## 2 Graphene Patch Antenna Design  
### Graphene's Conductivity and Dispersion Relation  
Graphene, as a carbon-based material, has been prepared with different techniques. A single layer of graphene was extracted from the bulk graphite crystal structure using a mechanical exfoliation approach. Chemical vapor deposition (CVD), thermal exfoliation, and carbon segregation [27] are other preparation methods for the graphene layers. At the THz frequency band, this exceptional material exhibits excellent mobility, transparency, flexibility, and environmental stability. As a result, it is the best choice that can be used in antenna applications. In comparison to other nano-plasmonic materials such as gold and silver, graphene is an excellent contender for promoting the propagation of SPP waves. The dispersion relation of Transverse Magnetic (TM) of SPP depends on the conductivity of graphene given by [28].  
\[-i\frac{\sigma_{\mathrm{s}}}{\omega\varepsilon_{0}}=\frac{\varepsilon_{r1}+ \varepsilon_{r2}\mathrm{coth}\left(K_{eff}t_{o}\right)}{K_{eff}} \tag{6}\]  
where \(\varepsilon_{r1}\) and \(\varepsilon_{r2}\) are the relative permittivity of air over the graphene layer and the dielectric constant of the substrate materials, respectively. \(K_{eff}\), is the wavenumber of the \(SPP\) wave which can be expressed by:  
\[K_{eff}=K_{0}\left(\eta_{eff}\right) \tag{7}\]  
where \(\eta_{eff}\) is the effective refractive index of SPP and \(K_{0}\) is the free space wavenumber \(K_{0}=\frac{2\pi}{\lambda}\). According to [PERSON]'s formula [29; 30]. The graphene conductivity is the sum of both intraband and interband conductivity, and can be described as:  
\[\sigma_{\mathrm{s}}=\sigma_{inter}+\sigma_{intra} \tag{8}\]  
\[\sigma_{\mathrm{s}}(\omega,\mu_{c},\tau,T)=\frac{je^{2}\left(\omega-j\tau^{-1 }\right)}{\pi\hbar^{2}}\times\left[\begin{array}{c}\frac{1}{\left(\omega-j \tau^{-1}\right)^{2}}\\ \int_{0}^{\infty}\mathcal{E}\left(\frac{\partial f_{d}\left(\mathcal{E} \right)}{\partial\mathcal{E}}-\frac{\partial f_{d}\left(-\mathcal{E}\right)}{ \partial\mathcal{E}}\right)d\mathcal{E}-\\ \int_{0}^{\infty}\frac{f_{d}\left(-\mathcal{E}\right)-f_{d}\left(\mathcal{E} \right)}{\left(\omega-j\tau^{-1}\right)^{2}-4\left(\varepsilon/\hbar\right)^{ 2}}d\mathcal{E}\end{array}\right] \tag{9}\]  
\[f_{d}\left(\mathcal{E}\right)=\frac{1}{\left[e^{\left(\mathcal{E}\right)/K_{B }T}+1\right]} \tag{10}\]  
where \(T\) is the temperature, \(K_{B}\) is the Boltzmann constant, \(\hbar\) is the reduced [PERSON]'s constant, \(\mu_{c}\) is the chemical potential, \(e\) is the charge of electron \(\tau\) is the relaxation time and \(f_{d}\left(\mathcal{E}\right)\) is the Fermi-Dirac distribution. The first term in the preceding equation denotes graphene's intraband conductivity, whereas the second term denotes graphene's interband conductivity. Intraband conductivity dominates in the MW and THz bands, whereas inter band conductivity dominates in the near-infrared and visible spectrum [31]. Graphene's intraband and interband conductivities can be expressed as:  
\[\sigma_{intra}(\omega,\mu_{c},\tau,T)=-j\frac{e^{2}K_{B}T}{\pi\hbar^{2}(\omega-j \tau^{-1})}\bigg{[}\frac{\mu_{c}}{K_{B}T}+2\ln\left(e^{-\mu_{c}/K_{B}T}+1\right) \bigg{]} \tag{11}\]  
\[\sigma_{inter}(\omega,\mu_{c},\tau,T)=-\frac{je^{2}}{4\pi\hbar}ln\left[\frac{2| \mu_{c}|-\hbar\left(\omega-j\tau^{-1}\right)}{2|\mu_{c}|+\hbar\left(\omega-j \tau^{-1}\right)}\right] \tag{12}\]