Review Article A Review of Computational Electromagnetic Methods for Graphene Modeling Yu Shao, Jing Jing Yang, and Ming Huang Wireless Innovation Lab of Yunnan University, School of Information Science and Engineering, Kunming, Yunnan 650091, China Correspondence should be addressed to Ming Huang; [email protected] Received 27 December 2015; Accepted 5 April 2016 Academic Editor: Rodolfo Araneo Copyright © 2016 Yu Shao et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Graphene is a very promising optoelectronic material and has gained more and more attention. To analyze its electromagnetic properties, several numerical methods have been developed for graphene simulation. In this paper, a review of application of graphene in electronic and photonic device is provided, as well as some widely used computational electromagnetic algorithms for graphene modeling. e advantages and drawbacks of each method are discussed and numerical examples of these methods are given to illustrate their performance and application. 1. Introduction Graphene consists of a monolayer of carbon atoms arranged on a two-dimensional honeycomb lattice which is made up of hexagons. Since its first discovery in 2004, graphene has attracted tremendous research interest in various fields due to its distinctive properties [1–3]. Graphene is a rising star not only in material science and condensed matter physics, but also in the electronic and photonic device communities [4]. Graphene has unique electronic band structure, and the electrons in it behave as massless Dirac fermion [5, 6]. Graphene acquires a pronounced electric field effect which means carrier concentration can be tuned by electrostatic gating. It has been shown that, in bilayer graphene, the electronic gap between conduction bands and the valence can be tuned between zero and midinfrared energies, which makes bilayer graphene the only known semiconductor with a tunable energy gap [7, 8]. Zhang et al. [9] presented a widely tunable electronic bandgap in electrically gated graphene, and they realized a gate-controlled, continuously tunable bandgap of up to 250 meV. Large-area graphene films of the order of centimeters on copper substrates were realized by chemical vapor deposition using methane [10], which facilitated the fabrication of graphene transistors. A good review of graphene transistors in both logic and radiofrequency applications is provided in [11]. Besides electron-device application, graphene has also been recognized as a novel optical material for pho- tonic application. Compared to traditional metal, graphene exhibits several favorable properties. Particularly, surface plasmon polaritons (SPPs), which are electromagnetic waves coupled charge excitations on the surface of a conductor, can be excited in graphene. e plasmons in graphene are tightly confined and the volumes of SPPs in graphene can be 10 6 times smaller than those in free space [12]. is property leads to strong light-matter interaction in graphene [13]. Additionally, the dielectric properties of graphene can be electrically or chemically tuned by changing the charge carrier density and Fermi energy [13–15]. Brar et al. [12] created and probed plasmons in graphene with ≤ 0 /100 and resonant energies as high as 310 meV for 15 nm nanores- onators. Fei et al. [16] showed common graphenes/SiO 2 /Si back-gated structure support propagating surface plasmons and they altered both the amplitude and the wavelength of these plasmons by varying the gate voltage. e graphene- based plasmonics may enable the novel optic devices working in different frequency ranges—from terahertz to the visible range—with extremely high speed, low driving voltage, low power consumption, and compact sizes [17]. Vakil and Engheta in [18] showed that graphene can be made into a one-atom-thick platform for infrared metamaterials and transformation optical devices by manipulating spatially Hindawi Publishing Corporation International Journal of Antennas and Propagation Volume 2016, Article ID 7478621, 9 pages http://dx.doi.org/10.1155/2016/7478621
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Review ArticleA Review of Computational Electromagnetic Methods forGraphene Modeling
Yu Shao Jing Jing Yang and Ming Huang
Wireless Innovation Lab of Yunnan University School of Information Science and Engineering Kunming Yunnan 650091 China
Correspondence should be addressed to Ming Huang huangmingynueducn
Received 27 December 2015 Accepted 5 April 2016
Academic Editor Rodolfo Araneo
Copyright copy 2016 Yu Shao et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Graphene is a very promising optoelectronic material and has gained more and more attention To analyze its electromagneticproperties several numerical methods have been developed for graphene simulation In this paper a review of application ofgraphene in electronic and photonic device is provided as well as some widely used computational electromagnetic algorithmsfor graphene modeling The advantages and drawbacks of each method are discussed and numerical examples of these methodsare given to illustrate their performance and application
1 Introduction
Graphene consists of a monolayer of carbon atoms arrangedon a two-dimensional honeycomb lattice which is made upof hexagons Since its first discovery in 2004 graphene hasattracted tremendous research interest in various fields dueto its distinctive properties [1ndash3] Graphene is a rising star notonly in material science and condensed matter physics butalso in the electronic and photonic device communities [4]
Graphene has unique electronic band structure and theelectrons in it behave as massless Dirac fermion [5 6]Graphene acquires a pronounced electric field effect whichmeans carrier concentration can be tuned by electrostaticgating It has been shown that in bilayer graphene theelectronic gap between conduction bands and the valencecan be tuned between zero and midinfrared energies whichmakes bilayer graphene the only known semiconductor witha tunable energy gap [7 8] Zhang et al [9] presenteda widely tunable electronic bandgap in electrically gatedgraphene and they realized a gate-controlled continuouslytunable bandgap of up to 250meV Large-area graphenefilms of the order of centimeters on copper substrates wererealized by chemical vapor deposition using methane [10]which facilitated the fabrication of graphene transistorsA good review of graphene transistors in both logic andradiofrequency applications is provided in [11]
Besides electron-device application graphene has alsobeen recognized as a novel optical material for pho-tonic application Compared to traditional metal grapheneexhibits several favorable properties Particularly surfaceplasmon polaritons (SPPs) which are electromagnetic wavescoupled charge excitations on the surface of a conductorcan be excited in graphene The plasmons in graphene aretightly confined and the volumes of SPPs in graphene canbe 106 times smaller than those in free space [12] Thisproperty leads to strong light-matter interaction in graphene[13] Additionally the dielectric properties of graphene canbe electrically or chemically tuned by changing the chargecarrier density and Fermi energy [13ndash15] Brar et al [12]created and probed plasmons in graphene with 120582
119901le 1205820100
and resonant energies as high as 310meV for 15 nm nanores-onators Fei et al [16] showed common graphenesSiO
2Si
back-gated structure support propagating surface plasmonsand they altered both the amplitude and the wavelength ofthese plasmons by varying the gate voltage The graphene-based plasmonicsmay enable the novel optic devices workingin different frequency rangesmdashfrom terahertz to the visiblerangemdashwith extremely high speed low driving voltage lowpower consumption and compact sizes [17] Vakil andEngheta in [18] showed that graphene can be made intoa one-atom-thick platform for infrared metamaterials andtransformation optical devices by manipulating spatially
Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2016 Article ID 7478621 9 pageshttpdxdoiorg10115520167478621
2 International Journal of Antennas and Propagation
inhomogeneous and nonuniform conductivity patterns Juet al [19] used metamaterial made up of periodic graphenemicroribbon arrays for terahertz plasmon excitations anddemonstrated that the plasmon resonance can be tuned overa broad terahertz frequency range by adjusting microribbonwidth and electrostatic doping
Our group has been dedicated to the study of molecularsensors for a long time Francescato et al [20] presenteda platform for broadband molecular spectroscopy based onthe propagation of strongly confined antibonding plasmonssupported by graphene sandwiches This novel scheme mea-sures the absorption spectrum of the molecule and shows abroadband capability and high sensitivity Recently Yang etal [21] proposed a cylindrical graphene plasmon waveguideand investigated its application in molecular sensing
The terahertz antenna based on graphene has promisingapplication in wireless communications in nanosystemsThe tunable property of graphene facilitates the design-ing of reconfigurable antenna Huang et al [22] presenteda beam reconfigurable antenna based on a switchablehigh-impedance surface using single-layer graphene Thegraphene-based antennas which have broad wavelength tun-ing range are proposed in [23 24]
In this paper the electromagnetic simulation of grapheneis introduced and some widely used computational electro-magnetic methods for graphene modeling are reviewed Theadvantages and drawbacks of each method are discussed andnumerical examples of these methods are given to illustratetheir performance and application
2 Electromagnetic Simulation of Graphene
To investigate the electromagnetic property of grapheneMaxwell equations need to be solved to simulate the wavepropagation in graphene In most cases Maxwell equationshave no analytical solutions except for a few simple canon-ical problems Therefore numerical methods are crucial tounderstand the wave guiding and scattering by grapheneThere are three popular computational electromagneticmethods to simulate graphene finite-difference time-domainmethod (FDTD) finite element method (FEM) and methodof moment (MoM) Each method has its own advantagesand disadvantages depending on the specific problem Manycommercial software packages based on these methods areavailable such as FDTD Solutions CST HFSS COMSOLIE3D and FEKO All the software packages can be usedto model graphene however none of these are developedspecifically for graphene which makes them less efficient indealing with this monoatomic layer device
Graphene is modeled as a thin surface with complexconductivity which can be tuned by external electric field biasor chemical doping The surface conductivity of graphene iscommonly described by Kubo formula [32]
120590119892=1198951198902(120596 minus 1198952Γ)
120587ℎ2[
1
(120596 minus 1198952Γ)2
sdot int
infin
0
120576 (120597119891119889 (120576)
120597120576minus120597119891119889 (minus120576)
120597120576) 119889120576
minus int
infin
0
119891119889 (minus120576) minus 119891119889 (120576)
(120596 minus 1198952Γ)2minus 4 (120576ℎ)
2119889120576]
(1)
where Γ is the phenomenological scattering rate 119890 is theelectron charge 120576 is the energy state and ℎ denotes thereduced Plank constantThe first term of (1) is correspondingto the intraband contribution while the second term is due tothe interband contribution That is
120590119892= 120590intra + 120590inter (2)
The intraband term in (2) can be evaluated by
120590intra =1198902119896119861119879
120587ℎ2 (2Γ + 119895120596)[120583119888
119896119861119879+ 2 ln (119890minus120583119888119896119861119879 + 1)] (3)
and the interband term in (2) is approximated by
120590inter = minus1198951198902
4120587ℎln(
210038161003816100381610038161205831198881003816100381610038161003816 minus (120596 minus 1198952Γ) ℎ
210038161003816100381610038161205831198881003816100381610038161003816 + (120596 minus 1198952Γ) ℎ
)
(119896119861119879 ≪
10038161003816100381610038161205831198881003816100381610038161003816)
(4)
where 120583119888is the chemical potential 119879 is the temperature and
119896119861is the Boltzmann constantIn electromagnetic simulation the 2D graphene surface
is usually approximated by a 3D dielectric slab whose 3Dconductivity is evaluated by the following equation
1205903D =
120590119892
119889 (5)
where 119889 is the thickness of the graphene layer The accuracyof 3D dielectric slab model degrades as 119889 increases It can beshown that using the 2D conductivity the SPP wavelength is[25]
120582SPP = 1205820 (1 minus (2
1205780120590)
2
)
minus05
(6)
The 3D dielectric slab can support even and odd modes Theodd mode has the wavelength of
120582odd = 2120587(minus2
119889cothminus1120576
3D)minus1
(7)
and (6) can be approximated by (7) only if the following threeconditions are satisfied [25]
119889
120582SPP≪ 1
|120590| ≪2
1205780
1003816100381610038161003816100381610038161003816
120590
119889
1003816100381610038161003816100381610038161003816gt 2120596120576
0
(8)
Figure 1 shows the relative error of using the dielectric slabmodel for graphene with respect to normalized 119889 and 120590 moredetails can be found in [25]
International Journal of Antennas and Propagation 3
1205763D gt minus1 (not allowed)
25
30
35
40
45
50
55
Im(120590)
(120583S)
300 400 500 600 700 800 900200
d1205820 (times106)
70
60
50
40
30
20
10
0
Figure 1 Relative error of the dielectric slab model with respect tonormalized 119889 and 120590 [25]
3 Finite-Difference Time-Domain Method
FDTD is time-domain method which has specific advantagein transient problem and wideband problemThe broadbandresults can be obtained in one simulation Compared tofrequency-domain method it takes less computing memoryand simulation time Additionally FDTD is an iterativescheme which eliminates solving large linear system henceit is simple robust and easy to implement
There are three general approaches to model graphenein FDTD (I) using standard FDTD method with very finemesh within the graphene sheet (II) using the subcell FDTDmethod (III) using surface resistive boundary condition
In the first approach graphene is modeled as thin layerwith finite thickness and the surface conductivity is trans-formed to volume permittivity According to the Courant-Friedrichs-Lewy (CFL) stability condition of FDTD finemesh leads to small time step so this approach is bothmemory- and time-consuming which reduces its efficiencyTo alleviate the time step constraint some unconditionallystable FDTD methods are developed for graphene simu-lation In [33] the locally one-dimensional (LOD) FDTDis applied for the efficient simulation of graphene deviceThey demonstrated that the LOD-FDTD can be 60 fasterthan standard FDTD with reasonable accuracy Wang et al[26] proposed an unconditionally stable one-step leapfrogalternating-direction-implicit (ADI) FDTD to study the SPPsin graphene structure In Figure 2 they showed the SPPspropagating along the spiral waveguide The one-step ADI-FDTD method gives accurate result which is comparable toconventional FDTDThey also showed that the conventionalFDTD took 17 times 105 s for this problem while ADI-FDTDonly took 08 times 105 s with CFLN = 10
In the second approach graphene is treated as thin layeroccupying a fraction of FDTD cell [27] The Yee cell for thesubcell FDTDmethod is shown in Figure 3 where119864
119911compo-
nent is split into119864119911119894and119864
119911119900in cells occupying graphene [27]
However this scheme is complex in mathematics and needsspecial type of PML to model infinitely thin sheets [34]
In the third approach graphene is modeled as a zero-thickness conductive sheet over which the fields satisfy thesurface boundary condition Nayyeri et al [28 35] presentedthis scheme in detail As shown in Figure 4(a) a conductivesheet locates at119870+ 12 The tangential component of119867 fieldis discontinuous over the sheet so 119867
119909is split into 1119867
119909and
2119867119909 which satisfy the following discretized Faraday law
1205831
1119867119899+12
119909minus1119867119899minus12
119909
Δ119905=119864119899
119910(119870 + 12) minus 119864
119899
119910(119870)
Δ1199092 (9a)
1205831
2119867119899+12
119909minus2119867119899minus12
119909
Δ119905=119864119899
119910(119870 + 1) minus 119864
119899
119910(119870 + 12)
Δ1199092 (9b)
The surface boundary condition is2119867119909minus1119867119909= 120590119904119864119910
(10)
from which 119864119899119910(119870 + 12) can be written as
119864119899
119910(119870 +
1
2) =
1
2120590119904
[(2119867119899+12
119909+2119867119899minus12
119909)
minus (1119867119899+12
119909+1119867119899minus12
119909)]
(11)
Substituting (11) into (9a) and (9b) 1119867119909and 2119867
119909can be
derived as1119867119899+12
119909=
1
1 minus 11988811198882
(119865119899
1+ 1198881119865119899
2) (12a)
2119867119899+12
119909=
1
1 minus 11988811198882
(119865119899
2+ 1198882119865119899
1) (12b)
The expressions of 1198881 1198882 1198651198991 and 119865
119899
2are shown in [28]
Figure 4(b) is a 3D FDTD cell containing conductive sheetat 119870 + 12 In this case tangential components of magneticfield (119867
119909 119867119910) and normal component of electric field (119864
119911) are
discontinuous over the sheet so they are split into two partson the two sides of the surface Their updating equations canbe derived in a similar way to 1D case and are presented in[28]
The surface boundary condition approach has been val-idated [28] Figure 5(a) shows a 2D problem in which aninfinite line source radiates above an infinite graphene sheetThe pattern of 119864
119911at the wavelength of 100 120583m is shown
in Figure 5(b) A strong agreement between the proposedFDTD result and that derived from semianalytic method isobtained In addition unlike the subcell FDTD method theclassical PML is applicable to the surface boundary conditionscheme
In time-domain simulation the conductivities ofgraphene shown in (2)ndash(4) need to be converted fromfrequency domain into time domain The intrabandconductivity which has a Drude-like expression can beeasily converted into time domain However the interbandconductivity has a complex logarithmic form which needsa vector fitting technique A summation of partial fractionsin terms of complex conjugate pole-residues is applied toapproximate the conductivityThe detailed description of thefitting technique can be found in [35ndash37]
4 International Journal of Antennas and Propagation
300nm
290
nmExciting line
120582SPP
x
y
02
0
minus02
(a)
Exciting line
x
y
02
0
minus02
(b)
Figure 2 The normalized 119864119910component of guided SPPs along the spiral graphene waveguide at time 119905 = 19258 ps calculated by (a)
conventional FDTD and (b) the proposed improved ADI-FDTD with CFLN = 10 [26]
H
E
Hzo
Ezo
Ezi
Hzi
Hz
(i j K)
(i j Klowast)(i + 1 j + 1 Kminus 1)
z
x
y
Figure 3Modified Yee cell for subcell FDTDmethod at the locationof graphene (shaded area) [27]
4 Finite Element Method
FEM is a full-wave numerical technique for electromagneticboundary-value problemsThe basic principle of this methodis to discretize the whole computing domain with a finitenumber of subdomains in which the unknown function isexpanded by simple interpolation functions with unknowncoefficients Then a system of algebraic equations of theseunknown coefficients is obtained by using Ritz variational orGalerkinrsquos method Finally by solving this linear system theapproximated solution of the entire domain can be obtained
