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electronics Article

Equivalent Resonant Circuit Modeling of a Graphene-Based Bowtie Antenna Bin Zhang 1,2 , Jingwei Zhang 1, *, Chengguo Liu 1 , Zhi P. Wu 1,2 and Daping He 1 1

2

*

Hubei Engineering Research Center of RF-Microwave Technology and Application, Wuhan University of Technology, Wuhan 430070, China; [email protected] (B.Z.); [email protected] (C.L.); [email protected] (Z.P.W.); [email protected] (D.H.) School of Information Engineering, Wuhan University of Technology, Wuhan 430070, China Correspondence: [email protected]

Received: 13 September 2018; Accepted: 26 October 2018; Published: 30 October 2018







Abstract: The resonance performance analysis of graphene antennas is a challenging problem for full-wave electromagnetic simulators due to the trade-off between the computer resource and the accuracy of results. In this paper, an equivalent circuit model is presented to provide a concise and fast way to analyze the graphene-based THz bowtie antenna. Based on the simulated results of the frequency responses of the antenna, a suitable equivalent circuit of Resistor-Inductor-Capacitor (RLC) series is proposed to describe the antenna. Then the RLC parameters are extracted by considering the graphene bowtie antenna as a one-port resonator. Parametric analyses, including chemical potential, arm length, relaxation time, and substrate thickness, are presented based on the proposed equivalent circuit model. Antenna input resistance R is a significant parameter in this model. Validation is performed by comparing the calculated R values with the ones from full-wave simulation. By applying different parameters to the graphene bowtie antenna, a set of R, L, and C values are obtained and analyzed comprehensively. A very good agreement is observed between the equivalent model and the numerical simulation. This work sheds light on the graphene-based bowtie antenna’s initial design and paves the way for future research and applications. Keywords: equivalent circuit; graphene antenna; THz antenna

1. Introduction In the past decades, terahertz (THz) communications have been envisioned as the candidate to provide the huge bandwidth and terabit-per-second (Tbps) data links, due to the fast-increasing terminal device numbers [1]. To maximize the use of the THz band, the ultramassive (UM) multi-input multi-output (MIMO) technique has been proposed [2]. The realization of this technique requires antennas to be densely embedded in a very small footprint, which is enabled by the graphene-based plasmonic antenna. Graphene is a rising star of electromagnetic materials. It has several unique electrical properties [3,4], which enables many potential applications in the field of electronics [5]. Among them, the graphene antenna is one of the potential applications which has drawn increasing attention in recent years. The first use of graphene in antenna application was simply taking it as a parasitic layer for improving the radiation pattern of metal antennas [6]. Then the transverse magnetic (TM) Surface Plasmon Polariton (SPP) waves were demonstrated to be supported on the monolayer graphene sheet in 2011 [7,8]. Due to the slow propagation of SPP waves, a miniature and low-profile graphene antenna could be achieved by adapting graphene sheets as the radiation part [9]. Consequently, a considerable amount of papers were published to show the theoretical results of graphene-based antennas [10–12]. Recently, a graphene-based antenna with a metamaterial substrate [13] and a few-layer (less than six) Electronics 2018, 7, 285; doi:10.3390/electronics7110285

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graphene antenna [14] were proposed for high radiation efficiency. Interested readers are encouraged to refer to the review article [15] for more details on the state of the art of the graphene-based THz antennas. Due to the highly dispersive, frequency-dependent properties and very small dimensions of the graphene-based THz bowtie antenna, there is no explicit formula and classic theory to build the connection between the bowtie antenna performance and related parameters like antenna dimensions and material properties. Full-wave numerical modeling is a way to characterize the performance of the graphene-based antenna. However, suffering from the one-atom thickness, full-wave simulation will cost a huge amount of computer resources if high accuracy is desired during the simulation. The regular simulation process is very intricate and can hardly demonstrate the theoretical principle of antenna resonant performance. Model-based parameter estimation (MBPE) [16] was used to analyze the carbon nanotube antenna with a high impedance [17], which still needs to do the large-size matrix calculation. To shed light on the parametric performance of the antenna, as well as the physical insight of antenna resonance property, the resonance circuit model for graphene-based antennas should be studied to deepen comprehension on such antennas. A simple circuit model based on transmission line for graphene plasmonic dipole was proposed for this purpose in 2014 [18]. In this model, input impedance and efficiency could be calculated from the dispersion relationship of SPPs. In 2016, a circuit model based on the partial element equivalent circuit (PEEC) [19] was presented for graphene antennas, which is a relatively complicated method. Some papers also presented circuit models for electromagnetic waves through graphene sheets [20], but not for graphene antennas. In this paper, from a different perspective, an equivalent series RLC resonant circuit model is proposed for terahertz graphene-based bowtie antennas. Since the designed graphene bowtie antenna in our work shows only one resonance mode with a sharp resonance, in this paper we mainly place the focus on the performance at the resonance point where the reflection coefficient of the antenna shows the minimum magnitude value. Therefore, a technique initially developed for studying the superconducting resonators [21] is borrowed in this paper due to the similarity between graphene patch antennas and superconducting resonators. For the graphene bowtie antenna, because of the edge effect, it can be regarded as a resonance cavity and the performance can be predicted by a proper circuit model. As to the technique for superconducting resonators, it is very simple and easy to apply in this case for graphene antennas. In this context, the concise and convenient technique is presented with the related formulas and the RLC parameters at resonance frequencies are extracted to study the resonant properties and understand the behavior of the graphene-based bowtie antenna. The effects of the applied chemical potential and different geometry dimensions of the antennas are considered and analyzed as well. The remainder of this paper is organized as follows: Section 2 presents the bowtie antenna design by a single-layer graphene sheet. Section 3 demonstrates the equivalent circuit model for the graphene-based antenna. The results are discussed in Section 4 and conclusions are drawn in Section 5. 2. Graphene Property and Bowtie Antenna Design As a two-dimensional material, graphene has excellent electrical properties due to its one-atom thickness. In this section, we firstly recall the surface conductivity model and the electromagnetic wave propagation properties of the monolayer graphene sheet. Then, a graphene-based plasmonic bowtie antenna working at THz band is proposed as the platform to study the corresponding equivalent circuit model. The full-wave simulation of this antenna will be presented in Section 4. Surface conductivity of single-layer graphene is dependent on other properties like frequency, chemical potential, and the scattering rate. In other words, graphene is a dispersive medium and its electrical performance varies with the change of frequency and other parameters. The graphene conductivity model can be calculated by means of Kubo’s formula [22]. It is composed of interband and intraband contributions, and the latter one dominates in the frequency region of interest (f < 5 THz)

