Graphene based nano patch antenna using photonic band gap insertion into substrate for applications at THz band

In this work, we present a numerical investigation of the properties of Graphene based Nano Patch Antenna (GNPA) with triangular periodic arrangements of Photonic Band Gap (PBG) insertion into substrate. In the proposed design, we studied the effect on the radiation characteristics, such as return loss, bandwidth, gain, directivity, voltage standing wave ratio (VSWR), and the radiation pattern using the Finite Integration Method (FIM). We analyzed three configurations of cylindrical air holes with different PBG insertion heights into silicon dioxide (SiO2) substrate, namely h1, h2 and h3, and also reference antennas. The best results obtained are S11=−31.5, 2.038 dB gain and 0.4 THz bandwidth for antenna h3. The investigated GNPA-PBG resonates around 1.5 THz, enabling the application at THz band.


Introduction
In recent years, the high demand for broader bandwidths, high data transmission and reception rates at low power in the wireless communication has motivated the development of systems operating at high frequencies, such as THz band. It has been implemented mainly in indoors mobile communications systems, where is necessary to 1 3 210 Page 2 of 13 miniaturize devices, such as filters and antennas. Nonetheless, devices miniaturization problems are commons especially on their performance Tadao 2011 andAkyildiz et al., 2014).
Patch antennas operating in the THz band have been used to solve most of the devices miniaturization problems. However, parameters such as bandwidth, gain and directivity present low performance due to surface waves caused mainly by substrate thickness. The antenna bandwidth decreases when the substrate thickness is relatively thin, however, when the substrate thickness is substantially thick the bandwidth increases, and also, increasing the losses by surface waves (Chang and Richards 1986;Vandenbosch and Van de Capelle, 1994).
Graphene-based microstrip antennas with PBG substrate (Photonic Band Gap) are considered important candidates to solve the problems described above, besides generating TE and TM band regions, it increases the antenna bandwidth and efficiency with the correct arrangement of the periodic lattice (Boutayeb and Denidni 2007;Mosallaei, and Rahmat-Samii 2003;Joannopoulos et al. 2011;Miranda et al. 2020;Miranda et al. 2021 andSousa et al. 2020). Considering the above, we propose in this paper a new graphene patch antenna with a triangular lattice of air holes PBG substrate operating in the THz band. Thus, the antenna radiation characteristics shows a significant improvement due to the use of PBG substrate for the patch antenna based on graphene. 13 Bala and Marwaha (2015a and b) present two antennas, a triangular GNPA operating in the range of 1-3 THz, which achieved a good combination of impedance in resonance frequencies with 5.6% of bandwidth, and another GNPA rectangle operating at a resonance frequency of 2.9 THz where they compare the results of five different substrates (hardwood, polyamide, quartz, silica and silicon). Thampy et al. (2015) point out that recent advances in the synthesis and characterization of graphene films indicate that this material is suitable for photoelectronic applications, and also, has optimum electrical and optical characteristics for nano-antenna systems that radiate in the THz band. George and Madhan (2017), proposed a microstrip antenna design, using as a patch, a very thin layer of graphene on a silica substrate with variable thickness, with the heights of 30, 33 and 37.5 µm only one transmission band was obtained, already for the heights of 43, 44 and 45 µm were obtained two transmission bands, these results were analyzed in terms of S11, gain, efficiency and bandwidth. Oliveira et al. (2018) modeled and simulated a microstrip antenna with a ceramic substrate of BiNbO 4 doped with V 2 O 5 in an air hole network using Ansys HFSS software. Parameters such as S11, radiation pattern, gain and directivity were analyzed, where they observed an improvement in performance when compared to antennas using non-ceramic substrates or with ceramics without periodic network. They stated that these ceramic antennas offer significant advantages in terms of compactness, weight, thermal stability, low production cost and they can also be used in microwave integrated circuits.
Recently Benlakehal et al. (2022), designed and analyzed an array of 2×2 microstrip patch antennas based on different substrates, including periodic, non-periodic and homogeneous photonic crystals, through the use of two different simulation techniques, CST Microwave Studio software based on the technique of finite integration and Ansys HFSS software based on the finite element technique. According to the researchers, these antennas could be applied in sensing and communication technologies, in this sense the set of antennas designed based on periodic photonic crystals had a better performance than the set of conventional antennas in terms of S11, bandwidth, VSWR, gain and radiation efficiency around 0.65 THz, however, these same parameters were considerably improved using non-periodic photonic crystal substrate.
Here we designed and simulated a numerical investigation by Ansys HFSS of a graphenebased GNPA with periodic triangular arrays of holes in the substrate in the SiO 2 substrate with different heights. The proposed antenna was designed for applications in the THz band. This paper is organized as follows: In Sect. 2 discusses the design of the GNPA. Section 3 presents the results and simulations. Parameters such as reflection parameter (S11), radiation pattern, gain, directivity, bandwidth and efficiency were analyzed. We proposed an innovative GNPA with a triangular lattice that allows double band operation. And finally in Sect. 4 the conclusions are made and finally the references.