In FDTD the computing space is discretized by orthogo-nal grid this staircase approximation will reduce the mod-eling fidelity when it comes to complex geometry whilein FEM where the triangles or tetrahedral elements areapplied arbitrary geometries can be modeled accurately Asa result FEM has advantage in complex and inhomogeneousproblems Furthermore FEM is a frequency domain method
which makes it efficient in dealing with narrow-band prob-lems
Brar et al [12] solved Maxwellrsquos equation by FEM andmodeled graphene as a thin sheet with 01 nm thicknessthe results suggested that graphene can increase light-matterinteractions at infrared energies Software packages such asHFSS and COMSOL are based on FEM and they are usedwidely in graphene simulation Recently we used COMSOLto investigate the transmission properties of a cylindricalgraphene plasmon waveguide [21] Tamagnone et al [29]simulated a reconfigurable graphene antenna with HFSS thestructure of the antenna is shown in Figure 6(a) the inputimpedance of the antenna can be tuned by changing thechemical potential as shown in Figure 6(b)
5 Method of Moments
In contrast to FDTD and FEM which solve differentialequations MoM is a technique used to solve electromag-netic surface or volume integral equations in the frequencydomain In MoM the quantities of interest are not the fieldsbut the electromagnetic sources (surface or volume current)so only the surface of the geometry needs to be discretizedThe surface current is discretized into wire segments andorsurface patches A linear system can be constructed by themethod of weighted residuals and the results of the linearequations give the surface current The far-field result canbe derived from the surface current by Greenrsquos functionBecause it only requires calculating the boundary valuesinstead of the values throughout the space MoM is highlyefficient for electrically large objects and is widely used insolving radiation and scattering problems However whenapplied to complex inhomogeneous cases it will be verycomputationally expensive and less efficient
The analytical expressions of dyadic Greenrsquos function forgraphene are derived in [38] Shapoval et al [39] proposedintegral equations based on surface-impedance boundary
International Journal of Antennas and Propagation 5
Conductive sheet
x
zK
Ey
K +1
2K + 1
Ey
1Hx2Hx
(a)
Conductivesheet
z
y
x
(i j K)
(i j K + 1)
(i j + 1 K)
(i + 1 j K)
Hz
ExEy
1Hx 1Hy
2Hx 2Hy
1Ez
2Ez
(b)
Figure 4 (a) 1D FDTD cell including a conductive sheet at grid 119870 + 12 (b) 3D FDTD cell with a conductive sheet at grid119870 + 12 [28]
PML
Line source Graphenesheet
20120583m
50120583m
Observation circlex
y
(a)
0
30
60
90
120
150
180
210
240
270
300
330
x
y
SemianalyticFDTD
1
08
06
04
02
(b)
Figure 5 (a) Line source scattering by a graphene sheet (b) Normalized pattern of 119864119911at the wavelength of 120582
0= 100 120583m [28]
condition to analyze plane wave scattering and absorptionby graphene-strip gratings The method of moments forgraphene nanoribbons was developed in [40ndash44] in whichthe issue of nonlocality of graphene conductivity was takeninto account Nonlocal effect arises from spatial dispersionof graphene which is nonnegligible when dealing with slowmodes supported by graphene nanoribbons The spatiallydispersive intraband conductivity tensor was derived in [41]
Software packages such as IE3D and FEKO are based onMoM IE3D is applied to simulate themicrowave propagationin a coplanar waveguide over graphene from 40MHz to110GHz [45] Cabellos-Aparicio et al [30] used FEKO tostudy the radiated power of a graphene plasmonic antennafed by photoconductive source The antenna structure isshown in Figure 7(a) and the radiated power with respect to
frequency is shown in Figure 7(b)The detailed parameters ofthe photoconductive antenna can be found in [30]
6 Discontinuous GalerkinTime-Domain Method
The graphene involved problems are often multiscaleGraphene is monoatomic and its thickness is much smallerthan wavelength so it is an electrically fine structure Incontrast the substrate belongs to electrically coarse structurebecause its dimension is much greater than wavelengthAs we have mentioned before time-domain methods havethe advantage that the broadband characterization can beobtained with only a single simulation However FDTD
6 International Journal of Antennas and Propagation
GrapheneAl2O3
GrapheneTHz photomixer
L WSubstrate (GaAs)
(a)
120583c
120583c
fr fr fr fr fr
minus400
minus200
0
200
400
Impe
danc
e (Ω
)
1 15 2 2505
Frequency (THz)
0 eV005 eV01 eV
015 eV02 eV
ReIm
(b)
Figure 6 (a) Structure of a reconfigurable graphene antenna (b) Input impedance of the antenna with respect to chemical potential [29]
Vminus
V+Vbias
BN
Si lens
fs laser pulse
5120583m
14nm
500120583m
GaAs
LT-GaAs
GrapheneContact pads
(120583m)10
20 30
40
(a)
1 2 3 40
Frequency (THz)
0
02
04
06
08
1
12
Radi
ated
pow
er (120583
W)
(b)
Figure 7 (a) Structure of graphene RF plasmonic antenna fed with a photoconductivematerial (b) Radiated power with respect to frequencyfor graphene antennas with lengths 10120583m (blue) 20 120583m (green) 30120583m (red) and 40120583m (black) [30]
method has a serious efficiency problem in multiscale sim-ulation because it requires high discretization density tomodel electrically fine structure due to its cartesian grid Thefinite element time-domain (FETD) method is capable ofmodeling complex and fine structures and achieving high-order accuracy with high-order basis functions The majordrawback is a global linear system of equations that needs tobe solved at each time step The multiscale problems usuallycontain a large number of unknowns FETDwill be very com-putationally expensive in this case Discontinuous Galerkintime-domain (DGTD) method is promising in multiscaleproblems [46] DGTD allows for domain decomposition Amultiscale structure is divided into several subdomains andeach subdomain can be discretized separately All operationsin DGTD are local so large global matrix is split into several
smaller matrices Unlike FETD the matrices of DGTD areinverted and stored before time marching and differenttime integration scheme can be used in different subdomainAdditionally DGTD is naturally adapted to parallel comput-ing Recently DGTD has been used in nanophotonics fieldand considered as a viable alternative to the well-establishedFDTD and FETD methods [47]
Li et al [31] proposed DGTD method with resistiveboundary condition for the electromagnetic analyzing ofgraphene They also applied this method to the magne-tized graphene from microwave to THz range where theanisotropic and disperse surface conductivity is involved[48] Figure 8(a) shows the reflection Γ
119877 transmission Γ
119879 and
absorption Γ119860coefficients of an infinitely large graphene sheet
under the illumination of a normally incident plane wave
International Journal of Antennas and Propagation 7
Exact
ΓT
ΓR
ΓA
0
02
04
06
08
1
Mag
nitu
de
2 4 6 8 100
Frequency (THz)
DGTD + RBC
(a)
Integral equation
0
1
2
3
4
5
6
7
8
ECS
05 1 15 2 25 3 35 40
Frequency (THz)
DGTD + RBC
(b)
Figure 8 (a) The magnitude of reflection Γ119877 transmission Γ
119879 and absorption Γ
119860coefficients calculated by DGFD as well as the theoretical
value (b) Normalized extinction cross section of a graphene patch calculated by DGTD and integral equation method [31]
the results of DGTD agree very well with the theoretical dataFigure 8(b) is the normalized extinction cross section of afreestanding graphene patch good consistency is achievedbetween DGTD and integral equation method More numer-ical examples can be found in [31]
7 Conclusion
Due to its intriguing properties graphene is used more andmore widely in electronic and photonic community Thispaper gives a brief review of application of graphene aswell asthe computational methods which can be used to investigateits electromagnetic properties Although a number of famouscommercial software packages are available for graphene sim-ulation none of them are developed specifically for graphenetherefore they suffer from inefficiency Each computationalmethod has its own advantages and drawbacks one shouldchoose the appropriate method according to the specificproblem FDTD is simple and easy to implement however itis less efficient in modeling fine structure FEM is flexible ingeometry modeling but solving large global system makes itcomputationally expensive DGTD is very suitable for multi-scale problem and we believe it will be more andmore widelyused in modeling and designing graphene-assisted device
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the National Natural Sci-ence Foundation of China (Grants nos 61161007 6126100261461052 and 11564044) the Specialized Research Fundfor the Doctoral Program of Higher Education (Grantsnos 20135301110003 and 20125301120009) China Postdoc-toral Science Foundation (Grants nos 2013M531989 and2014T70890) and the Key Program of Natural Science ofYunnan Province (Grants nos 2013FA006 and 2015FA015)
References
[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004
[2] A K Geim and K S Novoselov ldquoThe rise of graphenerdquo NatureMaterials vol 6 no 3 pp 183ndash191 2007
[3] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008
[4] P Avouris ldquoGraphene electronic and photonic properties anddevicesrdquo Nano Letters vol 10 no 11 pp 4285ndash4294 2010
[5] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005
[6] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009
[7] T Ohta A Bostwick T Seyller K Horn and E RotenbergldquoControlling the electronic structure of bilayer graphenerdquoScience vol 313 no 5789 pp 951ndash954 2006
[8] E V Castro K S Novoselov S VMorozov et al ldquoBiased bilayergraphene semiconductor with a gap tunable by the electric fieldeffectrdquo Physical Review Letters vol 99 no 21 Article ID 2168022007
[9] Y Zhang T-T Tang C Girit et al ldquoDirect observation of awidely tunable bandgap in bilayer graphenerdquo Nature vol 459no 7248 pp 820ndash823 2009
[10] X Li W Cai J An et al ldquoLarge-area synthesis of high-qualityand uniform graphene films on copper foilsrdquo Science vol 324no 5932 pp 1312ndash1314 2009
[11] F Schwierz ldquoGraphene transistorsrdquoNatureNanotechnology vol5 no 7 pp 487ndash496 2010
[12] VW BrarM S JangM Sherrott J J Lopez andH A AtwaterldquoHighly confined tunable mid-infrared plasmonics in graphenenanoresonatorsrdquoNano Letters vol 13 no 6 pp 2541ndash2547 2013
[13] F H L Koppens D E Chang and F J G de Abajo ldquoGrapheneplasmonics a platform for strong lightndashmatter interactionsrdquoNano Letters vol 11 no 8 pp 3370ndash3377 2011
8 International Journal of Antennas and Propagation
[14] J ChenM Badioli P A Gonzalez et al ldquoOptical nano-imagingof gate-tunable graphene plasmonsrdquoNature vol 487 pp 77ndash812012
[15] M Jablan H Buljan and M Soljacic ldquoPlasmonics in grapheneat infrared frequenciesrdquo Physical Review B vol 80 no 24Article ID 245435 2009
[16] Z Fei A S Rodin G O Andreev et al ldquoGate-tuningof graphene plasmons revealed by infrared nano-imagingrdquoNature vol 486 no 7405 pp 82ndash85 2012
[17] A N Grigorenko M Polini and K S Novoselov ldquoGrapheneplasmonicsrdquo Nature Photonics vol 6 no 11 pp 749ndash758 2012
[18] A Vakil and N Engheta ldquoTransformation optics usinggraphenerdquo Science vol 332 no 6035 pp 1291ndash1294 2011
[19] L Ju B Geng J Horng et al ldquoGraphene plasmonics for tunableterahertz metamaterialsrdquo Nature Nanotechnology vol 6 no 10pp 630ndash634 2010
[20] Y Francescato V Giannini J Yang M Huang and S A MaierldquoGraphene sandwiches as a platform for broadband molecularspectroscopyrdquo ACS Photonics vol 1 no 5 pp 437ndash443 2014
[21] J Yang J YangW Deng FMao andMHuang ldquoTransmissionproperties and molecular sensing application of CGPWrdquoOpticsExpress vol 23 no 25 pp 32289ndash32299 2015
[22] Y Huang L-S Wu M Tang and J Mao ldquoDesign of a beamreconfigurable thz antenna with graphene-based switchablehigh-impedance surfacerdquo IEEE Transactions on Nanotechnol-ogy vol 11 no 4 pp 836ndash842 2012
[23] I Llatser C Kremers A Cabellos-Aparicio J M Jornet EAlarcon and D N Chigrin ldquoGraphene-based nano-patchantenna for terahertz radiationrdquo Photonics and Nanostruc-turesmdashFundamentals and Applications vol 10 no 4 pp 353ndash358 2012
[24] Y Yao M A Kats P Genevet et al ldquoBroad electrical tuning ofgraphene-loaded plasmonic antennasrdquoNano Letters vol 13 no3 pp 1257ndash1264 2013
[25] E Forati G W Hanson A B Yakovlev and A Alu ldquoPlanarhyperlens based on a modulated graphene monolayerrdquo PhysicalReview B vol 89 no 8 Article ID 081410 2014
[26] X-H Wang W-Y Yin and Z Chen ldquoBroadband modelingsurface plasmon polaritons in optically pumped and curvedgraphene structures with an improved leapfrog ADI-FDTDmethodrdquo Optics Communications vol 334 pp 152ndash155 2015
[27] G D Bouzianas N V Kantartzis C S Antonopoulos and TD Tsiboukis ldquoOptimal modeling of infinite graphene sheets viaa class of generalized FDTD schemesrdquo IEEE Transactions onMagnetics vol 48 no 2 pp 379ndash382 2012
[28] V Nayyeri M Soleimani and O M Ramahi ldquoModelinggraphene in the finite-difference time-domain method using asurface boundary conditionrdquo IEEE Transactions on Antennasand Propagation vol 61 no 8 pp 4176ndash4182 2013
[29] M Tamagnone J S G Diaz J R Mosig and J P CarrierldquoReconfigurable terahertz plasmonic antenna concept using agraphene stackrdquo Applied Physics Letters vol 101 no 21 ArticleID 214102 2012
[30] A Cabellos-Aparicio I Llatser E Alarcon A Hsu and T Pala-cios ldquoUse of terahertz photoconductive sources to characterizetunable graphene RF plasmonic antennasrdquo IEEE Transactionson Nanotechnology vol 14 no 2 pp 390ndash396 2015
[31] P Li L J Jiang and H Bagc120580 ldquoA resistive boundary conditionenhancedDGTDscheme for the transient analysis of graphenerdquoIEEE Transactions on Antennas and Propagation vol 63 no 7pp 3065ndash3076 2015
[32] G W Hanson ldquoDyadic greenrsquos functions for an anisotropicnon-local model of biased graphenerdquo IEEE Transactions onAntennas and Propagation vol 56 no 3 pp 747ndash757 2008
[33] I Ahmed E H Khoo and E Li ldquoEfficient modeling andsimulation of graphene devices with the LOD-FDTD methodrdquoIEEEMicrowave andWireless Components Letters vol 23 no 6pp 306ndash308 2013
[34] X Yu and C D Sarris ldquoA perfectly matched layer for subcellFDTDand applications to themodeling of graphene structuresrdquoIEEE Antennas and Wireless Propagation Letters vol 11 pp1080ndash1083 2012
[35] V Nayyeri M Soleimani and O M Ramahi ldquoWidebandmodeling of graphene using the finite-difference time-domainmethodrdquo IEEE Transactions on Antennas and Propagation vol61 no 12 pp 6107ndash6114 2013
[36] H Lin M F Pantoja L D Angulo J Alvarez R G Martinand S G Garcia ldquoFDTD modeling of graphene devices usingcomplex conjugate dispersionmaterialmodelrdquo IEEEMicrowaveand Wireless Components Letters vol 22 no 12 pp 612ndash6142012
[37] D-W Wang W-S Zhao X-Q Gu W Chen and W-Y YinldquoWideband modeling of graphene-based structures at differenttemperatures using hybrid FDTD methodrdquo IEEE Transactionson Nanotechnology vol 14 no 2 pp 250ndash258 2015
[38] A Y Nikitin F J Garcia-Vidal and LMartin-Moreno ldquoAnalyt-ical expressions for the electromagnetic dyadic greenrsquos functionin graphene and thin layersrdquo IEEE Journal of Selected Topics inQuantum Electronics vol 19 no 3 Article ID 4600611 2013
[39] O V Shapoval J S Gomez-Diaz J Perruisseau-Carrier J RMosig and A I Nosich ldquoIntegral equation analysis of planewave scattering by coplanar graphene-strip gratings in the thzrangerdquo IEEE Transactions on Terahertz Science and Technologyvol 3 no 5 pp 666ndash674 2013
[40] R Araneo G Lovat and P Burghignoli ldquoGraphene nanostriplines dispersion and attenuation analysisrdquo in Proceedings of the16thWorkshop on Signal and Power Integrity (SPI rsquo12) pp 75ndash78IEEE Sorrento Italy May 2012
[41] G Lovat G W Hanson R Araneo and P Burghignoli ldquoSemi-classical spatially dispersive intraband conductivity tensor andquantum capacitance of graphenerdquo Physical Review B vol 87no 11 Article ID 115429 2013
[42] P Burghignoli R Araneo G Lovat and G Hanson ldquoSpace-domain method of moments for graphene nanoribbonsrdquo inProceedings of the 8th European Conference on Antennas andPropagation (EuCAP rsquo14) pp 666ndash669 IEEE The Hague TheNetherlands April 2014
[43] R Araneo P Burghignoli G Lovat and G W HansonldquoModal propagation and crosstalk analysis in coupled graphenenanoribbonsrdquo IEEE Transactions on Electromagnetic Compati-bility vol 57 no 4 pp 726ndash733 2015
[44] A Fallahi T Low M Tamagnone and J Perruisseau-CarrierldquoNonlocal electromagnetic response of graphene nanostruc-turesrdquo Physical Review B vol 91 Article ID 121405 2015
[45] M Dragoman D Neculoiu A Cismaru et al ldquoCoplanarwaveguide on graphene in the range 40MHz-110 GHzrdquoAppliedPhysics Letters vol 99 no 3 Article ID 033112 2011
[46] J Chen and Q H Liu ldquoDiscontinuous Galerkin Time-Domainmethods for multiscale electromagnetic simulations a reviewrdquoProceedings of the IEEE vol 101 no 2 pp 242ndash254 2013
International Journal of Antennas and Propagation 9
[47] S Descombes C Durochat S Lanteri L Moya C Scheid andJ Viquerat ldquoRecent advances on a DGTD method for time-domain electromagneticsrdquo Photonics and NanostructuresmdashFundamentals and Applications vol 11 no 4 pp 291ndash302 2013
[48] P Li and L J Jiang ldquoModeling of magnetized graphene frommicrowave to THz range by DGTD with a scalar RBC and anADErdquo IEEE Transactions on Antennas and Propagation vol 63no 10 pp 4458ndash4467 2015
International Journal of
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International Journal of
2 International Journal of Antennas and Propagation
inhomogeneous and nonuniform conductivity patterns Juet al [19] used metamaterial made up of periodic graphenemicroribbon arrays for terahertz plasmon excitations anddemonstrated that the plasmon resonance can be tuned overa broad terahertz frequency range by adjusting microribbonwidth and electrostatic doping
Our group has been dedicated to the study of molecularsensors for a long time Francescato et al [20] presenteda platform for broadband molecular spectroscopy based onthe propagation of strongly confined antibonding plasmonssupported by graphene sandwiches This novel scheme mea-sures the absorption spectrum of the molecule and shows abroadband capability and high sensitivity Recently Yang etal [21] proposed a cylindrical graphene plasmon waveguideand investigated its application in molecular sensing
The terahertz antenna based on graphene has promisingapplication in wireless communications in nanosystemsThe tunable property of graphene facilitates the design-ing of reconfigurable antenna Huang et al [22] presenteda beam reconfigurable antenna based on a switchablehigh-impedance surface using single-layer graphene Thegraphene-based antennas which have broad wavelength tun-ing range are proposed in [23 24]
In this paper the electromagnetic simulation of grapheneis introduced and some widely used computational electro-magnetic methods for graphene modeling are reviewed Theadvantages and drawbacks of each method are discussed andnumerical examples of these methods are given to illustratetheir performance and application
2 Electromagnetic Simulation of Graphene