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with electric field biasing on graphene film. Therefore, we can use the intraband part to approximately represent the total conductivity. The intraband conductivity is given by σs,intra

   q2e k B T µc −µc /(k B T ) = −j 2 × + 2 ln e +1 , kB T π } (ω − j2Γ)

(1)

where ω is angular frequency, kB is the Boltzmann constant, T denotes temperature in Kelvin, µc is the chemical potential, Γ is the scattering rate, which is related to the losses in graphene, } is the reduced Plank constant and qe represents the charge of an electron. It is noted that the surface impedance of graphene Zs = 1/σs can be calculated by employing boundary conditions [22]. Chemical potential is an essential factor that can be dynamically controlled by means of electrostatic biasing and chemical doping. It is noted that for monolayer graphene sheets, the chemical potential depends on the carrier density ns of the film. They are related by [22] 2 ns = π }2 v2F

Z∞

ε[ f d (ε − µc ) − f d (ε + µc )]dε,

(2)

0

  −1 where f d (ε) = e(ε−µc )/k0 T + 1 is the Fermi-Dirac distribution, ε donates energy and v F is Fermi velocity (assuming that v F is 106 m/s in this work). On the other hand, the carrier density is generally involved with the bias electrostatic field. As a result, the chemical potential is related to the biasing voltage through the carrier density ns [23]. Under such circumstance, the graphene complex conductivity model suggests that a monolayer graphene sheet or ribbon is seen as a tunable conductor at the frequency range of interest. Electromagnetic wave propagation is found to be supported on the single-layer graphene film. Indeed, it is confirmed that surface waves, known as the TM SPP waves, could propagate along the graphene sheet due to the inductive nature of graphene. For an infinite graphene layer placed between two different media, the dispersion relation of TM SPPs in the quasi-static regime is given by [24]
ε r1 + ε r2 coth k spp d σs , −i = ωε 0 k spp

(3)

where ε r1 and ε r2 are the permittivity of air and antenna substrate respectively, k spp is the guided complex propagation constant and d is the thickness of the substrate. In this context, graphene shows the potential to be utilized as the antenna radiator in the future Terahertz communication systems. In this paper, a graphene-based plasmonic THz bowtie antenna is designed to analyze the resonant performance and deduce the equivalent RLC resonant circuit. The schematic of the proposed antenna and the corresponding resonant circuit model are shown in Figure 1. The bowtie antenna patches are deposited on the quartz substrate with a thickness of 90 nm, which is practically achieved by the current fabrication technique [24,25]. The substrate permittivity is 3.75 and no loss is considered in this model. The length of the antenna Lant is set as 12 µm, the two arms are separated by a gap with Lgap = 2 µm and characterized by a short width W 2 = 1 µm and long width W 1 = 5 µm, then the arm length is 5 µm as well. The proposed antenna is fed by a THz continuous-wave (CW) photomixer [26] placed between the feeding gap. The feeding mechanism typically has a very high output impedance (of order 10 kΩ) [27,28]. As a result, a better impedance matching is achieved between the THz CW photomixer source and the graphene-based bowtie antenna. This is because the output impedance of the THz CW source lies in the same order of magnitude of the input impedance of the graphene antennas (several kiloohms) [26,29]. In practice, the photomixer source is a relatively complicated structure. To simplify the prototype model, we use a discrete port in Computer Simulation Technology (CST)

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microwave studio with a characteristic impedance of 10 kΩ to simulate the whole antenna structure in a practical way. Electronics 2018, 7, x FOR PEER REVIEW 4 of 15

(a)

(b)

Figure Figure 1. 1. (a) The schematic of the graphene-based bowtie antenna; (b) the corresponding equivalent resonant circuit.

3. RLC RLC Circuit Circuit Model Model 3. The graphene-based graphene-based bowtie bowtie antenna antenna resonates resonates at at THz THz band band due due to to the the metal-like metal-like property property of of The graphene. Moreover, over thethe substrate could be treated as aas mirror and graphene. Moreover,the theedge edgeofofthe thegraphene graphenesheet sheet over substrate could be treated a mirror then the graphene antenna would be seen as a resonator for TM SPP waves [12]. Usually, complicated and then the graphene antenna would be seen as a resonator for TM SPP waves [12]. Usually, theory and numerical methods are methods used to analyze resonant performance. For the antenna complicated theory and numerical are usedantenna to analyze antenna resonant performance. For with only one resonance mode, the performance at the resonance frequency could be described by the antenna with only one resonance mode, the performance at the resonance frequency could bea simple RLC circuit model. Incircuit this paper, weIn propose a simple RLC series circuit RLC to describe described byequivalent a simple RLC equivalent model. this paper, we propose a simple series the resonance performance of theperformance proposed graphene-based bowtie antenna. bowtie antenna. circuit to describe the resonance of the proposed graphene-based This model originally usedused to calculate surfacesurface resistance of a superconducting resonator. This model[21] [21]was was originally to calculate resistance of a superconducting Treating this antenna as a coupled resonator, then the working frequency of the antenna is the resonance resonator. Treating this antenna as a coupled resonator, then the working frequency of the antenna frequency of the equivalent that the antenna complex impedance is written as is the resonance frequencycircuit. of theAssume equivalent circuit. Assume thatinput the antenna complex input impedance is written as

  1 Zin = Rin + jXin = Rin + j ωLin − . (4)  1ωCin Z in  Rin  jX in  Rin  j   Lin  (4) . Cin   appears, At the resonance point, where the minimum S value the behavior of the graphene-based 11

