the propagation of SPPs on these structures is a key building block for future plasmonic-based devices. Graphene-based switches have already been proposed in the literature at DC [22

22. B. Standley, W. Bao, H. Zhang, J. Bruck, C. N. Lau, and M. Bockrath, “Graphene-based atomic-scale switches,” Appl. Phys. Lett. 8, 3345–3349 (2008).

, 23

23. T. Palacios, A. Hsu, and H. Wang, “Applications of graphene devices in rf communications,” IEEE Commun. Mag. 48, 122–128 (2010) [CrossRef] .

] and microwave frequencies [24

24. K. M. Milaninia, M. A. Baldo, A. Reina, and J. Kong, “All graphene electromechanical switch fabricated by chemical vapor deposition,” Appl. Phys. Lett. 95, 183105 (2009) [CrossRef] .

, 25

25. J. Perruisseau-Carrier, “Graphene for antenna applications: opportunities and challenges from microwaves to thz,” in Antennas and Propagation Conference (LAPC)Loughborough, UK (2012).

], based either on graphene electric field effect or exploiting the electromechanical properties of graphene. In the optical regime, [26

26. Y. V. Bludov, M. I. Vasilevskiy, and N. M. R. Peres, “Mechanism for graphene-based optoelectronic switches by tuning surface plasmon-polaritons in monolayer graphene,” Europhys. Lett. 92, 68001 (2010) [CrossRef] .

] proposed a structure able to switch the reflectance of a plane wave incoming from free-space between two different states, namely total reflection and total absorption. This is obviously a different functionality from the switching of a guided plasmonic wave as concerned here. Finally, in a recent work [27

27. P. Y. Chen, C. Argyropoulos, and A. Alu, “Terahertz antenna phase shifters using integrally-gated graphene transmission-lines,” IEEE T. Antenn. Propag. 61, 1528–1537 (2013) [CrossRef] .

], graphene-based longitudinally homogeneous parallel-plate waveguides were proposed to obtain switches and phase-shifter in the low terahertz band.

In this context, we propose and design series switches able to control the propagation of surface plasmons on finite graphene strips at near infrared frequencies. The structures are composed of a chemically-doped graphene strip, host waveguide of the switch, transferred onto a dielectric and of three polysilicon gating pads beneath the strip. The switch consists of the central section of the host waveguide, whereas the outer sections connect the switch to the input and output ports of the device. Switching is achieved by modifying the gate voltage of the central pad, which in turns controls the guiding properties of the strip in the area of the switch. In the ON state, the whole host waveguide has the same propagating characteristics and the structure behaves as a simple plasmonic transmission line (TL) propagating the incoming energy towards the output port. In the OFF state, the guiding properties of the central waveguide section are modified to provide large isolation between input and output ports. The structures are characterized by applying a transmission line approach and by using the commercial full-wave software HFSS. Note that [27

27. P. Y. Chen, C. Argyropoulos, and A. Alu, “Terahertz antenna phase shifters using integrally-gated graphene transmission-lines,” IEEE T. Antenn. Propag. 61, 1528–1537 (2013) [CrossRef] .

] recently studied graphene-based longitudinally inhomogeneous parallel-plate waveguide using solely TL techniques. Here, we present a rigorous comparison between the TL approach and a full-wave solver for the case of finite graphene-strip waveguides, demonstrating that while the former provides extremely fast results and physical insight into the structure, its mono-modal nature may lead to inaccuracies when characterizing the OFF state of the device. Finally, several devices, based on ideal 2D graphene surfaces and on realistic finite strips, are discussed and studied, evaluating their performance versus different parameters of the switches.

2. Implementation and modeling

This section details the concept, implementation and modeling of the proposed graphene-based switches. We first briefly review the characteristics of surface plasmons polaritons propagating on ideal 2D graphene surfaces [15

15. M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009) [CrossRef] .

] and on finite strips [18

18. 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 84, 161407 (2011) [CrossRef] .

, 20

20. J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. G. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6, 431–440 (2012) [CrossRef] .

]. Then, we describe the proposed switches, detailing their underlying operating principle and discussing its potential technological implementation. Finally, we address the modeling of the different structures, using both a TL approach and full-wave commercial software.

2.1. Characteristics of TM surface plasmon-polaritons

The characteristics of SPPs supported by graphene depend on the conductivity of the material and on the type of waveguide employed. In the case of ideal 2D graphene sheets, the dispersion relation of the propagating modes can be obtained as [15

15. M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009) [CrossRef] .

]
ωεr1ε0εr1k02kρ2ωεr2ε0εr2k02kρ2=σ,
(7)
where ε0 is the vacuum permittivity, εr1 and εr2 are the dielectric permittivities of the media surrounding graphene, k0 = ω/c is the free space wavenumber and kρ = β is the complex propagation constant of the SPP mode, being β and α the phase and atenuation constants, respectively. In addition, the characteristic impedance of the SPP may be expressed as [34

34. D. Pozar, Microwave Engineering(John Wiley and Sons, 2005).

]
ZC=kρωε0εeff,
(8)
where εeff is the effective permitivitty constant of the surrounding medium. Note that though the characteristic impedance is not often employed to model SPPs [2

2. J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007) [CrossRef] .

], this parameter will be useful to understand and optimize the behavior of the proposed switches.

