Short-channel MOSFET Thermal Noise Modeling

Using the inversion coefficient (Version 1)

Abstract

This notebook presents a model of the white noise generated by the MOS transistor that is valid in all regions of operation, from weak to strong inversion. It also describes an empirical model that accounts for the increase of the white noise due to short-channel effects.

Licensing

This document is licensed under the Creative Commons License CC BY-NC-SA

1 Introduction

Noise is important for analog IC design because it represents the fundamental lower limit for signal detection. At lower frequency, the flicker noise is dominant, particularly for advanced CMOS technologies. However, flicker noise can be reduced or even eliminated by circuit techniques such as autozero or chopper stabilization [1]. The remaining white noise then becomes the dominant noise. This white noise is mostly contributed by the MOS transistors. In this notebook we will model the white noise generated by the MOS transistor in all regions of operation, from weak to strong inversion. We also will develop a model that accounts for the increase of the white noise due to short-channel effects.

2 Channel thermal noise conductance

2.1 Long-channel (without VS)

The PSD of the drain current fluctuations due to thermal noise in the whole channel is obtained by integrating along the channel resulting in [2] \[\begin{equation} S_{\Delta I_D^2} = 4kT \cdot G_n, \end{equation}\] where \(G_n\) is the thermal noise conductance at the drain which is given by [3] [2] \[\begin{equation} G_n = \frac{1}{L^2} \cdot \int_0^L W \cdot \mu \cdot (-Q_i(x)) \cdot dx = \mu \cdot \frac{W}{L^2} \cdot \int_0^L -Q_i(x) \cdot dx = \frac{\mu}{L^2} \cdot |Q_I|, \end{equation}\] where it has been assumed that mobility is constant along the channel. We see that the thermal noise PSD and conductance at the drain is proportional to the total mobile (or inversion) charge stored in the channel \[\begin{equation} Q_I = W \cdot \int_0^L -Q_i(x) \cdot dx. \end{equation}\] Note that the above result only holds for a long-channel transistor. In the case of a short-channel device, the assumption of constant mobility does not hold anymore and a different approach is needed [4]. This will be discussed in more details below.

At low-frequency, the drain current fluctuations due to the thermal noise generated in the channel can therefore be modelled by a noiseless transistor and a noise current source connected between the drain and source of the transistor as illustrated in Figure 1.

Figure 1: Thermal noise modeled as a noisy current source between source and drain.

Of course the thermal noise conductance \(G_n\) depends on bias. To evaluate this bias dependence, we start by normalizing \(G_n\) to \(G_{spec}\) to evaluate it in terms of the normalized source and drain charges \(q_s\) and \(q_d\) using the charge-based model \[\begin{equation} g_n \triangleq \frac{G_n}{G_{spec}} = \int_0^1 q_i(\xi) \cdot d\xi = q_I, \end{equation}\] where \[\begin{equation}\label{eqn:gspec_def} G_{spec} \triangleq \frac{I_{spec}}{U_T} \end{equation}\] with \[\begin{equation}\label{eqn:ispec_def} I_{spec} \triangleq I_{spec\Box} \cdot \frac{W}{L} \end{equation}\] and where \(I_{spec\Box} \triangleq 2\,n\,\mu\,C_{ox}\) is an sEKV parameter [5] [6].

The charge-based drain current expression is given by [2] \[\begin{equation}\label{eqn:id_dqi_dxi} i_d \triangleq \frac{I_D}{I_{spec}} = -(2q_i+1) \cdot \frac{d q_i}{d\xi}, \end{equation}\] where \(q_i \triangleq Q_i/Q_{spec}\) is the inversion charge \(Q_i\) normalized to \(Q_{spec} \triangleq -2n U_T\,C_{ox}\) and \(\xi \triangleq x/L\) the distance along the channel normalized to the channel length \(L\) [2]. We can now express \(\xi\) in terms of the drain current according to \[\begin{equation} d\xi = -\frac{2q_i+1}{i_d} \cdot dq_i \end{equation}\] in order to perform the integration in the charge domain [2] \[\begin{align} g_n = q_I &= -\frac{1}{i_d} \cdot \int_{q_s}^{q_d} q_i \cdot (2q_i+1) \cdot dq_i = \frac{1}{i_d} \cdot \int_{q_d}^{q_s} q_i \cdot (2q_i+1) \cdot dq_i\\ &= \frac{1}{i_d} \cdot \frac{4 q_s^3+3 q_s^2-3 q_d^2-4 q_d^3}{6}\\\label{eqn:gn1} &= \frac{1}{i_d} \cdot \frac{(q_s-q_d)(4q_s^2+3q_s+4q_s q_d+3q_d+4q_d^2)}{6}. \end{align}\]

The normalized charge densities at the source \(q_s\) and at the drain \(q_d\) are defined as [2] \[\begin{align}\label{eqn:qs_qd_def} q_s &\triangleq q_i(\xi=0) = \frac{Q_i(x=0)}{Q_{spec}},\\ q_d &\triangleq q_i(\xi=1) = \frac{Q_i(x=L)}{Q_{spec}}. \end{align}\]

