MOSFET

Threshold Voltage

Dependent on the MOSFET material properties which are determined by the manufacture process.

\[V_t = 2\phi_b + \frac{\sqrt{2 \epsilon_{Si} q N_A 2\phi_b} }{C_{ox}} + V_{fb}\]

where \(\phi_b=V_T \ln{(\frac{N_A}{n_i})}\) is the bulk potential, \(N_A\) is carrier density in the doped substrate, \(n_i\) is the intrinsic carrier concentration in undoped silicon

Body Effect

\[V_t=V_{t0} - \gamma(\sqrt{\phi + V_{BS}}-\sqrt{\phi})\]

\[\gamma = \frac{\sqrt{2\epsilon_{Si}qN}}{C_{ox}}\]

Operating Regions

Sub-threshold

Also referred to as cutoff region, or operating in weak-inversion.

Conditional on \(V_{GS} < V_t\).

Simple

\[I_D = 0\]

More accurate

\[I_D = \mu C_{ox} \frac{W}{L} V_T^2 (n-1) e^{\frac{V_{GS}-V_t}{nV_T}} (1-e^{\frac{-V_{DS}}{V_T}})\]

where \(n = \frac{C_{ox} + C_{dep}}{C_{ox}} > 1\)

We can also define the subthreshold swing \(S\) , that is the change in \(V_{GS}\) needed for a 10 times change in current.

\[I_D \propto e^{\frac{V_{GS}}{nV_T}}\]

\[\Delta V_{GS} = n V_T \ln{10}\]

\[S = n \times 60 \text{mV per decade}\]

Linear region

Also known as the triode mode, or ohmic mode.

Conditional on \(V_{GS} > V_t\) and \(V_{DS} < V_{GS}-V_t\).

\[I_D = \mu C_{ox} \frac{W}{L} \left[(V_{GS} - V_t)V_{DS} - \frac{V_{DS}^2}{2}\right] (1+ \lambda V_{DS})\]

Saturation region

Conditional on \(V_{GS} > V_t\) and \(V_{DS} > V_{GS}-V_t\).

DC current:

\[I_D = \frac{1}{2} \mu C_{ox} \frac{W}{L} (V_{GS} - V_t)^2 (1+ \lambda V_{DS})\]

Small signal transconductance:

\[g_m=\]

Term Definitions and Useful Equations

\(V_{GS}-V_t\) is the overdrive voltage, also referred to as \(V_{DSsat}\), or \(V_{ov}\).

\(V_T\) is the thermal voltage \(V_T=kT/q\), where \(T\) is the absolute temperature in Kelvin, \(k\) is the Boltzman constant and \(q\) is the elementary charge.

\(V_T=25.86\) mV at 300K.

\(K'=\mu_0 C_{ox}\)

\(C_{ox}=\epsilon_{ox} / t_{ox}\)

\(\beta = K' \frac{W}{L}\)