Introduction

THz radiation has the advantages of numerous range of applications in the field of recent surveillance communication systems due to non-ionizing properties as well as large transmission capability [1]. In the electromagnetic (EM) spectrum with THz, frequency is placed between the microwave (mm) and infrared (IR) bands that possesses a particular property of the microwave and light waves. Hence, the terahertz (EM) waves are allowed to begin from both the electronic and photonic device [2, 3]. With the quick growth of terahertz detectors and sources, THz band has gained more attention in the field of wireless communication systems [4]. An important component of communication technology is the antenna that is used to transmit and receive information in the form of electromagnetic (EM) wave. The antennas used in the THz region should be capable of providing high gain, wide bandwidth (BW), compact, and low cost. This leads to make the microstrip patch antennas as a suitable choice for these applications [5, 6].

Additionally, the terahertz (THz) technology needs small size of the communication and electronic devices with enhancing the data speed. The nano-antennas are advance designed using good conductor materials like copper, silver, and gold, but these conductors in nano-antenna operate at THz frequency with very high energy losses and are not easy to tuning it for resonance frequency (fr) control [7]. To reduce these limitations, researchers are drawn towards the use of novel materials such as graphene as an alternative to aforementioned metals [8, 9].

Graphene is one layer of carbon atoms and formed in a honeycomb lattice that provides good optical characteristics that lead it to be suitable for plasmonic applications [7, 10]. Beyond that, graphene is currently popular with the interest of the research field because of its new electrical, mechanical, chemical, optical, and thermal properties [11,12,13]. With respect to its special properties, graphene offers wide possible uses in a variety of fields, such as transistors with ultra-high speed, antennas [14, 15], transparent solar cells [16], waveguides [17], demultiplexers [18, 19], resonators [20], absorbers [21, 22], filters [23], and modulator [24, 25]. Currently, several graphene-based antennas have been recommended involving the leaky-wave [26], Yagi-Uda [14, 27], reconfigurable [28,29,30], and reflect array [31] antennas that display widespread capabilities in the THz band [32, 33].

When graphene material is integrated into antenna systems, it displays improvement in the radiation performance compared to the typical conventional antennas at very high frequencies. This is because graphene has a good conductivity and its chemical potential (μc) can be changed via an external electrostatic direct current (DC) or by doping during manufacturing process [34]. Besides, the relationship between momentum and energy of electrons in graphene is linear rather than quadratic through a variety of energies and this allows extremely high carrier mobility to be obtained at room temperature [35]. These graphene extraordinary properties allow it to be very interesting to promote surface plasma polariton (SPP) mode, especially in the THz frequency range, as its resonance plasma frequency is exactly located in this band [35]. On the other hand, the graphene plasmonic behavior can be employed as a radiating patch or an artificial magnetic conductor (AMC) in miniaturization process for the design of antennas [36].

In this investigation, a novel plasmonic nano-antenna based on graphene is proposed and designed on the silicon nitride (SiO3N4) dielectric substrate using coupled gap feeding techniques operating at 0 to 5 THz. In the first step, the graphene surface conductivity (\({\sigma }_{\mathrm{g}}\)) is calculated in the range frequency of 1 to 10 THz with various chemical potential (μc) values. After that, the performance of the simulated antenna is computed under the impact change of graphene chemical potential. Finally, the chemical potential values, which maintain the dual and tri-band frequency operation of the various vibration modes, are specified. Also, their impacts on the overall suggested antenna parameters are determined.

The continuing section of the present work is arranged as follows: In the “Material and Simulation Technique” section, the numerical model for computing the electric conductivity of graphene and geometry of the proposed plasmonic antenna are provided in detail. The results and dissection of the proposed antenna parameters at various operational modes are described in the “Results and Discussions” section. The main conclusions are summarized in the “Conclusion” section.

Material and Simulation Technique

The graphene-based plasmonic antenna is designed and simulated using CST software techniques. In this article, a graphene radiation patch is built up on a dielectric substrate whose bias voltage (Vg) is connected between a single layer of graphene and antenna substrate that enhances the chemical potential (μc) and this in turn leads to varying graphene conductivity \(({\sigma }_{\mathrm{g}})\).