For a long-channel transistor, the normalized drain current is obtained by integrating \(\eqref{eqn:id_dqi_dxi}\) in the charge domain resulting in [2] \[\begin{equation}\label{eqn:id1} i_d = q_s^2-q_d^2+q_s-q_d = (q_s-q_d)(q_s+q_d+1). \end{equation}\] Replacing \(i_d\) from \(\eqref{eqn:id1}\) in \(\eqref{eqn:gn1}\) and simplifying results in \[\begin{equation}\label{eqn:qn_lin} g_n = \frac{1}{6} \cdot \frac{4q_s^2+3q_s+4q_s q_d+3q_d+4q_d^2}{q_s+q_d+1}. \end{equation}\] Eqn. \(\eqref{eqn:qn_lin}\) is valid for a long-channel transistor in all modes of operation, from weak to strong inversion and from linear to saturation. It also nicely illustrates the symmetry of the EKV model. Indeed, the expression of \(g_n\) remains unchanged when swapping \(q_s\) and \(q_d\).

In saturation, without accounting for velocity saturation (VS), \(g_n\) is then given by replacing \(q_d=0\) \[\begin{equation} g_n \cong \frac{2}{3} \cdot q_s \cdot \frac{q_s+3/4}{q_s+1} = \begin{cases} \frac{2}{3} q_s &\textsf{in strong inversion ($q_s \gg 1$)}\\ \frac{1}{2} q_s &\textsf{in weak inversion ($q_s \ll 1$)}. \end{cases} \end{equation}\]

In weak inversion, \(q_s,q_d \ll 1\) and the normalized thermal noise conductance given by \(\eqref{eqn:qn_lin}\) simplifies to \[\begin{equation}\label{eqn:gn_wi} g_n \cong \frac{q_s+q_d}{2}. \end{equation}\]

In strong inversion, \(q_s,q_d \gg 1\) and the normalized thermal noise conductance given by \(\eqref{eqn:qn_lin}\) simplifies to \[\begin{equation}\label{eqn:gn_si} g_n \cong \frac{2}{3} \cdot \frac{q_s^2+q_s q_d+4q_d^2}{q_s+q_d}. \end{equation}\]

We now will look at the impact of short-channel effects and mainly velocity saturation on the drain current thermal noise.

2.2 Short-channel (with VS)

As the drain voltage of a short-channel transistor biased in stong inversion is increased, at some point the longitudinal electric field \(E_x\) becomes equal to a critical field \(E_c\) and the drift velocity at the drain end of the channel starts to saturate. Since the drift velocity cannot increase anymore, if the gate voltage is increased a saturation charge builds up close to the drain in order to sustain the drain current despite the saturated velocity. The charge at the drain does not reduce to zero like it would for a long-channel transistor where \(E_x\) remains smaller than \(E_c\) avoiding any VS. This larger charge at the drain on one hand produces more noise but on the other since the mobility is strongly reduced it impact on the drain current fluctations is mitigated. In reality, there are many short-channel effects that influence the thermal noise as discussed in [4] [7]. In this notebook, we only will consider the effect of VS, ignoring the others. This will allow us to derive a simple empirical model which is very useful for circuit design. A similar approach was already sketched in [8] [7].

In order to obtain an expression valid in all regions of operation, we can proceed in the same way as for deriving the drain current. To obtain an expression of the drain current valid in all regions of operation, we can account for the effect of velocity saturation in strong inversion by replacing \(q_d\) by \(q_{d_{sat}}\) in the strong inversion part of the current expression and neglecting \(q_d\) compared to \(q_s\) in the weak inversion part. This allows to have the correct long-channel expression in weak inversion which is not affected by VS. This results in \[\begin{equation}\label{eqn:idsat} i_{d_{sat}} = q_s^2-q_{d_{sat}}^2+q_s, \end{equation}\] where \(q_{d_{sat}}\) is the normalized charge that is imposed at the drain because of VS. It depends on \(q_s\) and therefore the gate voltage according to \[\begin{equation}\label{eqn:qdsat_qs} q_{d_{sat}} = \frac{1}{\lambda_c} \cdot \left[\sqrt{1+\lambda_c^2\,(q_s^2+q_s)}-1\right] = \frac{\lambda_c\,(q_s^2+q_s)}{1+\sqrt{1+\lambda_c^2\,(q_s^2+q_s)}}. \end{equation}\]