bowtie can be point, described by athe resonance circuit for itsappears, S11 responses. Therefore, an equivalent At antenna the resonance where minimum S11 value the behavior of the grapheneresonance circuit with R , L , and C in series is used to describe the resonance property of the in bein described in by a resonance circuit for its S11 responses. Therefore, based bowtie antenna can an graphene-based antenna, as shown 1b.CTreating this a coupled work equivalent resonance circuit with Rinin,Figure Lin, and in in series is antenna used to as describe the resonator, resonancethen property frequency of the antenna is the resonance frequency ofTreating the equivalent circuit.asIna traditional antenna of the graphene-based antenna, as shown in Figure 1b. this antenna coupled resonator, analysis, the quality factor obeys the inverse relationship with the antenna bandwidth and has a limit then work frequency of the antenna is the resonance frequency of the equivalent circuit. In traditional within the antennas particular forms [30,31]. Here, we borrow the unloaded quality factor Qa and antenna analysis, theinquality factor obeys the inverse relationship with the antenna bandwidth the coupled quality factor Q , which are widely used in the filter design, to describe the resonance 0 has a limit within the antennas in particular forms [30,31]. Here, we borrow the unloaded quality behavior of the graphene-based bowtie antenna. factor Qa and the coupled quality factor Q0, which are√widely used in the filter design, to describe the With the angular frequencybowtie ω0 = 1/ Lin Cin , one can obtain the quality factor at the resonance behavior of resonance the graphene-based antenna. resonance to be Q = ω L /R . For the antenna seen as resonator in the equivalent circuit, 0 0 in in frequency   1 L Ca one-port With the angular resonance 0 in in , one can obtain the quality factor at the the coupling coefficient is β = Z0 /Rin . Therefore, the circuit parameters can be easily calculated by resonance to be Q0  0 Lin Rin . For the antenna seen as a one-port resonator in the equivalent circuit, Q0the Rin circuit parameters 1 the coupling coefficient is   Z0 Rin Z. 0Therefore, can be easily calculated by Rin = ,L = , Cin = . (5) β in ω0 Q0 Rin ω0 Z QR 1 Rin  0 , Lin  0 in , Cin  . (5)  0 Q0 Rin0 Based on the equations, quality factor Q0 and β are key for obtaining the circuit parameters, Rin, Lin, and Cin. Generally, according to [21], a quality factor Qa at arbitrary point (fa, S11 (fa)) in frequency response of S11 can be expressed by Qa   f0 2 f a  f0

1

,

(6)

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Based on the equations, quality factor Q0 and β are key for obtaining the circuit parameters, Rin , Lin , and Cin . Generally, according to [21], a quality factor Qa at arbitrary point (fa , S11 (fa )) in frequency response of S11 can be expressed by Q a = ( f 0 /2)| f a − f 0 |−1 ,

(6)

where f 0 is the resonance frequency. This is the parameter that directly reveals the bandwidth of the antenna. It is noteworthy that in this work, S11 (fa ) = −3 dB position is selected as the point to calculate the quality factor, as the point is basically far enough from the resonant point, and then the error can be reduced [21]. With Qa obtained at the selected point, the resonance quality factor Q0 can then be calculated simply by multiplying a correction coefficient “a” as Q0 = aQ a .

(7)

The coefficient “a” can be determined by reflection level at frequency fa and coupling coefficient β as [21] " #1/2 (1 + β)2 |S11 ( f a )|2 − (1 − β)2 a= , (8) 1 − |S11 ( f a )|2 with β determined by β=

1 ∓ |S11 ( f 0 )| , 1 ± |S11 ( f 0 )|

(9)

where S11 (f 0 ) is the S11 response at f 0 . Note that the sign in (9) is dependent on the phase variation of S11 (f ) around f 0 . The upper sign is used for the situation that phase derivative against frequency is larger than 0, and the lower sign for the negative derivative of the phase response. 4. Results and Discussions In this section, the designed antennas with different parameters were simulated to obtain frequency responses by using the time domain solver in CST microwave studio. In the numerical simulations, the mesh grid was carefully defined with a maximum cell size of 1 µm (about 1/20 SPP wavelength) and minimum cell size of 0.03 µm (about 1/3 of the minimum substrate thickness), which are fine enough to get the accurate results. The simulations were performed with a Gaussian pulse excitation to approximately model the realistic photomixer source. Then, the equivalent circuit parameters were extracted from the proposed model in Section 3 by employing software package MATLAB. Moreover, the input impedance values from the equivalent circuit model were compared to the numerical simulation results for validating the proposed model. The antenna performance is affected by the graphene material properties and the antenna geometry parameters. Here, we chose the graphene chemical potential, the relaxation time, the antenna arm length, and the substrate thickness to investigate the antenna performance and then analyze the equivalent circuit model. The initial parameter values were carefully defined based on existing publications [12,24]. The relaxation time of graphene was set as 0.5 ps, the temperature was 300 K, and the antenna arm length was 5 µm. The chemical potential of the graphene antenna was ranging from 0.1 eV to 0.6 eV. It should be noted that only the first resonance points of the S-parameters were used for the equivalent circuit modeling throughout this paper. 4.1. Chemical Potential Chemical potential, the energy level of the graphene sheet, is a crucial factor to control the graphene surface conductivity. In this part, the chemical potential varied from 0.1 eV to 0.6 eV with a step of 0.05 eV in the simulation and the further equivalent model calculation. The single layer

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graphene sheet can be modeled as a layer impedance which is characterized by complex surface conductivity model, as listed in Section 2. In this subsection, the relaxation time was assumed as 0.5 ps, which is widely used and based on the Raman spectra of graphene samples in the published papers [12,24]. For each chemical Electronics 2018, 7, x FORanalysis PEER REVIEW 6 of 15 potential, the simulated S11 parameters are shown in Figure 2. It is clearly seen that the antennas all resonated well and presented very sharp resonance with relatively narrow working bands, as shown in Figure 2. At the chemical potential level of 0.3 eV, the the minimum SS11 11 value was better than those at other chemical potentials, reaching − −51 dB, while the values for other chemical potentials remained 51 dB, while the values for less than −45 −45dB. dB.

Figure coefficient with chemical potential levels fromfrom 0.1 eV Figure 2. 2. Simulated Simulatedreflection reflection coefficient with chemical potential levels 0.1toeV0.6toeV. 0.6The eV. relaxation time is 0.5 ps and the temperature is 300 K. The relaxation time is 0.5 ps and the temperature is 300 K.

Based on the simulated simulated responses responses as as the the chemical chemical potential potential varied varied from from 0.1 0.1 to to 0.6 0.6 eV, eV, we can obtain the parameters of equivalent circuit circuit with with RRinin, Lin in,, and and C Cininin inseries seriesby byusing using the the Equations Equations (5)–(9) (5)–(9) in Section 3. Specially, Specially, the important results obtained from the equivalent circuit model, including ff00,, |S , are |S1111 (f(f0)| dB,and andRRinin , arelisted listedininTable Table1.1. The Thenumerical numericalresults results of of the the input input impedance impedance of of the 0 )|inindB, antenna antenna are listed in Table 1 as well. The data data in Table 1 are calculated and simulated based on the initial setup (the arm length was 5 µm µ m and chemical potential ranged ranged from from 0.1 0.1 to to 0.6 0.6 eV). eV). Table 1. potential ranging from 0.1 0.1 eV eV to 0.6 time Table 1. The Thesimulated simulateddata datawith withchemical chemical potential ranging from to eV. 0.6 The eV. relaxation The relaxation is setisasset 0.5asps, is 5 µm, of theofsubstrate is 90 nm. time 0.5the ps,arm the length arm length is 5the µ m,thickness the thickness the substrate is 90 nm. c/eV f 0 /THz f0/THzS11 (fS011 (f0)/dBRin /Ω Rin/Ω Lin /H Lin/H µcμ /eV )/dB

