## 3 Discussion: Two Mechanisms of THz Resonance Excitation for the Proposed Detector
### New THz Resonances for the Tunneling Effect  
Let us now investigate a new mechanism of resonance excitation in the THz range, associated to the tunneling effect. To this end, we will refer to the detector in Figure 1a, assuming the contacts to be made by two metallic A-GNRs, of width \(w\!=\!2\,\text{nm}\) and length \(l\!=\!50\) nm. In addition, a dielectric layer of permittivity \(\varepsilon_{r}\!=\!2.2\,\) and thickness \(0.1\,\upmu\)m is considered. According to criterion Equation (27), for such a GNR length no plasmon resonances fall in the THz range. To maximize the tunneling effect, we assume that the two GNR layers are separated by a minimum distance, namely the Van der Waals distance \(\delta\!=\!0.34\) nm: for such a case, a measured value of binding energy of \(h\omega_{t}\!\approx\!35\) meV was reported, which corresponds to a tunneling frequency of \(\omega_{t}\!\approx\!10^{3}\) rad/s [34]. The binding energy and, thus, the tunneling frequency, may be varied by changing  
Figure 4: Plasmon resonance in the input admittance of the detector in Figure 1, with a GNR of length \(1\,\upmu\)m, compared to the case of Cu electrodes short-circuited. the interlayer distance [39] or intercalating the layers with dopants, or even rotating the GNRs to form an angle, in order to excite the so-called chiral or Klein tunneling effect [40]. All these mechanism could be investigated in view of realizing multi-frequency detectors. For the sake of simplicity, we consider the impinging external field, \(\mathbf{E}^{i}\), to be a time-harmonic plane-wave with a wave-vector orthogonal to the \(z\)-axis. The general case of a field hitting the detector with any other angle may be handled by projecting the field in the plane of the detector. To solve MTL equations, let us focus on the effect of the additional term \(\mathbf{L}_{nn}\) Equation (20), assuming that the quantum capacitance is such that in Equation (14) it is \(\mathbf{U}+\mathbf{K}=\mathbf{U}\). Having assumed a length below 100 nm, in the THz range the line is electrically short and it is possible to disregard the spatial variation of the distribution of the currents along it. In other words, we can neglect the spatial dispersion introduced by Equation (20) and focus on the frequency dispersion. In addition, since \(\mathbf{v}\approx\)10\({}^{12}\) Hz, in the THz range it is also \(\omega^{\prime}\approx\omega\). As a consequence, the tunneling operator Equation (20) may be approximated as:  
\[\mathbf{L}_{nn}(\mathbf{I}(\omega))\approx\frac{1}{\pi}\Bigg{(}\frac{c}{v_{F}} \Bigg{)}^{2}\frac{\omega_{t}^{2}}{4\omega_{t}^{2}-\omega^{2}}\Bigg{[} \begin{array} []{cc}1&-1\\ -1&1\end{array} \Bigg{]}\mathbf{L}_{M}\mathbf{I}(\omega) \tag{28}\]  
Therefore, the tunneling effect introduces a singularity in operator (Equation (28), for a frequency \(\omega=2\omega_{t}\). This singularity is responsible for a resonance peak in the response of the device. For instance, Figure 5 shows the absolute value of the line transmission coefficient (_i.e._, the scattering parameter \(|\,S_{2_{1}}(\omega)\,|\)), assuming a matched load condition, in the absence of an external field. Without the tunneling effect, this coefficient exhibits the expected behavior for a matched load, but in the presence of tunneling there is a pronounced absorption peak corresponding to the abovementioned frequency \(f_{r}=2\omega_{t}\,/\,2\pi=3.2\) THz. The peak width can be modulated by the collision frequency, as shown in Figure 6.  
Figure 5: Transmission coefficient \(|\,S_{2_{1}}(\omega)\,|\) for GNR lines with matched loads, in the absence of external fields, with or without the tunneling effect. Such a behavior may be exploited for using the circuit in Figure 1a as a detector. To this end, let us assume that the external field \(\,\mathbf{E}^{i}\) induces currents only on the outermost GNR layer, being the inner one shielded by it. This is reasonable, if we take into account the extremely short distance between the two GNR layers.