Graphene model
A graphene antenna provides energy to the free electrons of the graphene sheet when exposed to an electromagnetic perpendicular wave, causing such electrons to participate in conduction of current. Due to this interaction, Surface Plasmon Polariton (SPP) modes between the graphene sheet and the substrate are also excited in the THz band. Therefore, to model the propagation of SPP waves along the graphene, its conductivity must be considered (George andMadhan 2017 andTamagnone et al. 2012).
The electrical conductivity of graphene is highly frequency dependent, in other words, it is directly related to the radiation properties of this material. Graphene can be represented by an infinitesimal plane characterized by a surface conductivity , c , , T given by the Eq. (1), as follows: which results from the Kubo's formula, where ω is the radiation frequency, c is the chemical potential, is the scattering rate, T is temperature, j = √ −1 , q e is the charge of eléctron, ℏ is the reduced Planck constant, is the energy and f d = e Kubo's formula (2) Kubo 1957): The contribution of the interband is given by (3):

A complete band gap for all polarization
The triangular periodic lattice is used to achieve polarization in both, TE and TM, modes simultaneously. To design a photonic crystal with full band gap (polarization in TE and TM (1) , it is necessary to consider that TM gaps are favored in regions with higher dielectric constants, whereas TE gaps are favored by a connected lattice. The connectivity of the veins ( Fig. 1) is the response to reach gaps in the TE band. A structure satisfying both requirements, found by and shown in Fig. 1, is a lattice of triangular holes of low dielectric constant (air) within a medium of higher dielectric constant (Plihal and Maradudin 1991). This complicated structure can be obtained by mechanically drilling cylindrical tubes into a dielectric material.

Graphene nano patch antenna using photonic band gap
The configurations and dimensions of GNPA-PBG proposed here are shown in Fig. 2 and Table 1 respectively. Figure 2a shows the design of this antenna, which has a total dimension Ls × Ws = 100 × 90 μm . In the top layer, the patch material is composed by a graphene sheet, which is excited by a feed port. The feeder is a microstrip transmission made of copper and just below is a silica substrate with air holes varying in three different heights (h1, h2 and h3), arranged in a triangular lattice. Such dimensions can be calculated by the following equations: (7) and (8). The resonant frequency of the antenna can be calculated by: where L p is the length of the rectangular patch, W p is the width of the patch, c is the speed of light, e is the dielectric constant and l extension variation in the lengh of the patch due to edge effects. e and l are given respectively by equations (5) and (6)  where Lp and Wp are respectively given by:  Fig. 2a is shown the antenna model, in which is possible to visualize the triangular periodic lattice as well as the graphene patch on the upper part. In Fig. 2b are presented the three configurations of hole heights (h1, h2 and h3). The height h1 represents the configuration in which the air holes are made throughout the thickness of the substrate with 0.7 µm. The height h2 represents the air holes made from top to half thickness of the substrate with 3.5 µm. And the height h3 represents the holes in the substrate made from bottom to half of the substrate,  also with 3.5 µm. Finally, the ground plane is defined with copper conductive material just below the silica substrate. The proposed GNPA-PBG is designed to operate in the THz band and the simulations were performed using commercial CST Studio software that uses the Numerical Method of Finite Integration (FIM). The chosen silica substrate has a thickness of 7 m and a relative permittivity of 11.9. The parameters of the GNPA-PBG were determined according to the references used, where Ls is the substrate length, Ws is the substrate width, w1 is the width of the feed microfine, L1 is the length of the microline, L p is the length of the patch, W p is the width of the patch, a is the periodicity of the triangular network and e r is the radius of the holes Tamagnone et al. 2012;Balanis 2015). The values of each of these parameters are described in Table 1.

