ft of a MOSFET


The frequency at which the small signal short circuit current gain of an intrinsic MOS transistor drops to unity is called transit frequency(fT). The parameter fT is used assess the speed of an intrinsic MOS transistor

Transit frequency of MOS transistor

The small-signal equivalent circuit of a MOS transistor to compute fT is shown in Figure 1. (Assumption: V_{bs}=0)

Figure 1. Small signal equivalent circuit to compute fT of a MOS transistor

Figure 1. Small signal equivalent circuit to compute fT of a MOS transistor


At input node \circled{A}

(1)   \begin{equation*} I_{in}(\omega) = j\omega (C_{gs} +  C_{gd}) V_{gs}(\omega) \end{equation*}

At output node \circled{B}

(2)   \begin{equation*} I_{out}(\omega)= (g_m - j\omega C_{gd}) V_{gs}(\omega) \end{equation*}

Current gain,

(3)   \begin{equation*} \left|{I_{out}(\omega) \over I_{in}(\omega)}\right| = \left|{g_m - j\omega C_{gd} \over  j\omega (C_{gs} +  C_{gd})}\right| = {\sqrt{g_m^2 + \omega^2 C_{gd}^2} \over \omega (C_{gs} +  C_{gd})} \end{equation*}

At low frequencies, g_m \gg \omega C_{gs}

(4)   \begin{equation*} \left|{I_{out}(\omega) \over I_{in}(\omega)}\right| \approx {g_m \over  \omega (C_{gs} +  C_{gd})} \end{equation*}

At high frequencies, g_m \ll \omega C_{gs}

(5)   \begin{equation*} \left|{I_{out}(\omega) \over I_{in}(\omega)}\right| \approx {C_{gs} \over (C_{gs} +  C_{gd})} \end{equation*}

At transit frequency (\omega_T), the current gain is unity. From Eq.(4), transit frequency is given by

(6)   \begin{equation*} \boxed{\omega_T = 2\pi f_T = {g_m \over C_{gs} + C_{gd}} \approx {g_m \over C_{gs}} } \end{equation*}

Substituting for gm in terms of gate bias voltage, g_m = \mu C_{ox} {W\over L} (V_{gs}-V_T) in Eq-(6)

(7)   \begin{equation*} \omega_T &\approx& {g_m \over C_{gs} } = {3\over 2} {\mu (V_{gs}-V_T) \over L^2} = {3 \over 2}{\mu E_{ch} \over L} = {3 \over 2}{v_d \over L}  = {3 \over 2}{1 \over \tau_t} \end{equation*}

\tau_t is the time required for the electron to transit from source to drain.

Impact of bias point on fT

From Eq-(7), transit frequency in terms of gate bias voltage and channel length is given by

(8)   \begin{equation*} \omega_T = {3\over 2} {\mu (V_{gs}-V_T) \over L^2} \end{equation*}

– Increasing gate drive increases transit frequency
– Increasing gate length decreases transit frequency. So technology scaling favor the transit frequency.

From Eq-(6), transit frequency in terms of drain current and channel length is obtained by substituting g_m = \sqrt{2I_D \mu C_{ox} W/L}

(9)   \begin{equation*} \omega_T = {{\sqrt{2I_D \mu C_{ox} W/L} \over C_{gs}} \end{equation*}

– Increasing bias current increases transit frequency (but square root dependence)
– Keeping bias current constant, decreasing length increases transit frequency ( \propto L^{-3/2})

Impact of temperature on fT

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