To investigate the electromagnetic property of grapheneMaxwell equations need to be solved to simulate the wavepropagation in graphene In most cases Maxwell equationshave no analytical solutions except for a few simple canon-ical problems Therefore numerical methods are crucial tounderstand the wave guiding and scattering by grapheneThere are three popular computational electromagneticmethods to simulate graphene finite-difference time-domainmethod (FDTD) finite element method (FEM) and methodof moment (MoM) Each method has its own advantagesand disadvantages depending on the specific problem Manycommercial software packages based on these methods areavailable such as FDTD Solutions CST HFSS COMSOLIE3D and FEKO All the software packages can be usedto model graphene however none of these are developedspecifically for graphene which makes them less efficient indealing with this monoatomic layer device
Graphene is modeled as a thin surface with complexconductivity which can be tuned by external electric field biasor chemical doping The surface conductivity of graphene iscommonly described by Kubo formula [32]
120590119892=1198951198902(120596 minus 1198952Γ)
120587ℎ2[
1
(120596 minus 1198952Γ)2
sdot int
infin
0
120576 (120597119891119889 (120576)
120597120576minus120597119891119889 (minus120576)
120597120576) 119889120576
minus int
infin
0
119891119889 (minus120576) minus 119891119889 (120576)
(120596 minus 1198952Γ)2minus 4 (120576ℎ)
2119889120576]
(1)
where Γ is the phenomenological scattering rate 119890 is theelectron charge 120576 is the energy state and ℎ denotes thereduced Plank constantThe first term of (1) is correspondingto the intraband contribution while the second term is due tothe interband contribution That is
120590119892= 120590intra + 120590inter (2)
The intraband term in (2) can be evaluated by
120590intra =1198902119896119861119879
120587ℎ2 (2Γ + 119895120596)[120583119888
119896119861119879+ 2 ln (119890minus120583119888119896119861119879 + 1)] (3)
and the interband term in (2) is approximated by
120590inter = minus1198951198902
4120587ℎln(
210038161003816100381610038161205831198881003816100381610038161003816 minus (120596 minus 1198952Γ) ℎ
210038161003816100381610038161205831198881003816100381610038161003816 + (120596 minus 1198952Γ) ℎ
)
(119896119861119879 ≪
10038161003816100381610038161205831198881003816100381610038161003816)
(4)
where 120583119888is the chemical potential 119879 is the temperature and
119896119861is the Boltzmann constantIn electromagnetic simulation the 2D graphene surface
is usually approximated by a 3D dielectric slab whose 3Dconductivity is evaluated by the following equation
1205903D =
120590119892
119889 (5)
where 119889 is the thickness of the graphene layer The accuracyof 3D dielectric slab model degrades as 119889 increases It can beshown that using the 2D conductivity the SPP wavelength is[25]
120582SPP = 1205820 (1 minus (2
1205780120590)
2
)
minus05
(6)
The 3D dielectric slab can support even and odd modes Theodd mode has the wavelength of
120582odd = 2120587(minus2
119889cothminus1120576
3D)minus1
(7)
and (6) can be approximated by (7) only if the following threeconditions are satisfied [25]
119889
120582SPP≪ 1
|120590| ≪2
1205780
1003816100381610038161003816100381610038161003816
120590
119889
1003816100381610038161003816100381610038161003816gt 2120596120576
0
(8)
Figure 1 shows the relative error of using the dielectric slabmodel for graphene with respect to normalized 119889 and 120590 moredetails can be found in [25]
International Journal of Antennas and Propagation 3
1205763D gt minus1 (not allowed)
25
30
35
40
45
50
55
Im(120590)
(120583S)
300 400 500 600 700 800 900200
d1205820 (times106)
70
60
50
40
30
20
10
0
Figure 1 Relative error of the dielectric slab model with respect tonormalized 119889 and 120590 [25]
3 Finite-Difference Time-Domain Method
FDTD is time-domain method which has specific advantagein transient problem and wideband problemThe broadbandresults can be obtained in one simulation Compared tofrequency-domain method it takes less computing memoryand simulation time Additionally FDTD is an iterativescheme which eliminates solving large linear system henceit is simple robust and easy to implement
There are three general approaches to model graphenein FDTD (I) using standard FDTD method with very finemesh within the graphene sheet (II) using the subcell FDTDmethod (III) using surface resistive boundary condition
In the first approach graphene is modeled as thin layerwith finite thickness and the surface conductivity is trans-formed to volume permittivity According to the Courant-Friedrichs-Lewy (CFL) stability condition of FDTD finemesh leads to small time step so this approach is bothmemory- and time-consuming which reduces its efficiencyTo alleviate the time step constraint some unconditionallystable FDTD methods are developed for graphene simu-lation In [33] the locally one-dimensional (LOD) FDTDis applied for the efficient simulation of graphene deviceThey demonstrated that the LOD-FDTD can be 60 fasterthan standard FDTD with reasonable accuracy Wang et al[26] proposed an unconditionally stable one-step leapfrogalternating-direction-implicit (ADI) FDTD to study the SPPsin graphene structure In Figure 2 they showed the SPPspropagating along the spiral waveguide The one-step ADI-FDTD method gives accurate result which is comparable toconventional FDTDThey also showed that the conventionalFDTD took 17 times 105 s for this problem while ADI-FDTDonly took 08 times 105 s with CFLN = 10
In the second approach graphene is treated as thin layeroccupying a fraction of FDTD cell [27] The Yee cell for thesubcell FDTDmethod is shown in Figure 3 where119864
119911compo-
nent is split into119864119911119894and119864
119911119900in cells occupying graphene [27]
However this scheme is complex in mathematics and needsspecial type of PML to model infinitely thin sheets [34]
In the third approach graphene is modeled as a zero-thickness conductive sheet over which the fields satisfy thesurface boundary condition Nayyeri et al [28 35] presentedthis scheme in detail As shown in Figure 4(a) a conductivesheet locates at119870+ 12 The tangential component of119867 fieldis discontinuous over the sheet so 119867
119909is split into 1119867
119909and
2119867119909 which satisfy the following discretized Faraday law
1205831
1119867119899+12
119909minus1119867119899minus12
119909
Δ119905=119864119899
119910(119870 + 12) minus 119864
119899
119910(119870)
Δ1199092 (9a)
1205831
2119867119899+12
119909minus2119867119899minus12
119909
Δ119905=119864119899
119910(119870 + 1) minus 119864
119899
119910(119870 + 12)
Δ1199092 (9b)
The surface boundary condition is2119867119909minus1119867119909= 120590119904119864119910
(10)
from which 119864119899119910(119870 + 12) can be written as
119864119899
119910(119870 +
1
2) =
1
2120590119904
[(2119867119899+12
119909+2119867119899minus12
119909)
minus (1119867119899+12
119909+1119867119899minus12
119909)]
(11)
Substituting (11) into (9a) and (9b) 1119867119909and 2119867
119909can be
derived as1119867119899+12
119909=
1
1 minus 11988811198882
(119865119899
1+ 1198881119865119899
2) (12a)
2119867119899+12
119909=
1
1 minus 11988811198882
(119865119899
2+ 1198882119865119899
1) (12b)
The expressions of 1198881 1198882 1198651198991 and 119865
119899
2are shown in [28]
Figure 4(b) is a 3D FDTD cell containing conductive sheetat 119870 + 12 In this case tangential components of magneticfield (119867
119909 119867119910) and normal component of electric field (119864
119911) are
discontinuous over the sheet so they are split into two partson the two sides of the surface Their updating equations canbe derived in a similar way to 1D case and are presented in[28]
The surface boundary condition approach has been val-idated [28] Figure 5(a) shows a 2D problem in which aninfinite line source radiates above an infinite graphene sheetThe pattern of 119864
119911at the wavelength of 100 120583m is shown
in Figure 5(b) A strong agreement between the proposedFDTD result and that derived from semianalytic method isobtained In addition unlike the subcell FDTD method theclassical PML is applicable to the surface boundary conditionscheme
In time-domain simulation the conductivities ofgraphene shown in (2)ndash(4) need to be converted fromfrequency domain into time domain The intrabandconductivity which has a Drude-like expression can beeasily converted into time domain However the interbandconductivity has a complex logarithmic form which needsa vector fitting technique A summation of partial fractionsin terms of complex conjugate pole-residues is applied toapproximate the conductivityThe detailed description of thefitting technique can be found in [35ndash37]
4 International Journal of Antennas and Propagation
300nm
290
nmExciting line
120582SPP
x
y
02
0
minus02
(a)
Exciting line
x
y
02
0
minus02
(b)
Figure 2 The normalized 119864119910component of guided SPPs along the spiral graphene waveguide at time 119905 = 19258 ps calculated by (a)
conventional FDTD and (b) the proposed improved ADI-FDTD with CFLN = 10 [26]
H
E
Hzo
Ezo
Ezi
Hzi
Hz
(i j K)
(i j Klowast)(i + 1 j + 1 Kminus 1)
z
x
y
Figure 3Modified Yee cell for subcell FDTDmethod at the locationof graphene (shaded area) [27]
4 Finite Element Method
FEM is a full-wave numerical technique for electromagneticboundary-value problemsThe basic principle of this methodis to discretize the whole computing domain with a finitenumber of subdomains in which the unknown function isexpanded by simple interpolation functions with unknowncoefficients Then a system of algebraic equations of theseunknown coefficients is obtained by using Ritz variational orGalerkinrsquos method Finally by solving this linear system theapproximated solution of the entire domain can be obtained
In FDTD the computing space is discretized by orthogo-nal grid this staircase approximation will reduce the mod-eling fidelity when it comes to complex geometry whilein FEM where the triangles or tetrahedral elements areapplied arbitrary geometries can be modeled accurately Asa result FEM has advantage in complex and inhomogeneousproblems Furthermore FEM is a frequency domain method
which makes it efficient in dealing with narrow-band prob-lems
Brar et al [12] solved Maxwellrsquos equation by FEM andmodeled graphene as a thin sheet with 01 nm thicknessthe results suggested that graphene can increase light-matterinteractions at infrared energies Software packages such asHFSS and COMSOL are based on FEM and they are usedwidely in graphene simulation Recently we used COMSOLto investigate the transmission properties of a cylindricalgraphene plasmon waveguide [21] Tamagnone et al [29]simulated a reconfigurable graphene antenna with HFSS thestructure of the antenna is shown in Figure 6(a) the inputimpedance of the antenna can be tuned by changing thechemical potential as shown in Figure 6(b)
5 Method of Moments
In contrast to FDTD and FEM which solve differentialequations MoM is a technique used to solve electromag-netic surface or volume integral equations in the frequencydomain In MoM the quantities of interest are not the fieldsbut the electromagnetic sources (surface or volume current)so only the surface of the geometry needs to be discretizedThe surface current is discretized into wire segments andorsurface patches A linear system can be constructed by themethod of weighted residuals and the results of the linearequations give the surface current The far-field result canbe derived from the surface current by Greenrsquos functionBecause it only requires calculating the boundary valuesinstead of the values throughout the space MoM is highlyefficient for electrically large objects and is widely used insolving radiation and scattering problems However whenapplied to complex inhomogeneous cases it will be verycomputationally expensive and less efficient
The analytical expressions of dyadic Greenrsquos function forgraphene are derived in [38] Shapoval et al [39] proposedintegral equations based on surface-impedance boundary
International Journal of Antennas and Propagation 5
Conductive sheet
x
zK
Ey
K +1
2K + 1
Ey
1Hx2Hx
(a)
Conductivesheet
z
y
x
(i j K)
(i j K + 1)
(i j + 1 K)
(i + 1 j K)
Hz
ExEy
1Hx 1Hy
2Hx 2Hy
1Ez
2Ez
(b)
Figure 4 (a) 1D FDTD cell including a conductive sheet at grid 119870 + 12 (b) 3D FDTD cell with a conductive sheet at grid119870 + 12 [28]
PML
Line source Graphenesheet
20120583m
50120583m
Observation circlex
y
(a)
0
30
60
90
120
150
180
210
240
270
300
330
x
y
SemianalyticFDTD
1
08
06
04
02
(b)
Figure 5 (a) Line source scattering by a graphene sheet (b) Normalized pattern of 119864119911at the wavelength of 120582
0= 100 120583m [28]
condition to analyze plane wave scattering and absorptionby graphene-strip gratings The method of moments forgraphene nanoribbons was developed in [40ndash44] in whichthe issue of nonlocality of graphene conductivity was takeninto account Nonlocal effect arises from spatial dispersionof graphene which is nonnegligible when dealing with slowmodes supported by graphene nanoribbons The spatiallydispersive intraband conductivity tensor was derived in [41]
Software packages such as IE3D and FEKO are based onMoM IE3D is applied to simulate themicrowave propagationin a coplanar waveguide over graphene from 40MHz to110GHz [45] Cabellos-Aparicio et al [30] used FEKO tostudy the radiated power of a graphene plasmonic antennafed by photoconductive source The antenna structure isshown in Figure 7(a) and the radiated power with respect to
frequency is shown in Figure 7(b)The detailed parameters ofthe photoconductive antenna can be found in [30]
6 Discontinuous GalerkinTime-Domain Method
The graphene involved problems are often multiscaleGraphene is monoatomic and its thickness is much smallerthan wavelength so it is an electrically fine structure Incontrast the substrate belongs to electrically coarse structurebecause its dimension is much greater than wavelengthAs we have mentioned before time-domain methods havethe advantage that the broadband characterization can beobtained with only a single simulation However FDTD
6 International Journal of Antennas and Propagation
GrapheneAl2O3
GrapheneTHz photomixer
L WSubstrate (GaAs)
(a)
120583c
120583c
fr fr fr fr fr
minus400
minus200
0
200
400
Impe
danc
e (Ω
)
1 15 2 2505
Frequency (THz)
0 eV005 eV01 eV
015 eV02 eV
ReIm
(b)
Figure 6 (a) Structure of a reconfigurable graphene antenna (b) Input impedance of the antenna with respect to chemical potential [29]
Vminus
V+Vbias
BN
Si lens
fs laser pulse
5120583m
14nm
500120583m
GaAs
LT-GaAs
GrapheneContact pads
(120583m)10
20 30
40
(a)
1 2 3 40
Frequency (THz)
0
02
04
06
08
1
12
Radi
ated
pow
er (120583
W)
(b)
Figure 7 (a) Structure of graphene RF plasmonic antenna fed with a photoconductivematerial (b) Radiated power with respect to frequencyfor graphene antennas with lengths 10120583m (blue) 20 120583m (green) 30120583m (red) and 40120583m (black) [30]
method has a serious efficiency problem in multiscale sim-ulation because it requires high discretization density tomodel electrically fine structure due to its cartesian grid Thefinite element time-domain (FETD) method is capable ofmodeling complex and fine structures and achieving high-order accuracy with high-order basis functions The majordrawback is a global linear system of equations that needs tobe solved at each time step The multiscale problems usuallycontain a large number of unknowns FETDwill be very com-putationally expensive in this case Discontinuous Galerkintime-domain (DGTD) method is promising in multiscaleproblems [46] DGTD allows for domain decomposition Amultiscale structure is divided into several subdomains andeach subdomain can be discretized separately All operationsin DGTD are local so large global matrix is split into several
smaller matrices Unlike FETD the matrices of DGTD areinverted and stored before time marching and differenttime integration scheme can be used in different subdomainAdditionally DGTD is naturally adapted to parallel comput-ing Recently DGTD has been used in nanophotonics fieldand considered as a viable alternative to the well-establishedFDTD and FETD methods [47]
Li et al [31] proposed DGTD method with resistiveboundary condition for the electromagnetic analyzing ofgraphene They also applied this method to the magne-tized graphene from microwave to THz range where theanisotropic and disperse surface conductivity is involved[48] Figure 8(a) shows the reflection Γ
119877 transmission Γ
119879 and
absorption Γ119860coefficients of an infinitely large graphene sheet
under the illumination of a normally incident plane wave
International Journal of Antennas and Propagation 7
Exact
ΓT
ΓR
ΓA
0
02
04
06
08
1
Mag
nitu
de
2 4 6 8 100
Frequency (THz)
DGTD + RBC
(a)
Integral equation
0
1
2
3
4
5
6
7
8
ECS
05 1 15 2 25 3 35 40
Frequency (THz)
DGTD + RBC
(b)
Figure 8 (a) The magnitude of reflection Γ119877 transmission Γ
119879 and absorption Γ
119860coefficients calculated by DGFD as well as the theoretical
value (b) Normalized extinction cross section of a graphene patch calculated by DGTD and integral equation method [31]
the results of DGTD agree very well with the theoretical dataFigure 8(b) is the normalized extinction cross section of afreestanding graphene patch good consistency is achievedbetween DGTD and integral equation method More numer-ical examples can be found in [31]
7 Conclusion
Due to its intriguing properties graphene is used more andmore widely in electronic and photonic community Thispaper gives a brief review of application of graphene aswell asthe computational methods which can be used to investigateits electromagnetic properties Although a number of famouscommercial software packages are available for graphene sim-ulation none of them are developed specifically for graphenetherefore they suffer from inefficiency Each computationalmethod has its own advantages and drawbacks one shouldchoose the appropriate method according to the specificproblem FDTD is simple and easy to implement however itis less efficient in modeling fine structure FEM is flexible ingeometry modeling but solving large global system makes itcomputationally expensive DGTD is very suitable for multi-scale problem and we believe it will be more andmore widelyused in modeling and designing graphene-assisted device
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the National Natural Sci-ence Foundation of China (Grants nos 61161007 6126100261461052 and 11564044) the Specialized Research Fundfor the Doctoral Program of Higher Education (Grantsnos 20135301110003 and 20125301120009) China Postdoc-toral Science Foundation (Grants nos 2013M531989 and2014T70890) and the Key Program of Natural Science ofYunnan Province (Grants nos 2013FA006 and 2015FA015)
References
[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004
[2] A K Geim and K S Novoselov ldquoThe rise of graphenerdquo NatureMaterials vol 6 no 3 pp 183ndash191 2007
[3] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008
[4] P Avouris ldquoGraphene electronic and photonic properties anddevicesrdquo Nano Letters vol 10 no 11 pp 4285ndash4294 2010
[5] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005
[6] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009
[7] T Ohta A Bostwick T Seyller K Horn and E RotenbergldquoControlling the electronic structure of bilayer graphenerdquoScience vol 313 no 5789 pp 951ndash954 2006
[8] E V Castro K S Novoselov S VMorozov et al ldquoBiased bilayergraphene semiconductor with a gap tunable by the electric fieldeffectrdquo Physical Review Letters vol 99 no 21 Article ID 2168022007
[9] Y Zhang T-T Tang C Girit et al ldquoDirect observation of awidely tunable bandgap in bilayer graphenerdquo Nature vol 459no 7248 pp 820ndash823 2009
[10] X Li W Cai J An et al ldquoLarge-area synthesis of high-qualityand uniform graphene films on copper foilsrdquo Science vol 324no 5932 pp 1312ndash1314 2009