Proceeding in the same way for the thermal noise conductance, we replace \(q_d\) by \(q_{d_{sat}}\) in the part \(4(q_s^3-q_d^3) \cong 4(q_s^3-q_{d_{sat}}^3)\) corresponding to strong inversion in \(\eqref{eqn:gn1}\) and neglecting \(q_d\) in the part \(3(q_s^2-q_d^3) \cong 3 q_s^2\) corresponding to weak inversion in \(\eqref{eqn:gn1}\). This leads to \[\begin{equation}\label{eqn:gnsat1} g_{n_{sat}} = \frac{1}{i_{d_{sat}}} \cdot \frac{4 q_s^3-4 q_{d_{sat}}^3+3 q_s^2}{6} \end{equation}\] Replacing \(i_{d_{sat}}\) in \(\eqref{eqn:gnsat1}\) by \(\eqref{eqn:idsat}\) results in \[\begin{equation}\label{eqn:gnsat2} g_{n_{sat}} = \frac{1}{6} \cdot \frac{4 q_s^3-4 q_{d_{sat}}^3+3 q_s^2}{q_s^2-q_{d_{sat}}^2+q_s}. \end{equation}\] Eqn. \(\eqref{eqn:gnsat2}\) gives the correct asymptote \(q_s/2\) in weak inversion. However the asymptote in strong inversion is \((1-\lambda_c/4)\,q_s\) instead of \(q_s\). We can modify \(\eqref{eqn:gnsat2}\) slightly to get the correct asymptotes in both weak and strong inversion according to \[\begin{equation}\label{eqn:gnsat3} g_{n_{sat}} = \frac{1}{6} \cdot \frac{4 (q_s^2+q_s\,q_{d_{sat}}+q_{d_{sat}}^2)+3 q_s}{q_s+q_{d_{sat}}+1}. \end{equation}\]

The normalized thermal noise conductance in strong inversion and saturation including VS simplifies to \[\begin{equation}\label{eqn:gnsat_si} g_{n_{sat}} \cong \frac{2}{3} \cdot \frac{q_s^2+q_s q_{d_{sat}}+4q_dsat^2}{q_s+q_{d_{sat}}}. \end{equation}\] where \(q_{d_{sat}}\) is given by \(\eqref{eqn:qdsat_qs}\).

In very strong inversion under heavy VS (\(\lambda_c\,q_s \gg 1\)), \(q_{d_{sat}}\) tends to \(q_s\) and \(\eqref{eqn:gnsat_si}\) reduces to \[\begin{equation}\label{eqn:} g_{n_{sat}} \cong q_s \; \textsf{for $q_s \gg 1$}. \end{equation}\]

3 Thermal noise parameter \(\delta_n\)

3.1 Definition

Several thermal noise factors can be defined according to the definitions introduced initially by van der Ziel [9]. The thermal noise parameter (TNP) \(\delta_n\) is defined as [7] [2] [10] \[\begin{equation}\label{eqn:deltan_def} \boxed{ \delta_n \triangleq \frac{G_n}{G_{dso}} = \frac{g_n}{g_{dso}} }, \end{equation}\] where \(G_{dso}\) is the drain-to-source conductance at \(V_{DS} = 0\) and \(g_{dso}\) its normalized form \[\begin{equation}\label{eqn:gdso} g_{dso} \triangleq \frac{G_{dso}}{G_{spec}} = g_{md} = g_{ms} = q_s, \end{equation}\] where \(g_{ms}\) and \(g_{md}\) are the normalized source and drain transconductances defined by \[\begin{align} g_{ms} &\triangleq \frac{G_{ms}}{G_{spec}} = -\frac{\partial i_d}{\partial v_s} = -\frac{\partial i_d}{\partial q_s} \cdot \frac{\partial q_s}{\partial v_s},\\ g_{md} & \triangleq \frac{G_{md}}{G_{spec}} = \frac{\partial i_d}{\partial v_d} = - \frac{\partial i_d}{\partial q_d} \cdot \frac{\partial q_d}{\partial v_d},\\ g_m & \triangleq \frac{G_m}{G_{spec}} = \frac{i_d}{v_g} = \frac{g_{ms} - g_{md}}{n}.\label{eqn:gmdef} \end{align}\]

Important

The \(\delta_n\) parameter shows how much the thermal noise of the active device deviates from the value it takes when it operates as a passive resistor of conductance \(G_{dso}\).

Since in the linear region (i.e. for \(V_{DS} = 0\)) the noise conductance \(G_n\) is equal to the channel conductance \(G_{dso}\), the noise parameter \(\delta_n\) is then equal to unity. In saturation, \(\delta_n\) is equal to \[\begin{equation} \delta_{n_{sat}} \triangleq \frac{g_{n_{sat}}}{G_{dso}} = \frac{g_{n_{sat}}}{g_{dso}} \end{equation}\] where \(g_{n_{sat}}\) and \(g_{n_{sat}}\) are the values of the thermal conductance in saturation.

Note

Note that Van der Ziel initially used \(\gamma\) for the thermal noise parameter defined by \(\eqref{eqn:deltan_def}\) and \(\alpha\) for the noise excess factor defined by \(\eqref{eqn:gamman_def}\) [9]. The most important noise excess factor from a circuit design point of view is the one given by \(\eqref{eqn:gamman_def}\), which has been called \(\gamma\) in many other circuit related paper instead of \(\alpha\) as it was defined initially by Van der Ziel in [9]. We will keep the circuit design definition of \(\gamma_n\) and rename the original Van der Ziel’s \(\gamma\) as \(\delta_n\).

