# Theory of Edge Effects and Conductance for Applications in Graphene-Based Nanoantennas
## 3 Numerical Results and Discussion  
The optical properties of graphene are generally defined by the geometrical size of the sample and the value of the electrochemical potential. These parameters may vary over a wide range and thus are of practical interest to nanoantenna design. The recent advances in graphene technology make it possible to produce graphene samples of size from few nanometers to few centimeters [6; 7; 22]. The electrochemical potential may also vary over the interval \(0<\mu<1.0\) eV, by doping the sample or by applying gate voltage [6; 7; 22]. In this section we present some numerical results of conductivity simulations for a wide range of physical parameters based on the theory developed above. One of the main results of the present study according to Equations (16) and (19), is that the non-homogeneity of conductance is mainly due to the edge effects. The conductance distribution is controlled by the parameter \(\vec{\mu}L=\mu L/\hbar v_{F}\), which defines the number of modes supported by the conductance value. Figures 3-5 present the normalized conductance distribution for a rather large length \(L=800\) nm and for different values of the electrochemical potential (increasing \(\mu\) corresponds to increasing number of modes). The qualitative behavior of the conductance is the same in all these  
The considered scenario changes dramatically with decreasing the dimension of the sheet, as shown in Figures 6-8 (each figure corresponds to a different value of the length for the same value of electrochemical potential). One can clearly see the enhancement of oscillations, which makes the Drude model invalid for such values of parameters. In fact, it becomes impossible to introduce the conductivity concept in its ordinary meaning. As it was mentioned above, the value defined by Equation (16) has the meaning of optical conductance, which strongly depend on the geometrical size of the sheet. It may be coupled with Drude conductivity by the Relation. \[\sigma^{Intra}(x)=G^{Drude}\frac{1}{L}\Theta(x), \tag{28}\]  
where  
\[\Theta(x)=\sum_{n=1}^{2N+1}\frac{1}{|k_{yn}(\widetilde{\mu})L-1|}\bigg{[}\sin^ {2}\!\bigg{(}\sqrt{\widetilde{\mu}^{2}-k_{yn}^{2}}\bigg{(}x-\frac{L}{2}\bigg{)} \bigg{)}+\sin^{2}\!\bigg{(}\sqrt{\widetilde{\mu}^{2}-k_{yn}^{2}}\bigg{(}x+ \frac{L}{2}\bigg{)}\bigg{)}\bigg{]} \tag{29}\]  
is an \(x\)-dependent dimensionless coefficient, in which the configuration of the sample manifests itself. This situation is rather similar to the electron transport in graphene in dc field (the concepts of conductance and conductivity discussed and compared in [17; 53]). The physical mechanism for the conductivity oscillations is the interference between the pseudo-spin modes due to the reflection from the sheet boundaries. The important features are demonstrated in the single-mode conductance (Figure 6a,b). For the rather small electrochemical potential the active mode is an edge type. The sign of the conductance is negative, which corresponds to its inductive origin. The increase in the electrochemical potential leads to the transformation from inductive to capacitive one (sign exchange) due to the transformation from the edge mode to the bulk one. As one can see, the conductivity of a graphene sheet changes its qualitative behavior for rather small values of length. However, graphene antennas generally exhibit a resonate behavior at much lower frequencies as well as their metallic counterparts, which is experimentally implemented in the THz range [54; 55; 56]. Thus, one can effectively exploit the electrically tunable conductance of graphene exactly for such small sizes, where the conventional models of conductivity become invalid due to the importance of edge effects. In summary, it is important to note that including edge effects in the physical modeling opens a new way for electrical controlling of resonant graphene antennas via the overturned electrochemical potential by means of the gate voltage.