In the case of finite graphene strips, the dispersion relation of propagating surface plasmons cannot be derived analytically and one has to resort to purely numerically full-wave solvers. There are two different types of SPP propagating along the strip [18

18. 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 84, 161407 (2011) [CrossRef] .

, 20

20. J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. G. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6, 431–440 (2012) [CrossRef] .

]: the waveguide type, which has the field concentrated along the whole strip, and the edge type, where the field is focused on the rims of the strip. Note that graphene relaxation time mainly control the propagation length of the modes, barely affecting to their field confinement. In addition, the characteristics of these modes can also be tuned by modifying the chemical potential of graphene.

Let us consider, for the sake of illustration, a graphene sheet transferred on a dielectric with permittivity εr = 4. The parameters of graphene are τ = 1/(2Γ) = 0.2 ps and T = 300 K, in agreement with measured data [35

35. J. Y. Kim, C. Lee, S. Bae, K. S. Kim, B. H. Hong, and E. J. Choi, “Far-infrared study of substrate-effect on large scale graphene,” Appl. Phys. Lett. 98, 201907 (2011) [CrossRef] .

]. The characteristics of a SPP propagating on the sheet are shown in Fig. 1 for different values of graphene chemical potential. We find that the propagation constant and characteristic impedance of the SPP mode can be tuned over a large range by varying μc. Focusing for instance in the range between 25 and 30 THz, the structure does not support the propagation of SPP when μc = 0.0 eV (because Im(σ) > 0, as demonstrated in [14

14. G. W. Hanson, “Dyadic green’s functions and guided surface waves for a surface conductivity of graphene,” J. Appl. Phys. 103, 064302 (2008) [CrossRef] .

]), it can support SPP propagating with large amount of losses (for instance with μc = 0.1 eV) or it can even support confined and low-loss propagating modes (μc = 0.5 eV). This rich variety of propagation characteristics provides unprecedent guiding opportunities in the field of plasmonics at near infrared frequencies, which are exploited below to proposed graphene-strip plasmonic waveguides with switching capabilities.

Fig. 1 Normalized dispersion relation (a), attenuation constant (b), and real and imaginary components (c–d) of the characteristic impedance of a SPP wave propagating on an air-graphene-dielectric interface versus graphene chemical potential μc computed using Eqs. (7) and (8). The dielectric permittivity is εr = 4.0 and graphene parameters are T = 300 K and τ = 0.2 ps.

2.2. Operation principle of graphene-based switches

Fig. 2 Proposed graphene-based 2D sheet plasmonic switch. The device comprises a monolayer graphene sheet transferred onto a dielectric (εr) and three polysilicon gating pads placed at a distance t below the sheet. The permittivity of the supporting substrate is also set to εr. The guiding properties of the SPP propagating along the sheet are controlled via the electric field effect by the DC bias applied to the gating pads. (a) Switch ON. Simulated results showing the z component of the electric field, Ez, of a SPP wave propagating along the sheet. The central and outer pads are biased with voltages Vout and Vin, chosen to provide the same chemical potential (μc = 0.5 eV) to the whole graphene sheet. (b) Switch OFF. Similar to (a) but here Vin is chosen to provide a chemical potential of μc = 0.1 eV to the inner surface of the graphene sheet. The parameters of the structure are εr = 4.0, L = 350 nm, in = 50 nm, t = 20 nm, T = 300 K, τ = 0.2 ps, and the operation frequency is set to 28 THz.
Fig. 3 Proposed graphene-based strip plasmonic switch. The device is similar to the switch shown in Fig. 2, but here the graphene sheet is replaced by a strip of width W. (a) Switch ON. Simulated results showing the z component of the electric field, Ez, of a SPP wave propagating along the strip. The voltages Vout and Vin are chosen to provide the same chemical potential (μc = 0.5 eV) to the whole graphene strip. (b) Switch OFF. Similar to (a) but here Vin is chosen to provide a chemical potential of μc = 0.1 eV to the inner section of the strip. The parameters of the structure are εr = 4.0, L = 350 nm, W = 150 nm, in = 50 nm, t = 20 nm, T = 300 K, τ = 0.2 ps, and the operation frequency is set to 28 THz.