3.2 Long-channel (without VS)

Since \(g_{dso}=q_s\), the thermal noise parameter \(\delta_n\) is given by \[\begin{equation} \delta_n \triangleq \frac{g_n}{g_{dso}} = \frac{1}{6} \cdot \frac{4q_s^2+3q_s+4q_sq_d+3q_d+4q_d^2}{q_s (q_s+q_d+1)}, \end{equation}\] which has the following asymptotes \[\begin{equation} \delta_n \cong \begin{cases} 1 &\textsf{in the linear region ($q_s = q_d$)}\\ \delta_{n_{sat}} = \frac{1}{6} \cdot \frac{4q_s+3}{q_s+1} &\textsf{in saturation ($q_s \gg q_d$)}. \end{cases}. \end{equation}\] In saturation without VS we have \[\begin{equation} \delta_{n_{sat}} \cong \frac{1}{6} \cdot \frac{4q_s+3}{q_s+1} = \begin{cases} \frac{2}{3} &\textsf{in strong inversion ($q_s \gg 1$)}\\ \frac{1}{2} &\textsf{in weak inversion ($q_s \ll 1$)}. \end{cases} \end{equation}\]

In strong inversion, \(g_n\) can be approximated by \(\eqref{eqn:gn_si}\) leading to \[\begin{equation}\label{eqn:deltan_si} \delta_{n(SI)} \cong \frac{2}{3} \cdot \frac{q_s^2+q_s q_d+q_d^2}{q_s \, (q_s+q_d)} = \frac{2}{3} \cdot \left[1+\frac{(q_d/q_s)^2}{1+q_d/q_s}\right]. \end{equation}\] For a long-channel transistor in saturation, \(q_d=0\) when \(v_d=v_p\) and \(\delta_{(SI)} \cong \frac{2}{3}\).

3.3 Short-channel (with VS)

The thermal noise parameter is plotted versus \(v_d\) in Figure 2. For a long channel transistor biased in strong inversion (\(v_p=40\) in Figure 2), \(\delta_n\) ranges from 1 when \(v_d=0\) to \(2/3\) when \(q_d=0\) as \(v_d=v_p\). When VS is included, \(q_d\) cannot go to zero and is clamped at \(q_{d_{sat}}\) as \(v_d=v_{d_{sat}}\). \(\delta_n\) is then also clamped to the value it takes for \(v_d=v_{d_{sat}}\) or \(q_d=q_{d_{sat}}\).

Figure 2: Thermal noise parameter \(\delta_n\) versus drain voltage \(v_d\) with and without VS.

We can obtain the value of \(\delta_n\) in saturation including the effect of VS from \(\eqref{eqn:gnsat3}\) \[\begin{equation}\label{eqn:deltansat1} \delta_{n_{sat}} = \frac{g_{n_{sat}}}{q_s} = \frac{1}{6} \cdot \frac{4 (q_s^2+q_s\,q_{d_{sat}}+q_{d_{sat}}^2)+3 q_s}{q_s(q_s+q_{d_{sat}}+1)}, \end{equation}\] where \(q_{d_{sat}}\) is given by \(\eqref{eqn:qdsat_qs}\).

In weak inversion, \(q_s \ll 1\) and \(\eqref{eqn:deltansat1}\) reduces to \(1/2\) as expected, because VS has no impact in weak inversion.

For a short-channel transistor in strong inversion including VS, \(\eqref{eqn:deltansat1}\) reduces to \[\begin{equation}\label{eqn:deltansat_si} \delta_{n_{sat(SI)}} \cong \frac{2}{3} \cdot \frac{q_s^2+q_s q_{d_{sat}}+q_{d_{sat}}^2}{q_s+q_{d_{sat}}} = \frac{2}{3} \cdot \left[1+\frac{(q_{d_{sat}}/q_s)^2}{1+q_{d_{sat}}/q_s}\right]. \end{equation}\]

As shown below, in strong inversion (\(q_s \gg 1\)), \(q_{d_{sat}}\) tends to \(q_s\) \[\begin{equation} q_{d_{sat}} \cong q_s \; \textsf{for $q_s \gg 1$}, \end{equation}\] and hence \(g_{n_{sat(SI)}}\) in strong inversion with VS also tends to \(q_s\) and therefore \(\delta_{n_{sat}}\) tends to unity.

The thermal noise parameters in saturation is plotted versus the inversion coefficient \(IC\) in Figure 3 for different values of \(\lambda_c\). We see that for \(\lambda_c = 0.5\), \(\delta_{n_{sat}}\) tends to unity for \(IC \gg 1\).

Figure 3: Thermal noise parameter \(\delta_{n_{sat}}\) versus inversion coefficient \(IC\) including short-channel effects.