[11] F Schwierz ldquoGraphene transistorsrdquoNatureNanotechnology vol5 no 7 pp 487ndash496 2010
[12] VW BrarM S JangM Sherrott J J Lopez andH A AtwaterldquoHighly confined tunable mid-infrared plasmonics in graphenenanoresonatorsrdquoNano Letters vol 13 no 6 pp 2541ndash2547 2013
[13] F H L Koppens D E Chang and F J G de Abajo ldquoGrapheneplasmonics a platform for strong lightndashmatter interactionsrdquoNano Letters vol 11 no 8 pp 3370ndash3377 2011
8 International Journal of Antennas and Propagation
[14] J ChenM Badioli P A Gonzalez et al ldquoOptical nano-imagingof gate-tunable graphene plasmonsrdquoNature vol 487 pp 77ndash812012
[15] M Jablan H Buljan and M Soljacic ldquoPlasmonics in grapheneat infrared frequenciesrdquo Physical Review B vol 80 no 24Article ID 245435 2009
[16] Z Fei A S Rodin G O Andreev et al ldquoGate-tuningof graphene plasmons revealed by infrared nano-imagingrdquoNature vol 486 no 7405 pp 82ndash85 2012
[17] A N Grigorenko M Polini and K S Novoselov ldquoGrapheneplasmonicsrdquo Nature Photonics vol 6 no 11 pp 749ndash758 2012
[18] A Vakil and N Engheta ldquoTransformation optics usinggraphenerdquo Science vol 332 no 6035 pp 1291ndash1294 2011
[19] L Ju B Geng J Horng et al ldquoGraphene plasmonics for tunableterahertz metamaterialsrdquo Nature Nanotechnology vol 6 no 10pp 630ndash634 2010
[20] Y Francescato V Giannini J Yang M Huang and S A MaierldquoGraphene sandwiches as a platform for broadband molecularspectroscopyrdquo ACS Photonics vol 1 no 5 pp 437ndash443 2014
[21] J Yang J YangW Deng FMao andMHuang ldquoTransmissionproperties and molecular sensing application of CGPWrdquoOpticsExpress vol 23 no 25 pp 32289ndash32299 2015
[22] Y Huang L-S Wu M Tang and J Mao ldquoDesign of a beamreconfigurable thz antenna with graphene-based switchablehigh-impedance surfacerdquo IEEE Transactions on Nanotechnol-ogy vol 11 no 4 pp 836ndash842 2012
[23] I Llatser C Kremers A Cabellos-Aparicio J M Jornet EAlarcon and D N Chigrin ldquoGraphene-based nano-patchantenna for terahertz radiationrdquo Photonics and Nanostruc-turesmdashFundamentals and Applications vol 10 no 4 pp 353ndash358 2012
[24] Y Yao M A Kats P Genevet et al ldquoBroad electrical tuning ofgraphene-loaded plasmonic antennasrdquoNano Letters vol 13 no3 pp 1257ndash1264 2013
[25] E Forati G W Hanson A B Yakovlev and A Alu ldquoPlanarhyperlens based on a modulated graphene monolayerrdquo PhysicalReview B vol 89 no 8 Article ID 081410 2014
[26] X-H Wang W-Y Yin and Z Chen ldquoBroadband modelingsurface plasmon polaritons in optically pumped and curvedgraphene structures with an improved leapfrog ADI-FDTDmethodrdquo Optics Communications vol 334 pp 152ndash155 2015
[27] G D Bouzianas N V Kantartzis C S Antonopoulos and TD Tsiboukis ldquoOptimal modeling of infinite graphene sheets viaa class of generalized FDTD schemesrdquo IEEE Transactions onMagnetics vol 48 no 2 pp 379ndash382 2012
[28] V Nayyeri M Soleimani and O M Ramahi ldquoModelinggraphene in the finite-difference time-domain method using asurface boundary conditionrdquo IEEE Transactions on Antennasand Propagation vol 61 no 8 pp 4176ndash4182 2013
[29] M Tamagnone J S G Diaz J R Mosig and J P CarrierldquoReconfigurable terahertz plasmonic antenna concept using agraphene stackrdquo Applied Physics Letters vol 101 no 21 ArticleID 214102 2012
[30] A Cabellos-Aparicio I Llatser E Alarcon A Hsu and T Pala-cios ldquoUse of terahertz photoconductive sources to characterizetunable graphene RF plasmonic antennasrdquo IEEE Transactionson Nanotechnology vol 14 no 2 pp 390ndash396 2015
[31] P Li L J Jiang and H Bagc120580 ldquoA resistive boundary conditionenhancedDGTDscheme for the transient analysis of graphenerdquoIEEE Transactions on Antennas and Propagation vol 63 no 7pp 3065ndash3076 2015
[32] G W Hanson ldquoDyadic greenrsquos functions for an anisotropicnon-local model of biased graphenerdquo IEEE Transactions onAntennas and Propagation vol 56 no 3 pp 747ndash757 2008
[33] I Ahmed E H Khoo and E Li ldquoEfficient modeling andsimulation of graphene devices with the LOD-FDTD methodrdquoIEEEMicrowave andWireless Components Letters vol 23 no 6pp 306ndash308 2013
[34] X Yu and C D Sarris ldquoA perfectly matched layer for subcellFDTDand applications to themodeling of graphene structuresrdquoIEEE Antennas and Wireless Propagation Letters vol 11 pp1080ndash1083 2012
[35] V Nayyeri M Soleimani and O M Ramahi ldquoWidebandmodeling of graphene using the finite-difference time-domainmethodrdquo IEEE Transactions on Antennas and Propagation vol61 no 12 pp 6107ndash6114 2013
[36] H Lin M F Pantoja L D Angulo J Alvarez R G Martinand S G Garcia ldquoFDTD modeling of graphene devices usingcomplex conjugate dispersionmaterialmodelrdquo IEEEMicrowaveand Wireless Components Letters vol 22 no 12 pp 612ndash6142012
[37] D-W Wang W-S Zhao X-Q Gu W Chen and W-Y YinldquoWideband modeling of graphene-based structures at differenttemperatures using hybrid FDTD methodrdquo IEEE Transactionson Nanotechnology vol 14 no 2 pp 250ndash258 2015
[38] A Y Nikitin F J Garcia-Vidal and LMartin-Moreno ldquoAnalyt-ical expressions for the electromagnetic dyadic greenrsquos functionin graphene and thin layersrdquo IEEE Journal of Selected Topics inQuantum Electronics vol 19 no 3 Article ID 4600611 2013
[39] O V Shapoval J S Gomez-Diaz J Perruisseau-Carrier J RMosig and A I Nosich ldquoIntegral equation analysis of planewave scattering by coplanar graphene-strip gratings in the thzrangerdquo IEEE Transactions on Terahertz Science and Technologyvol 3 no 5 pp 666ndash674 2013
[40] R Araneo G Lovat and P Burghignoli ldquoGraphene nanostriplines dispersion and attenuation analysisrdquo in Proceedings of the16thWorkshop on Signal and Power Integrity (SPI rsquo12) pp 75ndash78IEEE Sorrento Italy May 2012
[41] G Lovat G W Hanson R Araneo and P Burghignoli ldquoSemi-classical spatially dispersive intraband conductivity tensor andquantum capacitance of graphenerdquo Physical Review B vol 87no 11 Article ID 115429 2013
[42] P Burghignoli R Araneo G Lovat and G Hanson ldquoSpace-domain method of moments for graphene nanoribbonsrdquo inProceedings of the 8th European Conference on Antennas andPropagation (EuCAP rsquo14) pp 666ndash669 IEEE The Hague TheNetherlands April 2014
[43] R Araneo P Burghignoli G Lovat and G W HansonldquoModal propagation and crosstalk analysis in coupled graphenenanoribbonsrdquo IEEE Transactions on Electromagnetic Compati-bility vol 57 no 4 pp 726ndash733 2015
[44] A Fallahi T Low M Tamagnone and J Perruisseau-CarrierldquoNonlocal electromagnetic response of graphene nanostruc-turesrdquo Physical Review B vol 91 Article ID 121405 2015
[45] M Dragoman D Neculoiu A Cismaru et al ldquoCoplanarwaveguide on graphene in the range 40MHz-110 GHzrdquoAppliedPhysics Letters vol 99 no 3 Article ID 033112 2011
[46] J Chen and Q H Liu ldquoDiscontinuous Galerkin Time-Domainmethods for multiscale electromagnetic simulations a reviewrdquoProceedings of the IEEE vol 101 no 2 pp 242ndash254 2013
International Journal of Antennas and Propagation 9
[47] S Descombes C Durochat S Lanteri L Moya C Scheid andJ Viquerat ldquoRecent advances on a DGTD method for time-domain electromagneticsrdquo Photonics and NanostructuresmdashFundamentals and Applications vol 11 no 4 pp 291ndash302 2013
[48] P Li and L J Jiang ldquoModeling of magnetized graphene frommicrowave to THz range by DGTD with a scalar RBC and anADErdquo IEEE Transactions on Antennas and Propagation vol 63no 10 pp 4458ndash4467 2015
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
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Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 3
1205763D gt minus1 (not allowed)
25
30
35
40
45
50
55
Im(120590)
(120583S)
300 400 500 600 700 800 900200
d1205820 (times106)
70
60
50
40
30
20
10
0
Figure 1 Relative error of the dielectric slab model with respect tonormalized 119889 and 120590 [25]
3 Finite-Difference Time-Domain Method
FDTD is time-domain method which has specific advantagein transient problem and wideband problemThe broadbandresults can be obtained in one simulation Compared tofrequency-domain method it takes less computing memoryand simulation time Additionally FDTD is an iterativescheme which eliminates solving large linear system henceit is simple robust and easy to implement
There are three general approaches to model graphenein FDTD (I) using standard FDTD method with very finemesh within the graphene sheet (II) using the subcell FDTDmethod (III) using surface resistive boundary condition
In the first approach graphene is modeled as thin layerwith finite thickness and the surface conductivity is trans-formed to volume permittivity According to the Courant-Friedrichs-Lewy (CFL) stability condition of FDTD finemesh leads to small time step so this approach is bothmemory- and time-consuming which reduces its efficiencyTo alleviate the time step constraint some unconditionallystable FDTD methods are developed for graphene simu-lation In [33] the locally one-dimensional (LOD) FDTDis applied for the efficient simulation of graphene deviceThey demonstrated that the LOD-FDTD can be 60 fasterthan standard FDTD with reasonable accuracy Wang et al[26] proposed an unconditionally stable one-step leapfrogalternating-direction-implicit (ADI) FDTD to study the SPPsin graphene structure In Figure 2 they showed the SPPspropagating along the spiral waveguide The one-step ADI-FDTD method gives accurate result which is comparable toconventional FDTDThey also showed that the conventionalFDTD took 17 times 105 s for this problem while ADI-FDTDonly took 08 times 105 s with CFLN = 10
In the second approach graphene is treated as thin layeroccupying a fraction of FDTD cell [27] The Yee cell for thesubcell FDTDmethod is shown in Figure 3 where119864
119911compo-
nent is split into119864119911119894and119864
119911119900in cells occupying graphene [27]
However this scheme is complex in mathematics and needsspecial type of PML to model infinitely thin sheets [34]
In the third approach graphene is modeled as a zero-thickness conductive sheet over which the fields satisfy thesurface boundary condition Nayyeri et al [28 35] presentedthis scheme in detail As shown in Figure 4(a) a conductivesheet locates at119870+ 12 The tangential component of119867 fieldis discontinuous over the sheet so 119867
119909is split into 1119867
119909and
2119867119909 which satisfy the following discretized Faraday law
1205831
1119867119899+12
119909minus1119867119899minus12
119909
Δ119905=119864119899
119910(119870 + 12) minus 119864
119899
119910(119870)
Δ1199092 (9a)
1205831
2119867119899+12
119909minus2119867119899minus12
119909
Δ119905=119864119899
119910(119870 + 1) minus 119864
119899
119910(119870 + 12)
Δ1199092 (9b)
The surface boundary condition is2119867119909minus1119867119909= 120590119904119864119910
(10)
from which 119864119899119910(119870 + 12) can be written as
119864119899
119910(119870 +
1
2) =
1
2120590119904
[(2119867119899+12
119909+2119867119899minus12
119909)
minus (1119867119899+12
119909+1119867119899minus12
119909)]
(11)
Substituting (11) into (9a) and (9b) 1119867119909and 2119867
119909can be
derived as1119867119899+12
119909=
1
1 minus 11988811198882
(119865119899
1+ 1198881119865119899
2) (12a)
2119867119899+12
119909=
1
1 minus 11988811198882
(119865119899
2+ 1198882119865119899
1) (12b)
The expressions of 1198881 1198882 1198651198991 and 119865
119899
2are shown in [28]
Figure 4(b) is a 3D FDTD cell containing conductive sheetat 119870 + 12 In this case tangential components of magneticfield (119867
119909 119867119910) and normal component of electric field (119864
119911) are
discontinuous over the sheet so they are split into two partson the two sides of the surface Their updating equations canbe derived in a similar way to 1D case and are presented in[28]
The surface boundary condition approach has been val-idated [28] Figure 5(a) shows a 2D problem in which aninfinite line source radiates above an infinite graphene sheetThe pattern of 119864
119911at the wavelength of 100 120583m is shown
in Figure 5(b) A strong agreement between the proposedFDTD result and that derived from semianalytic method isobtained In addition unlike the subcell FDTD method theclassical PML is applicable to the surface boundary conditionscheme
In time-domain simulation the conductivities ofgraphene shown in (2)ndash(4) need to be converted fromfrequency domain into time domain The intrabandconductivity which has a Drude-like expression can beeasily converted into time domain However the interbandconductivity has a complex logarithmic form which needsa vector fitting technique A summation of partial fractionsin terms of complex conjugate pole-residues is applied toapproximate the conductivityThe detailed description of thefitting technique can be found in [35ndash37]
4 International Journal of Antennas and Propagation
300nm
290
nmExciting line
120582SPP
x
y
02
0
minus02
(a)
Exciting line
x
y
02
0
minus02
(b)
Figure 2 The normalized 119864119910component of guided SPPs along the spiral graphene waveguide at time 119905 = 19258 ps calculated by (a)
conventional FDTD and (b) the proposed improved ADI-FDTD with CFLN = 10 [26]
H
E
Hzo
Ezo
Ezi
Hzi
Hz
(i j K)
(i j Klowast)(i + 1 j + 1 Kminus 1)
z
x
y
Figure 3Modified Yee cell for subcell FDTDmethod at the locationof graphene (shaded area) [27]
4 Finite Element Method
FEM is a full-wave numerical technique for electromagneticboundary-value problemsThe basic principle of this methodis to discretize the whole computing domain with a finitenumber of subdomains in which the unknown function isexpanded by simple interpolation functions with unknowncoefficients Then a system of algebraic equations of theseunknown coefficients is obtained by using Ritz variational orGalerkinrsquos method Finally by solving this linear system theapproximated solution of the entire domain can be obtained
In FDTD the computing space is discretized by orthogo-nal grid this staircase approximation will reduce the mod-eling fidelity when it comes to complex geometry whilein FEM where the triangles or tetrahedral elements areapplied arbitrary geometries can be modeled accurately Asa result FEM has advantage in complex and inhomogeneousproblems Furthermore FEM is a frequency domain method
which makes it efficient in dealing with narrow-band prob-lems
Brar et al [12] solved Maxwellrsquos equation by FEM andmodeled graphene as a thin sheet with 01 nm thicknessthe results suggested that graphene can increase light-matterinteractions at infrared energies Software packages such asHFSS and COMSOL are based on FEM and they are usedwidely in graphene simulation Recently we used COMSOLto investigate the transmission properties of a cylindricalgraphene plasmon waveguide [21] Tamagnone et al [29]simulated a reconfigurable graphene antenna with HFSS thestructure of the antenna is shown in Figure 6(a) the inputimpedance of the antenna can be tuned by changing thechemical potential as shown in Figure 6(b)
5 Method of Moments
In contrast to FDTD and FEM which solve differentialequations MoM is a technique used to solve electromag-netic surface or volume integral equations in the frequencydomain In MoM the quantities of interest are not the fieldsbut the electromagnetic sources (surface or volume current)so only the surface of the geometry needs to be discretizedThe surface current is discretized into wire segments andorsurface patches A linear system can be constructed by themethod of weighted residuals and the results of the linearequations give the surface current The far-field result canbe derived from the surface current by Greenrsquos functionBecause it only requires calculating the boundary valuesinstead of the values throughout the space MoM is highlyefficient for electrically large objects and is widely used insolving radiation and scattering problems However whenapplied to complex inhomogeneous cases it will be verycomputationally expensive and less efficient
The analytical expressions of dyadic Greenrsquos function forgraphene are derived in [38] Shapoval et al [39] proposedintegral equations based on surface-impedance boundary
International Journal of Antennas and Propagation 5
Conductive sheet
x
zK
Ey
K +1
2K + 1
Ey
1Hx2Hx
(a)
Conductivesheet
z
y
x
(i j K)
(i j K + 1)
(i j + 1 K)
(i + 1 j K)
Hz
ExEy
1Hx 1Hy
2Hx 2Hy
1Ez
2Ez
(b)
Figure 4 (a) 1D FDTD cell including a conductive sheet at grid 119870 + 12 (b) 3D FDTD cell with a conductive sheet at grid119870 + 12 [28]
PML
Line source Graphenesheet
20120583m
50120583m
Observation circlex
y
(a)
0
30
60
90
120
150
180
210
240
270
300
330
x
y
SemianalyticFDTD
1
08
06
04
02
(b)
Figure 5 (a) Line source scattering by a graphene sheet (b) Normalized pattern of 119864119911at the wavelength of 120582
0= 100 120583m [28]
condition to analyze plane wave scattering and absorptionby graphene-strip gratings The method of moments forgraphene nanoribbons was developed in [40ndash44] in whichthe issue of nonlocality of graphene conductivity was takeninto account Nonlocal effect arises from spatial dispersionof graphene which is nonnegligible when dealing with slowmodes supported by graphene nanoribbons The spatiallydispersive intraband conductivity tensor was derived in [41]
Software packages such as IE3D and FEKO are based onMoM IE3D is applied to simulate themicrowave propagationin a coplanar waveguide over graphene from 40MHz to110GHz [45] Cabellos-Aparicio et al [30] used FEKO tostudy the radiated power of a graphene plasmonic antennafed by photoconductive source The antenna structure isshown in Figure 7(a) and the radiated power with respect to
frequency is shown in Figure 7(b)The detailed parameters ofthe photoconductive antenna can be found in [30]
6 Discontinuous GalerkinTime-Domain Method
The graphene involved problems are often multiscaleGraphene is monoatomic and its thickness is much smallerthan wavelength so it is an electrically fine structure Incontrast the substrate belongs to electrically coarse structurebecause its dimension is much greater than wavelengthAs we have mentioned before time-domain methods havethe advantage that the broadband characterization can beobtained with only a single simulation However FDTD
6 International Journal of Antennas and Propagation
GrapheneAl2O3
GrapheneTHz photomixer
L WSubstrate (GaAs)
(a)
120583c
120583c
fr fr fr fr fr
minus400
minus200
0
200
400
Impe
danc
e (Ω
)
1 15 2 2505
Frequency (THz)
0 eV005 eV01 eV
015 eV02 eV
ReIm
(b)
Figure 6 (a) Structure of a reconfigurable graphene antenna (b) Input impedance of the antenna with respect to chemical potential [29]
Vminus
V+Vbias
BN
Si lens
fs laser pulse
5120583m
14nm
500120583m
GaAs
LT-GaAs
GrapheneContact pads
(120583m)10
20 30
40
(a)
1 2 3 40
Frequency (THz)
0
02
04
06
08
1
12
Radi
ated
pow
er (120583
W)
(b)
Figure 7 (a) Structure of graphene RF plasmonic antenna fed with a photoconductivematerial (b) Radiated power with respect to frequencyfor graphene antennas with lengths 10120583m (blue) 20 120583m (green) 30120583m (red) and 40120583m (black) [30]
method has a serious efficiency problem in multiscale sim-ulation because it requires high discretization density tomodel electrically fine structure due to its cartesian grid Thefinite element time-domain (FETD) method is capable ofmodeling complex and fine structures and achieving high-order accuracy with high-order basis functions The majordrawback is a global linear system of equations that needs tobe solved at each time step The multiscale problems usuallycontain a large number of unknowns FETDwill be very com-putationally expensive in this case Discontinuous Galerkintime-domain (DGTD) method is promising in multiscaleproblems [46] DGTD allows for domain decomposition Amultiscale structure is divided into several subdomains andeach subdomain can be discretized separately All operationsin DGTD are local so large global matrix is split into several