0.10.1 0.15 0.15 0.20.2 0.25 0.25 0.30.3 0.35 0.35 0.40.4 0.45 0.45 0.5 0.550.5 0.55 0.6 0.6

1.8335 −38.125 −38.125 1.8335 2.2385 −43.636 −43.636 2.2385 2.5794 −48.305 −48.305 2.5794 2.8807 2.8807 −44.76 −44.76 3.1475 3.1475 −51.075 −51.075 3.3896 3.3896 −42.893 −42.893 3.6168 3.6168 −40.06 −40.06 3.8243 −36.28 3.8243 −36.28 4.0219 −33.968 4.0219 −33.968 4.2096 −31.359 4.2096 −29.526 −31.359 4.3825 4.3825 −29.526

C/F in/F Cin 9754.9 3907.3 1.93 × 10 −6−6 9754.9 3907.3 1.93 × 10 −6−6 9869.3 4208.9 4208.9 1.20 1.20 × 10 9869.3 × 10 −7−7 9923.4 4370.5 4370.5 8.71 8.71 × 10 9923.4 × 10 −7−7 9885 × 10 9885 4406.6 4406.6 6.93 6.93 × 10 −7−7 9944.3 × 10 9944.3 4477 4477 5.71 5.71 × 10 −7 9857.7 4501.7 4.90 × 10 9857.7 4501.7 4.90 × 10−7 −7 9803.3 × 10 9803.3 4478.4 4478.4 4.32 4.32 × 10−7 9697.7 4490.5 3.86 × 10−7−7 9697.7 4490.5 3.86 × 10 9607.3 4451.6 3.52 × 10−7−7 9607.3 4380.1 4451.6 3.26 3.52 × 10 −7 9473.4 × 10 9473.4 4371.1 4380.1 3.02 3.26 × 10 −7−7 9353.7 × 10 9353.7 4371.1 3.02 × 10−7

Xin in/Ω /Ω 0.00 0.00 00 0.00 0.00 −11 −−1.46 1.46 × × 10−11 0 0.00 0.00 −11 1.46 × 10−11 −−1.46 × 10 −11 −1.46 × 10−11 −1.46 × 10 0.00 −1.460.00 × 10−11 −11 × 10 −−1.46 1.46 × 10−11 −1.46 × 10−11

ββ 1.0251 1.0251 1.0132 1.0132 1.0077 1.0077 1.0116 1.0116 1.0056 1.0056 1.0144 1.0144 1.0201 1.0201 1.0312 1.0312 1.0409 1.0409 1.0556 1.0556 1.0691 1.0691

QQ a a 2.2734 2.2734 2.9725 2.9725 3.5468 3.5468 4.0017 4.0017 4.4288 4.4288 4.8169 4.8169 5.1274 5.1274 5.4662 5.4662 5.7259 5.7259 5.9395 5.9395 6.2112 6.2112

Q0Q0 Rin,CST /Ω /Ω Rin,CST 4.6143 9781 4.6143 9781 5.9983 5.9983 9871 9871 7.1378 7.1378 9925 9925 8.0688 9981 9981 8.0688 8.9035 9958 9958 8.9035 9.7259 9956.23 9956.23 9.7259 10.381 9805.3 9805.3 10.381 11.126 9711.14 11.126 9711.14 11.709 9607.7 11.709 12.229 9607.7 9480.8 12.229 12.868 9480.8 9355 12.868 9355

Obviously, the related to to thethe input impedance of the antenna. If the Obviously, theresonance resonancepoint pointmagnitude magnitudeisis related input impedance of the antenna. If antenna input impedance matches the source impedance well, then a very sharp resonance response the antenna input impedance matches the source impedance well, then a very sharp resonance will be obtained. The input The resistance of the antenna at antenna a chemical ofpotential 0.2 eV and response will be obtained. input resistance of the at apotential chemical of0.3 0.2eV eVwere and 9923 Ω and 9944 Ω, respectively, which are close to Z of 10 kΩ. It is not a monotonic linear relation 0 0.3 eV were 9923 Ω and 9944 Ω, respectively, which are close to Z0 of 10 kΩ. It is not a monotonic between the frequency magnitude andmagnitude the chemical potential level. That is a reason why is the linear relation betweenresponse the frequency response and the chemical potential level. That a reason why the magnitudes of 0.2 eV and 0.3 eV show lower position than others in Figure 2. The numerical results of the input resistance are given as well to verify the equivalent circuit model. As observed from the data list, the results from the circuit model and numerical simulation show a good agreement, which confirms that the proposed model is reliable in this work. The input resistance value of the proposed graphene-based bowtie antenna in this work is