4 Thermal noise excess factor \(\gamma_n\)

4.1 Definition

Note that the \(\delta_n\) thermal noise parameter compares the thermal noise conductance evaluated at a given operating point to that of the MOSFET when it is biased as resistor with \(V_{DS}=0\) having a condcutance \(G_{dso}\). This is not very usefull for circuit design and is mostly used for modeling comparing the actual noise to the noise produced by a resistor having a conductance \(G_{dso}\). For circuit design, it is more useful to define another figure-of-merit \(\gamma_n\), named the thermal noise excess factor (TNEF) and defined as [7] [2] [10] \[\begin{equation}\label{eqn:gamman_def} \boxed{ \gamma_n \triangleq \frac{G_n}{G_m} = \frac{g_n}{g_m} }. \end{equation}\] where \(g_m\) is the normalized gate transconductance defined by \(\eqref{eqn:gmdef}\).

The TNEF \(\gamma_n\) as a figure-of-merit (FoM)

The TNEF \(\gamma_n\) represents how much noise is generated in the drain current for a given transconductance.

Contrary to the TNP \(\delta_n\), the noise conductance and the gate transconductance used in the definition \(\eqref{eqn:gamman_def}\) are evaluated at the same operating point. \(\gamma_n\) has a direct impact on the noise performance of circuits. The smaller \(\gamma_n\), the better the noise performance of the circuit.

TNEF \(\gamma_n\) in the linear region

Note that \(\gamma_n\) can become quite large when the device operates in the linear region, i.e. for \(V_{DS} < V_P-V_S \cong (V_G-V_{T0})/n-V_S\). Indeed, in this region, the gate transconductance gets smaller as the drain voltage decreases, but the thermal noise conductance does not decrease, resulting in a degradation of the TNEF \(\gamma_n\). At the limit when \(V_{DS}\) becomes zero, the transconductance becomes zero and \(\gamma_n\) tends to infinity. In practice, the TNEF \(\gamma_n\) is mostly used for transistors biased in saturation.

The TNP \(\delta_n\) and the TNEF \(\gamma_n\) are related by \[\begin{equation}\label{eqn:gamman_deltan} \gamma_n = \frac{G_n}{G_{dso}} \cdot \frac{G_{dso}}{G_m} = \delta_n \cdot \frac{G_{dso}}{G_m} = \delta_n \cdot n \cdot \frac{G_{dso}}{G_{ms} - G_{md}} = \delta_n \cdot n \cdot \frac{g_{dso}}{g_{ms} - g_{md}}, \end{equation}\] which in saturation reduces to \[\begin{equation} \gamma_{n_{sat}} = \frac{g_{n_{sat}}}{g_{m_{sat}}} = n \cdot \frac{g_{n_{sat}}}{g_{ms}}, \end{equation}\]

4.2 Long-channel (without VS)

In saturation without VS, \(q_d=0\) and \(g_{n_{sat}}\) and \(g_{m_{sat}}\) reduce to \[\begin{align} g_{n_{sat}} &\cong q_s \cdot \frac{2}{3} \cdot \frac{q_s+3/4}{q_s+1},\\ g_{m_{sat}} &= \frac{g_{ms}}{n} \cong \frac{q_s}{n}. \end{align}\] The TNEF in saturation \(\gamma_{n_{sat}}\) in then given by \[\begin{equation} \gamma_{n_{sat}} \cong \frac{2 n}{3} \cdot \frac{q_s+3/4}{q_s+1} = \begin{cases} \frac{2 n}{3} &\textsf{in strong inversion ($q_s \gg 1$)}\\ \frac{n}{2} &\textsf{in weak inversion ($q_s \ll 1$)}. \end{cases} \end{equation}\]

The thermal noise excess factor for a long-channel transistor is plotted versus the inversion coefficient in Figure 4. The red curve assumes that the slope factor is constant, which is not exactly the case as shown by the blue dashed line. However, we will consider that \(n\) remains constant and independent of the inversion coefficient.

Figure 4: Thermal noise excess parameter \(\gamma_n\) versus inversion coefficient \(IC\).

4.3 Short-channel (with VS)

For short-channel devices in saturation, we need to account for VS \[\begin{equation}\label{eqn:gammansat} \gamma_{n_{sat}} = \frac{g_{n_{sat}}}{g_{m_{sat}}} = n \cdot \frac{g_{n_{sat}}}{g_{ms_{sat}}}, \end{equation}\] where the thermal noise conductance \(g_{n_{sat}}\) is given by \(\eqref{eqn:gnsat3}\) and the normalized source transconductance in saturation \(g_{ms_{sat}}\) and accounting for VS is given by \[\begin{equation}\label{eqn:eqn:gmssat} g_{ms_{sat}} = \frac{q_s}{\sqrt{1+\lambda_c^2\,(q_s^2+q_s)}}. \end{equation}\] The thermal noise excess factor in saturation is plotted version the inversion coefficient and represented by circles in Figure 5 for different values of \(\lambda_c\). We see that \(\gamma_{n_{sat}}\) increases dramatically with \(IC\) for \(\lambda_c=0.3\). Note that this happens for advanced technologies for which it becomes more an more difficult to reach inversion coefficient as high as 100 because of voltage scaling. So the TNEF increases to about 3.5 for \(IC=40\) which is consistent with measurements shown below.