Graphene Properties

In this section, the variation of graphene electric conductivity as a function of μc is studied. As indicated by [37], the graphene conductivity is complex and consists of intra-band (σintra) and inter-band (σinter) parts. Furthermore, graphene conductivity (\({\sigma }_{\mathrm{g}}\)) is dependent on the frequency which is described by Kubo’s formula and it is approximately expressed as [38]

$${\sigma }_{\mathrm{g}}\left(\omega ,{\mu }_{c}, s, T\right)= {\sigma }_{\mathrm{intra}}\left(\omega ,{\mu }_{c}, s,T\right)+{\sigma }_{\mathrm{inter}}(\omega ,{\mu }_{c}, s,T)$$
(1)

The conductivity of graphene due to intra-band (σintra) and inter-band (σinter) can be expressed, respectively, as given by [39, 40]

$$\sigma_{\mathrm{intra}}\left(\omega,\mu_c,s,T\right)=-j\frac{e^2k_BT}{\pi\hslash^2\left(\omega-2j{\Gamma}\right)}\left[\frac{\mu_c}{k_BT}2\ln\left(e^\frac{\mu_c}{k_BT}+1)\right)\right]$$
(2)
$${\sigma }_{\mathrm{inter}}\left(\omega ,{\mu }_{c}, s, T\right)= -j\frac{{e}^{2}}{4\pi \mathrm{\hbar }}\mathrm{ln}\left(\frac{{2|\mu }_{c}|-(\omega -2j{\Gamma})\mathrm{\hbar }}{{2|\mu }_{c}|+(\omega -2j{\Gamma})\mathrm{\hbar }}\right)$$
(3)

where ω is the angular frequency, T is the temperature, e is the charge of the electron, μc is the chemical potential, kB is Boltzmann’s constant, ℏ is reduced Planck’s constant, and г is a scattering rate. However, in the range of THz frequencies, the term of inter-band has little impact on the total surface conductivity of graphene (σg), and hence, the intra-band conductivity term will control the value of total surface conductivity (σg) [41, 42].

The numerical expression of Kubo formula, as mentioned in Eqs. (2) and (3), is handled to analyze the behavior conductivity of graphene in the terahertz (THz) band frequencies using various chemical potential values. The obtained results of the real component and imaginary component of the graphene surface conductivities are displayed in Fig. 1a and b, respectively. This figure displays that the conductivity of graphene is related to the chemical potential (μc) and it is obviously seen that σreal ≥ 0 while σimag ≤ 0 in the range frequency of 1 to 10 THz. It is also observed that the values of the imaginary component are greater in comparison to the values of the real component. Besides, in the frequency ranges 1–4 THz band, this figure indicates that the graphene conductivity is high compared to the other considered frequency ranges and it might be regarded as a good choice for some antenna design considerations.

Fig. 1
figure 1

Variation surface conductivity of graphene dependent on frequency with various values of μc using the Kubo model a real and b imaginary terms

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Therefore, the graphene conductivities can be controlled through the applied μc. Since graphene conductivity values are high relative to high values (μc) due to having a greater carrier density and this makes the graphene patch material to support (SPPs), which will be used to confine the incident EM wave. Moreover, the graphene material properties in terms of thermal, electrical, optical, and mechanical characteristics are displayed in Table 1.

Table 1 Thermal, electrical, optical, and mechanical characteristics of graphene material [43]
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Plasmonic Antenna Geometry

The microstrip antenna in simple form consists of dielectric substrate confinement between two parallel conductors [44]. In this work, a rectangular graphene radiation patch is placed at the upper part of the dielectric substrate of silicon nitride (SiO3N4). The opposed side of the antenna substrate is made up of graphene with the same dimensions of the substrate which is used like antenna ground plane as illustrated in Fig. 2. The thickness of the ground plane and the patch is considered to be the same and equal to 0.345 nm.