smaller matrices Unlike FETD the matrices of DGTD areinverted and stored before time marching and differenttime integration scheme can be used in different subdomainAdditionally DGTD is naturally adapted to parallel comput-ing Recently DGTD has been used in nanophotonics fieldand considered as a viable alternative to the well-establishedFDTD and FETD methods [47]
Li et al [31] proposed DGTD method with resistiveboundary condition for the electromagnetic analyzing ofgraphene They also applied this method to the magne-tized graphene from microwave to THz range where theanisotropic and disperse surface conductivity is involved[48] Figure 8(a) shows the reflection Γ
119877 transmission Γ
119879 and
absorption Γ119860coefficients of an infinitely large graphene sheet
under the illumination of a normally incident plane wave
International Journal of Antennas and Propagation 7
Exact
ΓT
ΓR
ΓA
0
02
04
06
08
1
Mag
nitu
de
2 4 6 8 100
Frequency (THz)
DGTD + RBC
(a)
Integral equation
0
1
2
3
4
5
6
7
8
ECS
05 1 15 2 25 3 35 40
Frequency (THz)
DGTD + RBC
(b)
Figure 8 (a) The magnitude of reflection Γ119877 transmission Γ
119879 and absorption Γ
119860coefficients calculated by DGFD as well as the theoretical
value (b) Normalized extinction cross section of a graphene patch calculated by DGTD and integral equation method [31]
the results of DGTD agree very well with the theoretical dataFigure 8(b) is the normalized extinction cross section of afreestanding graphene patch good consistency is achievedbetween DGTD and integral equation method More numer-ical examples can be found in [31]
7 Conclusion
Due to its intriguing properties graphene is used more andmore widely in electronic and photonic community Thispaper gives a brief review of application of graphene aswell asthe computational methods which can be used to investigateits electromagnetic properties Although a number of famouscommercial software packages are available for graphene sim-ulation none of them are developed specifically for graphenetherefore they suffer from inefficiency Each computationalmethod has its own advantages and drawbacks one shouldchoose the appropriate method according to the specificproblem FDTD is simple and easy to implement however itis less efficient in modeling fine structure FEM is flexible ingeometry modeling but solving large global system makes itcomputationally expensive DGTD is very suitable for multi-scale problem and we believe it will be more andmore widelyused in modeling and designing graphene-assisted device
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the National Natural Sci-ence Foundation of China (Grants nos 61161007 6126100261461052 and 11564044) the Specialized Research Fundfor the Doctoral Program of Higher Education (Grantsnos 20135301110003 and 20125301120009) China Postdoc-toral Science Foundation (Grants nos 2013M531989 and2014T70890) and the Key Program of Natural Science ofYunnan Province (Grants nos 2013FA006 and 2015FA015)
References
[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004
[2] A K Geim and K S Novoselov ldquoThe rise of graphenerdquo NatureMaterials vol 6 no 3 pp 183ndash191 2007
[3] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008
[4] P Avouris ldquoGraphene electronic and photonic properties anddevicesrdquo Nano Letters vol 10 no 11 pp 4285ndash4294 2010
[5] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005
[6] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009
[7] T Ohta A Bostwick T Seyller K Horn and E RotenbergldquoControlling the electronic structure of bilayer graphenerdquoScience vol 313 no 5789 pp 951ndash954 2006
[8] E V Castro K S Novoselov S VMorozov et al ldquoBiased bilayergraphene semiconductor with a gap tunable by the electric fieldeffectrdquo Physical Review Letters vol 99 no 21 Article ID 2168022007
[9] Y Zhang T-T Tang C Girit et al ldquoDirect observation of awidely tunable bandgap in bilayer graphenerdquo Nature vol 459no 7248 pp 820ndash823 2009
[10] X Li W Cai J An et al ldquoLarge-area synthesis of high-qualityand uniform graphene films on copper foilsrdquo Science vol 324no 5932 pp 1312ndash1314 2009
[11] F Schwierz ldquoGraphene transistorsrdquoNatureNanotechnology vol5 no 7 pp 487ndash496 2010
[12] VW BrarM S JangM Sherrott J J Lopez andH A AtwaterldquoHighly confined tunable mid-infrared plasmonics in graphenenanoresonatorsrdquoNano Letters vol 13 no 6 pp 2541ndash2547 2013
[13] F H L Koppens D E Chang and F J G de Abajo ldquoGrapheneplasmonics a platform for strong lightndashmatter interactionsrdquoNano Letters vol 11 no 8 pp 3370ndash3377 2011
8 International Journal of Antennas and Propagation
[14] J ChenM Badioli P A Gonzalez et al ldquoOptical nano-imagingof gate-tunable graphene plasmonsrdquoNature vol 487 pp 77ndash812012
[15] M Jablan H Buljan and M Soljacic ldquoPlasmonics in grapheneat infrared frequenciesrdquo Physical Review B vol 80 no 24Article ID 245435 2009
[16] Z Fei A S Rodin G O Andreev et al ldquoGate-tuningof graphene plasmons revealed by infrared nano-imagingrdquoNature vol 486 no 7405 pp 82ndash85 2012
[17] A N Grigorenko M Polini and K S Novoselov ldquoGrapheneplasmonicsrdquo Nature Photonics vol 6 no 11 pp 749ndash758 2012
[18] A Vakil and N Engheta ldquoTransformation optics usinggraphenerdquo Science vol 332 no 6035 pp 1291ndash1294 2011
[19] L Ju B Geng J Horng et al ldquoGraphene plasmonics for tunableterahertz metamaterialsrdquo Nature Nanotechnology vol 6 no 10pp 630ndash634 2010
[20] Y Francescato V Giannini J Yang M Huang and S A MaierldquoGraphene sandwiches as a platform for broadband molecularspectroscopyrdquo ACS Photonics vol 1 no 5 pp 437ndash443 2014
[21] J Yang J YangW Deng FMao andMHuang ldquoTransmissionproperties and molecular sensing application of CGPWrdquoOpticsExpress vol 23 no 25 pp 32289ndash32299 2015
[22] Y Huang L-S Wu M Tang and J Mao ldquoDesign of a beamreconfigurable thz antenna with graphene-based switchablehigh-impedance surfacerdquo IEEE Transactions on Nanotechnol-ogy vol 11 no 4 pp 836ndash842 2012
[23] I Llatser C Kremers A Cabellos-Aparicio J M Jornet EAlarcon and D N Chigrin ldquoGraphene-based nano-patchantenna for terahertz radiationrdquo Photonics and Nanostruc-turesmdashFundamentals and Applications vol 10 no 4 pp 353ndash358 2012
[24] Y Yao M A Kats P Genevet et al ldquoBroad electrical tuning ofgraphene-loaded plasmonic antennasrdquoNano Letters vol 13 no3 pp 1257ndash1264 2013
[25] E Forati G W Hanson A B Yakovlev and A Alu ldquoPlanarhyperlens based on a modulated graphene monolayerrdquo PhysicalReview B vol 89 no 8 Article ID 081410 2014
[26] X-H Wang W-Y Yin and Z Chen ldquoBroadband modelingsurface plasmon polaritons in optically pumped and curvedgraphene structures with an improved leapfrog ADI-FDTDmethodrdquo Optics Communications vol 334 pp 152ndash155 2015
[27] G D Bouzianas N V Kantartzis C S Antonopoulos and TD Tsiboukis ldquoOptimal modeling of infinite graphene sheets viaa class of generalized FDTD schemesrdquo IEEE Transactions onMagnetics vol 48 no 2 pp 379ndash382 2012
[28] V Nayyeri M Soleimani and O M Ramahi ldquoModelinggraphene in the finite-difference time-domain method using asurface boundary conditionrdquo IEEE Transactions on Antennasand Propagation vol 61 no 8 pp 4176ndash4182 2013
[29] M Tamagnone J S G Diaz J R Mosig and J P CarrierldquoReconfigurable terahertz plasmonic antenna concept using agraphene stackrdquo Applied Physics Letters vol 101 no 21 ArticleID 214102 2012
[30] A Cabellos-Aparicio I Llatser E Alarcon A Hsu and T Pala-cios ldquoUse of terahertz photoconductive sources to characterizetunable graphene RF plasmonic antennasrdquo IEEE Transactionson Nanotechnology vol 14 no 2 pp 390ndash396 2015
[31] P Li L J Jiang and H Bagc120580 ldquoA resistive boundary conditionenhancedDGTDscheme for the transient analysis of graphenerdquoIEEE Transactions on Antennas and Propagation vol 63 no 7pp 3065ndash3076 2015
[32] G W Hanson ldquoDyadic greenrsquos functions for an anisotropicnon-local model of biased graphenerdquo IEEE Transactions onAntennas and Propagation vol 56 no 3 pp 747ndash757 2008
[33] I Ahmed E H Khoo and E Li ldquoEfficient modeling andsimulation of graphene devices with the LOD-FDTD methodrdquoIEEEMicrowave andWireless Components Letters vol 23 no 6pp 306ndash308 2013
[34] X Yu and C D Sarris ldquoA perfectly matched layer for subcellFDTDand applications to themodeling of graphene structuresrdquoIEEE Antennas and Wireless Propagation Letters vol 11 pp1080ndash1083 2012
[35] V Nayyeri M Soleimani and O M Ramahi ldquoWidebandmodeling of graphene using the finite-difference time-domainmethodrdquo IEEE Transactions on Antennas and Propagation vol61 no 12 pp 6107ndash6114 2013
[36] H Lin M F Pantoja L D Angulo J Alvarez R G Martinand S G Garcia ldquoFDTD modeling of graphene devices usingcomplex conjugate dispersionmaterialmodelrdquo IEEEMicrowaveand Wireless Components Letters vol 22 no 12 pp 612ndash6142012
[37] D-W Wang W-S Zhao X-Q Gu W Chen and W-Y YinldquoWideband modeling of graphene-based structures at differenttemperatures using hybrid FDTD methodrdquo IEEE Transactionson Nanotechnology vol 14 no 2 pp 250ndash258 2015
[38] A Y Nikitin F J Garcia-Vidal and LMartin-Moreno ldquoAnalyt-ical expressions for the electromagnetic dyadic greenrsquos functionin graphene and thin layersrdquo IEEE Journal of Selected Topics inQuantum Electronics vol 19 no 3 Article ID 4600611 2013
[39] O V Shapoval J S Gomez-Diaz J Perruisseau-Carrier J RMosig and A I Nosich ldquoIntegral equation analysis of planewave scattering by coplanar graphene-strip gratings in the thzrangerdquo IEEE Transactions on Terahertz Science and Technologyvol 3 no 5 pp 666ndash674 2013
[40] R Araneo G Lovat and P Burghignoli ldquoGraphene nanostriplines dispersion and attenuation analysisrdquo in Proceedings of the16thWorkshop on Signal and Power Integrity (SPI rsquo12) pp 75ndash78IEEE Sorrento Italy May 2012
[41] G Lovat G W Hanson R Araneo and P Burghignoli ldquoSemi-classical spatially dispersive intraband conductivity tensor andquantum capacitance of graphenerdquo Physical Review B vol 87no 11 Article ID 115429 2013
[42] P Burghignoli R Araneo G Lovat and G Hanson ldquoSpace-domain method of moments for graphene nanoribbonsrdquo inProceedings of the 8th European Conference on Antennas andPropagation (EuCAP rsquo14) pp 666ndash669 IEEE The Hague TheNetherlands April 2014
[43] R Araneo P Burghignoli G Lovat and G W HansonldquoModal propagation and crosstalk analysis in coupled graphenenanoribbonsrdquo IEEE Transactions on Electromagnetic Compati-bility vol 57 no 4 pp 726ndash733 2015
[44] A Fallahi T Low M Tamagnone and J Perruisseau-CarrierldquoNonlocal electromagnetic response of graphene nanostruc-turesrdquo Physical Review B vol 91 Article ID 121405 2015
[45] M Dragoman D Neculoiu A Cismaru et al ldquoCoplanarwaveguide on graphene in the range 40MHz-110 GHzrdquoAppliedPhysics Letters vol 99 no 3 Article ID 033112 2011
[46] J Chen and Q H Liu ldquoDiscontinuous Galerkin Time-Domainmethods for multiscale electromagnetic simulations a reviewrdquoProceedings of the IEEE vol 101 no 2 pp 242ndash254 2013
International Journal of Antennas and Propagation 9
[47] S Descombes C Durochat S Lanteri L Moya C Scheid andJ Viquerat ldquoRecent advances on a DGTD method for time-domain electromagneticsrdquo Photonics and NanostructuresmdashFundamentals and Applications vol 11 no 4 pp 291ndash302 2013
[48] P Li and L J Jiang ldquoModeling of magnetized graphene frommicrowave to THz range by DGTD with a scalar RBC and anADErdquo IEEE Transactions on Antennas and Propagation vol 63no 10 pp 4458ndash4467 2015
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 International Journal of Antennas and Propagation
300nm
290
nmExciting line
120582SPP
x
y
02
0
minus02
(a)
Exciting line
x
y
02
0
minus02
(b)
Figure 2 The normalized 119864119910component of guided SPPs along the spiral graphene waveguide at time 119905 = 19258 ps calculated by (a)
conventional FDTD and (b) the proposed improved ADI-FDTD with CFLN = 10 [26]
H
E
Hzo
Ezo
Ezi
Hzi
Hz
(i j K)
(i j Klowast)(i + 1 j + 1 Kminus 1)
z
x
y
Figure 3Modified Yee cell for subcell FDTDmethod at the locationof graphene (shaded area) [27]
4 Finite Element Method
FEM is a full-wave numerical technique for electromagneticboundary-value problemsThe basic principle of this methodis to discretize the whole computing domain with a finitenumber of subdomains in which the unknown function isexpanded by simple interpolation functions with unknowncoefficients Then a system of algebraic equations of theseunknown coefficients is obtained by using Ritz variational orGalerkinrsquos method Finally by solving this linear system theapproximated solution of the entire domain can be obtained
In FDTD the computing space is discretized by orthogo-nal grid this staircase approximation will reduce the mod-eling fidelity when it comes to complex geometry whilein FEM where the triangles or tetrahedral elements areapplied arbitrary geometries can be modeled accurately Asa result FEM has advantage in complex and inhomogeneousproblems Furthermore FEM is a frequency domain method
which makes it efficient in dealing with narrow-band prob-lems
Brar et al [12] solved Maxwellrsquos equation by FEM andmodeled graphene as a thin sheet with 01 nm thicknessthe results suggested that graphene can increase light-matterinteractions at infrared energies Software packages such asHFSS and COMSOL are based on FEM and they are usedwidely in graphene simulation Recently we used COMSOLto investigate the transmission properties of a cylindricalgraphene plasmon waveguide [21] Tamagnone et al [29]simulated a reconfigurable graphene antenna with HFSS thestructure of the antenna is shown in Figure 6(a) the inputimpedance of the antenna can be tuned by changing thechemical potential as shown in Figure 6(b)
5 Method of Moments
In contrast to FDTD and FEM which solve differentialequations MoM is a technique used to solve electromag-netic surface or volume integral equations in the frequencydomain In MoM the quantities of interest are not the fieldsbut the electromagnetic sources (surface or volume current)so only the surface of the geometry needs to be discretizedThe surface current is discretized into wire segments andorsurface patches A linear system can be constructed by themethod of weighted residuals and the results of the linearequations give the surface current The far-field result canbe derived from the surface current by Greenrsquos functionBecause it only requires calculating the boundary valuesinstead of the values throughout the space MoM is highlyefficient for electrically large objects and is widely used insolving radiation and scattering problems However whenapplied to complex inhomogeneous cases it will be verycomputationally expensive and less efficient
The analytical expressions of dyadic Greenrsquos function forgraphene are derived in [38] Shapoval et al [39] proposedintegral equations based on surface-impedance boundary
International Journal of Antennas and Propagation 5
Conductive sheet
x
zK
Ey
K +1
2K + 1
Ey
1Hx2Hx
(a)
Conductivesheet
z
y
x
(i j K)
(i j K + 1)
(i j + 1 K)
(i + 1 j K)
Hz
ExEy
1Hx 1Hy
2Hx 2Hy
1Ez
2Ez
(b)
Figure 4 (a) 1D FDTD cell including a conductive sheet at grid 119870 + 12 (b) 3D FDTD cell with a conductive sheet at grid119870 + 12 [28]
PML
Line source Graphenesheet
20120583m
50120583m
Observation circlex
y
(a)
0
30
60
90
120
150
180
210
240
270
300
330
x
y
SemianalyticFDTD
1
08
06
04
02
(b)
Figure 5 (a) Line source scattering by a graphene sheet (b) Normalized pattern of 119864119911at the wavelength of 120582
0= 100 120583m [28]
condition to analyze plane wave scattering and absorptionby graphene-strip gratings The method of moments forgraphene nanoribbons was developed in [40ndash44] in whichthe issue of nonlocality of graphene conductivity was takeninto account Nonlocal effect arises from spatial dispersionof graphene which is nonnegligible when dealing with slowmodes supported by graphene nanoribbons The spatiallydispersive intraband conductivity tensor was derived in [41]
Software packages such as IE3D and FEKO are based onMoM IE3D is applied to simulate themicrowave propagationin a coplanar waveguide over graphene from 40MHz to110GHz [45] Cabellos-Aparicio et al [30] used FEKO tostudy the radiated power of a graphene plasmonic antennafed by photoconductive source The antenna structure isshown in Figure 7(a) and the radiated power with respect to
frequency is shown in Figure 7(b)The detailed parameters ofthe photoconductive antenna can be found in [30]
6 Discontinuous GalerkinTime-Domain Method
The graphene involved problems are often multiscaleGraphene is monoatomic and its thickness is much smallerthan wavelength so it is an electrically fine structure Incontrast the substrate belongs to electrically coarse structurebecause its dimension is much greater than wavelengthAs we have mentioned before time-domain methods havethe advantage that the broadband characterization can beobtained with only a single simulation However FDTD
6 International Journal of Antennas and Propagation
GrapheneAl2O3
GrapheneTHz photomixer
L WSubstrate (GaAs)
(a)
120583c
120583c
fr fr fr fr fr
minus400
minus200
0
200
400
Impe
danc
e (Ω
)
1 15 2 2505
Frequency (THz)
0 eV005 eV01 eV
015 eV02 eV
ReIm
(b)
Figure 6 (a) Structure of a reconfigurable graphene antenna (b) Input impedance of the antenna with respect to chemical potential [29]
Vminus
V+Vbias
BN
Si lens
fs laser pulse
5120583m
14nm
500120583m
GaAs
LT-GaAs
GrapheneContact pads
(120583m)10
20 30
40
(a)
1 2 3 40
Frequency (THz)
0
02
04
06
08
1
12
Radi
ated
pow
er (120583
W)
(b)
Figure 7 (a) Structure of graphene RF plasmonic antenna fed with a photoconductivematerial (b) Radiated power with respect to frequencyfor graphene antennas with lengths 10120583m (blue) 20 120583m (green) 30120583m (red) and 40120583m (black) [30]
method has a serious efficiency problem in multiscale sim-ulation because it requires high discretization density tomodel electrically fine structure due to its cartesian grid Thefinite element time-domain (FETD) method is capable ofmodeling complex and fine structures and achieving high-order accuracy with high-order basis functions The majordrawback is a global linear system of equations that needs tobe solved at each time step The multiscale problems usuallycontain a large number of unknowns FETDwill be very com-putationally expensive in this case Discontinuous Galerkintime-domain (DGTD) method is promising in multiscaleproblems [46] DGTD allows for domain decomposition Amultiscale structure is divided into several subdomains andeach subdomain can be discretized separately All operationsin DGTD are local so large global matrix is split into several
smaller matrices Unlike FETD the matrices of DGTD areinverted and stored before time marching and differenttime integration scheme can be used in different subdomainAdditionally DGTD is naturally adapted to parallel comput-ing Recently DGTD has been used in nanophotonics fieldand considered as a viable alternative to the well-establishedFDTD and FETD methods [47]
Li et al [31] proposed DGTD method with resistiveboundary condition for the electromagnetic analyzing ofgraphene They also applied this method to the magne-tized graphene from microwave to THz range where theanisotropic and disperse surface conductivity is involved[48] Figure 8(a) shows the reflection Γ