The TNEF is plotted with a linear x-axis for the same values of \(\lambda_c\) in Figure 6 which highlights that \(\gamma_{n_{sat}}\) actually increases linearly with \(IC\) in strong inversion. This is discussed in more details below.

Figure 5: Thermal noise excess factor \(\gamma_{n_{sat}}\) versus inversion coefficient \(IC\) including short-channel effects.

Figure 6: Thermal noise excess factor \(\gamma_{n_{sat}}\) versus inversion coefficient \(IC\) including short-channel effects.

4.4 Strong inversion approximation

In strong inversion and saturation \(g_{n_{sat}}\) and \(g_{m_{sat}}\) can be approximated by \[\begin{align}\label{eqn:gnsat_gmsat_si} g_{n_{sat}} &\cong \frac{2}{3} \cdot \frac{q_s^2+q_s\,q_{d_{sat}}+4q_dsat^2}{q_s+q_{d_{sat}}},\\ g_{m_{sat}} &= \frac{g_{ms}}{n} \cong \frac{q_s}{n \, \sqrt{1+(\lambda_c \, q_s)^2}} \end{align}\]

In very strong inversion under heavy VS, i.e. for \(\lambda_c\,q_s \gg 1\), \(\eqref{eqn:gnsat_gmsat_si}\) tend to \[\begin{align}\label{eqn:gnsat_gmsat_lim} g_{n_{sat}} &\cong q_s,\\ g_{m_{sat}} &= \frac{1}{n\,\lambda_c}. \end{align}\] The thermal noise excess factor in very strong inversion and under heavy VS has the following asymptote for \(\lambda_c\,q_s \gg 1\) \[\begin{equation} \gamma_{n_{sat}} \cong n \cdot \lambda_c \cdot q_s. \end{equation}\]

We can also express this asymptote in terms of the inversion coefficient \(IC\). Indeed, for \(1 \ll \lambda_c\,q_s\), \(i_{d_{sat}}\) simplifies to \(i_{d_{sat}} = IC \cong 2q_s/\lambda_c\). Hence \(q_s \cong \lambda_c/2 \cdot i_{d_{sat}}=\lambda_c/2 \cdot IC\). Therefore in very strong inversion under heavy VS, as illustrated in Figure 6, the TNEF becomes proportional to \(IC\) according to \[\begin{equation}\label{eqn:gammansat_si} \gamma_{n_{sat}} \cong \frac{n}{2} \cdot \lambda_c^2 \cdot IC. \end{equation}\]

4.5 Weak inversion approximation

In weak inversion \(q_d \ll 1\) and \(q_s \ll 1\) and \(g_n\) simplifies to \[\begin{equation}\label{eqn:gn_wi_mod1} g_{n} \cong \frac{q_s + q_d}{2}\quad\textsf{(weak inversion)} \end{equation}\] which in saturation \(q_d \ll q_s \ll 1\) simplifies to \[\begin{equation}\label{eqn:gnsat_wi_mod1} g_{n_{sat}} \cong \frac{q_s}{2}\quad\textsf{(weak inversion and saturation)}. \end{equation}\] On the other hand \(g_{m_{sat}}\) is given by \[\begin{equation}\label{eqn:gmsat_wi} g_{m_{sat}} \cong \frac{g_{ms_{sat}}}{n} = \frac{q_s}{n}. \end{equation}\] The thermal noise excess factor is therefore constant and equal to \[\begin{equation}\label{eqn:gammansat_wi_mod1} \gamma_{n_{sat}} \cong \frac{n}{2}\quad\textsf{(weak inversion and saturation)}. \end{equation}\]

4.6 Approximation valid from weak to strong inversion

The TNEF can be approximated by \[\begin{equation}\label{eqn:gammansat_approx} \gamma_{n_{sat}} \cong \left.\gamma_{n_{sat}}\right|_{\textsf{no VS}} \cdot \left(1 + \frac{3}{4} \cdot \lambda_c^2 \cdot IC \right), \end{equation}\] where \[\begin{equation} \left.\gamma_{n_{sat}}\right|_{\textsf{no VS}} = n \cdot \delta_{n_{sat}}, \end{equation}\] with \(\delta_{n_{sat}}\) given by \(\eqref{eqn:deltansat1}\). Eqn. \(\eqref{eqn:gammansat_approx}\) has the correct asymptotes, because in weak inversion \(\frac{3}{4} \cdot \lambda_c^2 \cdot IC \ll 1\) and hence \[\begin{equation} \gamma_{n_{sat}} \cong \left.\gamma_{n_{sat}}\right|_{\textsf{no VS}} = \frac{n}{2}. \end{equation}\] On the other hand, in strong inversion \(1 \ll \frac{3}{4} \cdot \lambda_c^2 \cdot IC\), resulting in \[\begin{equation} \gamma_{n_{sat}} \cong \left.\gamma_{n_{sat}}\right|_{\textsf{no VS}} \cdot \frac{3}{4} \cdot \lambda_c^2 \cdot IC = \frac{2\,n}{3} \cdot \frac{3}{4} \cdot \lambda_c^2 \cdot IC = \frac{n}{2} \cdot \lambda_c^2 \cdot IC. \end{equation}\] Eqn. \(\eqref{eqn:gammansat_approx}\) is plotted in Figure 5 and Figure 6 (represented by lines) and compared to the results from the full expression (represented by circles). We see a very good fit between the approximation \(\eqref{eqn:gammansat_approx}\) (lines) and the full expression \(\eqref{eqn:gammansat}\) (symbols).