Fig. 2
figure 2

Top and cross-sectional view of the graphene-based plasmonic microstrip antenna

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As the bias voltage (Vg) is put into the graphene patch, the electrons inside the patch are oscillating around their equilibrium position at THz range frequencies and this phenomenon is called SPP [45, 46]. The dispersion relation of the SPP wave vector (kSPP) or the transverse magnetic (TM) mode of the graphene sheet is evaluated in free space using the relation given by [47] as

$${k}_{\mathrm{SPP}}={k}_{\circ }\sqrt{1-{\left(\frac{2}{{\eta }_{\mathrm{eff}} {\sigma }_{g}}\right)}^{2}}\approx \frac{\hslash {\omega }^{2}}{2\alpha {\mu }_{c} c}$$
(4)
$${\eta }_{\mathrm{eff}}=\sqrt{1-\frac{4{\mu }_{\circ }}{{\varepsilon }_{\circ } {\left({\sigma }_{g}\right)}^{2}}}\approx \frac{\lambda }{{\lambda }_{\mathrm{SPP}}}$$
(5)

where λSPP, λ0, \(\alpha\), k0, and \({\eta }_{\mathrm{eff}}\) are, respectively, operational surface plasmonic polariton wavelength, free space wavelength, attenuation constant, wave vector, and effective intrinsic impedance of graphene material. Besides, the width (Wp) and length (Lp) of the simulated plasmonic antenna are determined through the formula expressions presented by [48] as

$${W}_{\mathrm{p}}=\frac{(2M+1)}{\sqrt{{\varepsilon }_{r}}}\times \frac{{\lambda }_{\circ }}{2}$$
(6)
$${L}_{\mathrm{p}}=\frac{(2N+1)}{\sqrt{{\upvarepsilon }_{\mathrm{eff}}}}\times \left(\frac{{\lambda }_{\mathrm{SPP}}}{2}\right)-2\Delta L$$
(7)

where M and N are positive integer numbers; εr and εeff are relative and effective permittivity of the dielectric material, respectively; \(\Delta L\) is the radiation patch length expansion as a result of the creation of fringing fields. Moreover, the width (Wf) of the microstrip line feed is determined from the mathematical relation of the input antenna impedance (Za) given by [48] as

$${W}_{\mathrm{f}}=\frac{11.96 {\lambda }_{\circ }}{{Z}_{a}}$$
(8)

It is worth mentioning that Za should be matched to characteristic impedance with the line feeding (Lf). Hence, the antenna ground and substrate dimensions are determined by using the expression given by [48] as

$${L}_{\mathrm{s}}={L}_{\mathrm{g}}={L}_{\mathrm{p}}+2{L}_{\mathrm{f}}$$
(9)
$${W}_{\mathrm{s}}={W}_{\mathrm{g}}={W}_{\mathrm{p}}+2{L}_{\mathrm{f}}$$
(10)

where Lg, Wg, Ls, and Ws are the length and width of antenna ground plane substrate material, respectively. According to the above equations, the dimensions of the proposed antennas computed are summarized in Table 2.

Table 2 Optimized plasmonic antenna dimensions (units in μm)
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At the same time, the selection of appropriate chemical potential (μc) values for graphene (μc) may significantly affect the impedance characteristics of the antenna and, therefore, the antenna resonating frequency. Furthermore, the chemical potential (μc) is able to provide variations in the charge carrier density in the radiation patch which in turn impacts the carrier mobility numbers and provides the graphene tunable. The specific details of the considered dielectric substrates with the implemented simulation and design procedure techniques are presented in Table 3.

Table 3 Simulation detailed information of recommended antenna
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With the use of the aforementioned specifications of the dielectric substrate and graphene patch dimensions, as well as their physical characteristics, the radiation performance of the plasmonic antennas is determined and discussed in the next section.

Results and Discussions

The radiation performances of the multiband plasmonic graphene patch antenna are investigated using different chemical potential (μc) by implementing CST at the operating frequency ranges from 0 to 5 THz. The characteristic of the multiband proposed antenna is analyzed in regard to reflection loss (S11), voltage standing wave ratio (VSWR), gain, efficiency, bandwidth (BW), fractional bandwidth (FBW), and far-field radiation pattern, as well as their E and H plane patterns.

Dual-Band Operation

The essential antenna characteristic is S11, and it could be lower than − 10 dB for suitable antenna impedance matching. Figures 3 and 4 display the proper operating frequency where S11 and VSWR parameters have minimum values with regard to the antenna bandwidth (BW). The change in the operating frequency (fr) dependent on the μc is immediately apparent from the S11 and VSWR parameter curve.

