# Simulation Modelling of Silicon Gated Field Emitter Based Electronic Circuits
## 3 Simulation Model  
The Fowler-Nordheim (F-N) model was used to establish the VT model [11; 20; 21; 22]. The F-N equation is given as  
\[\textbf{{I}}=\textbf{{a}}_{\text{FN}}\textbf{{V}_{G}}^{2}\textbf{{exp}}\left(- \frac{\textbf{{b}}_{\text{FN}}}{V_{G}}\right) \tag{1}\]  
Here, V\({}_{\text{G}}\) is applied gate potential in V, \(\textbf{{a}}_{\text{FN}}\) and b\({}_{\text{FN}}\) are F-N coefficients and are defined as follows:  
\[\textbf{{ln}}\left(\frac{\textbf{{I}}}{\textbf{{V}_{G}}^{2}}\right)=\textbf{{ ln}}\left(\textbf{{a}}_{\text{FN}}\right)-\frac{\textbf{{b}}_{\text{FN}}}{V_{G}} \tag{2}\]  
\[\textbf{{b}}_{\text{FN}}=\frac{\textbf{{0}}\textbf{{.95}}\textbf{{.6}}\textbf {{.83}}\times\textbf{{10}}^{7}\textbf{{.}}\varphi\frac{3}{2}}{\beta} \tag{3}\]  
where \(\varphi\) is the emitter work function (eV), and \(\beta\) is the field enhancement factor. Here, the work function is assumed to be 4.05 eV for Si. Further, to predict the model accurately, a hyperbolic tangent or tanh function was incorporated [17]. Figure 2: (**a**) Transfer characterization comparison between experiment and LTspice model (inset is the F-N plot) and gate current comparison. (**b**) Show the output characteristics and gate current comparison between experiment and model. Axis are marked with “Left” and “Right”. The model provides [17]  
\[\mathbf{I_{C,Model}}=\mathbf{I_{E,Model}}\cdot\mathbf{0.5}\cdot\left(\mathbf{1}+\tan\hbar\left( \frac{(V_{C}-\mathbf{d})}{\mathbf{e}}\right)\right) \tag{4}\]  
where the parameters d and e (two sets of coefficients for tanh) are dependent on V\({}_{\text{G}}\). Ic is = collector current (A), and I\({}_{\text{E}}\) is = emitter current (A). A detailed model development method can be found in our previous work [17]. Figure 3 shows the circuit diagram and the temporal response characteristics of a VT-based Wien oscillator circuit. The Wien oscillator [23,24] is based on a frequency-selective form of the Wheatstone bridge circuit [25]. The Wien Bridge Oscillator is a two-stage RC coupled amplifier circuit that has good stability at its resonant frequency and low distortion. The Wien Bridge Oscillator uses a feedback circuit consisting of a series RC circuit connected with a parallel RC of the same component values, producing a phase delay or phase advance circuit depending upon the frequency. Figure 3a shows the Wien oscillator circuit along with the values of the passive components, where V1 = 250 V and V\({}_{\text{BIAS}}\) = 37 V. Figure 3b shows the temporal response of the Wein oscillator simulation circuit. The observed oscillation frequency is \(\approx\)102 kHz with a V\({}_{\text{PP}}\) of \(\approx\)26 V. The circuit includes four VTs. Among the four VTs, the rightmost one is used as a load for the simulation. Figure 4 shows the schematic and temporal response characteristics of the VT-based Ring oscillator circuit. The Ring oscillator [26] is a combination of inverters connected in series with a feedback connection. The output of the final stage is again connected to the initial stage of the oscillator. The values for the passive components that can be seen in Figure 4a,b show the temporal response of the Ring oscillator simulation circuit. The circuit is comprised of three VTs and passive components (resistors and capacitors). The applied V\({}_{1}\) = 350 V and V\({}_{\text{BIAS}}\) = 38 V. Observed oscillation frequency is \(\approx\) 1.12 MHz with a V\({}_{\text{PP}}\) of  
Figure 3: (**a**) LTspice model of three-device Wien oscillator circuit plus output buffer (rightmost VT) and (**b**) output waveform. \(\approx\) 4.5 V. The V\({}_{\rm PP}\) of this Ring oscillator circuit could be limited by the RC delay caused by the device capacitance [17] and biasing resistors. A simple NOR gate [27] can be constructed using resistor-transistor logic (RTL) switches connected together with the inputs connected directly to the transistor gates.