5 Comparison with measurements

In this section, we will compare the simple model of the thermal noise excess factor in saturation \(\gamma_{n_{sat}}\) given by \(\eqref{eqn:gammansat_approx}\) with measurements made on different devices from various technologies [11], [12]. The slope factor is imposed and the VS parameter \(\lambda_c\) is extracted using simple curve fitting. The results are presented in Figure 7. The fit is OK for the 240nm, 180nm and 40nm technologies but there are some discrepancies at low \(IC\) for the 120nm and 100nm technologies. However, we have to remember that the model only uses a single parameter namely the VS parameter \(\lambda_c\).

Figure 7: Experimental validation of simple TNEF model.

Since the discrepancy appears at low \(IC\), we can modify the model making the value of \(\gamma_n\) in weak inversion, which is normally equal to \(n/2\), a fitting parameter. The proposed empirical model is then given by [13] \[\begin{equation}\label{eqn:gammansat_empirical} \gamma_{n_{sat}} \cong \gamma_{nwi} \cdot (1 + \alpha_n \cdot IC). \end{equation}\] From the asymptotic behavior in strong inversion given by \(\eqref{eqn:gammansat_si}\), \(\alpha_n\) should be equal to \[\begin{equation}\label{eqn:alphan} \alpha_n = \frac{n}{2\,\gamma_{nwi}} \cdot \lambda_c^2 \cong \frac{n}{2} \cdot \lambda_c^2. \end{equation}\] However, we now will use it as an independent fitting parameter.

The empirical model \(\eqref{eqn:gammansat_empirical}\) is compared to the same measured data in Figure 8 after extraction of parameters \(\gamma_{nwi}\) and \(\alpha_n\) by curve fitting. We now see a very good agreement between the simple empirical model \(\eqref{eqn:gammansat_empirical}\) and the measured data. We can observe that the fitting parameter \(\gamma_{nwi}\) is getting larger than the theoretical value \(n/2\). However it remains close to one.

Figure 8: Experimental validation of the TNEF empirical model.

We can extract the corresponding VS parameter from \(\eqref{eqn:alphan}\) resulting in \[\begin{equation}\label{eqn:lambdac_ext} \lambda_c = \sqrt{\frac{2 \, \gamma_{nwi} \, \alpha_n}{n}} \cong \sqrt{\frac{2 \, \alpha_n}{n}}. \end{equation}\]

The fitting parameters \(\gamma_{nwi}\) and \(\alpha_n\), the imposed slope factor \(n\) and the corresponding extracted value of \(\lambda_c\) are presented in Table 1. Figure 9 shows that \(\alpha_n\) is scaling as \(/1/L\).

Table 1: Fitting parameters.

Techno	\(\gamma_{{nwi}}\)	\(\alpha_n\)	\(n\)	\(\lambda_c\)
240nm	0.837	0.0138	1.3	0.133
180nm	0.952	0.0177	1.3	0.161
120nm	1.261	0.0217	1.3	0.205
100nm	1.266	0.0259	1.3	0.225
40nm	1.005	0.0679	1.45	0.307

Figure 9: Scaling of the \(\alpha_n\) parameter with \(1/L\).

6 Application to LNA design

6.1 Input-referred thermal noise resistance

The input-referred thermal noise resistance is defined as \[\begin{equation} R_n \triangleq \frac{\gamma_{n_{sat}}}{g_{m_{sat}}} \end{equation}\] and its normalized form is given by \[\begin{equation} r_n \triangleq G_{spec} \cdot R_n = \frac{\gamma_{n_{sat}}}{g_{m_{sat}}}. \end{equation}\]

Without VS, \(\gamma_{n_{sat}}\) remains constant and \(g_{m_{sat}}\) increases proportionally to the inversion coefficient \(IC\) in weak inversion and then as \(\sqrt{IC}\) in strong inversion. Therefore, the term \(\gamma_{n_{sat}}/g_{m_{sat}}\) decreases with respect to \(IC\) improving the noise performance. When VS is present, \(\gamma_{n_{sat}}\) depends on \(IC\) and increases as \(\alpha_n \cdot IC\) in strong inversion and saturation whereas \(g_{m_{sat}}\) saturates to its maximum value \(G_{spec}/(n \lambda_c)\). The \(\gamma_{n_{sat}}/g_{m_{sat}}\) ratio therefore increases now in strong inversion and saturation. The normalized \(\gamma_{n_{sat}}/g_{m_{sat}}\) ratio is then given by \[\begin{equation} r_n = \frac{\gamma_{n_{sat}}}{g_{m_{sat}}} \cong \frac{n}{2} \cdot \lambda_c^2 \cdot IC \cdot n\,\lambda_c = \frac{n^2}{2} \cdot \lambda_c^3 \cdot IC \quad \textsf{in SI and sat. with VS}. \end{equation}\]

