Noise in MOS Transistors

MOSFET Channel Thermal Noise

For MOS devices operating in saturation region the channel noise can be modeled by a current source connected between the drain and source terminals and expressed as,

(1)   \begin{equation*} \overline{i_{nd}^{\tiny 2} \over \Delta f} = 4kT \gamma g_{do} \end{equation*}

where,
\gamma is {2}/{3} for long channel devices in saturation, and 2 or even higher for sub-micron devices.
g_{do} is zero bias drain conductance.

Thermal noise of the channel when referred to input, can be represented by a voltage source in series with gate. The voltage spectral density is

(2)   \begin{equation*} \overline{v_{n,d}^{\tiny 2} \over \Delta f} = {8 \over 3} {kT g_{do} \over g_m^2} \end{equation*}

where g_m transconductance of transistor in saturation

Flicker Noise

As charge carriers mover at the interface the random charge trapping by the energy states introduces a noise in the drain current called flicker noise. Depending on the impurities of the oxide-silicon interface this noise may vary from process to process. Flicker noise is modelled by a current source across the drain and source and expressed as

(3)   \begin{equation*}\frac{\overline{i_{nf}^{\tiny 2}}}{\Delta f}=\frac{K_f}{f}\frac{g_m^{\tiny 2}}{ C_{ox}WL}\end{equation*}

When referred to input, the noise is represented by a voltage source between gate and source. The voltage spectral density is,

(4)   \begin{equation*}\frac{\overline{v_{nf}^{\tiny 2}}}{\Delta f}=\frac{K_f}{f}\frac{1}{ C_{ox}WL}\end{equation*}

where,
K_f \rightarrow a device-specific constant.
W, L \rightarrow effective width and length of a MOS device.

K_f of PMOS devices is an order lower than that of NMOS devices. Therefore PMOS devices are believed to exhibit less flicker noise than NMOS devices. Flicker noise has inverse dependence on frequency, so it is also called 1/f noise. Yet another name pink noise.

 

Flicker noise reduces with increasing frequency and at a point it starts falling much below the thermal noise. The frequency at which flicker noise is equal to thermal noise is called corner frequency (f_C) of flicker noise.

(5)   \begin{equation*} \frac{\overline{i_{nd}^{\tiny 2}(f_C)}}{\Delta f}=\frac{\overline{i_{nf}^{\tiny 2}(f_C)}}{\Delta f} \Rightarrow 4kT\gamma g_m =\frac{K_f}{f_C}\frac{g_m^{\tiny 2}}{ C_{ox}WL}\end{equation*}

Substituting \omega_{\tiny T}= {g_m}/{C_{gs}} \approx g_m/(C_{ox}WL) in the above equation and solving for f_{C}, we get

(6)   \begin{equation*} f_{\tiny C} = \frac{K_{\tiny f}/\gamma}{4kT}\omega_{\tiny T} \end{equation*}

From Eq-(6), flicker noise is directly proportional to \omega_T. With technology scaling \omega_T increases, thus flicker noise also.

Though flicker noise is small at RF frequencies, its effect is considerable in mixers and oscillators due to nonlinearity or time variance of those circuits.

MOSFET Gate Induced Thermal Noise

Gate induced thermal noise model of a MOS transistor

Figure 3. Gate induced thermal noise model of a MOS transistor

The fluctuations in the channel charge in the inversion region will induce a noisy current in the gate due to capacitive coupling. According to Van der Ziel, a gate circuit model that represents gate induced noise is illustrated in Figure 1.

(7)   \begin{equation*} {\overline{i_{ng}^{\tiny 2}} \over \Delta f} = 4kT\delta g_g \end{equation*}

where, g_g = \frac{\omega^{\tiny 2} C_{gs}^{\tiny 2}}{5g_{do}} and \delta = 4/3 for long channel devices

Thevenin equivalent circuit representation of gate induced noise shown in Figure 2 is obtained through parallel to series impedance transformation.

(8)   \begin{equation*} \frac{\overline{v_{ng}^{\tiny 2}}}{\Delta f} = 4kT\delta r_g \end{equation*}

where, g_g = \frac{1}{5g_{do}}

The above two circuits are interchangeable under the condition, Q_{Cgs}=5g_{do}/\omega C_{gs} \gg 1 or \omega \ll 5\omega_T ~g_{do}/g_m.

The spectral density of gate induced noise is proportional to f^2, and hence the gate induced thermal noise is not a white noise source.

Gate induced noise is partially correlated with the drain noise, with a complex correlation coefficient given by

(9)   \begin{equation*} c={{\overline{i_{ng}.i_{nd}^*}}\over{\sqrt{\overline{i_{ng}^{\tiny 2}}~\overline{i_{nd}^{\tiny 2}}}}} \end{equation*}

where, c \approx j0.395 for long channel devices.

Using the correlation coefficient, the gate noise can be expressed in terms of correlated and uncorrelated components as,

(10)   \begin{equation*} \frac{\overline{i_{ng}^{\tiny 2}}}{\Delta f} = \underbrace{4kT\delta g_g(1-|c|^{\tiny 2})}_{\mbox{uncorrelated}} + \underbrace{4kT\delta g_g|c|^{\tiny 2}}_{\mbox{correlated}} \end{equation*}

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