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magnitudes of 0.2 eV and 0.3 eV show lower position than others in Figure 2. The numerical results of the input resistance are given as well to verify the equivalent circuit model. As observed from the data list, the results from the circuit model and numerical simulation show a good agreement, which confirms that the proposed model is reliable in this work. The input resistance value of the proposed graphene-based bowtie antenna in this work is approximately 10 kΩ, which is higher than that of the graphene antenna reported in previous work, the latter one2018, usually in the range from several hundred to more than 3 kΩ from the Electronics 7, x FORlying PEER REVIEW 7 ofexisting 15 work [24,26,29]. This mainly resulted from the very small value of the arm width (1 µm) and the the latter one usually lying range from several hundred to more thandesign 3 kΩ from the existing substrate thickness (90 nm) of in thethe graphene-based bowtie antenna in our [18,24]. work [24,26,29]. Thisthe mainly resulted from the very small value of the arm of width (1 µ m) and the In these responses, chemical potential variation leads to the change surface conductivity or substrate thickness (90 nm) of the graphene-based bowtie antenna in our design [18,24]. the impedance of the monolayer graphene sheet. Consequently, the graphene antenna shows to have In these responses, the chemical potential variation leads to the change of surface conductivity its resonance performance related to µc . As shown in Table 1, with the increase of chemical potential, or the impedance of the monolayer graphene sheet. Consequently, the graphene antenna shows to both β and Q0 rise, which indicates the reduction in Rin and increase in impedance bandwidth. have its resonance performance related to μc. As shown in Table 1, with the increase of chemical It should be noted that quality is obtained from the −in3 and dB bandwidth the proposed potential, both β and Q0the rise, whichfactor indicates the reduction in R increase in of impedance antenna in our work. The graphene-based bowtie antennas all show relatively narrow working band bandwidth. (−10 dB bandwidth less than 10%), except theisantenna chemical potential of proposed 0.1 eV due to It should be noted that the quality factor obtained with from the the −3 dB bandwidth of the the very lowin energy level. from the S-parameters Figure 2, wenarrow can seeworking there isband only one antenna our work. TheMoreover, graphene-based bowtie antennas all in show relatively (−10 dB bandwidth less than 10%), the antenna with the chemical of 0.1bowtie eV dueantenna to resonance mode for each antenna withexcept a different chemical potential level.potential The metallic the very low energy level. Moreover, from the S-parameters in Figure 2, we can see there is only one usually exhibits the broadband performance due to the involvement of more incoherent diffractions mode each of antenna with apatches different[32]. chemical potential from level. the The current metallic distribution bowtie fromresonance the corners andfor edges the bowtie This resulted antenna usually exhibits the broadband performance due to the involvement of more incoherent along the antenna edge. However, the current vector on the proposed graphene-based bowtie antenna diffractions from the corners and edges of the bowtie patches [32]. This resulted from the current mainly propagates along the source direction due to the dielectric nature of the graphene with a low distribution along the antenna edge. However, the current vector on the proposed graphene-based energy levelantenna as well mainly as the dimensions of thethe antenna. proposed bowtie antenna bowtie propagates along source The direction due tographene-based the dielectric nature of the is utilized as the platform to apply proposed equivalentofcircuit model.The proposed graphenegraphene with a low energy levelthe as well as the dimensions the antenna. Further investigation the impact varioustochemical will circuit be discussed based bowtie antenna isof utilized as theof platform apply thepotential proposed levels equivalent model. with the Further investigation of the impact of various chemical levels subsections. will be discussed with aid of other graphene properties and antenna parameters in potential the following the aid of other graphene properties and antenna parameters in the following subsections.

4.2. Antenna Arm Length 4.2. Antenna Arm Length

The bowtie antenna is a quasi-dipole antenna, which can be controlled to show different resonance Thechanging bowtie antenna is a quasi-dipole which be controlled show points with arm length. In this part, antenna, the antenna armcan lengths ranging to from 5 todifferent 10 µm were resonance points with changing arm length. In this part, the antenna arm lengths ranging from 5 to considered and results were obtained as shown in Figures 3–5. We only demonstrate the S-parameter 10 µ m were considered and results were obtained as shown in Figures 3–5. We only demonstrate the curves when the chemical potential was 0.2 eV, as shown in Figure 3. As the arm length increased, S-parameter curves when the chemical potential was 0.2 eV, as shown in Figure 3. As the arm length the resonance point shiftedpoint to lower frequencies. At the same time, due to due the impedance variation increased, the resonance shifted to lower frequencies. At the same time, to the impedance resulting from the arm length change, the resonance performances became worse, from − variation resulting from the arm length change, the resonance performances became worse, from 48 −48dB to −16 dB. The S-parameters based on other chemical potential levels showed similar trends, which dB to −16 dB. The S-parameters based on other chemical potential levels showed similar trends, which are not shown this part. are notin shown in this part.

Figure 3. Simulated reflection coefficientwith with arm arm length 5 µ5mµm to 10 m.µm. The relaxation Figure 3. Simulated reflection coefficient lengthranging rangingfrom from toµ10 The relaxation time is 0.5 ps and the temperature is 300 K. time is 0.5 ps and the temperature is 300 K.

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(a) (a)

(b) (b)

Figure results versus chemical potential with arm length 5–10 µµm. Figure 4. 4. Calculated parameter 5–10 µm. (a)(a) β Calculated circuit circuit parameter parameterresults resultsversus versuschemical chemicalpotential potentialwith witharm armlength length 5–10 m. (a) βand and (b) Q 0 curves with the arm length ranging from 5 µ m to 10 µ m over different chemical potentials. (b) β and (b)QQ curveswith withthe thearm armlength lengthranging rangingfrom from55µm µ mto to10 10µm µ m over different different chemical potentials. 0 0curves

(a) (a)

(b) (b)

(c) (c)

Figure of 5. Equivalent parameters. (a) (a) R in,,, (b) (b) LLinin in,,,and and (c) C the equivalent Figure 5. 5. Equivalent circuit circuit parameters. parameters. Rinin and (c) Cinin of the equivalent circuit circuit against against in of chemical potential with different arm lengths. The solid lines are the results from the equivalent chemical potential with different arm lengths. The solid lines are the results from the equivalent chemical potential with different arm lengths. The solid lines are the results from the equivalent circuit circuit model the scatter denotes the data. relaxation ps circuit and model and thedenotes scatterthe denotes the numerical numerical data. The Thetime relaxation time is 0.5 0.5 ps and and the the model theand scatter numerical data. The relaxation is 0.5 pstime and is the temperature is temperature is temperature is 300 300 K. K. 300 K.

Using the Using the equivalent equivalent circuit circuit model model proposed proposed in in this this paper, paper, the the calculated calculated coupling coupling coefficient coefficient β, β, quality factor factorQ Q , as well as the R , L , and C parameters are also shown for several arm lengths quality 0 , as well as the R in , L in , and C in parameters are also shown for several arm lengths with Q00, as well as the Rin,inLin, in and Cin parameters are also shown for several arm lengths with in with increasing energy in Figure 5a–c, respectively. As can be seen4, Figure 4, increasing energy levels in 4a,b and 5a–c, respectively. As in for longer increasing energy levelslevels in Figures Figures 4a,b4a,b and and 5a–c,Figure respectively. As can can be be seen seen in Figure Figure 4,in for longer for arm length, the graphene-based bowtie antenna toaashow β value a smaller arm length, the graphene-based bowtie tended to large and smaller Q armlonger length, the graphene-based bowtie antenna antenna tended tended to show show largea βlarge β value value and aaand smaller Q00,, Q suggests input impedance shrinks whilst impedancebandwidth bandwidthdecreases. decreases.Also, Also, the the ββ which suggests that input impedance shrinks whilst impedance which suggests thatthat input impedance shrinks whilst impedance bandwidth decreases. Also, 0 , which showed showed aa relatively relatively gentle gentle rising rising comparing comparing to to the the curves curves of of the the quality quality factor, factor, which which indicates indicates chemical chemical potential potential mainly mainly affects affects the the antenna antenna resonance resonance frequency frequency but but not not the the bandwidth. bandwidth. In In Figure Figure