119877 transmission Γ
119879 and
absorption Γ119860coefficients of an infinitely large graphene sheet
under the illumination of a normally incident plane wave
International Journal of Antennas and Propagation 7
Exact
ΓT
ΓR
ΓA
0
02
04
06
08
1
Mag
nitu
de
2 4 6 8 100
Frequency (THz)
DGTD + RBC
(a)
Integral equation
0
1
2
3
4
5
6
7
8
ECS
05 1 15 2 25 3 35 40
Frequency (THz)
DGTD + RBC
(b)
Figure 8 (a) The magnitude of reflection Γ119877 transmission Γ
119879 and absorption Γ
119860coefficients calculated by DGFD as well as the theoretical
value (b) Normalized extinction cross section of a graphene patch calculated by DGTD and integral equation method [31]
the results of DGTD agree very well with the theoretical dataFigure 8(b) is the normalized extinction cross section of afreestanding graphene patch good consistency is achievedbetween DGTD and integral equation method More numer-ical examples can be found in [31]
7 Conclusion
Due to its intriguing properties graphene is used more andmore widely in electronic and photonic community Thispaper gives a brief review of application of graphene aswell asthe computational methods which can be used to investigateits electromagnetic properties Although a number of famouscommercial software packages are available for graphene sim-ulation none of them are developed specifically for graphenetherefore they suffer from inefficiency Each computationalmethod has its own advantages and drawbacks one shouldchoose the appropriate method according to the specificproblem FDTD is simple and easy to implement however itis less efficient in modeling fine structure FEM is flexible ingeometry modeling but solving large global system makes itcomputationally expensive DGTD is very suitable for multi-scale problem and we believe it will be more andmore widelyused in modeling and designing graphene-assisted device
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the National Natural Sci-ence Foundation of China (Grants nos 61161007 6126100261461052 and 11564044) the Specialized Research Fundfor the Doctoral Program of Higher Education (Grantsnos 20135301110003 and 20125301120009) China Postdoc-toral Science Foundation (Grants nos 2013M531989 and2014T70890) and the Key Program of Natural Science ofYunnan Province (Grants nos 2013FA006 and 2015FA015)
References
[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004
[2] A K Geim and K S Novoselov ldquoThe rise of graphenerdquo NatureMaterials vol 6 no 3 pp 183ndash191 2007
[3] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008
[4] P Avouris ldquoGraphene electronic and photonic properties anddevicesrdquo Nano Letters vol 10 no 11 pp 4285ndash4294 2010
[5] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005
[6] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009
[7] T Ohta A Bostwick T Seyller K Horn and E RotenbergldquoControlling the electronic structure of bilayer graphenerdquoScience vol 313 no 5789 pp 951ndash954 2006
[8] E V Castro K S Novoselov S VMorozov et al ldquoBiased bilayergraphene semiconductor with a gap tunable by the electric fieldeffectrdquo Physical Review Letters vol 99 no 21 Article ID 2168022007
[9] Y Zhang T-T Tang C Girit et al ldquoDirect observation of awidely tunable bandgap in bilayer graphenerdquo Nature vol 459no 7248 pp 820ndash823 2009
[10] X Li W Cai J An et al ldquoLarge-area synthesis of high-qualityand uniform graphene films on copper foilsrdquo Science vol 324no 5932 pp 1312ndash1314 2009
[11] F Schwierz ldquoGraphene transistorsrdquoNatureNanotechnology vol5 no 7 pp 487ndash496 2010
[12] VW BrarM S JangM Sherrott J J Lopez andH A AtwaterldquoHighly confined tunable mid-infrared plasmonics in graphenenanoresonatorsrdquoNano Letters vol 13 no 6 pp 2541ndash2547 2013
[13] F H L Koppens D E Chang and F J G de Abajo ldquoGrapheneplasmonics a platform for strong lightndashmatter interactionsrdquoNano Letters vol 11 no 8 pp 3370ndash3377 2011
8 International Journal of Antennas and Propagation
[14] J ChenM Badioli P A Gonzalez et al ldquoOptical nano-imagingof gate-tunable graphene plasmonsrdquoNature vol 487 pp 77ndash812012
[15] M Jablan H Buljan and M Soljacic ldquoPlasmonics in grapheneat infrared frequenciesrdquo Physical Review B vol 80 no 24Article ID 245435 2009
[16] Z Fei A S Rodin G O Andreev et al ldquoGate-tuningof graphene plasmons revealed by infrared nano-imagingrdquoNature vol 486 no 7405 pp 82ndash85 2012
[17] A N Grigorenko M Polini and K S Novoselov ldquoGrapheneplasmonicsrdquo Nature Photonics vol 6 no 11 pp 749ndash758 2012
[18] A Vakil and N Engheta ldquoTransformation optics usinggraphenerdquo Science vol 332 no 6035 pp 1291ndash1294 2011
[19] L Ju B Geng J Horng et al ldquoGraphene plasmonics for tunableterahertz metamaterialsrdquo Nature Nanotechnology vol 6 no 10pp 630ndash634 2010
[20] Y Francescato V Giannini J Yang M Huang and S A MaierldquoGraphene sandwiches as a platform for broadband molecularspectroscopyrdquo ACS Photonics vol 1 no 5 pp 437ndash443 2014
[21] J Yang J YangW Deng FMao andMHuang ldquoTransmissionproperties and molecular sensing application of CGPWrdquoOpticsExpress vol 23 no 25 pp 32289ndash32299 2015
[22] Y Huang L-S Wu M Tang and J Mao ldquoDesign of a beamreconfigurable thz antenna with graphene-based switchablehigh-impedance surfacerdquo IEEE Transactions on Nanotechnol-ogy vol 11 no 4 pp 836ndash842 2012
[23] I Llatser C Kremers A Cabellos-Aparicio J M Jornet EAlarcon and D N Chigrin ldquoGraphene-based nano-patchantenna for terahertz radiationrdquo Photonics and Nanostruc-turesmdashFundamentals and Applications vol 10 no 4 pp 353ndash358 2012
[24] Y Yao M A Kats P Genevet et al ldquoBroad electrical tuning ofgraphene-loaded plasmonic antennasrdquoNano Letters vol 13 no3 pp 1257ndash1264 2013
[25] E Forati G W Hanson A B Yakovlev and A Alu ldquoPlanarhyperlens based on a modulated graphene monolayerrdquo PhysicalReview B vol 89 no 8 Article ID 081410 2014
[26] X-H Wang W-Y Yin and Z Chen ldquoBroadband modelingsurface plasmon polaritons in optically pumped and curvedgraphene structures with an improved leapfrog ADI-FDTDmethodrdquo Optics Communications vol 334 pp 152ndash155 2015
[27] G D Bouzianas N V Kantartzis C S Antonopoulos and TD Tsiboukis ldquoOptimal modeling of infinite graphene sheets viaa class of generalized FDTD schemesrdquo IEEE Transactions onMagnetics vol 48 no 2 pp 379ndash382 2012
[28] V Nayyeri M Soleimani and O M Ramahi ldquoModelinggraphene in the finite-difference time-domain method using asurface boundary conditionrdquo IEEE Transactions on Antennasand Propagation vol 61 no 8 pp 4176ndash4182 2013
[29] M Tamagnone J S G Diaz J R Mosig and J P CarrierldquoReconfigurable terahertz plasmonic antenna concept using agraphene stackrdquo Applied Physics Letters vol 101 no 21 ArticleID 214102 2012
[30] A Cabellos-Aparicio I Llatser E Alarcon A Hsu and T Pala-cios ldquoUse of terahertz photoconductive sources to characterizetunable graphene RF plasmonic antennasrdquo IEEE Transactionson Nanotechnology vol 14 no 2 pp 390ndash396 2015
[31] P Li L J Jiang and H Bagc120580 ldquoA resistive boundary conditionenhancedDGTDscheme for the transient analysis of graphenerdquoIEEE Transactions on Antennas and Propagation vol 63 no 7pp 3065ndash3076 2015
[32] G W Hanson ldquoDyadic greenrsquos functions for an anisotropicnon-local model of biased graphenerdquo IEEE Transactions onAntennas and Propagation vol 56 no 3 pp 747ndash757 2008
[33] I Ahmed E H Khoo and E Li ldquoEfficient modeling andsimulation of graphene devices with the LOD-FDTD methodrdquoIEEEMicrowave andWireless Components Letters vol 23 no 6pp 306ndash308 2013
[34] X Yu and C D Sarris ldquoA perfectly matched layer for subcellFDTDand applications to themodeling of graphene structuresrdquoIEEE Antennas and Wireless Propagation Letters vol 11 pp1080ndash1083 2012
[35] V Nayyeri M Soleimani and O M Ramahi ldquoWidebandmodeling of graphene using the finite-difference time-domainmethodrdquo IEEE Transactions on Antennas and Propagation vol61 no 12 pp 6107ndash6114 2013
[36] H Lin M F Pantoja L D Angulo J Alvarez R G Martinand S G Garcia ldquoFDTD modeling of graphene devices usingcomplex conjugate dispersionmaterialmodelrdquo IEEEMicrowaveand Wireless Components Letters vol 22 no 12 pp 612ndash6142012
[37] D-W Wang W-S Zhao X-Q Gu W Chen and W-Y YinldquoWideband modeling of graphene-based structures at differenttemperatures using hybrid FDTD methodrdquo IEEE Transactionson Nanotechnology vol 14 no 2 pp 250ndash258 2015
[38] A Y Nikitin F J Garcia-Vidal and LMartin-Moreno ldquoAnalyt-ical expressions for the electromagnetic dyadic greenrsquos functionin graphene and thin layersrdquo IEEE Journal of Selected Topics inQuantum Electronics vol 19 no 3 Article ID 4600611 2013
[39] O V Shapoval J S Gomez-Diaz J Perruisseau-Carrier J RMosig and A I Nosich ldquoIntegral equation analysis of planewave scattering by coplanar graphene-strip gratings in the thzrangerdquo IEEE Transactions on Terahertz Science and Technologyvol 3 no 5 pp 666ndash674 2013
[40] R Araneo G Lovat and P Burghignoli ldquoGraphene nanostriplines dispersion and attenuation analysisrdquo in Proceedings of the16thWorkshop on Signal and Power Integrity (SPI rsquo12) pp 75ndash78IEEE Sorrento Italy May 2012
[41] G Lovat G W Hanson R Araneo and P Burghignoli ldquoSemi-classical spatially dispersive intraband conductivity tensor andquantum capacitance of graphenerdquo Physical Review B vol 87no 11 Article ID 115429 2013
[42] P Burghignoli R Araneo G Lovat and G Hanson ldquoSpace-domain method of moments for graphene nanoribbonsrdquo inProceedings of the 8th European Conference on Antennas andPropagation (EuCAP rsquo14) pp 666ndash669 IEEE The Hague TheNetherlands April 2014
[43] R Araneo P Burghignoli G Lovat and G W HansonldquoModal propagation and crosstalk analysis in coupled graphenenanoribbonsrdquo IEEE Transactions on Electromagnetic Compati-bility vol 57 no 4 pp 726ndash733 2015
[44] A Fallahi T Low M Tamagnone and J Perruisseau-CarrierldquoNonlocal electromagnetic response of graphene nanostruc-turesrdquo Physical Review B vol 91 Article ID 121405 2015
[45] M Dragoman D Neculoiu A Cismaru et al ldquoCoplanarwaveguide on graphene in the range 40MHz-110 GHzrdquoAppliedPhysics Letters vol 99 no 3 Article ID 033112 2011
[46] J Chen and Q H Liu ldquoDiscontinuous Galerkin Time-Domainmethods for multiscale electromagnetic simulations a reviewrdquoProceedings of the IEEE vol 101 no 2 pp 242ndash254 2013
International Journal of Antennas and Propagation 9
[47] S Descombes C Durochat S Lanteri L Moya C Scheid andJ Viquerat ldquoRecent advances on a DGTD method for time-domain electromagneticsrdquo Photonics and NanostructuresmdashFundamentals and Applications vol 11 no 4 pp 291ndash302 2013
[48] P Li and L J Jiang ldquoModeling of magnetized graphene frommicrowave to THz range by DGTD with a scalar RBC and anADErdquo IEEE Transactions on Antennas and Propagation vol 63no 10 pp 4458ndash4467 2015
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 5
Conductive sheet
x
zK
Ey
K +1
2K + 1
Ey
1Hx2Hx
(a)
Conductivesheet
z
y
x
(i j K)
(i j K + 1)
(i j + 1 K)
(i + 1 j K)
Hz
ExEy
1Hx 1Hy
2Hx 2Hy
1Ez
2Ez
(b)
Figure 4 (a) 1D FDTD cell including a conductive sheet at grid 119870 + 12 (b) 3D FDTD cell with a conductive sheet at grid119870 + 12 [28]
PML
Line source Graphenesheet
20120583m
50120583m
Observation circlex
y
(a)
0
30
60
90
120
150
180
210
240
270
300
330
x
y
SemianalyticFDTD
1
08
06
04
02
(b)
Figure 5 (a) Line source scattering by a graphene sheet (b) Normalized pattern of 119864119911at the wavelength of 120582
0= 100 120583m [28]
condition to analyze plane wave scattering and absorptionby graphene-strip gratings The method of moments forgraphene nanoribbons was developed in [40ndash44] in whichthe issue of nonlocality of graphene conductivity was takeninto account Nonlocal effect arises from spatial dispersionof graphene which is nonnegligible when dealing with slowmodes supported by graphene nanoribbons The spatiallydispersive intraband conductivity tensor was derived in [41]
Software packages such as IE3D and FEKO are based onMoM IE3D is applied to simulate themicrowave propagationin a coplanar waveguide over graphene from 40MHz to110GHz [45] Cabellos-Aparicio et al [30] used FEKO tostudy the radiated power of a graphene plasmonic antennafed by photoconductive source The antenna structure isshown in Figure 7(a) and the radiated power with respect to
frequency is shown in Figure 7(b)The detailed parameters ofthe photoconductive antenna can be found in [30]
6 Discontinuous GalerkinTime-Domain Method
The graphene involved problems are often multiscaleGraphene is monoatomic and its thickness is much smallerthan wavelength so it is an electrically fine structure Incontrast the substrate belongs to electrically coarse structurebecause its dimension is much greater than wavelengthAs we have mentioned before time-domain methods havethe advantage that the broadband characterization can beobtained with only a single simulation However FDTD
6 International Journal of Antennas and Propagation
GrapheneAl2O3
GrapheneTHz photomixer
L WSubstrate (GaAs)
(a)
120583c
120583c
fr fr fr fr fr
minus400
minus200
0
200
400
Impe
danc
e (Ω
)
1 15 2 2505
Frequency (THz)
0 eV005 eV01 eV
015 eV02 eV
ReIm
(b)
Figure 6 (a) Structure of a reconfigurable graphene antenna (b) Input impedance of the antenna with respect to chemical potential [29]
Vminus
V+Vbias
BN
Si lens
fs laser pulse
5120583m
14nm
500120583m
GaAs
LT-GaAs
GrapheneContact pads
(120583m)10
20 30
40
(a)
1 2 3 40
Frequency (THz)
0
02
04
06
08
1
12
Radi
ated
pow
er (120583
W)
(b)
Figure 7 (a) Structure of graphene RF plasmonic antenna fed with a photoconductivematerial (b) Radiated power with respect to frequencyfor graphene antennas with lengths 10120583m (blue) 20 120583m (green) 30120583m (red) and 40120583m (black) [30]
method has a serious efficiency problem in multiscale sim-ulation because it requires high discretization density tomodel electrically fine structure due to its cartesian grid Thefinite element time-domain (FETD) method is capable ofmodeling complex and fine structures and achieving high-order accuracy with high-order basis functions The majordrawback is a global linear system of equations that needs tobe solved at each time step The multiscale problems usuallycontain a large number of unknowns FETDwill be very com-putationally expensive in this case Discontinuous Galerkintime-domain (DGTD) method is promising in multiscaleproblems [46] DGTD allows for domain decomposition Amultiscale structure is divided into several subdomains andeach subdomain can be discretized separately All operationsin DGTD are local so large global matrix is split into several
smaller matrices Unlike FETD the matrices of DGTD areinverted and stored before time marching and differenttime integration scheme can be used in different subdomainAdditionally DGTD is naturally adapted to parallel comput-ing Recently DGTD has been used in nanophotonics fieldand considered as a viable alternative to the well-establishedFDTD and FETD methods [47]
Li et al [31] proposed DGTD method with resistiveboundary condition for the electromagnetic analyzing ofgraphene They also applied this method to the magne-tized graphene from microwave to THz range where theanisotropic and disperse surface conductivity is involved[48] Figure 8(a) shows the reflection Γ
119877 transmission Γ
119879 and
absorption Γ119860coefficients of an infinitely large graphene sheet
under the illumination of a normally incident plane wave
International Journal of Antennas and Propagation 7
Exact
ΓT
ΓR
ΓA
0
02
04
06
08
1
Mag
nitu
de
2 4 6 8 100
Frequency (THz)
DGTD + RBC
(a)
Integral equation
0
1
2
3
4
5
6
7
8
ECS
05 1 15 2 25 3 35 40
Frequency (THz)
DGTD + RBC
(b)
Figure 8 (a) The magnitude of reflection Γ119877 transmission Γ
119879 and absorption Γ
119860coefficients calculated by DGFD as well as the theoretical
value (b) Normalized extinction cross section of a graphene patch calculated by DGTD and integral equation method [31]
the results of DGTD agree very well with the theoretical dataFigure 8(b) is the normalized extinction cross section of afreestanding graphene patch good consistency is achievedbetween DGTD and integral equation method More numer-ical examples can be found in [31]
7 Conclusion
Due to its intriguing properties graphene is used more andmore widely in electronic and photonic community Thispaper gives a brief review of application of graphene aswell asthe computational methods which can be used to investigateits electromagnetic properties Although a number of famouscommercial software packages are available for graphene sim-ulation none of them are developed specifically for graphenetherefore they suffer from inefficiency Each computationalmethod has its own advantages and drawbacks one shouldchoose the appropriate method according to the specificproblem FDTD is simple and easy to implement however itis less efficient in modeling fine structure FEM is flexible ingeometry modeling but solving large global system makes itcomputationally expensive DGTD is very suitable for multi-scale problem and we believe it will be more andmore widelyused in modeling and designing graphene-assisted device
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the National Natural Sci-ence Foundation of China (Grants nos 61161007 6126100261461052 and 11564044) the Specialized Research Fundfor the Doctoral Program of Higher Education (Grantsnos 20135301110003 and 20125301120009) China Postdoc-toral Science Foundation (Grants nos 2013M531989 and2014T70890) and the Key Program of Natural Science ofYunnan Province (Grants nos 2013FA006 and 2015FA015)
References
[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004
[2] A K Geim and K S Novoselov ldquoThe rise of graphenerdquo NatureMaterials vol 6 no 3 pp 183ndash191 2007
[3] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008
[4] P Avouris ldquoGraphene electronic and photonic properties anddevicesrdquo Nano Letters vol 10 no 11 pp 4285ndash4294 2010
[5] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005
[6] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009
[7] T Ohta A Bostwick T Seyller K Horn and E RotenbergldquoControlling the electronic structure of bilayer graphenerdquoScience vol 313 no 5789 pp 951ndash954 2006
[8] E V Castro K S Novoselov S VMorozov et al ldquoBiased bilayergraphene semiconductor with a gap tunable by the electric fieldeffectrdquo Physical Review Letters vol 99 no 21 Article ID 2168022007
[9] Y Zhang T-T Tang C Girit et al ldquoDirect observation of awidely tunable bandgap in bilayer graphenerdquo Nature vol 459no 7248 pp 820ndash823 2009
[10] X Li W Cai J An et al ldquoLarge-area synthesis of high-qualityand uniform graphene films on copper foilsrdquo Science vol 324no 5932 pp 1312ndash1314 2009
[11] F Schwierz ldquoGraphene transistorsrdquoNatureNanotechnology vol5 no 7 pp 487ndash496 2010
[12] VW BrarM S JangM Sherrott J J Lopez andH A AtwaterldquoHighly confined tunable mid-infrared plasmonics in graphenenanoresonatorsrdquoNano Letters vol 13 no 6 pp 2541ndash2547 2013
[13] F H L Koppens D E Chang and F J G de Abajo ldquoGrapheneplasmonics a platform for strong lightndashmatter interactionsrdquoNano Letters vol 11 no 8 pp 3370ndash3377 2011
8 International Journal of Antennas and Propagation
[14] J ChenM Badioli P A Gonzalez et al ldquoOptical nano-imagingof gate-tunable graphene plasmonsrdquoNature vol 487 pp 77ndash812012
