1 Introduction

Terahertz (THz) technology requires miniaturization of the communication devices to enhance the speed of communication [1]. One of the key components used in the communication system is antenna. The direct scaling down of these antennas leads to several problems such as reduced mobility [2], losses at THz frequency. At this high frequency, oscillations are due to Surface Plasmon Polaritons (SPP). The plasmonic/nano/optical antennas support transmission of the SPP oscillations to enable THz communication. The nano-antennas are preferably designed using noble metals such as gold and silver. These metallic nanostructures resonate at very high THz frequencies with high losses. Another problem is these structures are associated with difficult tuning mechanism for resonance frequency control [3]. To mitigate these limitations recently graphene has gained attention due to its desirable optical properties.

Graphene is first invented in 2004 by Novoselov from Graphite [1, 4]. Graphene has very good optical properties makes it suitable for plasmonic applications. One of them is its tunable optical conductivity. The surface conductivity of the graphene is finite and directly proportional to its chemical potential [5]. Mathematically graphene conductivity is modeled using Kubo formalism [4, 6] which includes inter and intra band conductivities of the graphene at low THz band. The negative imaginary part of the conductivity influences the negative real permittivity of the graphene. Hence graphene supports plasmonic oscillations at THz frequency. Initially Jornet et al. demonstrated that graphene nano patches can support SPPand resonate at THz frequency [5]. The resonance frequency of the graphene is mainly depending on its conductivity. The prominent advantage of the Graphene is its tunable conductivity which makes graphene popular in the plasmonic applications [7]. Due to this property, graphene is also used in gold and silver nano-antennas for the resonance tuning. Mehta et al. demonstrated the control of resonance frequency of gold nano dipole antenna by placing a graphene sheet on top of the structure [8]. The graphene-metal hybrid structures are gaining popularity since implementation of the graphene antenna is not so easy because of its very small thickness. Yu Yao et al. designed and analysed the graphene-metal structure for resonance frequency tuning [3]. The resonance of graphene nano-antenna falls in Infra Red (IR) and visible frequency region [9]. The analysis on the graphene antennas has been performed using numerical and mathematical techniques by various researchers/scientists and it is observed that most of the researches are on GHz frequency range.

In this paper graphene nano patch antenna is proposed and designed on Silicon Dioxide substrate. The conductivity model of the graphene is analyzed mathematically and numerically. The complex conductivity of the graphene is measured at different values of the chemical potential and dispersive characteristic of the graphene is analyzed using Drude dispersive model. The tuning mechanism for the resonance frequency control is demonstrated using graphene’s chemical potential. The variation of the chemical potential makes the graphene nano patch resonating at multiple frequencies and also changes half power beam width of antenna. The mechanism of enhancing the gain of the graphene nano patch antenna is demonstrated with the help of L-shaped nano patch. The proposed graphene nano patch antenna is resonating at 30 THz frequency and it is applicable for THz communication. The design of graphene nano patch antenna, complex conductivity model and results are discussed in the next sections.

2 Graphene nano patch antenna design

The Fig. 1a, b shows 2D and 3D view of the graphene nano patch antenna structure. The designed patch antenna has dimension W = 900 nm, L = 900 nm and thickness of Tp = 3 nm and is placed on 1800 × 1800 × 36 nm Silicon Dioxide (Ghosh)(optical) substrate. This substrate has various properties such as dispersive characteristic density ρ = 2270 kg/m3, thermal conductivity σt = 0.32 W/K/m, heat capacity hc = 1 kJ/K/kg and diffusivity ν = 1.40969 × 10–7 m2/s. The graphene material is modeled using surface conductivity characteristics and is explained in section III. The graphene parameters are defined as thickness TG = 10 nm, chemical potential μc = 0.1 eV and temperature T = 300° K. The Fig. 1c shows the side view of the proposed graphene based patch antenna it has a ground plane of dimension 1800 × 1800 × 3 nm and the proposed nano-antenna is excited by a waveguide port of 50 Ω.