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showed a relatively gentle rising comparing to the curves of the quality factor, which indicates chemical potential mainly affects the antenna resonance frequency but not the bandwidth. In Figure 5a, the input resistance Rin decreased as the chemical potential increased from 0.1 to 0.6 eV, and so was the loss on the antenna. When µc = 0.3 eV and the arm length was 5 µm, Rin was close to the source Electronics 2018, 7, x FOR PEER REVIEW 9 of 15 impedance. The arm length of the graphene antenna is related to the SPP wavelength at working frequency. length comes a reduced impedance. It is noted that the proposed 5a, the The inputincreasing resistance arm Rin decreased as thewith chemical potential increased from 0.1 to 0.6 eV, and so model is validated to theμcimpedance values calculated by 5the numerical method in CST. was the loss onby thecomparing antenna. When = 0.3 eV and the arm length was µ m, Rin was close to the source impedance. The arm length of the graphene antenna is related to the SPP wavelength at The impedance values from CST were also calculated from S parameters, but they were based on the working frequency. The increasing arm length a reduced impedance. It is noted thatmethods. the full-wave numerical algorithm. It is shown thatcomes there with is a good agreement between the two proposed model is validated by comparing to the impedance values calculated by the numerical As shown in Figure 5b, from the perspective of chemical potential, Lin increased as the chemical method in CST. The impedance values from CST were also calculated from S parameters, but they potential improved, and so did the magnetic energy in the antenna. However, in Figure 5c, Cin shows a were based on the full-wave numerical algorithm. It is shown that there is a good agreement between different trend when chemical potential increased from 0.1 eV to 0.6 eV, and the electric energy stored the two methods. As shown in Figure 5b, from the perspective of chemical potential, Lin increased as at thethe antenna saw a decline. The equation resonance frequency of this in ) = ωL in , defines chemical potential improved, and so did1/(ωC the magnetic energy in the the antenna. However, in Figure circuit, is consistent with when the curves in Figure 5. increased At the resonance frequency, resistance 5c,which Cin shows a different trend chemical potential from 0.1 eV to 0.6 eV,the andinput the electric Rin dominates, and the imaginary part of impedance should be zero. Also, it should be a larger energy stored at the antenna saw a decline. The equation 1/(ωCin) = ωLin, defines the resonance Lin if therefrequency is a very of small Cin . From theis perspective of the armcurves length, Lin and showed different trends as this circuit, which consistent with in Figure 5. C Atinthe resonance frequency, input Rin going dominates, andmagnetic the imaginary part of impedance shouldwas be zero. Also, but it the well.the With theresistance arm length up, the energy stored in the antenna enriched should a larger Lin if there is a very small Cin. From the perspective of arm length, Lin and Cin showed electric one be reduced. different trends as well. With the arm length going up, the magnetic energy stored in the antenna was enriched but the electric one reduced. 4.3. Relaxation Time

TheRelaxation relaxation time of graphene is one of the most fundamental quantities, which is related to the 4.3. Time electron scattering rate and the Fermi energy level of graphene. In this part, we analyze the antenna The relaxation time of graphene is one of the most fundamental quantities, which is related to performance and equivalent circuit parameters with several typical and widely used relaxation time the electron scattering rate and the Fermi energy level of graphene. In this part, we analyze the values, from 0.1 ps [12], 0.2 [33], to 0.5 ps [24,34]. The listed valuestypical were carefully chosen antenna performance andpsequivalent circuit parameters with several and widely used to be approximately the values, same as the0.1 realistic measured experimental is noted that the relaxation time from ps [12],values 0.2 ps [33], to 0.5 psin[24,34]. The listedwork. values It were carefully larger values not considered this work thevalues graphene with in large relaxationwork. timeIttends to chosen to are be approximately theinsame as the since realistic measured experimental is noted thathigher the larger values are not considered in this result work since the graphene and withlow largeefficiency relaxationof the demonstrate impedance values, which would in mismatching time tends to demonstrate higher impedance values, which would result in mismatching and low proposed antenna in this work. efficiency of the proposed antenna in thiscoefficients work. Figure 6 demonstrates the reflection versus frequency with graphene relaxation time Figure 6 demonstrates the reflection coefficients versus frequency with graphene relaxation time varying from 0.1 ps to 0.5 ps. The chemical potential level of 0.2 eV was chosen to address the varying from 0.1 ps to 0.5 ps. The chemical potential level of 0.2 eV was chosen to address the impact impact from the variation of the relaxation time. The S-parameter curves based on other chemical from the variation of the relaxation time. The S-parameter curves based on other chemical potential potential levels are expectable and not shown here. It can be seen in Figure 6 that the antenna with levels are expectable and not shown here. It can be seen in Figure 6 that the antenna with longer longer relaxation time tended to show a sharper resonance. magnitudes the resonance curves of relaxation time tended to show a sharper resonance. The The magnitudes of theofresonance curves of relaxation time time 0.1 ps0.1 and ps 0.2 were than 10 than dB, which the impedance mismatching relaxation ps0.2 and psboth wereless both less 10 dB,indicates which indicates the impedance in these two cases. mismatching in these two cases.

Figure 6. Simulated reflectioncoefficient coefficient with with different times, 0.1 0.1 ps, 0.2 ps,0.4 ps, Figure 6. Simulated reflection differentrelaxation relaxation times, ps, ps, 0.20.3 ps,ps, 0.30.4ps, and 0.5 ps. The arm length is 5 µ m and the temperature is 300 K. and 0.5 ps. The arm length is 5 µm and the temperature is 300 K.

The equivalent circuit parameters were extracted from the simulated S-parameters, as shown in Figures 7 and 8. The coupling coefficients based on different relaxation time values slightly went up,

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The2018, equivalent circuit parameters Electronics 7, Electronics 2018, 7, xx FOR FOR PEER PEER REVIEW REVIEW

were extracted from the simulated S-parameters, as shown in 10 10 of of 15 15 Figures 7 and 8. The coupling coefficients based on different relaxation time values slightly went up, except for for the the results results obtained obtained with with the the relaxation relaxation time time of of 0.1 0.1 ps. ps. The The relaxation relaxation time time 0.1 0.1 ps ps for for the the except proposed graphene notnot large enough to drive the antenna. On the On contrary, the graphene proposed grapheneantenna antennawas was large enough to drive the antenna. the contrary, the bowtie antenna on 0.5 ps relaxation time worked as shown graphene bowtiebased antenna based on 0.5 ps relaxation timewell, worked well, in as Figure shown 7. in Figure 7.