[15] M Jablan H Buljan and M Soljacic ldquoPlasmonics in grapheneat infrared frequenciesrdquo Physical Review B vol 80 no 24Article ID 245435 2009
[16] Z Fei A S Rodin G O Andreev et al ldquoGate-tuningof graphene plasmons revealed by infrared nano-imagingrdquoNature vol 486 no 7405 pp 82ndash85 2012
[17] A N Grigorenko M Polini and K S Novoselov ldquoGrapheneplasmonicsrdquo Nature Photonics vol 6 no 11 pp 749ndash758 2012
[18] A Vakil and N Engheta ldquoTransformation optics usinggraphenerdquo Science vol 332 no 6035 pp 1291ndash1294 2011
[19] L Ju B Geng J Horng et al ldquoGraphene plasmonics for tunableterahertz metamaterialsrdquo Nature Nanotechnology vol 6 no 10pp 630ndash634 2010
[20] Y Francescato V Giannini J Yang M Huang and S A MaierldquoGraphene sandwiches as a platform for broadband molecularspectroscopyrdquo ACS Photonics vol 1 no 5 pp 437ndash443 2014
[21] J Yang J YangW Deng FMao andMHuang ldquoTransmissionproperties and molecular sensing application of CGPWrdquoOpticsExpress vol 23 no 25 pp 32289ndash32299 2015
[22] Y Huang L-S Wu M Tang and J Mao ldquoDesign of a beamreconfigurable thz antenna with graphene-based switchablehigh-impedance surfacerdquo IEEE Transactions on Nanotechnol-ogy vol 11 no 4 pp 836ndash842 2012
[23] I Llatser C Kremers A Cabellos-Aparicio J M Jornet EAlarcon and D N Chigrin ldquoGraphene-based nano-patchantenna for terahertz radiationrdquo Photonics and Nanostruc-turesmdashFundamentals and Applications vol 10 no 4 pp 353ndash358 2012
[24] Y Yao M A Kats P Genevet et al ldquoBroad electrical tuning ofgraphene-loaded plasmonic antennasrdquoNano Letters vol 13 no3 pp 1257ndash1264 2013
[25] E Forati G W Hanson A B Yakovlev and A Alu ldquoPlanarhyperlens based on a modulated graphene monolayerrdquo PhysicalReview B vol 89 no 8 Article ID 081410 2014
[26] X-H Wang W-Y Yin and Z Chen ldquoBroadband modelingsurface plasmon polaritons in optically pumped and curvedgraphene structures with an improved leapfrog ADI-FDTDmethodrdquo Optics Communications vol 334 pp 152ndash155 2015
[27] G D Bouzianas N V Kantartzis C S Antonopoulos and TD Tsiboukis ldquoOptimal modeling of infinite graphene sheets viaa class of generalized FDTD schemesrdquo IEEE Transactions onMagnetics vol 48 no 2 pp 379ndash382 2012
[28] V Nayyeri M Soleimani and O M Ramahi ldquoModelinggraphene in the finite-difference time-domain method using asurface boundary conditionrdquo IEEE Transactions on Antennasand Propagation vol 61 no 8 pp 4176ndash4182 2013
[29] M Tamagnone J S G Diaz J R Mosig and J P CarrierldquoReconfigurable terahertz plasmonic antenna concept using agraphene stackrdquo Applied Physics Letters vol 101 no 21 ArticleID 214102 2012
[30] A Cabellos-Aparicio I Llatser E Alarcon A Hsu and T Pala-cios ldquoUse of terahertz photoconductive sources to characterizetunable graphene RF plasmonic antennasrdquo IEEE Transactionson Nanotechnology vol 14 no 2 pp 390ndash396 2015
[31] P Li L J Jiang and H Bagc120580 ldquoA resistive boundary conditionenhancedDGTDscheme for the transient analysis of graphenerdquoIEEE Transactions on Antennas and Propagation vol 63 no 7pp 3065ndash3076 2015
[32] G W Hanson ldquoDyadic greenrsquos functions for an anisotropicnon-local model of biased graphenerdquo IEEE Transactions onAntennas and Propagation vol 56 no 3 pp 747ndash757 2008
[33] I Ahmed E H Khoo and E Li ldquoEfficient modeling andsimulation of graphene devices with the LOD-FDTD methodrdquoIEEEMicrowave andWireless Components Letters vol 23 no 6pp 306ndash308 2013
[34] X Yu and C D Sarris ldquoA perfectly matched layer for subcellFDTDand applications to themodeling of graphene structuresrdquoIEEE Antennas and Wireless Propagation Letters vol 11 pp1080ndash1083 2012
[35] V Nayyeri M Soleimani and O M Ramahi ldquoWidebandmodeling of graphene using the finite-difference time-domainmethodrdquo IEEE Transactions on Antennas and Propagation vol61 no 12 pp 6107ndash6114 2013
[36] H Lin M F Pantoja L D Angulo J Alvarez R G Martinand S G Garcia ldquoFDTD modeling of graphene devices usingcomplex conjugate dispersionmaterialmodelrdquo IEEEMicrowaveand Wireless Components Letters vol 22 no 12 pp 612ndash6142012
[37] D-W Wang W-S Zhao X-Q Gu W Chen and W-Y YinldquoWideband modeling of graphene-based structures at differenttemperatures using hybrid FDTD methodrdquo IEEE Transactionson Nanotechnology vol 14 no 2 pp 250ndash258 2015
[38] A Y Nikitin F J Garcia-Vidal and LMartin-Moreno ldquoAnalyt-ical expressions for the electromagnetic dyadic greenrsquos functionin graphene and thin layersrdquo IEEE Journal of Selected Topics inQuantum Electronics vol 19 no 3 Article ID 4600611 2013
[39] O V Shapoval J S Gomez-Diaz J Perruisseau-Carrier J RMosig and A I Nosich ldquoIntegral equation analysis of planewave scattering by coplanar graphene-strip gratings in the thzrangerdquo IEEE Transactions on Terahertz Science and Technologyvol 3 no 5 pp 666ndash674 2013
[40] R Araneo G Lovat and P Burghignoli ldquoGraphene nanostriplines dispersion and attenuation analysisrdquo in Proceedings of the16thWorkshop on Signal and Power Integrity (SPI rsquo12) pp 75ndash78IEEE Sorrento Italy May 2012
[41] G Lovat G W Hanson R Araneo and P Burghignoli ldquoSemi-classical spatially dispersive intraband conductivity tensor andquantum capacitance of graphenerdquo Physical Review B vol 87no 11 Article ID 115429 2013
[42] P Burghignoli R Araneo G Lovat and G Hanson ldquoSpace-domain method of moments for graphene nanoribbonsrdquo inProceedings of the 8th European Conference on Antennas andPropagation (EuCAP rsquo14) pp 666ndash669 IEEE The Hague TheNetherlands April 2014
[43] R Araneo P Burghignoli G Lovat and G W HansonldquoModal propagation and crosstalk analysis in coupled graphenenanoribbonsrdquo IEEE Transactions on Electromagnetic Compati-bility vol 57 no 4 pp 726ndash733 2015
[44] A Fallahi T Low M Tamagnone and J Perruisseau-CarrierldquoNonlocal electromagnetic response of graphene nanostruc-turesrdquo Physical Review B vol 91 Article ID 121405 2015
[45] M Dragoman D Neculoiu A Cismaru et al ldquoCoplanarwaveguide on graphene in the range 40MHz-110 GHzrdquoAppliedPhysics Letters vol 99 no 3 Article ID 033112 2011
[46] J Chen and Q H Liu ldquoDiscontinuous Galerkin Time-Domainmethods for multiscale electromagnetic simulations a reviewrdquoProceedings of the IEEE vol 101 no 2 pp 242ndash254 2013
International Journal of Antennas and Propagation 9
[47] S Descombes C Durochat S Lanteri L Moya C Scheid andJ Viquerat ldquoRecent advances on a DGTD method for time-domain electromagneticsrdquo Photonics and NanostructuresmdashFundamentals and Applications vol 11 no 4 pp 291ndash302 2013
[48] P Li and L J Jiang ldquoModeling of magnetized graphene frommicrowave to THz range by DGTD with a scalar RBC and anADErdquo IEEE Transactions on Antennas and Propagation vol 63no 10 pp 4458ndash4467 2015
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 International Journal of Antennas and Propagation
GrapheneAl2O3
GrapheneTHz photomixer
L WSubstrate (GaAs)
(a)
120583c
120583c
fr fr fr fr fr
minus400
minus200
0
200
400
Impe
danc
e (Ω
)
1 15 2 2505
Frequency (THz)
0 eV005 eV01 eV
015 eV02 eV
ReIm
(b)
Figure 6 (a) Structure of a reconfigurable graphene antenna (b) Input impedance of the antenna with respect to chemical potential [29]
Vminus
V+Vbias
BN
Si lens
fs laser pulse
5120583m
14nm
500120583m
GaAs
LT-GaAs
GrapheneContact pads
(120583m)10
20 30
40
(a)
1 2 3 40
Frequency (THz)
0
02
04
06
08
1
12
Radi
ated
pow
er (120583
W)
(b)
Figure 7 (a) Structure of graphene RF plasmonic antenna fed with a photoconductivematerial (b) Radiated power with respect to frequencyfor graphene antennas with lengths 10120583m (blue) 20 120583m (green) 30120583m (red) and 40120583m (black) [30]
method has a serious efficiency problem in multiscale sim-ulation because it requires high discretization density tomodel electrically fine structure due to its cartesian grid Thefinite element time-domain (FETD) method is capable ofmodeling complex and fine structures and achieving high-order accuracy with high-order basis functions The majordrawback is a global linear system of equations that needs tobe solved at each time step The multiscale problems usuallycontain a large number of unknowns FETDwill be very com-putationally expensive in this case Discontinuous Galerkintime-domain (DGTD) method is promising in multiscaleproblems [46] DGTD allows for domain decomposition Amultiscale structure is divided into several subdomains andeach subdomain can be discretized separately All operationsin DGTD are local so large global matrix is split into several
smaller matrices Unlike FETD the matrices of DGTD areinverted and stored before time marching and differenttime integration scheme can be used in different subdomainAdditionally DGTD is naturally adapted to parallel comput-ing Recently DGTD has been used in nanophotonics fieldand considered as a viable alternative to the well-establishedFDTD and FETD methods [47]
Li et al [31] proposed DGTD method with resistiveboundary condition for the electromagnetic analyzing ofgraphene They also applied this method to the magne-tized graphene from microwave to THz range where theanisotropic and disperse surface conductivity is involved[48] Figure 8(a) shows the reflection Γ
119877 transmission Γ
119879 and
absorption Γ119860coefficients of an infinitely large graphene sheet
under the illumination of a normally incident plane wave
International Journal of Antennas and Propagation 7
Exact
ΓT
ΓR
ΓA
0
02
04
06
08
1
Mag
nitu
de
2 4 6 8 100
Frequency (THz)
DGTD + RBC
(a)
Integral equation
0
1
2
3
4
5
6
7
8
ECS
05 1 15 2 25 3 35 40
Frequency (THz)
DGTD + RBC
(b)
Figure 8 (a) The magnitude of reflection Γ119877 transmission Γ
119879 and absorption Γ
119860coefficients calculated by DGFD as well as the theoretical
value (b) Normalized extinction cross section of a graphene patch calculated by DGTD and integral equation method [31]
the results of DGTD agree very well with the theoretical dataFigure 8(b) is the normalized extinction cross section of afreestanding graphene patch good consistency is achievedbetween DGTD and integral equation method More numer-ical examples can be found in [31]
7 Conclusion
Due to its intriguing properties graphene is used more andmore widely in electronic and photonic community Thispaper gives a brief review of application of graphene aswell asthe computational methods which can be used to investigateits electromagnetic properties Although a number of famouscommercial software packages are available for graphene sim-ulation none of them are developed specifically for graphenetherefore they suffer from inefficiency Each computationalmethod has its own advantages and drawbacks one shouldchoose the appropriate method according to the specificproblem FDTD is simple and easy to implement however itis less efficient in modeling fine structure FEM is flexible ingeometry modeling but solving large global system makes itcomputationally expensive DGTD is very suitable for multi-scale problem and we believe it will be more andmore widelyused in modeling and designing graphene-assisted device
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the National Natural Sci-ence Foundation of China (Grants nos 61161007 6126100261461052 and 11564044) the Specialized Research Fundfor the Doctoral Program of Higher Education (Grantsnos 20135301110003 and 20125301120009) China Postdoc-toral Science Foundation (Grants nos 2013M531989 and2014T70890) and the Key Program of Natural Science ofYunnan Province (Grants nos 2013FA006 and 2015FA015)
References
[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004
[2] A K Geim and K S Novoselov ldquoThe rise of graphenerdquo NatureMaterials vol 6 no 3 pp 183ndash191 2007
[3] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008
[4] P Avouris ldquoGraphene electronic and photonic properties anddevicesrdquo Nano Letters vol 10 no 11 pp 4285ndash4294 2010
[5] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005
[6] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009
[7] T Ohta A Bostwick T Seyller K Horn and E RotenbergldquoControlling the electronic structure of bilayer graphenerdquoScience vol 313 no 5789 pp 951ndash954 2006
[8] E V Castro K S Novoselov S VMorozov et al ldquoBiased bilayergraphene semiconductor with a gap tunable by the electric fieldeffectrdquo Physical Review Letters vol 99 no 21 Article ID 2168022007
[9] Y Zhang T-T Tang C Girit et al ldquoDirect observation of awidely tunable bandgap in bilayer graphenerdquo Nature vol 459no 7248 pp 820ndash823 2009
[10] X Li W Cai J An et al ldquoLarge-area synthesis of high-qualityand uniform graphene films on copper foilsrdquo Science vol 324no 5932 pp 1312ndash1314 2009
[11] F Schwierz ldquoGraphene transistorsrdquoNatureNanotechnology vol5 no 7 pp 487ndash496 2010
[12] VW BrarM S JangM Sherrott J J Lopez andH A AtwaterldquoHighly confined tunable mid-infrared plasmonics in graphenenanoresonatorsrdquoNano Letters vol 13 no 6 pp 2541ndash2547 2013
[13] F H L Koppens D E Chang and F J G de Abajo ldquoGrapheneplasmonics a platform for strong lightndashmatter interactionsrdquoNano Letters vol 11 no 8 pp 3370ndash3377 2011
8 International Journal of Antennas and Propagation
[14] J ChenM Badioli P A Gonzalez et al ldquoOptical nano-imagingof gate-tunable graphene plasmonsrdquoNature vol 487 pp 77ndash812012
[15] M Jablan H Buljan and M Soljacic ldquoPlasmonics in grapheneat infrared frequenciesrdquo Physical Review B vol 80 no 24Article ID 245435 2009
[16] Z Fei A S Rodin G O Andreev et al ldquoGate-tuningof graphene plasmons revealed by infrared nano-imagingrdquoNature vol 486 no 7405 pp 82ndash85 2012
[17] A N Grigorenko M Polini and K S Novoselov ldquoGrapheneplasmonicsrdquo Nature Photonics vol 6 no 11 pp 749ndash758 2012
[18] A Vakil and N Engheta ldquoTransformation optics usinggraphenerdquo Science vol 332 no 6035 pp 1291ndash1294 2011
[19] L Ju B Geng J Horng et al ldquoGraphene plasmonics for tunableterahertz metamaterialsrdquo Nature Nanotechnology vol 6 no 10pp 630ndash634 2010
[20] Y Francescato V Giannini J Yang M Huang and S A MaierldquoGraphene sandwiches as a platform for broadband molecularspectroscopyrdquo ACS Photonics vol 1 no 5 pp 437ndash443 2014
[21] J Yang J YangW Deng FMao andMHuang ldquoTransmissionproperties and molecular sensing application of CGPWrdquoOpticsExpress vol 23 no 25 pp 32289ndash32299 2015
[22] Y Huang L-S Wu M Tang and J Mao ldquoDesign of a beamreconfigurable thz antenna with graphene-based switchablehigh-impedance surfacerdquo IEEE Transactions on Nanotechnol-ogy vol 11 no 4 pp 836ndash842 2012
[23] I Llatser C Kremers A Cabellos-Aparicio J M Jornet EAlarcon and D N Chigrin ldquoGraphene-based nano-patchantenna for terahertz radiationrdquo Photonics and Nanostruc-turesmdashFundamentals and Applications vol 10 no 4 pp 353ndash358 2012
[24] Y Yao M A Kats P Genevet et al ldquoBroad electrical tuning ofgraphene-loaded plasmonic antennasrdquoNano Letters vol 13 no3 pp 1257ndash1264 2013
[25] E Forati G W Hanson A B Yakovlev and A Alu ldquoPlanarhyperlens based on a modulated graphene monolayerrdquo PhysicalReview B vol 89 no 8 Article ID 081410 2014
[26] X-H Wang W-Y Yin and Z Chen ldquoBroadband modelingsurface plasmon polaritons in optically pumped and curvedgraphene structures with an improved leapfrog ADI-FDTDmethodrdquo Optics Communications vol 334 pp 152ndash155 2015
[27] G D Bouzianas N V Kantartzis C S Antonopoulos and TD Tsiboukis ldquoOptimal modeling of infinite graphene sheets viaa class of generalized FDTD schemesrdquo IEEE Transactions onMagnetics vol 48 no 2 pp 379ndash382 2012
[28] V Nayyeri M Soleimani and O M Ramahi ldquoModelinggraphene in the finite-difference time-domain method using asurface boundary conditionrdquo IEEE Transactions on Antennasand Propagation vol 61 no 8 pp 4176ndash4182 2013
[29] M Tamagnone J S G Diaz J R Mosig and J P CarrierldquoReconfigurable terahertz plasmonic antenna concept using agraphene stackrdquo Applied Physics Letters vol 101 no 21 ArticleID 214102 2012
[30] A Cabellos-Aparicio I Llatser E Alarcon A Hsu and T Pala-cios ldquoUse of terahertz photoconductive sources to characterizetunable graphene RF plasmonic antennasrdquo IEEE Transactionson Nanotechnology vol 14 no 2 pp 390ndash396 2015
[31] P Li L J Jiang and H Bagc120580 ldquoA resistive boundary conditionenhancedDGTDscheme for the transient analysis of graphenerdquoIEEE Transactions on Antennas and Propagation vol 63 no 7pp 3065ndash3076 2015
[32] G W Hanson ldquoDyadic greenrsquos functions for an anisotropicnon-local model of biased graphenerdquo IEEE Transactions onAntennas and Propagation vol 56 no 3 pp 747ndash757 2008
[33] I Ahmed E H Khoo and E Li ldquoEfficient modeling andsimulation of graphene devices with the LOD-FDTD methodrdquoIEEEMicrowave andWireless Components Letters vol 23 no 6pp 306ndash308 2013
[34] X Yu and C D Sarris ldquoA perfectly matched layer for subcellFDTDand applications to themodeling of graphene structuresrdquoIEEE Antennas and Wireless Propagation Letters vol 11 pp1080ndash1083 2012
[35] V Nayyeri M Soleimani and O M Ramahi ldquoWidebandmodeling of graphene using the finite-difference time-domainmethodrdquo IEEE Transactions on Antennas and Propagation vol61 no 12 pp 6107ndash6114 2013
[36] H Lin M F Pantoja L D Angulo J Alvarez R G Martinand S G Garcia ldquoFDTD modeling of graphene devices usingcomplex conjugate dispersionmaterialmodelrdquo IEEEMicrowaveand Wireless Components Letters vol 22 no 12 pp 612ndash6142012
[37] D-W Wang W-S Zhao X-Q Gu W Chen and W-Y YinldquoWideband modeling of graphene-based structures at differenttemperatures using hybrid FDTD methodrdquo IEEE Transactionson Nanotechnology vol 14 no 2 pp 250ndash258 2015
[38] A Y Nikitin F J Garcia-Vidal and LMartin-Moreno ldquoAnalyt-ical expressions for the electromagnetic dyadic greenrsquos functionin graphene and thin layersrdquo IEEE Journal of Selected Topics inQuantum Electronics vol 19 no 3 Article ID 4600611 2013
[39] O V Shapoval J S Gomez-Diaz J Perruisseau-Carrier J RMosig and A I Nosich ldquoIntegral equation analysis of planewave scattering by coplanar graphene-strip gratings in the thzrangerdquo IEEE Transactions on Terahertz Science and Technologyvol 3 no 5 pp 666ndash674 2013
[40] R Araneo G Lovat and P Burghignoli ldquoGraphene nanostriplines dispersion and attenuation analysisrdquo in Proceedings of the16thWorkshop on Signal and Power Integrity (SPI rsquo12) pp 75ndash78IEEE Sorrento Italy May 2012
[41] G Lovat G W Hanson R Araneo and P Burghignoli ldquoSemi-classical spatially dispersive intraband conductivity tensor andquantum capacitance of graphenerdquo Physical Review B vol 87no 11 Article ID 115429 2013
[42] P Burghignoli R Araneo G Lovat and G Hanson ldquoSpace-domain method of moments for graphene nanoribbonsrdquo inProceedings of the 8th European Conference on Antennas andPropagation (EuCAP rsquo14) pp 666ndash669 IEEE The Hague TheNetherlands April 2014
[43] R Araneo P Burghignoli G Lovat and G W HansonldquoModal propagation and crosstalk analysis in coupled graphenenanoribbonsrdquo IEEE Transactions on Electromagnetic Compati-bility vol 57 no 4 pp 726ndash733 2015
[44] A Fallahi T Low M Tamagnone and J Perruisseau-CarrierldquoNonlocal electromagnetic response of graphene nanostruc-turesrdquo Physical Review B vol 91 Article ID 121405 2015
[45] M Dragoman D Neculoiu A Cismaru et al ldquoCoplanarwaveguide on graphene in the range 40MHz-110 GHzrdquoAppliedPhysics Letters vol 99 no 3 Article ID 033112 2011
