# 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 diagonal matrix elements of the operator \(\hat{A}^{ab}\) are defined by  
\[\left(\hat{A}^{ab}\right)_{ss}=\sum_{s^{\prime}}\bigl{(}\hat{j}_{a}(t,\mathbf{x} )\bigr{)}_{ss^{\prime}}\bigl{(}\hat{x}_{b}\bigl{(}0,\mathbf{x}^{\prime}\bigr{)} \bigr{)}_{s^{\prime}s^{\prime}}, \tag{5}\]  
and following Equation (4) we have  
\[\left\langle\hat{A}^{ab}\right\rangle=\sum_{s}\sum_{s^{\prime}}\bigl{(}\hat{j} _{a}(t,\mathbf{x})\bigr{)}_{ss^{\prime}}\cdot\bigl{(}\hat{x}_{b}\bigl{(}0, \mathbf{x}^{\prime}\bigr{)}\bigr{)}_{s^{\prime}s}\rho_{0 ss}. \tag{6}\]  
where \(s=\bigl{\{}p,k_{y}\bigr{\}}\) represents the combinative discrete-continuous index (\(p\) is the discrete number of the state and \(k_{y}\) is its continuous wave number over the \(y\)-axis). The temporal behavior within the linear approximation can be also expressed as \(\bigl{(}\hat{j}_{a}(t,\mathbf{x})\bigr{)}_{ss^{\prime}}=\bigl{(}\hat{j}_{a}(0,\mathbf{x})\bigr{)}_{ss^{\prime}}e^{-i(\varepsilon_{s}-\varepsilon_{s^{ \prime}})t/\hbar}\), and as a result we obtain  
\[\left\langle\hat{A}^{ab}\right\rangle=\sum_{s}\sum_{s^{\prime}}\bigl{(}\hat{j} _{a}(0,\mathbf{x})\bigr{)}_{ss^{\prime}}\cdot\bigl{(}\hat{x}_{b}\bigl{(}0, \mathbf{x}^{\prime}\bigr{)}\bigr{)}_{s^{\prime}s}\rho_{0 ss}e^{-\frac{i}{\hbar }(\varepsilon_{s}-\varepsilon_{s^{\prime}})t}. \tag{7}\]  
A similar relation may be also obtained for \(\hat{B}^{ab}\), namely  
\[\left\langle\hat{B}^{ab}\right\rangle=\sum_{s}\sum_{s^{\prime}}\bigl{(}\hat{x }_{b}\bigl{(}0,\mathbf{x}^{\prime}\bigr{)}\bigr{)}_{ss^{\prime}}\cdot\bigl{(} \hat{j}_{a}(0,\mathbf{x})\bigr{)}_{s^{\prime}s^{\prime}}\rho_{0 ss}e^{\frac{i}{ \hbar}(\varepsilon_{s}-\varepsilon_{s^{\prime}})t}. \tag{8}\]  
Next, combining Equations (7) and (8) with Equation (3), renders  
\[ \begin{split} K_{ab}&(\omega;\mathbf{x},\mathbf{x}^{ \prime})=\frac{i}{4\hbar\omega}\sum_{s}\sum_{s^{\prime}}\cdot\\ &\int\limits_{0}^{\infty}e^{i\omega t}\Bigl{\{}\bigl{(}\hat{j}_{a} (0,\mathbf{x})\bigr{)}_{ss^{\prime}}(\hat{x}_{b}(0,\mathbf{x}^{\prime}))_{s^{ \prime}s}e^{-\frac{i}{\hbar}(\varepsilon_{s}-\varepsilon_{s^{\prime}})t}-( \hat{x}_{b}(0,\mathbf{x}^{\prime}))_{ss^{\prime}}\bigl{(}\hat{j}_{a}(0,\mathbf{ x})\bigr{)}_{s^{\prime}s^{\prime}}e^{\frac{i}{\hbar}(\varepsilon_{s}-\varepsilon_{s^{ \prime}})t}\Bigr{\}}\rho_{0 ss}dt.\end{split}  \tag{9}\]  
For shortness, we omit the initial value at time \(t=0\) in the matrix elements of the current operator (\(\bigl{(}\hat{j}_{a}(0,\mathbf{x})\bigr{)}_{ss^{\prime}}\to\bigl{(}\hat{j}_{a} (\mathbf{x})\bigr{)}_{ss^{\prime}}\), etc. ), which by using elementary integration yields the following expression:  
\[K_{ab}\bigl{(}\omega;\mathbf{x},\mathbf{x}^{\prime}\bigr{)}=-\frac{ie}{\hbar} \sum_{s}\sum_{s^{\prime}}\Bigg{\{}\frac{\bigl{(}\hat{j}_{a}(\mathbf{x})\bigr{)} _{ss^{\prime}}(\hat{x}_{b}(\mathbf{x}^{\prime}))_{s^{\prime}s}}{(\hbar\omega-( \varepsilon_{s}-\varepsilon_{s^{\prime}}))}-\frac{(\hat{x}_{b}(\mathbf{x}^{ \prime}))_{ss^{\prime}}\bigl{(}\hat{j}_{a}(\mathbf{x})\bigr{)}_{s^{\prime}s}}{ (\hbar\omega+(\varepsilon_{s}-\varepsilon_{s^{\prime}}))}\Bigg{\}}\rho_{0 ss}. \tag{10}\]  
By introducing the following change \(s\to s^{\prime},s^{\prime}\to s\) in the second term of Equation (10) and using the relation \(\bigl{(}\hat{j}_{a}(\mathbf{x})\bigr{)}_{ss^{\prime}}=ie(\varepsilon_{s^{ \prime}}-\varepsilon_{s})(\hat{x}_{a}(\mathbf{x}))_{ss^{\prime}}/\hbar\), the latter can be written as  
\[K_{ab}\bigl{(}\omega;\mathbf{x},\mathbf{x}^{\prime}\bigr{)}=-i\hbar\sum_{s} \sum_{s^{\prime}}\frac{\bigl{(}\hat{j}_{a}(\mathbf{x})\bigr{)}_{ss^{\prime}} \bigl{(}\hat{j}_{b}(\mathbf{x}^{\prime})\bigr{)}_{s^{\prime}s}}{(\hbar\omega-( \varepsilon_{s}-\varepsilon_{s^{\prime}}))}(\rho_{0 ss}-\rho_{0s^{\prime}s^{ \prime}})\frac{1}{(\varepsilon_{s}-\varepsilon_{s^{\prime}})}, \tag{11}\]  
where \(\rho_{0 ss}=f(\varepsilon)=1/\Bigl{(}e^{(\varepsilon-\mu)/k_{B}T}+1\Bigr{)}\) denotes the Fermi distribution and \(\mu\) is the electrochemical potential.