(a)

(b)

Figure Figure 7. The calculated circuit parameter results versus chemical potential with different relaxation Figure 7. 7. The The calculated calculated circuit circuit parameter parameter results results versus versus chemical chemical potential potential with with different different relaxation relaxation times. (a) β and (b) Q 00 curves with the graphene relaxation time from 0.1 ps to 0.5 ps over different times. (a) β and (b) Q curves with the graphene relaxation time from 0.1 ps to 0.5 ps over different times. (a) β and (b) 0 curves with the graphene relaxation time from 0.1 ps to 0.5 chemical chemical potentials. potentials.

(a)

(b)

(c)

Figure Figure 8. (a) R in,,, (b) (b) and (c) Cininin of of the equivalent circuit against against 8. Equivalent Equivalent circuit circuit parameters. parameters. (a) Rinin (b) LLininin,,,and and (c) (c) C of the the equivalent equivalent circuit chemical chemical potential potential with with different different relaxation ps. relaxation times: times: 0.1 0.1 ps, ps, 0.2 0.2 ps, ps, 0.3 0.3 ps, ps, 0.4 0.4 ps, ps, and and 0.5 0.5 ps. ps.

In Figure coefficient andand the the coupled quality factorfactor curvescurves are presented for various In Figure 7,7,the thecoupling coupling coefficient coupled quality are presented for relaxation time values. At the relaxation time oftime 0.1 ps, theps, β shows a distinct trend,trend, increasing from various relaxation time values. At the relaxation of 0.1 the β shows a distinct increasing from approximately 2 to more than 9, which was larger than other curves. The quality factor at relaxation time of 0.1 ps, shows a consistent trend with the coupling coefficient. The quality factor value was much lower than the ones with other relaxation time values, which indicates a broader working bandwidth. However, since the antenna barely resonates with relaxation time of 0.1 ps, as

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approximately 2 to more than 9, which was larger than other curves. The quality factor at relaxation time of 0.1 ps, shows a consistent trend with the coupling coefficient. The quality factor value was much lower than the ones with other relaxation time values, which indicates a broader working bandwidth. However, since the antenna barely resonates with relaxation time of 0.1 ps, as shown in Figure 5, the large quality factor does not mean this antenna would behave better than others. The RLC parameters were also extracted from the equivalent circuit model, as depicted in Figure 8. The input resistances of the proposed antennas showed slight downward trends with all evaluated relaxation times. However, at the relaxation time of 0.5 ps, the graphene antennas with energy level of 0.2 and 0.3 eV demonstrated larger input resistance values (close to 10 kΩ). Moreover, the input resistance increased with relaxation time ranging from 0.1 ps to 0.5 ps. This is consistent with the curves in Figure 7; the graphene antenna with relaxation time of 0.5 ps showed better resonance performance because the input impedance of this antenna matched well with the source impedance of 10 kiloohms. All the designed points were validated by the numerical results obtained from the full-wave simulation solver. The analytical results and the numerical results agree well with each other with different chemical potential levels and relaxation time values. The curves in Figure 8b suggest that the stored magnetic energy remains with the chemical potential variation while the relaxation time has a significant impact on the magnetic energy of the graphene bowtie antenna. The electric energy is related to the Cin of the extracted circuit model. The antenna with high relaxation time shows relatively low electric energy storage. It is noted that the max value of the Cin was only around 16 µF, which is negligible compared to the Lin values. As a result, the graphene antenna shows more of an inductive nature than a capacitive one. 4.4. Substrate Thickness The substrate thickness in this paper is selected as 90 nm since this is the value that can maximize the visibility of graphene paved on the quartz substrate [25]. In this part, we also investigate the impacts from the benchmark thickness, 300 nm, and the other typical values, 200 nm and 500 nm. It should be noted that the relaxation time was set as 0.5 ps throughout this part. The frequency responses of the proposed graphene-based bowtie antenna are presented in Figure 9 with several selected substrate thickness values. Based on the 90-nm-thick substrate, the antenna displayed sharper resonance performance than those with thicker substrates. With such wild change of the substrate thickness, the antenna still worked well to resonate below −10 dB. It should be noted that the resonance points of the antennas demonstrated a shift towards the lower frequencies. As the substrate thickness increases, the fringing fields increase accordingly. This results in a longer effective electrical length of the antenna. In addition, the dispersion relationship of the SPP waves on the graphene sheet can be affected by the substrate thickness, as described in Equation (3). With the very thin substrate and the wild change (from 90 nm to 500 nm) of the substrate thickness, the SPP wavelength would become longer and then the working frequency would shift a lot to the lower frequencies. As can be observed in Figure 10a, the coupling coefficient, similarly, showed gentle trends with the rise of chemical potential of the graphene-based bowtie antenna. Also, the coupling coefficient went up from approximately 1 to 1.5 when the antenna rested on a thicker substrate with thickness up to 500 nm. In Figure 10b, the quality factor increased with chemical potential rises, while the antenna with a thicker substrate tended to demonstrate higher quality values, which would lower the bandwidth of the antenna.

As the substrate thickness increases, the fringing fields increase accordingly. This results in a longer effective electrical length of the antenna. In addition, the dispersion relationship of the SPP waves on the graphene sheet can be affected by the substrate thickness, as described in Equation (3). With the very thin substrate and the wild change (from 90 nm to 500 nm) of the substrate thickness, the SPP wavelength lower Electronics 2018, would 7, 285 become longer and then the working frequency would shift a lot to the 12 of 15 frequencies.

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As can be observed in Figure 10a, the coupling coefficient, similarly, showed gentle trends with the rise of chemical potential of the graphene-based bowtie antenna. Also, the coupling coefficient went up from approximately 1 to 1.5 when the antenna rested on a thicker substrate with thickness up to 500 nm. In Figure 10b, the quality factor increased with chemical potential rises, while the Figure 9. with different substrate values, 90which nm, 200 nm, nm, 300 antenna with a thicker reflection substrate tended to demonstrate higherthickness quality values, would lower Figure 9. Simulated Simulated reflectioncoefficient coefficient with different substrate thickness values, 90 nm, 200 nm, and 500 nm. The arm length is 5 µ m, the temperature is 300 K, and the relaxation time is 0.5 ps. the bandwidth the antenna. 300 nm, andof 500 nm. The arm length is 5 µm, the temperature is 300 K, and the relaxation time is 0.5 ps.