[46] J Chen and Q H Liu ldquoDiscontinuous Galerkin Time-Domainmethods for multiscale electromagnetic simulations a reviewrdquoProceedings of the IEEE vol 101 no 2 pp 242ndash254 2013
International Journal of Antennas and Propagation 9
[47] S Descombes C Durochat S Lanteri L Moya C Scheid andJ Viquerat ldquoRecent advances on a DGTD method for time-domain electromagneticsrdquo Photonics and NanostructuresmdashFundamentals and Applications vol 11 no 4 pp 291ndash302 2013
[48] P Li and L J Jiang ldquoModeling of magnetized graphene frommicrowave to THz range by DGTD with a scalar RBC and anADErdquo IEEE Transactions on Antennas and Propagation vol 63no 10 pp 4458ndash4467 2015
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 7
Exact
ΓT
ΓR
ΓA
0
02
04
06
08
1
Mag
nitu
de
2 4 6 8 100
Frequency (THz)
DGTD + RBC
(a)
Integral equation
0
1
2
3
4
5
6
7
8
ECS
05 1 15 2 25 3 35 40
Frequency (THz)
DGTD + RBC
(b)
Figure 8 (a) The magnitude of reflection Γ119877 transmission Γ
119879 and absorption Γ
119860coefficients calculated by DGFD as well as the theoretical
value (b) Normalized extinction cross section of a graphene patch calculated by DGTD and integral equation method [31]
the results of DGTD agree very well with the theoretical dataFigure 8(b) is the normalized extinction cross section of afreestanding graphene patch good consistency is achievedbetween DGTD and integral equation method More numer-ical examples can be found in [31]
7 Conclusion
Due to its intriguing properties graphene is used more andmore widely in electronic and photonic community Thispaper gives a brief review of application of graphene aswell asthe computational methods which can be used to investigateits electromagnetic properties Although a number of famouscommercial software packages are available for graphene sim-ulation none of them are developed specifically for graphenetherefore they suffer from inefficiency Each computationalmethod has its own advantages and drawbacks one shouldchoose the appropriate method according to the specificproblem FDTD is simple and easy to implement however itis less efficient in modeling fine structure FEM is flexible ingeometry modeling but solving large global system makes itcomputationally expensive DGTD is very suitable for multi-scale problem and we believe it will be more andmore widelyused in modeling and designing graphene-assisted device
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by the National Natural Sci-ence Foundation of China (Grants nos 61161007 6126100261461052 and 11564044) the Specialized Research Fundfor the Doctoral Program of Higher Education (Grantsnos 20135301110003 and 20125301120009) China Postdoc-toral Science Foundation (Grants nos 2013M531989 and2014T70890) and the Key Program of Natural Science ofYunnan Province (Grants nos 2013FA006 and 2015FA015)
References
[1] K S Novoselov A K Geim S V Morozov et al ldquoElectric fieldeffect in atomically thin carbon filmsrdquo Science vol 306 no5696 pp 666ndash669 2004
[2] A K Geim and K S Novoselov ldquoThe rise of graphenerdquo NatureMaterials vol 6 no 3 pp 183ndash191 2007
[3] C Lee X Wei J W Kysar and J Hone ldquoMeasurementof the elastic properties and intrinsic strength of monolayergraphenerdquo Science vol 321 no 5887 pp 385ndash388 2008
[4] P Avouris ldquoGraphene electronic and photonic properties anddevicesrdquo Nano Letters vol 10 no 11 pp 4285ndash4294 2010
[5] K S Novoselov A K Geim S V Morozov et al ldquoTwo-dimensional gas of massless Dirac fermions in graphenerdquoNature vol 438 no 7065 pp 197ndash200 2005
[6] A H Castro Neto F Guinea N M R Peres K S Novoselovand A K Geim ldquoThe electronic properties of graphenerdquoReviews of Modern Physics vol 81 no 1 pp 109ndash162 2009
[7] T Ohta A Bostwick T Seyller K Horn and E RotenbergldquoControlling the electronic structure of bilayer graphenerdquoScience vol 313 no 5789 pp 951ndash954 2006
[8] E V Castro K S Novoselov S VMorozov et al ldquoBiased bilayergraphene semiconductor with a gap tunable by the electric fieldeffectrdquo Physical Review Letters vol 99 no 21 Article ID 2168022007
[9] Y Zhang T-T Tang C Girit et al ldquoDirect observation of awidely tunable bandgap in bilayer graphenerdquo Nature vol 459no 7248 pp 820ndash823 2009
[10] X Li W Cai J An et al ldquoLarge-area synthesis of high-qualityand uniform graphene films on copper foilsrdquo Science vol 324no 5932 pp 1312ndash1314 2009
[11] F Schwierz ldquoGraphene transistorsrdquoNatureNanotechnology vol5 no 7 pp 487ndash496 2010
[12] VW BrarM S JangM Sherrott J J Lopez andH A AtwaterldquoHighly confined tunable mid-infrared plasmonics in graphenenanoresonatorsrdquoNano Letters vol 13 no 6 pp 2541ndash2547 2013
[13] F H L Koppens D E Chang and F J G de Abajo ldquoGrapheneplasmonics a platform for strong lightndashmatter interactionsrdquoNano Letters vol 11 no 8 pp 3370ndash3377 2011
8 International Journal of Antennas and Propagation
[14] J ChenM Badioli P A Gonzalez et al ldquoOptical nano-imagingof gate-tunable graphene plasmonsrdquoNature vol 487 pp 77ndash812012
[15] M Jablan H Buljan and M Soljacic ldquoPlasmonics in grapheneat infrared frequenciesrdquo Physical Review B vol 80 no 24Article ID 245435 2009
[16] Z Fei A S Rodin G O Andreev et al ldquoGate-tuningof graphene plasmons revealed by infrared nano-imagingrdquoNature vol 486 no 7405 pp 82ndash85 2012
[17] A N Grigorenko M Polini and K S Novoselov ldquoGrapheneplasmonicsrdquo Nature Photonics vol 6 no 11 pp 749ndash758 2012
[18] A Vakil and N Engheta ldquoTransformation optics usinggraphenerdquo Science vol 332 no 6035 pp 1291ndash1294 2011
[19] L Ju B Geng J Horng et al ldquoGraphene plasmonics for tunableterahertz metamaterialsrdquo Nature Nanotechnology vol 6 no 10pp 630ndash634 2010
[20] Y Francescato V Giannini J Yang M Huang and S A MaierldquoGraphene sandwiches as a platform for broadband molecularspectroscopyrdquo ACS Photonics vol 1 no 5 pp 437ndash443 2014
[21] J Yang J YangW Deng FMao andMHuang ldquoTransmissionproperties and molecular sensing application of CGPWrdquoOpticsExpress vol 23 no 25 pp 32289ndash32299 2015
[22] Y Huang L-S Wu M Tang and J Mao ldquoDesign of a beamreconfigurable thz antenna with graphene-based switchablehigh-impedance surfacerdquo IEEE Transactions on Nanotechnol-ogy vol 11 no 4 pp 836ndash842 2012
[23] I Llatser C Kremers A Cabellos-Aparicio J M Jornet EAlarcon and D N Chigrin ldquoGraphene-based nano-patchantenna for terahertz radiationrdquo Photonics and Nanostruc-turesmdashFundamentals and Applications vol 10 no 4 pp 353ndash358 2012
[24] Y Yao M A Kats P Genevet et al ldquoBroad electrical tuning ofgraphene-loaded plasmonic antennasrdquoNano Letters vol 13 no3 pp 1257ndash1264 2013
[25] E Forati G W Hanson A B Yakovlev and A Alu ldquoPlanarhyperlens based on a modulated graphene monolayerrdquo PhysicalReview B vol 89 no 8 Article ID 081410 2014
[26] X-H Wang W-Y Yin and Z Chen ldquoBroadband modelingsurface plasmon polaritons in optically pumped and curvedgraphene structures with an improved leapfrog ADI-FDTDmethodrdquo Optics Communications vol 334 pp 152ndash155 2015
[27] G D Bouzianas N V Kantartzis C S Antonopoulos and TD Tsiboukis ldquoOptimal modeling of infinite graphene sheets viaa class of generalized FDTD schemesrdquo IEEE Transactions onMagnetics vol 48 no 2 pp 379ndash382 2012
[28] V Nayyeri M Soleimani and O M Ramahi ldquoModelinggraphene in the finite-difference time-domain method using asurface boundary conditionrdquo IEEE Transactions on Antennasand Propagation vol 61 no 8 pp 4176ndash4182 2013
[29] M Tamagnone J S G Diaz J R Mosig and J P CarrierldquoReconfigurable terahertz plasmonic antenna concept using agraphene stackrdquo Applied Physics Letters vol 101 no 21 ArticleID 214102 2012
[30] A Cabellos-Aparicio I Llatser E Alarcon A Hsu and T Pala-cios ldquoUse of terahertz photoconductive sources to characterizetunable graphene RF plasmonic antennasrdquo IEEE Transactionson Nanotechnology vol 14 no 2 pp 390ndash396 2015
[31] P Li L J Jiang and H Bagc120580 ldquoA resistive boundary conditionenhancedDGTDscheme for the transient analysis of graphenerdquoIEEE Transactions on Antennas and Propagation vol 63 no 7pp 3065ndash3076 2015
[32] G W Hanson ldquoDyadic greenrsquos functions for an anisotropicnon-local model of biased graphenerdquo IEEE Transactions onAntennas and Propagation vol 56 no 3 pp 747ndash757 2008
[33] I Ahmed E H Khoo and E Li ldquoEfficient modeling andsimulation of graphene devices with the LOD-FDTD methodrdquoIEEEMicrowave andWireless Components Letters vol 23 no 6pp 306ndash308 2013
[34] X Yu and C D Sarris ldquoA perfectly matched layer for subcellFDTDand applications to themodeling of graphene structuresrdquoIEEE Antennas and Wireless Propagation Letters vol 11 pp1080ndash1083 2012
[35] V Nayyeri M Soleimani and O M Ramahi ldquoWidebandmodeling of graphene using the finite-difference time-domainmethodrdquo IEEE Transactions on Antennas and Propagation vol61 no 12 pp 6107ndash6114 2013
[36] H Lin M F Pantoja L D Angulo J Alvarez R G Martinand S G Garcia ldquoFDTD modeling of graphene devices usingcomplex conjugate dispersionmaterialmodelrdquo IEEEMicrowaveand Wireless Components Letters vol 22 no 12 pp 612ndash6142012
[37] D-W Wang W-S Zhao X-Q Gu W Chen and W-Y YinldquoWideband modeling of graphene-based structures at differenttemperatures using hybrid FDTD methodrdquo IEEE Transactionson Nanotechnology vol 14 no 2 pp 250ndash258 2015
[38] A Y Nikitin F J Garcia-Vidal and LMartin-Moreno ldquoAnalyt-ical expressions for the electromagnetic dyadic greenrsquos functionin graphene and thin layersrdquo IEEE Journal of Selected Topics inQuantum Electronics vol 19 no 3 Article ID 4600611 2013
[39] O V Shapoval J S Gomez-Diaz J Perruisseau-Carrier J RMosig and A I Nosich ldquoIntegral equation analysis of planewave scattering by coplanar graphene-strip gratings in the thzrangerdquo IEEE Transactions on Terahertz Science and Technologyvol 3 no 5 pp 666ndash674 2013
[40] R Araneo G Lovat and P Burghignoli ldquoGraphene nanostriplines dispersion and attenuation analysisrdquo in Proceedings of the16thWorkshop on Signal and Power Integrity (SPI rsquo12) pp 75ndash78IEEE Sorrento Italy May 2012
[41] G Lovat G W Hanson R Araneo and P Burghignoli ldquoSemi-classical spatially dispersive intraband conductivity tensor andquantum capacitance of graphenerdquo Physical Review B vol 87no 11 Article ID 115429 2013
[42] P Burghignoli R Araneo G Lovat and G Hanson ldquoSpace-domain method of moments for graphene nanoribbonsrdquo inProceedings of the 8th European Conference on Antennas andPropagation (EuCAP rsquo14) pp 666ndash669 IEEE The Hague TheNetherlands April 2014
[43] R Araneo P Burghignoli G Lovat and G W HansonldquoModal propagation and crosstalk analysis in coupled graphenenanoribbonsrdquo IEEE Transactions on Electromagnetic Compati-bility vol 57 no 4 pp 726ndash733 2015
[44] A Fallahi T Low M Tamagnone and J Perruisseau-CarrierldquoNonlocal electromagnetic response of graphene nanostruc-turesrdquo Physical Review B vol 91 Article ID 121405 2015
[45] M Dragoman D Neculoiu A Cismaru et al ldquoCoplanarwaveguide on graphene in the range 40MHz-110 GHzrdquoAppliedPhysics Letters vol 99 no 3 Article ID 033112 2011
[46] J Chen and Q H Liu ldquoDiscontinuous Galerkin Time-Domainmethods for multiscale electromagnetic simulations a reviewrdquoProceedings of the IEEE vol 101 no 2 pp 242ndash254 2013
International Journal of Antennas and Propagation 9
[47] S Descombes C Durochat S Lanteri L Moya C Scheid andJ Viquerat ldquoRecent advances on a DGTD method for time-domain electromagneticsrdquo Photonics and NanostructuresmdashFundamentals and Applications vol 11 no 4 pp 291ndash302 2013
[48] P Li and L J Jiang ldquoModeling of magnetized graphene frommicrowave to THz range by DGTD with a scalar RBC and anADErdquo IEEE Transactions on Antennas and Propagation vol 63no 10 pp 4458ndash4467 2015
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 International Journal of Antennas and Propagation
[14] J ChenM Badioli P A Gonzalez et al ldquoOptical nano-imagingof gate-tunable graphene plasmonsrdquoNature vol 487 pp 77ndash812012
[15] M Jablan H Buljan and M Soljacic ldquoPlasmonics in grapheneat infrared frequenciesrdquo Physical Review B vol 80 no 24Article ID 245435 2009
[16] Z Fei A S Rodin G O Andreev et al ldquoGate-tuningof graphene plasmons revealed by infrared nano-imagingrdquoNature vol 486 no 7405 pp 82ndash85 2012
[17] A N Grigorenko M Polini and K S Novoselov ldquoGrapheneplasmonicsrdquo Nature Photonics vol 6 no 11 pp 749ndash758 2012
[18] A Vakil and N Engheta ldquoTransformation optics usinggraphenerdquo Science vol 332 no 6035 pp 1291ndash1294 2011
[19] L Ju B Geng J Horng et al ldquoGraphene plasmonics for tunableterahertz metamaterialsrdquo Nature Nanotechnology vol 6 no 10pp 630ndash634 2010
[20] Y Francescato V Giannini J Yang M Huang and S A MaierldquoGraphene sandwiches as a platform for broadband molecularspectroscopyrdquo ACS Photonics vol 1 no 5 pp 437ndash443 2014
[21] J Yang J YangW Deng FMao andMHuang ldquoTransmissionproperties and molecular sensing application of CGPWrdquoOpticsExpress vol 23 no 25 pp 32289ndash32299 2015
[22] Y Huang L-S Wu M Tang and J Mao ldquoDesign of a beamreconfigurable thz antenna with graphene-based switchablehigh-impedance surfacerdquo IEEE Transactions on Nanotechnol-ogy vol 11 no 4 pp 836ndash842 2012
[23] I Llatser C Kremers A Cabellos-Aparicio J M Jornet EAlarcon and D N Chigrin ldquoGraphene-based nano-patchantenna for terahertz radiationrdquo Photonics and Nanostruc-turesmdashFundamentals and Applications vol 10 no 4 pp 353ndash358 2012
[24] Y Yao M A Kats P Genevet et al ldquoBroad electrical tuning ofgraphene-loaded plasmonic antennasrdquoNano Letters vol 13 no3 pp 1257ndash1264 2013
[25] E Forati G W Hanson A B Yakovlev and A Alu ldquoPlanarhyperlens based on a modulated graphene monolayerrdquo PhysicalReview B vol 89 no 8 Article ID 081410 2014
[26] X-H Wang W-Y Yin and Z Chen ldquoBroadband modelingsurface plasmon polaritons in optically pumped and curvedgraphene structures with an improved leapfrog ADI-FDTDmethodrdquo Optics Communications vol 334 pp 152ndash155 2015
[27] G D Bouzianas N V Kantartzis C S Antonopoulos and TD Tsiboukis ldquoOptimal modeling of infinite graphene sheets viaa class of generalized FDTD schemesrdquo IEEE Transactions onMagnetics vol 48 no 2 pp 379ndash382 2012
[28] V Nayyeri M Soleimani and O M Ramahi ldquoModelinggraphene in the finite-difference time-domain method using asurface boundary conditionrdquo IEEE Transactions on Antennasand Propagation vol 61 no 8 pp 4176ndash4182 2013
[29] M Tamagnone J S G Diaz J R Mosig and J P CarrierldquoReconfigurable terahertz plasmonic antenna concept using agraphene stackrdquo Applied Physics Letters vol 101 no 21 ArticleID 214102 2012
[30] A Cabellos-Aparicio I Llatser E Alarcon A Hsu and T Pala-cios ldquoUse of terahertz photoconductive sources to characterizetunable graphene RF plasmonic antennasrdquo IEEE Transactionson Nanotechnology vol 14 no 2 pp 390ndash396 2015
[31] P Li L J Jiang and H Bagc120580 ldquoA resistive boundary conditionenhancedDGTDscheme for the transient analysis of graphenerdquoIEEE Transactions on Antennas and Propagation vol 63 no 7pp 3065ndash3076 2015
[32] G W Hanson ldquoDyadic greenrsquos functions for an anisotropicnon-local model of biased graphenerdquo IEEE Transactions onAntennas and Propagation vol 56 no 3 pp 747ndash757 2008
[33] I Ahmed E H Khoo and E Li ldquoEfficient modeling andsimulation of graphene devices with the LOD-FDTD methodrdquoIEEEMicrowave andWireless Components Letters vol 23 no 6pp 306ndash308 2013
[34] X Yu and C D Sarris ldquoA perfectly matched layer for subcellFDTDand applications to themodeling of graphene structuresrdquoIEEE Antennas and Wireless Propagation Letters vol 11 pp1080ndash1083 2012
[35] V Nayyeri M Soleimani and O M Ramahi ldquoWidebandmodeling of graphene using the finite-difference time-domainmethodrdquo IEEE Transactions on Antennas and Propagation vol61 no 12 pp 6107ndash6114 2013
[36] H Lin M F Pantoja L D Angulo J Alvarez R G Martinand S G Garcia ldquoFDTD modeling of graphene devices usingcomplex conjugate dispersionmaterialmodelrdquo IEEEMicrowaveand Wireless Components Letters vol 22 no 12 pp 612ndash6142012
[37] D-W Wang W-S Zhao X-Q Gu W Chen and W-Y YinldquoWideband modeling of graphene-based structures at differenttemperatures using hybrid FDTD methodrdquo IEEE Transactionson Nanotechnology vol 14 no 2 pp 250ndash258 2015
[38] A Y Nikitin F J Garcia-Vidal and LMartin-Moreno ldquoAnalyt-ical expressions for the electromagnetic dyadic greenrsquos functionin graphene and thin layersrdquo IEEE Journal of Selected Topics inQuantum Electronics vol 19 no 3 Article ID 4600611 2013
[39] O V Shapoval J S Gomez-Diaz J Perruisseau-Carrier J RMosig and A I Nosich ldquoIntegral equation analysis of planewave scattering by coplanar graphene-strip gratings in the thzrangerdquo IEEE Transactions on Terahertz Science and Technologyvol 3 no 5 pp 666ndash674 2013
[40] R Araneo G Lovat and P Burghignoli ldquoGraphene nanostriplines dispersion and attenuation analysisrdquo in Proceedings of the16thWorkshop on Signal and Power Integrity (SPI rsquo12) pp 75ndash78IEEE Sorrento Italy May 2012
[41] G Lovat G W Hanson R Araneo and P Burghignoli ldquoSemi-classical spatially dispersive intraband conductivity tensor andquantum capacitance of graphenerdquo Physical Review B vol 87no 11 Article ID 115429 2013
[42] P Burghignoli R Araneo G Lovat and G Hanson ldquoSpace-domain method of moments for graphene nanoribbonsrdquo inProceedings of the 8th European Conference on Antennas andPropagation (EuCAP rsquo14) pp 666ndash669 IEEE The Hague TheNetherlands April 2014
[43] R Araneo P Burghignoli G Lovat and G W HansonldquoModal propagation and crosstalk analysis in coupled graphenenanoribbonsrdquo IEEE Transactions on Electromagnetic Compati-bility vol 57 no 4 pp 726ndash733 2015
[44] A Fallahi T Low M Tamagnone and J Perruisseau-CarrierldquoNonlocal electromagnetic response of graphene nanostruc-turesrdquo Physical Review B vol 91 Article ID 121405 2015
[45] M Dragoman D Neculoiu A Cismaru et al ldquoCoplanarwaveguide on graphene in the range 40MHz-110 GHzrdquoAppliedPhysics Letters vol 99 no 3 Article ID 033112 2011
[46] J Chen and Q H Liu ldquoDiscontinuous Galerkin Time-Domainmethods for multiscale electromagnetic simulations a reviewrdquoProceedings of the IEEE vol 101 no 2 pp 242ndash254 2013
International Journal of Antennas and Propagation 9
[47] S Descombes C Durochat S Lanteri L Moya C Scheid andJ Viquerat ldquoRecent advances on a DGTD method for time-domain electromagneticsrdquo Photonics and NanostructuresmdashFundamentals and Applications vol 11 no 4 pp 291ndash302 2013
[48] P Li and L J Jiang ldquoModeling of magnetized graphene frommicrowave to THz range by DGTD with a scalar RBC and anADErdquo IEEE Transactions on Antennas and Propagation vol 63no 10 pp 4458ndash4467 2015
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Antennas and Propagation 9
[47] S Descombes C Durochat S Lanteri L Moya C Scheid andJ Viquerat ldquoRecent advances on a DGTD method for time-domain electromagneticsrdquo Photonics and NanostructuresmdashFundamentals and Applications vol 11 no 4 pp 291ndash302 2013
[48] P Li and L J Jiang ldquoModeling of magnetized graphene frommicrowave to THz range by DGTD with a scalar RBC and anADErdquo IEEE Transactions on Antennas and Propagation vol 63no 10 pp 4458ndash4467 2015
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International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
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