In weak inversion the situation is different since \(\gamma_{n_{sat}} \cong \gamma_{nwi} \cong 1\) and \(g_{m_{sat}} \propto IC\). The term \(\gamma_{n_{sat}}/g_{m_{sat}}\) is now decreasing as \(1/IC\) in weak inversion and saturation. In normalized form this gives \[\begin{equation} r_n = \frac{\gamma_{n_{sat}}}{g_{m_{sat}}} \cong \gamma_{nwi} \cdot \frac{n}{IC} = \frac{n}{IC} \quad \textsf{in WI and sat.} \end{equation}\]

This means that there is an optimum value of \(IC\) for which the term \(\gamma_{n_{sat}}/g_{m_{sat}}\) reaches a minimum. An approximation of this optimum inversion coefficient is found by equating the two asymptotes and solving for \(IC\). This results in \[\begin{equation}\label{eqn:ICopt} IC_{opt} \cong \sqrt{\frac{2}{n\,\lambda_c^3}}. \end{equation}\] Note that this optimum \(IC\) only depends on \(\lambda_c\) according to \(\eqref{eqn:ICopt}\) and is often in the upper side of moderate inversion.

The normalized input-referred thermal noise resistance \(r_n\) in saturation is plotted versus the inversion coefficient \(IC\) for a 40nm transistor in Figure 10.

Figure 10: Normalized input-referred thermal noise resistance versus \(IC\) including VS.

For a common-source low-noise amplifier (LNA) at RF we need to add the transistor gate resistance \(R_G\) to the input-referred thermal noise resistance according to \[\begin{equation} R_n \cong \frac{\gamma_n}{G_m} + R_G. \end{equation}\] Although this gate resistance is usually small, it plays a fundamental role for noise at RF particularly in advanced CMOS technologies for which it has significantly increased after the the introduction of high-K dieletric and metal gates.

The input-referred noise resistance \(R_n\) has been measured on an RF transistor from a 40nm bulk CMOS technology at 10 and 14 GHz [12]. The measured \(R_n\) normalized to \(Z_0=50\,\Omega\) are plotted in Figure 11 versus the inversion coefficient \(IC\) and compared to the analytical model taking the values of \(\gamma_{nwi}\) and \(\alpha_n\) extracted above for the same 40nm device. The good match between the model and the experimental data validates this simple model. The optimum inversion coefficient for which \(R_n\) is minimum is about equal to 10.

Figure 11: Measured input-referred noise resistance versus \(IC\).

6.2 Minimum noise figure

It can be shown that the minimum noise factor accounting for the effects of the gate resistance and the induced-gate noise (but without the correlation) is given by [2] \[\begin{equation} F_{min} \cong 1 + 2\frac{\gamma_{n_{sat}}}{g_{m_{sat}}} \, \omega \, C_{GS} \, \sqrt{\alpha_G + b_n}, \end{equation}\] where \(C_{GS}\) is the gate-to-source capacitance, \(b_n=2/(5n^2)\) and \(\alpha_G\) is the thermal noise contribution of the gate resistance normalized to that of the channel \[\begin{equation}\label{eq:alphag} \alpha_G \triangleq \frac{R_G}{\gamma_{n_{sat}}/g_{m_{sat}}}. \end{equation}\] The minimum noise figure \(NF_{min}\) has been measured on the same device at 10 GHz and 14 GHz and is plotted versus the inversion coefficient in Figure 12. The minimum noise figure \(NF_{min}\) is sensitive to many parameters and it is therefore not that easy to get a simple but accurate model. We can consider therefore that the match between the experimental data and the simple model is acceptable accounting for the simplicity of the proposed model. The optimum inversion coefficient providing the minimum \(NF_{min}\) is again close 10.

Figure 12: Minimum noise figure measured at 10 and 14 GHz versus \(IC\).

7 Conclusion

This notebook has presented a simple model of the thermal noise of short-channel transistors. It has been shown that the thermal noise excess factor in saturation accounting for short-channel effects (actually velocity saturation) actually increases proportionnally to the inversion coefficient \(IC\) in strong inversion while the impact of velocity saturation in weak inversion remains negligible. The proposed model has been validated with experimental data from various CMOS bulk technologies down to 40nm. The model shows that the input-referred thermal noise resistance of a common-source transistor shows a minimum in the upper side of moderate inversion. The input-referred noise resistance of a common-source transistor has been measured at 10 and 14 GHz and confirms the existence of a minimum. The model including an additional gate resistance matches the measured input-referred noise resistance very well. A model of the minimum noise figure has been proposed and has been successfully compared to measurements made at 10 and 14 GHz. The proposed model is simple enough for correctly choosing the best operating point choosing the optimum inversion coefficient providing a minimum input-referred noise resistance and minimum noise figure.

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