**6. Evaluating surface impedance models for terahertz applications**

The design of the terahertz antenna demands particular attention on the choice of materials to fulfill the microfabrication process.

As illustrated in **Figure 20**, a resonant antenna element for the terahertz application proposed here consists of three multilayers, from top to bottom: a gold or graphene dipole, polydimethylsiloxane (PDMS) substrate, and a ground plane. Gold and graphene are good conductive and are not susceptible to oxidation in air, while PDMS has comparatively low losses in the terahertz range.

#### **Figure 20.**

*Dipole-like antenna made of two graphenes (or gold) strips on a substrate and integrated with a located gap source in the center (example of the photonic mixer).*

Various metals are selected for the top and bottom layer metallizations due to their selectivity for patterning, as they respond to various etchants. If the same metal is employed, the leakage of the etchant through the PDMS will damage the ground layer during the modeling of the upper metallization.

In the simulations stage, the material properties of the upper metals are obtained from a Drude model to evaluate the surface impedance *ZSR* [20]. More details are given in [21] knowing that *ZSR* is expressed by:

$$Z\_{\rm SR} = \sqrt{\frac{j o \mu\_0 \mu\_r}{\sigma\_r + j o \nu\_0}} \text{With } \sigma\_r = \frac{\sigma\_0}{1 + j o \nu} \tag{23}$$

where angular frequency *ω* ¼ 2*πf*, and *f* = frequency of the driving electric field, *μ*<sup>0</sup> = permeability of free space, *μ<sup>r</sup>* = relative permeability, *ε*<sup>0</sup> = permittivity of free, *σ*<sup>0</sup> = intrinsic bulk conductivity at DC, and *τ* = phenomenological scattering relaxation time for the free electrons (i.e., mean time between collisions).

Except for the graphene model, as proposed in [22, 23], a single graphene layer can be modeled by a 2D surface conductivity. Other models (with a classical, semi-classical, and quantum mechanical treatment) are proposed to evaluate the surface impedance, as provided in [24]. We utilize the frequency-dependent conductivity of a monolayer of graphene based on Kubo's formula [22, 23].

$$\sigma(\boldsymbol{\alpha}, \boldsymbol{\mu}\_c, \Gamma, T) \approx -j \frac{q\_c k\_B T}{\pi \hbar^2 (\boldsymbol{\alpha} - 2j\Gamma)} \left( \frac{\mu\_c}{k\_B \Gamma} + 2 \ln \left( e^{-\frac{\mu\_c}{k\_B \Gamma}} + 1 \right) \right) \tag{24}$$

Where *ω* is the angular frequency; *μ<sup>c</sup>* is the chemical potential; Γ ¼ 1*=*2*τ*; *τ* is the transport relaxation time; *T* is the temperature; *kB* is the Boltzmann constant, and ℏ is the reduced Planck constant. It is important to note that only the intra-band term is considered in relation to Eq. (24), which gives good and exact results for frequencies limited to a few THz.

The implementation of surface impedance in the formulation of a simplified equivalent circuit is described in Eq. (34) and is detailed in the given articles [23, 25]. The terahertz antenna is studied based on graphene nanoribbon using the MoM-GEC formulation [23, 25]. For this purpose, we propose **Figure 21**, which depicts the radiated electric field for several *z* planes (near field). For a minimal value of *Z*, the electric field distribution is perturbed by the evanescence modes in the vicinity of the discontinuity surface, which gives information about the antenna structure. For a distance so far, it gives the fundamental mode of the waveguide.
