## 2 Electrical Response of a Bi-Layer Graphene Nanoribbon THz Detector
### A Glimpse at the Graphene Nanoribbon Band Structure  
The transport model presented in [25; 26; 27; 28] is based on the knowledge of the band structure of the GNRs, which can be expressed in terms of the energy dispersion relation of the conduction electrons (the so-called \(\pi\)-electrons) in graphene [29]:  
\[E^{(\pm)}(\textbf{k})=\pm\gamma\Bigg{[}1+4\cos\Bigg{(}\frac{\sqrt{3}k_{x}a_{ 0}}{2}\Bigg{)}\cos\Bigg{(}\frac{k_{x}a_{0}}{2}\Bigg{)}+4\cos^{2}\Bigg{(}\frac {k_{y}a_{0}}{2}\Bigg{)}\Bigg{]}^{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!  
where \(E^{(\pm)}\) is the energy (signs \(+\) and \(-\) denote the conduction and valence band, respectively), \(\gamma\) = 2.7 eV is the carbon-carbon interaction energy, and \(a_{0}\) = \(\sqrt{3}b\) is a lattice constant, being \(b\) = 1.42 A the interatomic distance.  
At equilibrium, the energy distribution function of \(\pi\)-electrons is given by the Dirac-Fermi function:  
\[F\left[E^{(\pm)}\right]=\frac{1}{e^{E^{(\pm)}/\,k_{B}T_{0}}\,+1} \tag{2}\]  
where \(k_{B}\) is the Boltzmann constant and \(T_{0}\) is the absolute temperature.  
Graphene nanoribbons are obtained by cutting a graphene layer in ribbons of a transverse width \(w\) much smaller than the longitudinal length (\(w<<l\), see Figure 1b). Two basic shapes may be obtained, \"armchair\" (A-GNR) or \"zig-zag\" (Z-GNR): they differ in the edges, which show a 30\({}^{\circ}\) difference in their orientation within the graphene sheet.  
The width \(w\) of a graphene nanoribbon is directly related to the number of \(N\) dimers (two carbon sites) for the A-GNRs, and the number \(N\) of zig-zag lines for the Z-GNRs. The unit cell has translational length \(T\) = \(\sqrt{3}a_{0}\) (\(T=a_{0}\)) and width \(w\) = \(Na_{0}\) / 2 (\(w\) = \(\sqrt{3}Na_{0}\) / 2) for A-GNRs (Z-GNRs).  
In the nearest-neighbor tight-binding approximation, the dispersion relation of A-GNRs for the generic \(\mu\)-th band is given by [30, 31]:  
\[E_{\mu}(k)=\gamma\sqrt{1+2\varepsilon_{\mu}\cos(kT\,/\,2)+\varepsilon_{\mu}^{ 2}}\quad\text{for}\ \ -\frac{\pi}{T}\leq k\leq\frac{\pi}{T} \tag{3}\]  
where \(\varepsilon_{\mu}\) = 2 cos(\(p_{\mu}\)), and the transverse wavenumber \(p_{\mu}\) is determined by the edge boundary condition:  
\[p_{\mu}=\mu\pi\,/\,(N+1),\ \ \ \mu=1,2,...N. \tag{4}\]  
For Z-GNRs the energy spectrum may not be given in closed form. However, in the nearest-neighbor tight-binding approximation, it can be derived by solving the eigenvalue problem [30, 31]:  
\[H(k)\mathbf{u}=E(k)\mathbf{u} \tag{5}\]  
where \(H\) is a \(2N\times 2N\) symmetric sparse matrix that has nonzero elements only in the first diagonals below and above the main diagonal, defined as:  
\[H_{J,J+1}=\gamma\begin{cases}2\cos(a_{0}k\,/\,2)&j\ \ \text{odd}\\ 1&j\ \ \text{even}\end{cases} \tag{6}\]  
Note that A-GNRs are metallic only for \(N\) = \(3q-1\), with \(q\) = 0,1,2,.., whereas for other values they are semiconducting. As for Z-GNRs, they have partially flat bands owing to the degeneration of edge states at \(k\) = \(\pm\pi\) / \(T\); hence, their bandgap is always zero and their behavior is always metallic [31].  
Figure 2 shows the typical band structures for A-GNRs, either in metallic and semiconducting cases. Figure 3, instead, reports the case of Z-GNRs.