# Theory of Edge Effects and Conductance for Applications in Graphene-Based Nanoantennas
## Appendix A Derivation of Equation (16)  
In this Appendix we discuss the boundary value problem for pseudo-spin defined by Equation (14). As it was mentioned above, the pseudo-spin satisfies the Dirac equation with the following boundary conditions; \(u_{s}(-L/2)=v_{s}(L/2)=0\)[26]. It may be also transformed into the Helmholtz equation with two special sets of boundary conditions. We have the _Dirichlet_ condition at the left-hand side and the impedance condition \(\left(k_{y}-\partial_{x}\right)u\big{|}_{x=L/2}=0\) at the right-hand side for the components \(u(x)\) (determined by the Dirac equation). The situation is precisely inverted for the second type, namely a _Dirichlet_ condition at the right-hand side and an impedance condition \(\left(k_{y}+\partial_{x}\right)v\big{|}_{x=-L/2}=0\) at the left-hand side. These problems are Hermitian, whereby the eigenmodes form a complete basis. The components \(u(x)\) and \(v(x)\) are both separately orthogonal, but are mutually non-orthogonal, due to their coupling over the electron motion between the atoms of A and B sublattices. The property of orthogonality is shown at Appendix C. Using completeness, orthogonality, and normalization conditions \(\int_{-L/2}^{L/2}\big{|}u_{p}(x)\big{|}^{2}dx+\int_{-L/2}^{L/2}\big{|}v_{p}(x) \big{|}^{2}dx=1\), we obtain  
\[2\underset{p^{\prime}}{\overset{\sum}{\underset{p^{\prime}}{\sum}}}u_{p^{ \prime}}^{*}(x)u_{p^{\prime}}(x^{\prime})=\delta(x-x^{\prime}), \tag{17}\]  
\[2\underset{p^{\prime}}{\overset{\sum}{\underset{p^{\prime}}{\sum}}}v_{p^{ \prime}}^{*}(x)v_{p^{\prime}}(x^{\prime})=\delta(x-x^{\prime}). \tag{18}\]Starting from the \(xx\)-component and using the basis relation (11) for \(a=x\), \(b=x\), the matrix element of the current density operator, one obtains  
\[\left(\hat{j}_{x}(\mathbf{x},0)\right)_{ss^{\prime}}=\frac{1}{l}ev_{F}(u_{s^{ \prime}}^{*}(x)v_{s}(x)+v_{s^{\prime}}^{*}(x)u_{s}(x))e^{i(k_{y}-k_{y^{\prime} })y}. \tag{13}\]  
Next we examine the limit of \(l\to\infty\) by making the exchange \(l^{-1}\!\!\sum_{s}\to\left(2\pi\right)^{-1}\!\!\int_{-\infty}^{\infty}\sum_{n}\) and the same for \(s^{\prime},n^{\prime}\). Summing over \(s\),\(s^{\prime}\) and taking into account both electrons and holes \(\left(\pm\mathrm{in}\,v_{n}(x)\,\right)\) as well and the charge carriers in two valleys K and K\({}^{\prime}\), leads to:  
\[ \begin{array}{c}K_{xx}(\mathbf{x},\mathbf{x}^{\prime};\omega)\approx-\frac{ i\,e^{2}p_{F}^{2}}{4\pi^{2}\left(\omega+i0\right)}\int\limits_{-\infty}^{ \infty}\int\limits_{0}^{\infty}\int\limits_{x_{y}}^{\infty}dk_{y}dk^{\prime} \cdot e^{i(k_{y}-k^{\prime})(y-y^{\prime})}.\\ \left.\sum\limits_{n}\frac{\partial f(\epsilon)}{\partial\epsilon}\right|_{ \epsilon=\epsilon_{n}(k_{y})}\cdot\sum\limits_{n^{\prime}}\Lambda_{nn^{\prime }}^{\left(xx\right)}\left(\mathbf{x}\mathbf{x}^{\prime};k_{y}\right),\end{array}  \tag{14}\]  
where  
\[ \begin{array}{c}\sum\limits_{p^{\prime}}\Lambda_{pp^{\prime}}^{\left(xx \right)}\left(x,x^{\prime};k_{y}\right)=\\ v_{p}(x)v_{p}^{*}(x^{\prime})\!\!\sum\limits_{p^{\prime}}u_{p^{\prime}}(x)u_ {p^{\prime}}^{*}(x^{\prime})+\\ u_{p}(x)u_{p}^{*}(x^{\prime})\!\!\sum\limits_{p^{\prime}}v_{p^{\prime}}(x)v_ {p^{\prime}}^{*}(x^{\prime})+\\ u_{p}(x)v_{p}^{*}(x^{\prime})\!\!\sum\limits_{p^{\prime}}v_{p^{\prime}}(x)u_ {p^{\prime}}^{*}(x^{\prime})+\\ v_{p}(x)u_{p}^{*}(x^{\prime})\!\!\sum\limits_{p^{\prime}}u_{p^{\prime}}(x)v_ {p^{\prime}}^{*}(x^{\prime})\end{array} . \tag{15}\]  
Note that the summation in (15), means including the contribution from both electrons and holes and the valleys K, K\({}^{\prime}\). The sum over electrons and holes may be transformed to the sum over the electron states by means of their doubling. The sum of the two other terms is zero due to the opposite sign of \(v_{n}(x)\) (subject to the same \(u_{n}(x)\)).