Fig. 3
figure 3

Changing of S11 versus frequency of the plasmonic antenna with distinct μc

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Fig. 4
figure 4

Changing of VSWR versus frequency of the plasmonic antenna with distinct μc

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This dependence indicates that the plasmonic antenna can be tuned within a frequency range, which shifts from 0.795 to 0.855 THz in the first excited vibration mode (TM10), while shifting from 3.605 to 3.790 THz in the second excited vibration mode (TM20) and from 4.647 to 4.670 THz in the third excited vibration mode at TM11. The plasmonic nano-antenna is simulated in the μc value which varies from 0.0 to 1.0 eV, which can be achieved simply through an external DC voltage source. Besides, for every chemical potential, the S11, VSWR, operating frequency, gain, efficiency, bandwidth (BW), and fractional bandwidth (FBW) of the proposed antenna are measured and the results are demonstrated in Table 4. It is clearly seen from this table that the minimum values of S11 and VSWR are − 42.374 dB and 1.015, respectively, which are recorded at the μc value of 0.5 eV and at the resonance frequency of 0.855.

Table 4 Plasmonic graphene-based patch antenna parameters for various chemical potential values
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Moreover, the antennas with both μc of 0.0 and 0.1 eV are resonating at dual-band frequencies, which are 0.795 and 0.820 THz at the first excited mode (TM10) and 3.605 and 3.705 THz for the second excited mode (TM20) with S11 values of − 41.484 and − 39.559 and − 20.683 and − 31.018 dB, respectively. Besides, the implementation of other studied chemical potential values makes the proposed plasmonic antenna to operate at triple-band frequencies with acceptable values of S11 and VSWR. Furthermore, the wider fractional antenna bandwidth of 32.40% is also observed with the μc value of 0.5 eV for the first excited vibration modes (TM10). In addition, this table also displays that the second excited vibration modes (TM20) provide a higher antenna gain value of 7.69 dB with the use of all considered chemical potential values. On the other side, the 2D view electric (E) and magnetic(H) planes of far-field radiation pattern for the first (TM10), second (TM20), and third excitation mode (TM11) of the plasmonic antenna for each chemical potentials are also measured and displayed in Fig. 5.

Fig. 5
figure 5

2D representation of the recommended plasmonic graphene-based antenna radiation pattern for E plane a TM10, b TM20, and c TM11 and for H plane d TM10, e TM20, and f TM11 modes

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These figures demonstrate a directional E plane while a bidirectional H plane for the first excited vibration mode. However, in the second resonant modes, both E and H planes show a semi-omnidirectional pattern with some back and side lobes. Besides, the radiation figures for the third resonant frequencies display generally an approximate bidirectional E plane and H plane.

Change of μc for Tri-band Operation

In this section, the influence of μc on the antenna parameter printed on SiO3N4 as a dielectric substrate is investigated, which is widely employed in solar cell technologies. In addition, it can be supported multiband resonant that develop it suitable for designing new smart antennas. As explained by the results presented in Table 4, it is clear to see that the recommended antenna operates at a dual-band frequency at 0.795 and 3.605 THz and 0.820 and 3.705 THz, respectively, for the μc values of 0.0 and 0.1 eV. However, with an increase in μc from 0.3 to 1.0 eV, a third mode of vibration appears at frequencies 4.647, 4.650, 4.665, 4.605, and 4.670 THz, respectively. The excitation of the third vibration modes can be attributed to the fact that as the chemical potential increases fringe line fields will be produced along the edge of the antenna patch. Consequently, the stored energy in the plasmonic nano-antenna resonator reduces. Therefore, the antenna quality factor (Qc) decreases, and thus, the antenna bandwidth (BW) of the proposed plasmonic antenna is changed.

As a result, the fundamental vibration mode is transvers magnetic mode (TM10) [49]. However, when the chemical potential becomes higher, sufficient carrier charge density is formed within the antenna patch and this causes the antenna to operate with a high-order vibration mode, for example, TM10, TM20, and TM30, to propagate due to the creation of SPP at THz frequencies. The above modes will produce resonance equivalent to TM10 but in different frequency bands due to a different phase angle. Therefore, with the appropriate chemical potential values, the plasmonic patch antenna can be adjusted to resonate at tri-band frequencies.

