# A High-Gain CNTFET-Based LNA Developed Using a Compact Design-Oriented Device Model
## 2 CNTFET Extrinsic Noise Model  
As a first step to designing a low-noise amplifier, a suitable model of the noise sources within the employed active devices is necessary. We assume that noise can be modeled by adding stochastic current sources in parallel to the main small-signal elements of the standard FET equivalent circuit and to the contact resistances [25], as shown in Figure 1, which is valid in the saturation region of the transistor. Here, \(g_{\text{m}}\) and \(g_{\text{o}}\) are the intrinsic FET transconductance and output conductance, respectively, \(R_{\text{d}}\) and \(R_{\text{s}}\) represent the channel resistance at the drain and source terminals, respectively, while \(C_{\text{gs}}\) and \(C_{\text{gd}}\) model the capacitive effects of the gate. The noise sources \(\overline{i_{n,x}^{2}}\), where \(x\) can denote the noise associated with the source (Rs), drain (Rd) or the intrinsic device (int), can be assumed to have a power spectral density proportional to the transistor drain current \(I_{\text{D}}\):  
\[\overline{i_{n,x}^{2}}=2\,q\,I_{\text{D}}\,F_{x} \tag{1}\]  
where \(q\) is the elementary charge and \(F_{x}\) is the appropriate Fano factor. According to [25], it is common to assume \(F_{\text{Rs}}=F_{\text{Rd}}=0.3\) if the contact resistances are mostly due to the doped nanotube extensions, while \(F_{\text{int}}\) might depend both on technological details of the nanotubes and transistor bias. However, from [38], \(F_{\text{int}}\) can be assumed to be almost constant and smaller than 0.1 for a wide range of typical bias currents. For simplicity, we will conservatively assume \(F_{\text{int}}=0.1\) in our simulations. From that, we aim to derive a simplified extrinsic model composed of just the transconductance element, the output conductance, and a single noise current source, all in parallel. From a rapid inspection of the equivalent circuit reported in Figure 1, the extrinsic transconductance \(g_{\text{m}}^{*}\) can be expressed as:  
\[g_{\text{m}}^{*}=\frac{g_{\text{m}}}{1+g_{\text{m}}\cdot R_{\text{s}}} \tag{2}\]  
and the extrinsic output conductance \(g_{\text{o}}^{*}\) can also be expressed as:  
\[g_{\text{o}}^{*}=\frac{g_{\text{o}}}{1+g_{\text{m}}\cdot R_{\text{s}}+g_{\text {o}}\cdot(R_{\text{s}}+R_{\text{d}})} \tag{3}\]  
where obviously both of them can be approximated by \(g_{\text{m}}\) and \(g_{\text{o}}\), respectively, when the contact resistances \(R_{\text{s}}\) and \(R_{\text{d}}\) are vanishing. The total noise density \(\overline{i_{n,\text{TOT}}^{2}}\) in the saturation region can be calculated as below:  
Figure 1: CNTFET noise model in the saturation region. \[\overline{i_{\text{n,TOT}}^{2}}=\left[\frac{(g_{\text{m}}+g_{\text{o}})\cdot R_{ \text{s}}}{1+g_{\text{m}}\cdot R_{\text{s}}+g_{\text{o}}\cdot(R_{\text{s}}+R_{ \text{d}})}\right]^{2}\cdot\overline{i_{\text{n,Rs}}^{2}}+\left[\frac{1}{1+g_{ \text{m}}\cdot R_{\text{s}}+g_{\text{o}}\cdot(R_{\text{s}}+R_{\text{d}})} \right]^{2}\cdot\overline{i_{\text{n,int}}^{2}} \tag{4}\] \[+\left[\frac{g_{\text{o}}\cdot R_{\text{d}}}{1+g_{\text{m}}\cdot R _{\text{s}}+g_{\text{o}}\cdot(R_{\text{s}}+R_{\text{d}})}\right]^{2}\cdot \overline{i_{\text{n,Rd}}^{2}}\]  
which is in parallel to the equivalent output conductance \(g_{\text{o}}^{*}\). For completeness, the CNTFET simplified extrinsic model with noise sources in the triode region can be derived similarly. The small-signal equivalent circuit is reported in Figure 2, and the total noise in the triode region can be calculated as below:  
\[\overline{i_{\text{n,TOT}}^{2}}=\left[\frac{R_{\text{s}}}{R_{\text{s}}+R_{ \text{ds}}+R_{\text{d}}}\right]^{2}\cdot\overline{i_{\text{n,Rs}}^{2}}+ \left[\frac{R_{\text{ds}}}{R_{\text{s}}+R_{\text{ds}}+R_{\text{d}}}\right]^{ 2}\cdot\overline{i_{\text{n,int}}^{2}}+\left[\frac{R_{\text{d}}}{R_{\text{s}} +R_{\text{ds}}+R_{\text{d}}}\right]^{2}\cdot\overline{i_{\text{n,Rd}}^{2}} \tag{5}\]  
where \(R_{\text{ds}}\) is the on-state resistance of the transistor in the triode region.