# Theory of Edge Effects and Conductance for Applications in Graphene-Based Nanoantennas
## 2 Optical Conductance of a Terminated Graphene Sheet  
### Kubo Approach for Optical Conductance of Graphene  
The statement of the problem is depicted in Figure 1. In the following we use the synthesis of the Dirac fermion model [17; 49; 50] with Kubo approach for conductance calculation. The Kubo approach is considered an exact universal technique which couples the generalized forces of arbitrary physical origin with the responses of correspondent origin via general susceptibilities [48]. The integral for general susceptibility is calculated by deflecting the integration contour in the complex plane of time upwards or downwardsaccording to the sign of frequency. Towards this goal it is convenient to define the forces as tangential components of electric field and responses as components of current densities. Thus, the general susceptibilities are defined by a \(2\times 2\) matrix where  
\[\vec{j}_{a,\omega}(\mathbf{x})=-\sum\limits_{b=1,2}\int K_{ab}\big{(}\omega; \mathbf{x},\mathbf{x}^{\prime}\big{)}E_{b,\omega}(\mathbf{x}^{\prime})d \mathbf{x}^{\prime}, \tag{1}\]  
and  
\[ \begin{array}{c}K_{ab}(\omega;\mathbf{x},\mathbf{x}^{\prime})=\\ \frac{e^{2}\omega}{\hbar}\int\limits_{0}^{\infty}e^{i\omega t}\big{\langle} \hat{x}_{a}(t,\mathbf{x})\hat{x}_{b}(0,\mathbf{x}^{\prime})-\hat{x}_{b}(0, \mathbf{x}^{\prime})\hat{x}_{a}(t,\mathbf{x})\big{\rangle}dt.\end{array}  \tag{2}\]  
Here \(\vec{j}_{a}(\mathbf{x})=\big{\langle}\hat{j}_{a}(\mathbf{x})\big{\rangle}\), and \(\hat{j}_{a}(t,\mathbf{x})=i\omega e\hat{x}_{a}(t,\mathbf{x})\), with \(\big{\langle}\hat{j}_{a}(\mathbf{x})\big{\rangle}\) representing the observable current density, \(\hat{x}_{a}(t,\mathbf{x})\) is an operator of charge displacement per unit area. The general susceptibility tensor \(K_{ab}(\omega;\mathbf{x},\mathbf{x}^{\prime})\) depends on the geometry of the sample as well as the electronic properties of the medium. Therefore, it has a physical meaning of non-local optical conductance (in the units \(e^{2}/\hbar\)) in a similar manner to the dc conductance of graphene in [17]. The next step is the transformation of Equation (2) to a form that is convenient for applications in graphene electrodynamics. Thus, one can write  
\[K_{ab}\big{(}\omega;\mathbf{x},\mathbf{x}^{\prime}\big{)}=-\frac{ie}{\hbar} \int\limits_{0}^{\infty}e^{i\omega t}\Big{(}\big{\langle}\hat{A}^{ab}\big{\rangle} -\big{\langle}\hat{B}^{ab}\big{\rangle}\Big{)}dt, \tag{3}\]  
\[\text{where }\Big{\langle}\hat{A}^{ab}\Big{\rangle}=\big{\langle}\hat{j}_{a}(t,\mathbf{x})\hat{x}_{b}(0,\mathbf{x}^{\prime})\big{\rangle},\Big{\langle}\hat {B}^{ab}\Big{\rangle}=\big{\langle}\hat{x}_{b}(0,\mathbf{x}^{\prime})\hat{j}_ {a}(t,\mathbf{x})\big{\rangle}\text{ and }\]  
\[\Big{\langle}\hat{A}^{ab}\Big{\rangle}=Tr\Big{(}\hat{A}^{ab}\hat{\rho}_{0} \Big{)}=\sum\limits_{s}\Big{(}\hat{A}^{ab}\Big{)}_{ss}\rho_{0 ss}. \tag{4}\]  
Figure 1: Problem statement: (**a**) Geometry of a graphene sheet with zigzag edge configuration. A-atoms and B-atoms form two sub-lattices coupled in every spinor modes. Zigzag edges produced via removing A- or B-atoms out of the sheet at the edge (the places of removed atoms are marked via dash lines); (**b**) Fermi–Dirac conical valleys for the model of pseudo-spin dispersion. The self-crossing points K and K\({}^{\prime}\) correspond to the pair of Fermi points in the Brillouin zone. where \(\theta_{0}\) denotes is the density matrix of free motion.