We propose two different alternatives, illustrated in Fig. 4, for the technological implementation of the switches. In both cases, the ON state is obtained by providing high chemical potential to the whole graphene surface while in the OFF state the chemical potential of the central section is highly reduced. The first approach, shown in Fig. 4(a), uses uniformly highly doped graphene sheets/strips. Note that recent fabrication techniques have demonstrated large control of graphene chemical doping [36

36. X. Wang, X. Li, L. Zhang, Y. Yoon, P. K. Weber, H. Wang, J. Guo, and H. Dai, “N-doping of graphene through electrothermal reactions with ammonia,” Science 324, 768–771 (2009) [CrossRef] [PubMed] .

38

38. A. Reina, X. Jia, J. Z. Ho, D. Nezich, H. Son, V. Bulovic, M. Dresselhaus, and J. Kong, “Large area, few-layer graphene films on arbitrary substrates by chemical vapor deposition,” Nano Lett. 9, 30–35 (2009) [CrossRef] .

], and measured values around 0.4 eV have already been reported [39

39. S. Bae, K. Heongkeun, Y. Lee, X. Xu, J. S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. R. Kim, Y. I. Song, Y. J. Kim, K. S. Kim, B. Özyilmaz, J. H. Ahn, B. H. Hong, and S. Iijima, “Roll-to-roll production of 30-inch graphene films for transparent electrodes,” Nat Nano 5, 574–578 (2010) [CrossRef] .

]. In this case, the ON state is obtained by applying a low DC voltage (Vout = Vin = VL) to the gating pads, which provide to the whole graphene area the required chemical potential. On the other hand, the OFF state is obtained by applying a negative voltage (−VH) to the central gating pad. Indeed, due to the ambipolarity property of graphene [8

8. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Mater. 6, 183–91 (2007) [CrossRef] .

, 40

40. K. Geim, “Graphene: status and prospects,” Science 324, 1530–1532 (2009) [CrossRef] [PubMed] .

], an applied negative DC voltage decreases the chemical potential of graphene. Therefore, the central and outer gating pads are biased with voltages VL and −VH, respectively. The second approach [see Fig. 4(b)] relays on tailoring the chemical doping of the different graphene regions. In this way, the outer graphene surfaces are highly chemically doped, whereas the inner surface is slightly doped. The ON state is obtained providing a low DC bias voltage to the outer gating pads (Vout = VL) and a larger bias to the central one (Vin = VH), while in the OFF state the voltage applied to the central pad is reduced (Vin = VL). Note that this second approach requires a more complicated fabrication process due to the tailoring of the chemical doping applied to the graphene area.

Fig. 4 Cross section of the proposed switch and chemical potential profile along the ‘x’ axis of the graphene area for the ON and OFF states of the device. The different contributions to the chemical potential of graphene (solid line), namely chemical doping (dotted line) and elecrostatic DC bias (dashed line), are also shown. (a) Uniformly highly chemically doped graphene. The OFF state is obtained by applying a negative DC bias to the central gating pad. (b) Non-uniformly chemically doped graphene. Outer and inner surfaces of graphene are highly and slightly chemically doped, respectively. The ON state is obtained by applying a positive DC bias to the central gating pad.

2.3. Electromagnetic modeling

The electromagnetic modeling of the proposed graphene-based switches is performed using two different techniques, namely a TL formalism combined with an ABCD transfer-matrix approach [34

34. D. Pozar, Microwave Engineering(John Wiley and Sons, 2005).

] and a commercial full-wave software (HFSS) [41

41. Ansoft Corporation, “High frequency structure simulator (HFSS) v.14.,” (2012).

] based on the finite element method.

Fig. 5 Equivalent transmission line model of the proposed graphene-based switches shown in Fig. 2 and in Fig. 3.

In the case of the graphene-based 2D sheet switch shown in Fig. 2, the propagating constant and characteristic impedance of the equivalent transmission lines are directly obtained using Eqs. (7) and (8), respectively. In the case of graphene-based strip switches, depicted in Fig. 3, no analytical formulas are available. Here, we employ full-wave simulations to extract the propagation constant of SPP propagating on various infinitely-long graphene strips. Then, the extracted wavenumbers are included in the TL formalism to characterize the complete switch.

3. Numerical results

In this section, we investigate the characteristics of the proposed switches in terms of their scattering parameters. These parameters are commonly employed in the microwaves and terahertz frequency ranges [34

34. D. Pozar, Microwave Engineering(John Wiley and Sons, 2005).

] and are perfectly suited to evaluate the behavior of a switch. Note that an ideal switch in its ON state propagates all input energy towards the output port (i.e. S21 ≈ 1), while no energy is transmitted in its OFF state (i.e. S21 ≈ 0). First, we verify that the two numerical techniques employed to characterize plasmon propagation on graphene waveguides, namely the TL approach and the commercial software HFSS, leads to similar results. Then, we focus for simplicity on switches suspended on free space. There, we study the characteristics of the switches in their ON and OFF states, and we present a parametric study of the switches performance as a function of their features. Finally, we extend this analysis to consider realistic graphene-based switches, taking into account the presence of a substrate. In our study, we consider a relaxation time of 0.2 ps, in agreement with measured values of graphene carrier mobility [35

35. J. Y. Kim, C. Lee, S. Bae, K. S. Kim, B. H. Hong, and E. J. Choi, “Far-infrared study of substrate-effect on large scale graphene,” Appl. Phys. Lett. 98, 201907 (2011) [CrossRef] .