Fig. 1
figure 1

Graphene nano patch antenna a 2D view b 3D view c side view

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3 Complex conductivity model of graphene

The frequency dependent complex conductivity of the graphene can be tuned using chemical potential, temperature and scattering rate (relaxation time) [7, 11]. The conductivity of graphene is given as,

$$ \sigma \left( \omega \right) = \, \sigma ^{\prime}\left( \omega \right) \, + {\text{ j }}\sigma ^{\prime\prime}\left( \omega \right) $$
(1)

σ'(ω) and σ''(ω) in Eq. (1) are real and imaginary parts of conductivity respectively. At the THz frequency complex conductivity of the graphene can be modeled using Kubo formula [12,13,14] which includes both inter and intra band transitions.

$$\sigma \left(\omega ,{\mu }_{c}, \Gamma , T\right)={\sigma }_{intraband}\left(\omega ,{\mu }_{c}, \Gamma , T\right)+{\sigma }_{interband}\left(\omega ,{\mu }_{c}, \Gamma , T\right)$$
(2)
$${\sigma }_{intraband}\left(\omega ,{\mu }_{c}, \Gamma , T\right)=\frac{-i{e}^{2}{k}_{B}T}{\pi {\hbar }^{2}\left(\omega -i2\Gamma \right)}\left( \frac{{\mu }_{c}}{{k}_{B}T}+2\mathit{ln}\left({e}^{\frac{{\mu }_{c}}{{k}_{B}T}}+1\right)\right)$$
(3)

The inter band conductivity is not significant up to the frequency ω = 2\({\mu }_{c}\)/ħ, however it can be neglected for ω < 2\({\mu }_{c}\)/ħ. It can be approximated as a logarithmic function for low temperature values in the range \({k}_{B}T\) << \(\left|{\mu }_{c}\right|,\) ħ \(\omega \) [10].

$${\sigma }_{interband}\left(\omega ,{\mu }_{c}, \Gamma , 0\right)=\frac{-i{e}^{2}}{4\pi {\hbar }^{2}}ln\left( \frac{2\left|{\mu }_{c}\right|-(\omega -i2\Gamma )\hbar }{2\left|{\mu }_{c}\right|+(\omega -i2\Gamma ){\hbar }}\right)$$
(4)

In the above model [10], \({k}_{B}\) is Boltzman constant, ħ is reduced Planks constant given by ħ = h/2π, \({\mu }_{c}\) is the chemical potential, \({\omega }\) is the operating frequency, \(\Gamma \) is the scattering rate which also equal to \({{\tau }^{-1}}\) is the relaxation time, T is the temperature. At lower THz frequency intra band transitions are significant over inter band transitions [7]. The inter band conductivity is significant at near infra-red and visible region [4]. The variation of chemical potential directly increases intra band conductivity of the graphene material [3] and changes the resonance characteristics of the graphene nano-antenna.

Traditionally, the chemical potential tuning of the graphene is achieved by using electrostatic gate (bias) voltage or with chemical doping. The variation of the bias voltage results in change in the conductivity of graphene. In this manuscript, graphene patch antenna is designed on a substrate whose gate voltage (bias voltage) which is connected in between the substrate and graphene sheet can modify the chemical potential and thus changing conductivity.

The mathematical analysis of Kubo model as given in Eqs. (2) and (3) is performed to investigate the conductivity of the graphene. The real and imaginary parts of the conductivities are shown in Fig. 2a, b respectively. The analysis is executed for the different values of the chemical potential of the graphene ranging from 0.1 eV to 1.3 eV. The conductivity plots reveal that as the chemical potential is increased, carrier concentration increases. This affects the resonance behavior of the nanostructure. It is observed from Fig. 2a that the maximum conductivity of 2.436 × 10–15 S and minimum value of 1.894 × 10–16 S occurs at \({\mu }_{c}\) = 1.3 eV and 0.1 eV respectively. It is observed from Fig. 2b that imaginary part of the conductivity is negative and hence graphene nano-antenna can support plasmonic oscillation at the THz frequency. The quality of plasmonic oscillation in the graphene increases as the chemical potential of the graphene increases. Further it is also noticed that the graphene shows stable characteristics after 2 THz frequency.

