# Theory of Edge Effects and Conductance for Applications in Graphene-Based Nanoantennas
## 2 Optical Conductance of a Terminated Graphene Sheet
### Zigzag Edges  
In this subsection we apply the general methodology to an edge of a zigzag-type. The electronic properties of the ribbon are described by the model of Dirac fermion corresponding to a tight-binding model on a two-dimensional honeycomb lattice [17; 26; 51]. We use the zigzag-type boundary conditions for Dirac fermion on a terminated (finite) lattice derived in [17; 26; 51], since it was demonstrated that this type of boundary condition can be generally applied to a terminated honeycomb lattice in the case of electron-hole symmetry [26]. The A- and B-atoms are coupled in every spinor mode. From a physical point of view, the interpretation of electronic transport may be considered as a motion from A-atoms to B-atoms and vice versa, while A-A and B-B motions are forbidden. The electron transport for a zigzag edge in valleys K and K\({}^{\prime}\) is independent and is considered separately. The wave function is expressed as a linear combination of the eigen 4D spinors. The pseudo-spinor modes for valleys K and K\({}^{\prime}\) have the following form:  
\[\boldsymbol{\Psi}_{Ks}(\mathbf{x},t)= \begin{pmatrix}\boldsymbol{\Psi}_{A,s}( \mathbf{x},t)\\ \boldsymbol{\Psi}_{B,s}(\mathbf{x},t)\\ 0\\ 0\end{pmatrix} = \begin{pmatrix}u_{s}(x)\\ v_{s}(x)\\ 0\\ 0\end{pmatrix} e^{i(k_{y}y-\omega_{s}t)}, \tag{14a}\] \[\boldsymbol{\Psi}_{K^{\prime}s}(\mathbf{x},t)= \begin{pmatrix}0\\ 0\\ -\widetilde{\boldsymbol{\Psi}}_{B,s}(\mathbf{x},t)\end{pmatrix} = \begin{pmatrix} 0\\ 0\\ -\widetilde{u}_{s}(x)\\ -\widetilde{v}_{s}(x)\end{pmatrix} e^{i(k_{y}y-\omega_{s}t)}. \tag{14b}\]  
The functions \(u_{s}(x),v_{s}(x)(\widetilde{u}_{s}(x),\widetilde{v}_{s}(x))\) in the valley K (K\({}^{\prime}\)) take the following form for bulk and edge states  
\[ \begin{split} u_{s}^{Bulk}(x)&=i\sqrt{\frac{1}{2L}B_{p}^{ Bulk}(k_{y})}\sin\left[{\kappa_{px}(k_{y})\left(x+\frac{L}{2}\right)}\right]\\ u_{s}^{Edge}(x)&=-\sqrt{\frac{1}{2L}B^{Edge}(k_{y})}\sinh\left[{\eta \left(k_{y}\right)\left(x+\frac{L}{2}\right)}\right]\\ v_{s}^{Bulk}(x)&=\pm\sqrt{\frac{1}{2L}B_{p}^{ Bulk}(k_{y})}\sin\left[{\kappa_{px}(k_{y})\left(x-\frac{L}{2}\right)} \right]\\ v_{s}^{Edge}(x)&=\pm i\sqrt{\frac{1}{2L}B^{Edge}(k_{y})}\sinh\left[{\eta \left(k_{y}\right)\left(x-\frac{L}{2}\right)}\right]\end{split} , \tag{14c}\]  
where  
\[B_{p}^{Bulk}(k_{y})=\left(1-\frac{\sin\left(2\kappa_{px}(k_{y})L\right)}{2 \kappa_{px}(k_{y})L}\right)^{-1}, \tag{14d}\]  
\[B^{Edge}(k_{y})=\left(\frac{\sinh\left(2\eta(k_{y})L\right)}{2\eta\left(k_{y} \right)L}-1\right)^{-1}, \tag{14e}\]  
where \(l\) represents the normalization length in the y-direction (which goes to infinity in the final result), \(B_{p}^{Bulk}(k_{y})\), \(B^{Edge}(k_{y})\), is a normalization constants, \(k_{y},k_{xp}=k_{xp}(k_{y})\) are the corresponding wavenumbers which are defined by the dispersive relation \(k_{y}-ik_{xp}=\left(k_{y}+ik_{xp}\right)e^{-2 ik_{xp}L}\), and \(s=\left\{p,k_{y}\right\}\) is the combinative discrete-continuous index (\(p\) is the discrete number with respect to the \(x\)-axis and \(k_{y}\) is the continuous wave-number over the \(y\)-axis). The above dispersion relation has an infinite number of real roots single accumulation point at infinity (confined modes) and one imaginary root that corresponds to the edge state [17]. Transformation to the edge state from the confined modes in Equations (14a-e) and characteristic equation may be performed via the exchange \(\kappa_{xx}\rightarrow\eta\). The two signs in (14) correspond to electrons and holes respectively. As one can see, Equations (14a,b) satisfy the Dirac equation and are subjected to the following boundary conditions; \(u_{s}(-L/2)=v_{s}(L/2)=0\)[17, 26, 51]. The components \(u(x)\) and \(v(x)\) are separately orthogonal (mutually non-orthogonal) due to their coupling over electron motion between the atoms of A and B sublattices.