], and a temperature of T = 300° K (room temperature). For simplicity, we neglect the possible fluctuations of graphene relaxation time due to optical phonons [15

15. M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009) [CrossRef] .

].

In order to assess the accuracy of the numerical methods employed to study graphene-based switches, we first consider the propagation of surface plasmons on the structure shown in Fig. 2. The parameters of the structure are εr = 1, L = 3 μm and in = 1 μm, and the chemical potential of the central and outer graphene waveguide sections is set to 0.2 eV and 0.15 eV, respectively. The scattering parameters of the structure, computed using the TL approach and HFSS, are shown in Fig. 6. Very good agreement is found between the methods, verifying that the propagation of surface plasmons on the graphene waveguide is correctly modeled. Note that the accurate results provided by the TL approach are due to the negligible influence of higher order modes in this structure, which are barely excited in the weak discontinuity between the different waveguide sections.

Fig. 6 Scattering parameters of the structure shown in Fig. 2, with εr = 1, L = 3 μm and in = 1 μm, computed using the transmission line approach and the commercial software HFSS. The chemical potential of the outer and central graphene waveguide sections are set to μcout = 0.2 eV and μcin = 0.15 eV.

Fig. 7 presents the scattering parameters, obtained with the commercial software HFSS, of the proposed graphene-based switches suspended in free-space (see Fig. 2 and Fig. 3, with εr = 1). The length of the total device and the switch section are set to L = 1.75 μm and in = 0.5 μm. In the ON state, a DC bias voltage is applied to all gating pads to provide a chemical potential of μc = 0.5 eV to the whole graphene area, while in the OFF state the DC bias applied to the central pad is reduced, leading to a chemical potential of 0.1 eV for the switch area. Figures 7(a)–7(b) show the performance of the proposed graphene-based 2D sheet and strip switches. In the ON state, the switches behave as a transmission line and the input energy propagates towards the output port (i.e. S21 ≈ 1). Note that the extremely low value of S11, which indicates that the device is very well matched, has been obtained by renormalizing the scattering parameters to the characteristic impedance of the outer waveguide sections (Zout, see Section 2.3). This state also provides some dissipation losses, about 4 dB in both switches, which are due to the intrinsic characteristics of graphene and directly depends on the total length of the device. In the OFF state, the graphene-based 2D surface and strip switches provide large isolation in the whole frequency band, around 37 and 50 dB respectively. Importantly, the use of realistic graphene strips, instead of ideal 2D sheets, leads to devices with higher isolation levels. This is due to the field confinement of surface plasmons propagating on strips, which is much larger than in case of 2D sheets [18

18. 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 84, 161407 (2011) [CrossRef] .

, 20

20. J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. G. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6, 431–440 (2012) [CrossRef] .

].

Fig. 7 Simulated scattering parameters of the proposed graphene-based switches, suspended in free-space, at their ON and OFF states. The parameters of the device are L = 1.75 μm and in = 0.5 μm. (a) Graphene-based 2D sheet switch, see Fig. 2. (b) Graphene-based strip switch with W = 0.2 μm, see Fig. 3.

Fig. 8 Power transmitted, reflected, and dissipated in the graphene-based strip plasmonic switch shown in Fig. 7(b). The superscripts ON and OFF are related to the operation state of the switch, and the subscripts T, R, and D refer to the power transmitted towards the output port, reflected into the input port, and dissipated in the structure, respectively.

In order to further study the isolation performance of the proposed switches, Fig. 9 reports a parametric study of S21 as a function of the length and chemical potential of the central waveguide section, at the fixed frequency of 28 THz. The length of the devices (L = 1.75 μm) is kept constant in all cases. Note that the S21 parameter allows to evaluate the overall performance of the switches, implicitly providing information about the losses and power reflected back to the input port [see Eqs. (9)(11)]. The results have been obtained using both the TL approach (dashed line) and the full-wave solver HFSS (solid line), and are shown in Fig. 9(a) and Fig. 9(b) for ideal 2D sheet and strip-based graphene switches, respectively. The length of the inner (switch) waveguide section in is swept from 20 to 500 nm, avoiding values below 20 nm where quantum effects may be non-negligible [20

20. J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. G. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6, 431–440 (2012) [CrossRef] .

