## 2 Quasi-Static Analysis Using the Proposed Equivalent Circuit Model
### Analysis Using a Small-Signal Method at The Subthreshold Region  
A FET-based resistive self-mixing with a zero-biased channel is analytically derived for the drain current, which is represented in previous studies by the channel conductance and applied  
Figure 2: Schematic diagram by using the proposed superposition method at the input interconnection lines of the plasmon detector. mixing signal for a single device of strong inversion (\(V_{gs}>V_{th}\)) for purposes of simplicity [19; 25]. The time-independent drain current in the proposed detector model is derived based on previous studies as follows:  
\[i_{d}(t)=g_{c}(t)\cdot v_{IN}(t). \tag{9}\]  
The plasmonic conversion transconductance \(g_{c}(t)\) of the detector operating at strong inversion is approximated as follows:  
\[g_{c}(t)=k_{n}\bigg{(}v_{gs}(t)+V_{GS}-V_{th}-\frac{v_{ds}(t)}{2}\bigg{)}, \tag{10}\]  
where \(k_{n}\) is the transconductance parameter of the FET. The parameter \(k_{n}\) denotes the product of the process transconductance parameter \(\mu C_{ox}\) and the aspect ratio of the transistor \(W/L\) where \(\mu\) is electron mobility, \(C_{ox}\) is the oxide capacitance, \(W\) and \(L\) are the channel width and length, and \(V_{th}\) is the threshold voltage of the detector device. The self-mixing detector is simultaneously applied with the same AC signal to each port of the resistive mixer and all AC signals are equal [19]. The relationship between all ac signals and the terahertz input signal received from the integrated antenna is as follows:  
\[v_{THz}(t)=v_{gs}(t)=v_{ds}(t), \tag{11}\]  
\[v_{THz}(t)=V_{THz}\sin(\omega t). \tag{12}\]  
Modifying (10) using (11) and (12) and substituting into (9), the current flowing through the drain node of the detector device is modified as  
\[i_{d}(t)=\frac{1}{4}k_{n}V_{THz}^{2}+k_{n}(V_{GS}-V_{th})V_{THz}\sin(\omega t) -\frac{1}{4}k_{n}V_{THz}^{2}\cos(2\omega t), \tag{13}\]  
where the first term denotes the DC component current, and the second and third terms denote the ac component current. All ac currents at the output node are filtered by the RC filter of the drain node that consists of the resistor \(R_{DS}\) and parasitic capacitor \(C_{DS}\). Only the dc current is extracted as  
\[I_{D}=\frac{1}{4}k_{n}V_{THz}^{2}. \tag{14}\]  
The performance of the detector is predicted from the detected DC output current. The voltage responsivity \(R_{V}\) and noise equivalent power NEP are indicators of the detector performance and are expressed as follows:  
\[R_{V}=\frac{I_{D}}{P_{in}}.\frac{1}{g_{c}}=\frac{R_{in}}{4(V_{GS}-V_{th})}, \tag{15}\]  
\[NEP=\frac{\sqrt{N_{v}}}{R_{V}}=\frac{\sqrt{4 KT/Sc}}{R_{V}}=\sqrt{\frac{4^{3} KT(V_{GS}-V_{th})}{k_{n}\cdot{R_{in}}^{2}}}, \tag{16}\]  
where the input power \(P_{in}\) is \(V_{THz}^{2}/R_{in}\), and \(N_{v}\) denotes the noise spectral density. The transconductance \(g_{c}\) given only dc term by the ac signals filtered from the output node is \(k_{n}(V_{GS}-V_{th})\). The voltage responsivity and NEP appear as a function of the gate-to-source voltage. The maximum voltage responsivity and minimum NEP are obtained as the gate-to-source voltage approaches the threshold voltage in a way similar to the theoretical analysis in previous studies. In contrast to conventional analysis methods, the detector device for the subthreshold region is analyzed by small-signal modeling as shown in Figure 3. The total current flowing to the detector device is obtained by applying Kirchhoff's rules of current to the output node as follows:  
\[i_{d}(t)=g_{c}(t)\cdot v_{IN}(t). \tag{9}\]0The voltage at the output node denotes the sum of the detected dc output voltage and AC voltage, where the AC voltage is the signal presented by the capacitor from the gate of the detector.