Fig. 2
figure 2

Conductivity of the graphene using Kubo model a real part of conductivity b imaginary part of conductivity

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The permittivity of the graphene material is evaluated using Drude dispersive model [15] given in Eq. (5).

$$\upepsilon \left(\upomega \right)={\upepsilon }_{\infty }-\frac{{{\upomega }_{\mathrm{p}}}^{2}}{\upomega (\upomega -{\mathrm{jv}}_{\mathrm{c }})}$$
(5)

where \({\epsilon }_{\infty }\) is permittivity at \(\omega =\infty \), \({v}_{c}\) is collision frequency and ωp is plasma frequency. The mathematical analysis of Drude model is shown in Fig. 3. It is observed from Fig. 3a that the real part of graphene permittivity is negative and this is the desired criteria for the propagation of the SPP in the nano materials. It is observed from Fig. 3b that real part of the graphene permittivity is more negative at lower THz frequency and negativity decreases as the frequency increases. The high value of the imaginary permittivity indicates high losses occurring in the graphene material so Fig. 3b reveals that high loses are occurring till 2 THz frequency and low losses are occurring after this point in the graphene nano-antenna.

Fig. 3
figure 3

Permittivity of the graphene using Drude model a real part of permittivity b Imaginary part of permittivity

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4 Results and discussions

The simulation results of the graphene nano Patch antenna and the approach to increase the gain of the designed antenna is demonstrated in this section. The special property of graphene is its tunable conductivity and this can be used to vary the resonating characteristics of the proposed graphene nano patch antenna. The designed structure is analyzed on the CST Microwave Studio platform using time domain solver with Finite Integration Technique (FIT). This solver enhances the speed of execution as compared to frequency domain solvers.

The Fig. 4 depicts the variation of reflection coefficient with the frequency of proposed graphene nano patch antenna. the graphene is modeled with chemical potential of 0.1 eV, relaxation time of 0.1 psec at the temperature 300° K. The proposed nano patch antenna resonates at 30 THz with—28.38 dB and shows a good S11 characteristic which is suitable for the THz optical communication. It has a frequency band of bandwidth 16.83 THz and percentage bandwidth of 75.38%. Further it is observed that few other frequency bands are near − 20 dB but they are not supporting good quality plasmonic oscillations according optical reflection coefficient threshold. This characteristic of proposed antenna may be because of molecular structure of the graphene.

Fig. 4
figure 4

S11 parameters of graphene nano patch antenna

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The proposed graphene nano patch antenna is fed by a waveguide port through the strip line. The graphene nano-antenna is excited by a 60 femto sec Gaussian pulse with the peak amplitude of 1 V as shown in Fig. 5. The transient response of the designed graphene nano patch antenna is as shown in Fig. 5. The peak output amplitude of the antenna is observed as 0.15 V. The transient response of the graphene nano patch antenna is good enough for THz communication.

Fig. 5
figure 5

Transient response of the Graphene nano patch antenna

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The Fig. 6a, b shows the E and H field distribution of the graphene nano patch antenna respectively. It is observed that the maximum electric field is 3.49 × 109 V/m and the maximum magnetic field is 3.09 × 108 A/m at the resonating frequency of 30 THz. It is observed that there is a large amplification in the field strength of the graphene patch nano-antenna as compared to the radio frequency antenna. This is the prominent difference between radio frequency antenna and nano-antenna. Since there is a large enhancement in the field distribution nano-antennas are also called as “hotspots”.

Fig. 6
figure 6

Field distribution a E field b H field

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The polar plots of the E plane (sweep Theta and Phi = 0) and H plane (sweep Phi and Theta = 90°) patterns of the graphene nano patch antenna is given in Fig. 7a, b respectively. It is observed that the designed nano patch antenna has a beam width of 90.4° for both E and H plane and gain of 3.52 dB at 30 THz and shown in Fig. 8. The radiation pattern of the graphene nano patch antenna has same amount of back lobe as that of main lobe therefore it is radiating in both the directions. The proposed graphene nano patch antenna is compared with the graphene nano-antennas in the literature and it is tabulated in Table 1.