]. A standing wave appears within the structure and its interference pattern varies versus in thus explaining the oscillatory behavior of the S21 parameter observed in the figures. As expected, the isolation levels increase with in. As explained in Section 2.2, when the chemical potential of the inner graphene waveguide is different from the one of the outer sections of the device, the energy propagating along the structure finds a discontinuity due to the different impedances and propagation characteristics of the plasmon modes supported by each region. Specifically, decreasing the chemical potential of the central graphene waveguide from 0.5 eV to 0.175 eV leads to switches with low isolation levels. This is because the plasmon impedance is “weakly” affected by the chemical potential in this range (see Fig. 1). However, decreasing μc to lower values such as 0.125 eV or 0.1 eV leads to high isolation levels for almost any length of the inner waveguide section. Besides, it is observed that the isolation level converges when the chemical potential is further decreased. This behavior suggests that in the OFF state isolation is not governed by the fundamental plasmonic mode, but by higher order evanescent modes excited at the discontinuities between the different waveguide sections. Consequently, the transmission line approach -which by definition only consider the fundamental mode- is not able to provide accurate results in this state. However, these effects are obviously accounted for by full-wave simulations Thus, in the case of high isolation, the TL modes slightly overestimate the performance of the switch. In general, this parametric study demonstrates the excellent capabilities of graphene as a material for developing plasmonic-based switches, allowing isolation levels better than 40 dB using graphene sections of about 500 nm.

Fig. 9 Parametric study of the isolation (S21) provided by the proposed graphene-based switches as a function of the length (in) and chemical potential (μcin) of their central waveguide section at the fixed frequency of 28 THz. The length of the devices (L = 1.75 μm) is kept constant in all cases. (a) Graphene-based 2D sheet switch, see Fig. 2. (b) Graphene-based strip switch with W = 0.2 μm, see Fig. 3.

Fig. 10 Simulated scattering parameters of the proposed graphene-based switches at their states ON and OFF. The parameters of the structure are εr = 4.0, L = 0.7 μm and in = 0.2 μm. (a) Graphene-based 2D sheet switch, see Fig. 2. (b) Graphene-based strip switch with W = 0.2 μm, see Fig. 3.
Fig. 11 Parametric study of the isolation (S21) provided by the proposed graphene-based switches as a function of the length (in) and chemical potential (μcin) of their central waveguide section at the fixed frequency of 28 THz. The length of the devices (L = 1.75 μm) is kept constant in all cases. The dielectric permittivity is set to εr = 4.0. (a) Graphene-based 2D sheet switch, see Fig. 2. (b) Graphene-based strip switch with W = 0.2 μm, see Fig. 3.

4. Conclusions

We have proposed and designed series switches able to dynamically control the propagation of plasmons on graphene surfaces at near infrared frequencies. Several configurations, based on 2D graphene surfaces and strips, have been analyzed and their performance have been evaluating versus different parameters of the structures. Two different techniques, namely a transmission line approach and full-wave simulations, have been employed to characterize the switches. It has been shown that the former method provides fast results and physical insight into the problem, but it lacks of accuracy to characterize the OFF state of the devices.

Our results have demonstrated that controlling the properties of very reduced graphene areas provides extremely large isolation levels between the input and output ports. For example, isolation levels larger that 80 dB have been achieved by using a graphene strip of just 500 nm. In addition, it has been shown that increasing the permittivity of the surrounding media allows to increase the isolation level of the switches. These interesting features can be used to further develop guided graphene-based devices at near infrared frequencies, leading to functionalities similar to current nanophotonic plasmonic-based devices at optics.

Acknowledgments

This work was supported by by the Swiss National Science Foundation (SNSF) under grant 133583 and by the EU FP7 Marie-Curie IEF grant “Marconi", with ref. 300966.

References and links

1.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004) [CrossRef] [PubMed] .

2.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007) [CrossRef] .

3.

Y. Wang, E. W. Plummer, and K. Kempa, “Foundations of plasmonics,” Adv. Phys. 60, 799–898 (2011) [CrossRef] .

4.

J. Elser, A. A. Govyadinov, I. Avrutsky, I. Salakhutdinov, and V. A. Podolskiy, “Plasmonic nanolayer composites: Coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation,” J. Nanomaterials 2007, 79469 (2007) [CrossRef] .

5.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003) [CrossRef] [PubMed] .

6.

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Sensor Actuat. B. 3, 388–394 (2009).

7.

J. Homola, S. S. Yeea, and G. Gauglitzb, “Surface plasmon resonance sensors: review,” Sensor Actuat. B. 54, 3–15 (1999) [CrossRef] .

8.

K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Mater. 6, 183–91 (2007) [CrossRef] .

9.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant faraday rotation in single and multilayer graphene,” Nature Phys. 7, 48–51 (2010) [CrossRef] .

