## 2 Electrical Response of a Bi-Layer Graphene Nanoribbon THz Detector
### Transport Model along Graphene Nanoribbons, without the Tunneling Effect  
The cross-section sizes of GNRs are typically large enough (at least 1 nm in the quantum confined directions) to have local crystal structures and to allow using the semi-classical transport model. Thus, the electrons are regarded as classical particles, which move under the action of a longitudinal electric field according to Boltzmann transport equation. In the THz range we can assume that the following conditions hold:  
1. the GNR width \(w\) is electrically small;
2. the transverse currents may be neglected; and
3. only intraband transitions are considered, whereas interband ones are not allowed. Figure 3: Typical band structure for Z-GNRs _vs._ normalized wavenumber \(k\)’ = \(kT\) / \(\pi\). Figure 2: Typical band structure for A-GNRs _vs._ normalized wavenumber \(k\)’ = \(kT\) / \(\pi\) : (**a**) metallic; and (**b**) semi-conducting. Conditions (i) and (iii) are fully satisfied in the THz range, since the GNR width \(w\) is far below the minimum wavelength (0.03 mm at 10 THz), and the interband transitions occur at optical frequencies. Condition (ii) is fulfilled for large aspect-ratio GNRs. Let us consider time-harmonic variation for the fields, expressed as \(X(z,t)=\mathrm{Re}\Big{\{}\hat{X}_{z}\exp[i(\omega t-\beta z)\Big{\}}\), where \(\beta\) is the wavenumber, and let us indicate with \(J_{z}(z,t)\) and \(E_{z}(z,t)\) the longitudinal component of the current density and the electric field, respectively. The latter is written as:  
\[E_{z}(z,t)=E_{z}^{i}(z,t)+E_{z}^{s}(z,t) \tag{7}\]  
If the tunneling effect is neglected, the current density along the \(n\)-th signal line (\(n=1\),2) only depends on the electric field on the same line, and the following generalized [PERSON]'s law may be written in the wavenumber domain [32,33]:  
\[\hat{J}_{z,n}(\beta,\omega)=\hat{\sigma}_{n}(\beta,\omega)\hat{E}_{z,n}(\beta, \omega)\ \ \ n=1,2 \tag{8}\]  
where \(\hat{\sigma}_{n}(\beta,\omega)\) is the generalized longitudinal conductivity, which may be expressed as:  
\[\hat{\sigma}_{n}(\beta,\omega)\equiv-i\frac{2e^{2}}{\pi^{2}w}\frac{v_{F}}{ \hbar}\frac{1}{\omega-i\
u_{n}}M_{n}\Bigg{[}1+\alpha_{n}\Bigg{(}\frac{v_{F} \beta}{\omega-i\
u}\Bigg{)}^{2}\Bigg{]} \tag{9}\]  
where \(e\) is the electron charge, \(\hbar\) is the Planck constant, \(v_{F}\) the Fermi velocity, usually of the order of \(10^{6}\) m/s, and \(\
u_{n}\) is the collision frequency. The functions \(M_{n}\) and \(\alpha_{n}\) appearing in Equation (9) are expressed in terms of the energy bands Equation (1), \(E_{\mu,n}^{(\pm)}\) and by their associated velocities \(\
u_{\mu,n}^{(\pm)}=dE_{\mu,n}^{(\pm)}/d(\hbar k)\), according to:  
\[M_{n}=\frac{\hbar}{\
u_{F}}\sum_{i=1}^{N}\int\limits_{0}^{\pi/}\Bigg{[}\
u_{ \mu,n}^{(+)2}\Bigg{(}-\frac{dF}{dE_{\mu,n}^{(+)}}\Bigg{)}+\
u_{\mu,n}^{(-)2} \Bigg{(}-\frac{dF}{dE_{\mu,n}^{(-)}}\Bigg{)}\Bigg{]}dk \tag{10}\]  
\[\alpha_{n}=\frac{\hbar}{M_{n}v_{F}^{3}}\sum_{\
u=0}^{N-1}\int\limits_{0}^{\pi/ }\Bigg{[}\
u_{\mu,n}^{(+)4}\Bigg{(}-\frac{dF}{dE_{\mu,n}^{(+)}}\Bigg{)}+\
u_ {\mu,n}^{(-)4}\Bigg{(}-\frac{dF}{dE_{\mu,n}^{(-)}}\Bigg{)}\Bigg{]}dk \tag{11}\]  
where \(F\) s the Dirac-Fermi distribution defined by Equation (2), \(M_{n}\) represents the equivalent number of conducting channels, which depends on GNR chirality, width, and temperature, too, as shown in [27]. In the following, we assume that the two GNRs have the same chirality and width; therefore, all the parameters in Equations (8)-(11) are the same, in particular: \(M_{1}=M_{2}\), \(\alpha_{1}=\alpha_{2}\), and \(\
u_{1}=\
u_{2}\).