Therefore, the total antenna radiation parameters that are summarized in Table 4 imply that the reliable chemical potential values of the graphene patch for tri-band operation appear to be at 0.3 eV and 0.5 eV. On the other hand, whenever μc is greater than the aforementioned limit, the ability of the presented plasmonic graphene patch antenna will be reduced due to the creation of surface waves. Generally, the advised plasmonic nano-antenna operates within a dual-band frequency as μc are below 0.1 eV, while beyond this value it resonates with a tri-band frequency. In addition, the variance of S11 and VSWR in tri-band operations is calculated again and the results are shown in Fig. 6. These two figures indicate that all considered chemical potential values provide acceptable S11 and VSWR values.

Fig. 6
figure 6

The S11 and VSWR depended on frequency with various μc at TM11

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Furthermore, the 2D and 3D view of the radiation pattern for the tri-band antenna with μc of 0.5 eV is computed and the results are illustrated in Fig. 7. This figure demonstrates omnidirectional pattern behavior of the plasmonic nano-antenna in the first excited vibration mode at 0.855 THz, contrarily a bidirectional pattern for the second excited mode at 3.760 THz and an approximate bidirectional pattern for the third resonant mode at 4.650 THz.

Fig. 7
figure 7

3D view of the radiation far field at a 0.855 THz, b 3.760 THz, and c 4.650 THz and 2D view of the E plane and H plane at d 0.855 THz, e 3.760 THz, and f 4.650 THz for TM10, TM20, and TM11 mode with μc of 0.5 eV

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Moreover, the directivity and realized gain for the advised plasmonic nano-antenna are also measured in the considered frequency bands and the achieved results are demonstrated in Fig. 8. This figure indicates that the recommended antenna resonated at triple-band frequencies of 0.855, 3.760, and 4.650 THz with a gain of 3.06, 7.65, and 7.53 dB, respectively.

Fig. 8
figure 8

Directivity and gain relative to frequency of the proposed plasmonic antenna with a μc value of 0.5 eV

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Lastly, the calculated gain, bandwidth (BW), and size of the simulated tri-band antenna with the use of a μc value of 0.5 eV are compared to the earlier achieved by other research teams and the achievements are listed in Table 5. It is clearly seen from this table that the bandwidth, gain, and size of the proposed antennas agree with the recent available published works cited here at dual-band and tri-band operating frequencies. Additionally, the results have advantages of miniaturizing antenna size and operates with a higher gain and broad bandwidth (BW) values.

Table 5 Tri-band plasmonic antenna compared to previously published work
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Conclusion

This article involves the design and analyses of a high-performance multiband plasmonic antenna. It is consists of graphene patch placed on the silicon nitride (SiO3N4) dielectric substrate to be used in terahertz wireless communication systems. The design procedure is performed by utilizing CST simulation technique and investigating different graphene chemical potential values in the range of 0.0 to 1.0 eV. Generally, the calculated results reveal that the operating frequency of the graphene patch antenna can be simply tuned through its chemical potential.

In addition, the simulation result implies that, as chemical potentials raised to more than 0.3 eV, the proposed antenna resonates with a triple-band frequency. Meanwhile, better values of gain are 3.06, 7.69, and 6.53 dB and fractional bandwidths are 32.4%, 5.21%, and 4.24% when μc is equal to 0.5 eV, at the frequencies of 0.766, 3.285, and 4.510 THz, respectively. On the other hand, the computed far-field radiation pattern displays an omnidirectional, a bidirectional, and an approximate bidirectional pattern behavior for the first, second, and third excited resonant modes, respectively. The reliability of the proposed antenna relies on multiband operating frequencies; it is smaller in size and provides high gain and a wider bandwidth compared to the corresponding values recently achieved by other researchers using another dielectric substrate with different feeding techniques. Therefore, it can be said that the proposed plasmonic antenna is suitable for high-speed indoor wireless application systems especially in the field of medical imaging, material identification, and security scanning in the terahertz frequency range.