Simulation, results and discussion
The analysis of the results of the triangular GNPA-PBG simulations were performed in terms of the following parameters: bandwidth, S11, radiation pattern, gain and directivity, for the three configurations of air hole heights on the silica substrate of the chemical potential of the graphene patch (0.1, 0.2 and 0.3 eV). The chemical potential of a graphene sheet can be changed through the chemical method or applying a uniform field by gate tension (Hason 2008;Li et al. 2014).
The return loss graphs shown in Figs. 3, 4 and 5 are represented by different colors for each of the heights of the holes: h1 (red), h2 (green) and h3 (blue). That is, three different configurations of the triangular GNPA-PBG with varying chemical potential were simulated. The abscissa axis represents the frequency in THz and the ordinate axis is the parameter S11 (reflection parameter) given in dB. From now on will be agreed in the text of this article: antenna h1, antenna h2 and antenna h3, when we refer to each of the three configurations of holes of GNPA-PBG. Figure 3 shows the return loss graphs for antennas h1, h2 and h3, both with the same chemical potential value of 0.1 eV. By analyzing the Fig. 3, we showed that the h3 antenna  Figure 3 also shows that in relation to the frequency the antenna h1 has resonance in 1.6 THz, the antenna h2 has resonance in 1.5 THz and the antenna h3 has resonance in 1.4 THz, thus characterizing a delay in frequency of the antenna h2 in relation to h1 and antenna h3 in relation to antenna h1 and antenna h2. Table 2 presents the details of the graphs of Fig. 3, where the parameters: bandwidth, resonance point and S11 of antennas h1, h2 and h3 can be seen in detail. The operating range is also shown for each of the antennas, including those that obtained two transmission bands. In this case the chemical potential is 0.1 eV and the antenna that obtained the best performance according to the parameters previously reported was the h3 antenna because it has lower S11 values and bandwidth of 0.4 THz.  Fig. 4, again the S11 graphs for the three antennas h1, h2 and h3 are displayed, but now with a chemical potential value of 0.2 eV. Through the analysis of the graphs we showed that the h3 antenna produced better reflection parameter values with a resonance point later than antennas h1 and h2. In the same way as in Fig. 3, it can be observed that in the results, in relation to the operating frequency, a right shift occurred at the three resonance points, but in these cases with the chemical potential of 0.2 eV, there was also the formation of a second transmission band for antennas h1, h2, and h3, that is, they have two operating ranges. Table 3 presents the details of the graphs of Fig. 4, the same parameters of Table 2 are shown and analyzed. For this case, the chemical potential equal to 0.2 eV, the antennas h1, h2 and h3 obtained two resonance points. All simulated antennas obtained bandwidth of 0.2 THz at the first resonance point. For the second resonance point, the antenna that obtained the best result in terms of this parameter was h1 with the same 0.2 THz of bandwidth. Taking the return loss as the analyzed parameter, the antenna that presented the better S11 value in the first transmission band was the antenna h3 with −14.6 dB, and for the second transmission band, the best result was obtained for the antenna h2 with −1.7 dB of reflection parameter.