(a)

(b)

Figure 10. The calculated circuit parameter results versus chemical potential. (a) β β and (b) Q00 curves with different substrate thickness values, 90 nm, 200 nm, 300 nm, and 500 nm. The relaxation time is set as 0.5 ps, the arm length is 5 µµm m and the temperature is 300 K.

in the subsections, the RLC presented with the parameters Similar to tothe theresults results in previous the previous subsections, themodel RLCismodel is presented with the of interest. In Figure 11a, the input from the proposed equivalentequivalent circuit model as parameters of interest. In Figure 11a,resistance the input values resistance values from the proposed circuit well as the numerical simulation results are demonstrated with different substrate thickness values. model as well as the numerical simulation results are demonstrated with different substrate thickness The impact thefrom chemical potential potential has been well discussed the above We will focus onwill the values. The from impact the chemical has been well in discussed intexts. the above texts. We influence brought by the variousby antenna substrate thickness values in this subsection. As observed in focus on the influence brought the various antenna substrate thickness values in this subsection. Figure 11a, the in graphene-based antenna with 90-nm-thick substrate demonstrated higher input As observed Figure 11a, bowtie the graphene-based bowtie antenna with 90-nm-thick substrate impedance than otherinput antennas, reaching 10 kΩ. Asantennas, a result, the higher10 input better demonstrated higher impedance than other reaching kΩ.impedance As a result,led thetohigher matching and sharper input impedance led toresonance. better matching and sharper resonance. Figure11a,b, 11a,b,the theLinLin increased with both chemical potential substrate thickness rising. In Figure increased with both chemical potential andand substrate thickness rising. The The electric capacity of the antenna at the resonance point wasatata avery verylow low level, level, only only several several electric capacity of the antenna at the resonance point was microfarads. The Thesubstrate substratewould wouldtrap trap the energy radiated from antenna, which would lower microfarads. the energy radiated from thethe antenna, which would lower the the antenna efficiency. In this case, with 500-nm-thicksubstrate, substrate,the theantenna antennadisplayed displayed aa more antenna efficiency. In this case, with thethe 500-nm-thick inductive manner manner than than the the one with a thinner substrate. The The stored magnetic magnetic energy dominated dominated the inductive consistent with the the previous previous analysis. With With a thicker thicker substrate, substrate, the the antenna antenna showed showed antenna, which is consistent higher capacity, as seen in Figure 11c. The substrate substrate thickness thickness should should be be carefully carefully chosen chosen to to make make higher sure that the antenna antenna resonates resonates well well at at the the desired desired frequency frequency band. band. For For aa certain certain substrate substratethickness, thickness, dimensions, so the antenna could have a better impedance we should carefully adjust the antenna dimensions, values. matching with the source, which leads to relatively low S11 11 values.

inductive manner than the one with a thinner substrate. The stored magnetic energy dominated the antenna, which is consistent with the previous analysis. With a thicker substrate, the antenna showed higher capacity, as seen in Figure 11c. The substrate thickness should be carefully chosen to make sure that the antenna resonates well at the desired frequency band. For a certain substrate thickness, Electronics 2018, 7, 285 13 of 15 we should carefully adjust the antenna dimensions, so the antenna could have a better impedance matching with the source, which leads to relatively low S11 values.

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(a)

(c)

Figure Equivalentcircuit circuit model model parameters potentials. (a) (a) Rin,R(b) in, and Figure 11.11.Equivalent parametersover overchemical chemical potentials. (b) Lin , (c) andCin(c)ofCin in , L against chemical potential withwith different substrate thickness values,values, 90 nm,90 200nm, of the theequivalent equivalentcircuit circuit against chemical potential different substrate thickness nm, 300 nm, and 500 nm. The relaxation time is 0.5 ps, the temperature is 300 K, and the arm length 200 nm, 300 nm, and 500 nm. The relaxation time is 0.5 ps, the temperature is 300 K, and the arm length 5 µ m. is 5is µm.

5. 5. Conclusions Conclusions summary,an an equivalent equivalent resonance model with R, L,R, and connected in series beenhas InInsummary, resonancecircuit circuit model with L, Cand C connected inhas series proposed to describe the the graphene-based THzTHz bowtie antenna for for thethe insight of of thethe resonance been proposed to describe graphene-based bowtie antenna insight resonance behaviorofofthe thegraphene grapheneantenna. antenna. Several Several significant significant parameters, thethe antenna behavior parameters,chemical chemicalpotential, potential, antenna arm length, relaxation time, and substrate thickness, are considered in the modeling and analysis. By By arm length, relaxation time, and substrate thickness, are considered in the modeling and analysis. using the equivalentcircuit circuitmodeling, modeling, it it has has been asas thethe chemical using the equivalent been shown shownthat thatthe theRRvalues valuesdecrease decrease chemical potential and the arm length increase. Also, C values show a similar trend with tuning the energy potential and the arm length increase. Also, C values show a similar trend with tuning the energy level. Antennadimension dimensionalso also shows shows influence influence on L and C values level. Antenna on the theextracted extractedcircuit circuitparameters. parameters. L and C values remain a balance to keep a very small value of the imaginary part of input impedance. This indicates remain a balance to keep a very small value of the imaginary part of input impedance. This indicates physically that the high energy level tends to reduce the loss resistance and the increasing arm length physically that the high energy level tends to reduce the loss resistance and the increasing arm tends to drain the magnetic energy stored in this set of antennas. Electric energy varies with magnetic length tends to drain the magnetic energy stored in this set of antennas. Electric energy varies with energy to keep a balance. The rise of relaxation time leads to higher input resistance, which would magnetic energy to keep a balance. The rise of relaxation time leads to higher input resistance, influence the resonance performance. The substrate thickness affects both the resonance frequency which would influence the resonance performance. The substrate thickness affects both the resonance and magnitude. The graphene-based bowtie antenna shows more an inductive property than a frequency magnitude. Themodel graphene-based antenna shows inductive property capacitiveand one. The proposed is validated bowtie by the numerical results.more This an work sheds light on than capacitive one.bowtie The proposed modeland is validated by the results. This sheds theagraphene-based antenna design paves the way fornumerical further investigation andwork potential light on the graphene-based bowtie antenna design and paves the way for further investigation and applications. potential applications. Author Contributions: B.Z. performed the simulation and wrote the paper. J.Z. and Z.P.W. proposed the idea Author Contributions: B.Z. performed the simulation and wroteand the paper paper.revision. J.Z. and Z.P.W. proposed the idea and C.L., J.Z., D.H., and Z.P.W. helped with the results discussion and C.L., J.Z., D.H. and Z.P.W. helped with the results discussion and paper revision. Funding: This research was funded by the Fundamental Research Funds for the Central Universities, grant Funding: This research was funded by the Fundamental Research Funds for the Central Universities, grant number WUT: 2017IB015. number WUT: 2017IB015. Conflicts of Interest: The authors declare no conflict of interest. Conflicts of Interest: The authors declare no conflict of interest.

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