Fig. 7
figure 7

Radiation pattern a E plane b H plane

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Fig. 8
figure 8

Radiation pattern of the graphene nano patch

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Table 1 Comparison of graphene nano-antennas with proposed nano patch antenna
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It is observed that on varying the chemical potential of the graphene, patch antenna will resonate at multiple frequencies. This is because of the hexagonal structure of graphene material. The Fig. 9 shows reflection coefficient of the graphene nano patch at μc = 1.3 eV. At this value of chemical potential graphene nano patch is resonating at multiple frequencies such as 30 THz, 115 THz and 176 THz with S11 characteristics − 30.05 dB, − 20.8 dB and − 22.4 dB respectively. All the three frequencies exhibit good resonance characteristics (S11 < − 20 dB) and hence suitable for THz optical applications. The three frequency bands in S11 characteristics have bandwidths of 2.48 THz, 5 THz, 28.7 THz and percentage bandwidths of 8.28%, 4.428%, 17.40% respectively.

Fig. 9
figure 9

S11 parameters of the graphene nano patch at μc = 1.3 eV

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The polar plots of the E and H plane at 115 THz and 176 THz are shown in Fig. 10a, b respectively. It is observed that E plane does not contain any side lobes whereas H plane contains side lobe. The half power beam width for the E and H plane are 88.2° and 94.8° respectively at the frequency of 115 THz. At 176 THz half power beam width observed is 92.4° and 97.3° for E and H plane respectively. It is observed that beam width of H field wider than the E plane. It is observed that as the resonance frequency increases half power beam width also increases as shown in Table 2.

Fig. 10
figure 10

E and H plane at μc = 1.3 eV a 115 THz b 176 THz

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Table 2 Half power beam width of graphene nano patch antenna
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4.1 L-shaped graphene nano patch antenna

The Fig. 11a, b depicts the 3D view and side view of L-shaped graphene nano patch antenna respectively. In this section the effect of changing the shape of the designed graphene nano patch is demonstrated. The S11 characteristics of the L-shaped graphene nano patch is shown in Fig. 12. This new patch antenna is resonating at 30 THz frequency with reflection coefficient of − 44.55 dB, having bandwidth of 3 THz and percentage bandwidth of 10.08%. The S11 characteristic is having improved characteristics as compared with the square graphene patch antenna.

Fig. 11
figure 11

a 3D view b side view of L-shaped Grahene nano patch antenna

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Fig. 12
figure 12

S11 parameters of L-shaped graphene nano Patch

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The antenna design shown in Fig: 13, contains an L-shaped patch whose aim is to increase the gain of the antenna. This L-shaped graphene structure helps in transitions of electrons from lower to upper levels. Further the graphene patch, substrate and ground graphene sheet acting like layered structure which helps in the producing optical waves. The overall structure increases the intensity of the wave which leads to increase in the gain of the optical/nano-antenna.

The Fig. 13a, b shows the radiation patterns for E and H plane of L-shaped graphene nano patch antenna respectively. At the resonating frequency of 30 THz it is observed that beam width of E plane is 89.9° and H plane is 55.1°. It is observed from the radiation pattern beam width of the E and H plane is reduced as compared with the square shaped graphene patch antenna. Further it is also observed that the new structure shows enhanced antenna gain. The H plane shows multiple side lobes in addition to the main lobe.

Fig. 13
figure 13

Polar plot of L-shaped graphene nano patch a E plane b H plane

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The Fig. 14 depicts the radiation pattern of the L-shaped graphene nano patch antenna which is resonating at 30 THz frequency. It is observed that gain of the proposed antenna is 9.512 dB. The gain of graphene antenna is enhanced from 3.52 to 9.512 dB with L-shaped structure. It is observed that there is approximately three times enhancement in the antenna gain as compared with the square shaped structure.

Fig. 14
figure 14

Radiation pattern of L shaped graphene nano-antenna

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5 Conclusion

The analysis and mathematical modeling of graphene nano patch antenna is discussed using Kubo and Drude dispersive models. Drude dispersive model exhibits the negative real permittivity of the graphene which is essential for plasmonic oscillations. It is noticed that resonance frequency of the graphene can be easily tuned using its chemical potential. Further tuning of chemical potential of the graphene affects the half power beam width of antenna. It is observed that for higher value of chemical potential the graphene is behaving like a multiband plasmonic antenna. It was noticed that on changing the patch shape from square to L-shape, there is increase in gain from 3.52 to 9.512 dB. It is concluded that graphene nano patch shows good optical antenna characteristics for the THz applications.