## 3 Transport Model
### Transport Model for the Graphene Layer  
The \(\pi\)-electrons in an infinite graphene layer behave as a two-dimensional electron gas moving on the surface under the action of the electric field. We disregard the Lorentz force term due to the magnetic field, because it does not contribute to the linear response of the electron gas. The electron gases are described by the distribution functions \(\
ot{f}^{\pm})=\
ot{f}^{\pm})(**r**,**k**,_t_), where **r** is the vector position in the direct space with the origin at \(O\), **k** is the electron wave-vector restricted to the first Brillouin zone and \(\sum_{\mathrm{BZ}}\) and \(t\) represents the time variable. Here, the \(+\) sign denotes the conduction band, the \(-\) sign denotes the valence band. Each of these distribution functions satisfies the quasi-classic Boltzmann kinetic equation,  
Figure 5: The energy band structure of a zigzag nanoribbon _versus_ the normalized wavenumber \(k^{\prime}=kl/\pi\). (**a**) \(N=5\); (**b**) \(N=20\). \[\frac{\partial f^{(\pm)}}{\partial t}+\mathbf{v}^{(\pm)}\cdot\frac{\partial f^{(\pm)} }{\partial\mathbf{r}}+\frac{e}{h}\mathbf{E}_{\cdot}\cdot\frac{\partial f^{(\pm) }}{\partial\mathbf{k}}=-\
u\Big{(}f^{(\pm)}-f_{0}^{(\pm)}\Big{)} \tag{4}\]  
where \(e\) is the electron charge, \(\mathbf{E}_{t}=\mathbf{E}_{t}(\mathbf{r},t)\) represents the tangential component of the electric field at the graphene surface, \(\mathbf{v}\) is the relaxation frequency ( \(\
u\equiv 5\cdot 10^{(11}s^{-1)}\) and \(\mathbf{v}^{(\pm)}=\mathbf{v}^{(\pm)}(\mathbf{k})\) is the velocity of the \(\pi\)-electrons in the conduction/valence band given by:  
\[\mathbf{v}^{(\pm)}(\mathbf{k})=\frac{1}{h}\frac{\partial E^{(\pm)}}{\partial \mathbf{k}} \tag{5}\]  
The symbols, \(\partial/\partial\mathbf{r}\) and \(\partial/\partial\mathbf{k}\), denote the gradients with respect to the variables, \(\mathbf{r}\) and \(\mathbf{k}\), respectively. The distribution functions of the \(\pi\)-electrons at equilibrium are:  
\[f_{0}^{(\pm)}(\mathbf{k})=\frac{1}{2\pi^{2}}\,F\Big{[}E^{(\pm)}(\mathbf{k}) \Big{]} \tag{6}\]  
Let us consider a time-harmonic field, in the form \(\mathbf{E}_{\cdot}(\mathbf{r},t)=\mathrm{Re}\big{[}\hat{\mathbf{E}}_{\cdot} \exp\big{[}i\big{(}\omega t-\mathbf{\beta}\cdot\mathbf{r}\big{)}\big{]}\big{]}\), where \(\mathbf{\beta}\) is the wave-vector defined in the two-dimensional space. Let us set \(f^{(\pm)}=f_{0}^{(\pm)}(\mathbf{k})+\delta f^{(\pm)}(\mathbf{r},\mathbf{k},t)\) and consider a time-harmonic perturbation \(\delta f^{(\pm)}(\mathbf{r},\mathbf{k},t)=\mathrm{Re}\big{\{}\delta f_{1}^{( \pm)}(\mathbf{k};\omega,\mathbf{\beta})\exp\big{[}i\big{(}\omega t-\mathbf{ \beta}\cdot\mathbf{r}\big{)}\big{]}\big{]}\), where \(\delta f_{1}^{(\pm)}\) is a small quantity to be found: using Equation (4) and retaining only the first order terms, we obtain:  
\[\delta f_{1}^{(\pm)}=i\frac{e}{h}\frac{1}{\Big{(}\omega-\mathbf{v}^{(\pm)} \cdot\mathbf{\beta}-i\
u\Big{)}}\hat{\mathbf{E}}_{\cdot}\cdot\frac{\partial f _{0}^{(\pm)}}{\partial\mathbf{k}} \tag{7}\]  
Let us now introduce the time harmonic surface current density \(\mathbf{j}(\mathbf{r},t)=\mathrm{Re}\big{[}\hat{\mathbf{J}}(\omega,\mathbf{ \beta})\exp\big{[}i\big{(}\omega t-\mathbf{\beta}\cdot\mathbf{r}\big{)}\big{]} \big{]}\): its amplitude, \(\mathbf{j}(\omega,\mathbf{\beta})\), can be computed by summing the contributions from all the sub-bands in the first Brillouin zone:  
\[\hat{\mathbf{J}}(\omega,\mathbf{\beta})=\int_{z_{\mathrm{zz}}}\Big{[}e^{\
u ^{(\pm)}}(\mathbf{k})\delta f_{1}^{(\pm)}(\mathbf{k};\omega,\mathbf{\beta})+e ^{\
u^{(\pm)}}(\mathbf{k})\delta f_{1}^{(\pm)}(\mathbf{k};\omega,\mathbf{ \beta})\Big{]}d^{\pm}\mathbf{k} \tag{8}\]  
Here, the first term is related to the valence bands and the second one to the conduction bands.