10.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011) [CrossRef] [PubMed] .

11.

M. Tamagnone, J. S. Gómez-Díaz, J. R. Mosig, and J. Perruisseau-Carrier, “Reconfigurable thz plasmonic antenna concept using a graphene stack,” Appl. Phys. Lett. 101, 214102 (2012) [CrossRef] .

12.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nature Photon. 6, 749–758 (2012) [CrossRef] .

13.

F. H. Koppens, D. E. Chang, and F. J. G. de Abajo, “Graphene plasmonics: A plaftform for strong light-matter interactions,” Nano Lett. 11, 3370–3377 (2011) [CrossRef] [PubMed] .

14.

G. W. Hanson, “Dyadic green’s functions and guided surface waves for a surface conductivity of graphene,” J. Appl. Phys. 103, 064302 (2008) [CrossRef] .

15.

M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009) [CrossRef] .

16.

A. Ferreira, N. M. R. Peres, and A. H. C. Neto, “Confined magneto-optical waves in graphene,” Phys. Rev. B 85, 205426 (2012) [CrossRef] .

17.

J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Propagation of hybrid transverse magnetic-transverse electric plasmons on magnetically-biased graphene sheets,” J. Appl. Phys. 112, 124906 (2012) [CrossRef] .

18.

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 84, 161407 (2011) [CrossRef] .

19.

D. L. Sounas and C. Caloz, “Edge surface modes in magnetically biased chemically doped graphene strips,” Appl. Phys. Lett. 99, 231902 (2011) [CrossRef] .

20.

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. G. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6, 431–440 (2012) [CrossRef] .

21.

E. H. Hwang and J. D. Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B 75, 205418 (2007) [CrossRef]

22.

B. Standley, W. Bao, H. Zhang, J. Bruck, C. N. Lau, and M. Bockrath, “Graphene-based atomic-scale switches,” Appl. Phys. Lett. 8, 3345–3349 (2008).

23.

T. Palacios, A. Hsu, and H. Wang, “Applications of graphene devices in rf communications,” IEEE Commun. Mag. 48, 122–128 (2010) [CrossRef] .

24.

K. M. Milaninia, M. A. Baldo, A. Reina, and J. Kong, “All graphene electromechanical switch fabricated by chemical vapor deposition,” Appl. Phys. Lett. 95, 183105 (2009) [CrossRef] .

25.

J. Perruisseau-Carrier, “Graphene for antenna applications: opportunities and challenges from microwaves to thz,” in Antennas and Propagation Conference (LAPC)Loughborough, UK (2012).

26.

Y. V. Bludov, M. I. Vasilevskiy, and N. M. R. Peres, “Mechanism for graphene-based optoelectronic switches by tuning surface plasmon-polaritons in monolayer graphene,” Europhys. Lett. 92, 68001 (2010) [CrossRef] .

27.

P. Y. Chen, C. Argyropoulos, and A. Alu, “Terahertz antenna phase shifters using integrally-gated graphene transmission-lines,” IEEE T. Antenn. Propag. 61, 1528–1537 (2013) [CrossRef] .

28.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbo filts,” Science 306, 666–669 (2004) [CrossRef] [PubMed] .

29.

V. P. Gusynin, S. G. Sharapov, and J. B. Carbotte, “On the universal ac optical background in graphene,” New J. Physics 11, 095013 (2009) [CrossRef] .

30.

L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56, 281–284 (2007) [CrossRef] .

31.

J. S. Gómez-Díaz, J. R. Mosig, and J. Perruisseau-Carrier, “Effect of spatial dispersion on surfaces waves propagating along graphene sheets,” arXiv:1301.1337 (2012).

32.

Z. Chen and J. Appenzeller, “Mobility extraction and quantum capacitance impact in high performance graphene field-effect transistor devices,” in IEEE International Electron Devices Meeting (IEDM)San Francisco, USA (2008)

33.

D. Berdebes, T. Low, and M. Lundstrom, “Low bias transport in graphene: An introduction,” in Proc. NCN@Purdue Summer Sch.-Electronics from the Bottom Up (2011)

34.

D. Pozar, Microwave Engineering(John Wiley and Sons, 2005).

35.

J. Y. Kim, C. Lee, S. Bae, K. S. Kim, B. H. Hong, and E. J. Choi, “Far-infrared study of substrate-effect on large scale graphene,” Appl. Phys. Lett. 98, 201907 (2011) [CrossRef] .

36.

X. Wang, X. Li, L. Zhang, Y. Yoon, P. K. Weber, H. Wang, J. Guo, and H. Dai, “N-doping of graphene through electrothermal reactions with ammonia,” Science 324, 768–771 (2009) [CrossRef] [PubMed] .

37.