In Fig. 5 are shown the same three antennas configurations with chemical potential of 0.3 eV. Through the analysis of the graph we showed that the antenna h3 produced better S11 values with a resonance point later than the other configurations, and the antenna h3 was the only one that obtained double transmission band, that is, the same antenna operates into two different bands, characterizing one of the advantages of this configuration. Table 4 presents the details of the graphs of Fig. 5, where again the parameters of bandwidth, resonance point, S11 and operating range can be seen in detail, but for the chemical potential it was 0.3 eV. In this case, the only antenna that presented two resonance points was antenna h3, but the bandwidth of antennas h1 and h2 were larger than that of antenna h3. Thus, the antenna that obtained the best reflection parameter result was h2 with a value equal to −13.6 dB. Figures 6, 7 and 8 present the proposed antenna radiation diagrams for each of the height configurations (h1, h2 and h3) and chemical potentials (0.1, 0.2 and 0.3 eV), respectively. The magnitude of the main lobe gradually increases as the chemical potential c increases. In the case of c = 0.26 eV, the antenna reaches its maximum magnitude of the main lobe of 7.8 dBi. It is also observed that the angular width of the main lobe narrows from 80.7 to 64° when c increases progressively from 0.16 to 0.24 eV. As the surface current distributions in the patch change somewhat to different chemical potential, the beam variation at the second resonant frequency is small. Table 5 presents the gain and the directivity in relation to the chemical potential of each of the three antenna configurations proposed here. It can be observed that antennas h1, h2 and h3 with chemical potential of 0.3 eV obtained the highest values of gain. While the antennas h1, h2 and h3 with chemical potential of 0.1 eV were the most directive. Table 6 compares the proposed antennas that obtained better results in relation to the following parameters: reflection parameter (S11), size, resonance frequency and bandwidth. In this case we chose the antennas h3 ( c 0.1 eV), h1 ( c 0.2 eV) and h3 ( c 0.3 eV) that had better results and compared them with other path antennas described in the works of the references ; George and Madhan (2017) and Kazemi and Mokhtari(2017).
As shown in Table 6, the h3 antenna, with a chemical potential of 0.1 eV, obtained greater loss of return and greater bandwidth compared to the references ; George and Madhan (2017) and and Kazemi and Mokhtari (2017). The antennas h1 ( c = 0.2 eV) and h3 ( c = 0.3 eV), both dual band, obtained better results in relation to the bandwidth when comparing the references George and Madhan (2017) and and Kazemi and Mokhtari, (2017). And the antenna h1 ( c = 0.2 eV) obtained better bandwidth relative to the reference . In general the antenna h3 with ( c = 0.1 eV) and ( c = 0.3 eV), obtained better results in relation to the parameters: bandwidth and S11, when compared with all antennas presented in Table 6.

Conclusion
In this work we present the study of a nano patch antenna of graphene with a triangular periodic network of air holes in three different configurations in the silica substrate and with three different chemical potential values of the graphene. Through the modeling and    computational simulations, the values of the following parameters were obtained: reflection parameter, radiation pattern, gain and directivity. It was observed that the antenna denominated h3 obtained better results in comparison to the antennas of the references ; George and Madhan (2017) and and Kazemi and Mokhtari (2017). The frequency shift of the resonance point of the antenna was also verified along with the formation of a second transmission band, that is, the antenna went from one to two resonance points with the gradual increase of the chemical potential. These characteristics are important for antennas operating in the THz band, since most antennas that are designed to operate in this frequency spectrum have low efficiency and low gain and can be used for many applications, for example Graphene-based RFID tag sensors Singh, et al. (2017). Thus, the antenna proposed here is an important contribution to the antennas patch that operates in the THz band