# A High-Gain CNTFET-Based LNA Developed Using a Compact Design-Oriented Device Model
## 3 DC CNTFET Simplified Model  
In order to simplify the design process of the amplifier, a compact, designer-friendly model of the behavior of the drain current in a CNTFET that is suitable for estimating operating points can also be useful. To develop such a model, it is important to understand the differences in behavior between a conventional MOSFET and a CNTFET of the type analyzed in [37], which, unlike many other studied structures, exhibits a drain current relationship with respect to biasing voltages that is almost exactly a separable function of the gate and drain voltages. This can be seen in Figure 3, which shows the input and output characteristics on a normalized vertical axis (i.e., individual current curves had been divided by their mean value). For what regards the input characteristics, it can be seen that the drain current is almost linearly dependent on the gate overdrive voltage above the threshold voltage and that the curve shape does not depend at all on drain voltage. On the other hand, for what concerns the CNTFET output characteristics, there is a very weak, and negligible, dependence of their shape on the gate voltage, but unlike conventional MOSFETs, the saturation region, where the characteristic becomes almost linear, can be assumed to start from a fixed voltage that it is not bound to the gate overdrive voltage, thus allowing a great simplification of both the empirical model and of the design procedure. It is thus reasonable to employ a simple empirical model with separated variables, such as:  
\[I_{\text{D}}=g_{\text{m}}\,f_{\text{G}}(V_{\text{GS}}-V_{\text{TH}})\,f_{ \text{D}}(V_{\text{DS}}/V_{\text{P}})\,(1+\lambda\,V_{\text{DS}}) \tag{6}\]  
where \(f_{\text{G}}\) should mimic the shape on the left panel of Figure 3, and \(f_{\text{D}}\) that on the right panel. Figure 2: CNTFET noise model in triode region. If maximum simplicity is sought, and the circuit only operates well above threshold, a simple, piecewise linear model for \(f_{\mathrm{G}}\) might suffice, such as:  
\[f_{\mathrm{G}}(v)=\min(0,v) \tag{7}\]  
otherwise, a slightly more smoothed version can better approximate sub-threshold and near-threshold behavior:  
\[f_{\mathrm{G}}(v)=\left\{ \begin{array}{ll}V_{\sigma}\,\log\Bigl{(}1+\exp( \frac{v}{V_{\sigma}})\Bigr{)}&v>0\\ V_{\sigma}\,\log\Bigl{(}1+\exp(\frac{v}{KV_{\sigma}})\Bigr{)}&v<0\end{array} \right. \tag{8}\]  
where \(V_{\sigma}\) is a parameter that determines the width of the smoothed transition region, and \(k\) is a parameter that can be used to adjust the sub-threshold transconductance. It can be noted that Equation (7) is the limit of Equation (8) as \(V_{\sigma}\to 0\), and that \(\partial f_{\mathrm{G}}(v)/\partial v|_{v\gg 0}=1\) so that \(g_{\mathrm{m}}\) retains its normal meaning. On the other hand, \(f_{\mathrm{D}}(x)\) can be assumed as a saturating power function to describe the dependence on \(V_{\mathrm{DS}}\), such as:  
\[f_{\mathrm{D}}(x)=\left\{ \begin{array}{ll}\left(n\,x-x^{n}\right)/\left(n-1 \right)&0<x\leq 1\\ 1&x>1\end{array} \right. \tag{9}\]  
which satisfies the properties \(\forall n\
eq 1\), \(f_{\mathrm{D}}(0)=0\), \(f_{\mathrm{D}}(1)=1\), \(\partial f_{\mathrm{D}}(x)/\partial x|_{x=1}=0\). We thus have a total of seven fitting parameters: \(g_{\mathrm{m}}\) and \(V_{\mathrm{TH}}\) are the slope and threshold voltage of the input characteristics, respectively, while \(V_{\sigma}\) controls the near-threshold behavior and \(k\) the sub-threshold transconductance. For the output characteristics, \(V_{\mathrm{p}}\) is the equivalent of the \"pinch-off\" voltage that denotes the starting of the saturation region, \(\lambda\) defines the output resistance (\(g_{\mathrm{o}}\simeq\lambda I_{\mathrm{D}}\)), and the exponent \(n\) controls the \"steepness\" of the transition between the triode and saturation regions. These parameters can be fitted to match the simplified model to the full-fledged simulation, as shown in Figure 4 for a \(W=1\,\mathrm{\SIUnitSymbolMicro m}\) CNTFET (100 nanotubes).