K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J. H. Ahn, P. Kim, J. Y. Choi, and B. H. Hong, “Large-scale pattern growth of graphene films for stretchable transparent electrodes,” Nature 457, 706–710 (2009) [CrossRef] [PubMed] .

38.

A. Reina, X. Jia, J. Z. Ho, D. Nezich, H. Son, V. Bulovic, M. Dresselhaus, and J. Kong, “Large area, few-layer graphene films on arbitrary substrates by chemical vapor deposition,” Nano Lett. 9, 30–35 (2009) [CrossRef] .

39.

S. Bae, K. Heongkeun, Y. Lee, X. Xu, J. S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. R. Kim, Y. I. Song, Y. J. Kim, K. S. Kim, B. Özyilmaz, J. H. Ahn, B. H. Hong, and S. Iijima, “Roll-to-roll production of 30-inch graphene films for transparent electrodes,” Nat Nano 5, 574–578 (2010) [CrossRef] .

40.

K. Geim, “Graphene: status and prospects,” Science 324, 1530–1532 (2009) [CrossRef] [PubMed] .

41.

Ansoft Corporation, “High frequency structure simulator (HFSS) v.14.,” (2012).

42.

R. E. Collin, Field theory of guided waves(IEEE, Piscataway, 1991)

43.

J. Jin, The finite element method in electromagnetic(Wiley, New York, 1993)

44.

J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite element method for electromagnetics: antennas, microwave circuits, and scattering applications(IEEE, Piscataway, 1998) [CrossRef] .

OCIS Codes
(130.2790) Integrated optics : Guided waves
(240.6680) Optics at surfaces : Surface plasmons
(250.6715) Optoelectronics : Switching

ToC Category:
Integrated Optics

History
Original Manuscript: February 1, 2013
Revised Manuscript: June 12, 2013
Manuscript Accepted: June 12, 2013
Published: June 21, 2013

Citation
J. S. Gómez-Díaz and J. Perruisseau-Carrier, "Graphene-based plasmonic switches at near infrared frequencies," Opt. Express 21, 15490-15504 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-13-15490


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References

  1. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science305, 847–848 (2004). [CrossRef] [PubMed]
  2. J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys.70, 1–87 (2007). [CrossRef]
  3. Y. Wang, E. W. Plummer, and K. Kempa, “Foundations of plasmonics,” Adv. Phys.60, 799–898 (2011). [CrossRef]
  4. J. Elser, A. A. Govyadinov, I. Avrutsky, I. Salakhutdinov, and V. A. Podolskiy, “Plasmonic nanolayer composites: Coupled plasmon polaritons, effective-medium response, and subdiffraction light manipulation,” J. Nanomaterials2007, 79469 (2007). [CrossRef]
  5. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature424, 824–830 (2003). [CrossRef] [PubMed]
  6. S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Sensor Actuat. B.3, 388–394 (2009).
  7. J. Homola, S. S. Yeea, and G. Gauglitzb, “Surface plasmon resonance sensors: review,” Sensor Actuat. B.54, 3–15 (1999). [CrossRef]
  8. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Mater.6, 183–91 (2007). [CrossRef]
  9. I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B. Kuzmenko, “Giant faraday rotation in single and multilayer graphene,” Nature Phys.7, 48–51 (2010). [CrossRef]
  10. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science332, 1291–1294 (2011). [CrossRef] [PubMed]
  11. M. Tamagnone, J. S. Gómez-Díaz, J. R. Mosig, and J. Perruisseau-Carrier, “Reconfigurable thz plasmonic antenna concept using a graphene stack,” Appl. Phys. Lett.101, 214102 (2012). [CrossRef]
  12. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nature Photon.6, 749–758 (2012). [CrossRef]
  13. F. H. Koppens, D. E. Chang, and F. J. G. de Abajo, “Graphene plasmonics: A plaftform for strong light-matter interactions,” Nano Lett.11, 3370–3377 (2011). [CrossRef] [PubMed]
  14. G. W. Hanson, “Dyadic green’s functions and guided surface waves for a surface conductivity of graphene,” J. Appl. Phys.103, 064302 (2008). [CrossRef]
  15. M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B80, 245435 (2009). [CrossRef]
  16. A. Ferreira, N. M. R. Peres, and A. H. C. Neto, “Confined magneto-optical waves in graphene,” Phys. Rev. B85, 205426 (2012). [CrossRef]
  17. J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Propagation of hybrid transverse magnetic-transverse electric plasmons on magnetically-biased graphene sheets,” J. Appl. Phys.112, 124906 (2012). [CrossRef]
  18. 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. B84, 161407 (2011). [CrossRef]
  19. D. L. Sounas and C. Caloz, “Edge surface modes in magnetically biased chemically doped graphene strips,” Appl. Phys. Lett.99, 231902 (2011). [CrossRef]
  20. J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. G. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano6, 431–440 (2012). [CrossRef]
  21. E. H. Hwang and J. D. Sarma, “Dielectric function, screening, and plasmons in two-dimensional graphene,” Phys. Rev. B75, 205418 (2007) [CrossRef]
  22. B. Standley, W. Bao, H. Zhang, J. Bruck, C. N. Lau, and M. Bockrath, “Graphene-based atomic-scale switches,” Appl. Phys. Lett.8, 3345–3349 (2008).
  23. T. Palacios, A. Hsu, and H. Wang, “Applications of graphene devices in rf communications,” IEEE Commun. Mag.48, 122–128 (2010). [CrossRef]
  24. K. M. Milaninia, M. A. Baldo, A. Reina, and J. Kong, “All graphene electromechanical switch fabricated by chemical vapor deposition,” Appl. Phys. Lett.95, 183105 (2009). [CrossRef]
  25. J. Perruisseau-Carrier, “Graphene for antenna applications: opportunities and challenges from microwaves to thz,” in Antennas and Propagation Conference (LAPC)Loughborough, UK (2012).
  26. Y. V. Bludov, M. I. Vasilevskiy, and N. M. R. Peres, “Mechanism for graphene-based optoelectronic switches by tuning surface plasmon-polaritons in monolayer graphene,” Europhys. Lett.92, 68001 (2010). [CrossRef]
  27. P. Y. Chen, C. Argyropoulos, and A. Alu, “Terahertz antenna phase shifters using integrally-gated graphene transmission-lines,” IEEE T. Antenn. Propag.61, 1528–1537 (2013). [CrossRef]
  28. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbo filts,” Science306, 666–669 (2004). [CrossRef] [PubMed]
  29. V. P. Gusynin, S. G. Sharapov, and J. B. Carbotte, “On the universal ac optical background in graphene,” New J. Physics11, 095013 (2009). [CrossRef]
  30. L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B56, 281–284 (2007). [CrossRef]
  31. J. S. Gómez-Díaz, J. R. Mosig, and J. Perruisseau-Carrier, “Effect of spatial dispersion on surfaces waves propagating along graphene sheets,” arXiv:1301.1337 (2012).
  32. Z. Chen and J. Appenzeller, “Mobility extraction and quantum capacitance impact in high performance graphene field-effect transistor devices,” in IEEE International Electron Devices Meeting (IEDM)San Francisco, USA (2008)
  33. D. Berdebes, T. Low, and M. Lundstrom, “Low bias transport in graphene: An introduction,” in Proc. NCN@Purdue Summer Sch.-Electronics from the Bottom Up (2011)
  34. D. Pozar, Microwave Engineering(John Wiley and Sons, 2005).
  35. J. Y. Kim, C. Lee, S. Bae, K. S. Kim, B. H. Hong, and E. J. Choi, “Far-infrared study of substrate-effect on large scale graphene,” Appl. Phys. Lett.98, 201907 (2011). [CrossRef]
  36. X. Wang, X. Li, L. Zhang, Y. Yoon, P. K. Weber, H. Wang, J. Guo, and H. Dai, “N-doping of graphene through electrothermal reactions with ammonia,” Science324, 768–771 (2009). [CrossRef] [PubMed]
  37. K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S. Kim, J. H. Ahn, P. Kim, J. Y. Choi, and B. H. Hong, “Large-scale pattern growth of graphene films for stretchable transparent electrodes,” Nature457, 706–710 (2009). [CrossRef] [PubMed]
  38. A. Reina, X. Jia, J. Z. Ho, D. Nezich, H. Son, V. Bulovic, M. Dresselhaus, and J. Kong, “Large area, few-layer graphene films on arbitrary substrates by chemical vapor deposition,” Nano Lett.9, 30–35 (2009). [CrossRef]
  39. S. Bae, K. Heongkeun, Y. Lee, X. Xu, J. S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. R. Kim, Y. I. Song, Y. J. Kim, K. S. Kim, B. Özyilmaz, J. H. Ahn, B. H. Hong, and S. Iijima, “Roll-to-roll production of 30-inch graphene films for transparent electrodes,” Nat Nano5, 574–578 (2010). [CrossRef]
  40. K. Geim, “Graphene: status and prospects,” Science324, 1530–1532 (2009). [CrossRef] [PubMed]
  41. Ansoft Corporation, “High frequency structure simulator (HFSS) v.14.,” (2012).
  42. R. E. Collin, Field theory of guided waves(IEEE, Piscataway, 1991)
  43. J. Jin, The finite element method in electromagnetic(Wiley, New York, 1993)
  44. J. L. Volakis, A. Chatterjee, and L. C. Kempel, Finite element method for electromagnetics: antennas, microwave circuits, and scattering applications(IEEE, Piscataway, 1998). [CrossRef]

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