MOS Capacitor

Name	Formula	Related Formulae	Variables/Constants
Oxide capacitance (per unit area)	$C_{ox}=\frac{{\epsilon }_{ox}}{t_{ox}}$		$C_{ox}:gateoxidecapacitanceperunitarea$ ${\epsilon }_{ox}:oxidepermittivity(\approx 3.9{\epsilon }_{0}forSi{O}_2)$ $t_{ox}:oxidethickness$
Semiconductor work function	${\varphi }_S={\chi }_s+\frac{E_C-E_F}{q}$	${\varphi }_F=\frac{k_{B}T}{q}\ln \left(\frac{N_A}{n_i}\right)$	${\varphi }_S:semiconductorworkfunction$ ${\chi }_s:electronaffinity$ $\begin{array}{r}{\varphi }_F:bulkFermipotential\\ (p-typeconventionshown)\end{array}$ $k_B:Boltzmannconstant$
Flat-band voltage	$V_{FB}={\varphi }_{MS}-\frac{Q_{ox}}{C_{ox}}$	${\varphi }_{MS}={\varphi }_M-{\varphi }_S$	$\begin{array}{r}{V}_{FB}:gatevoltageatwhichthebands\\ areflatatthesurface\end{array}$ ${\varphi }_M:metal(orgate)workfunction$ ${\varphi }_{MS}:metal-semiconductorworkfunctiondifference$ $Q_{ox}:effectiveoxidechargeperunitarea$
Surface potential at strong inversion (threshold condition)	${\psi }_s(inv)=2{\varphi }_F$	${\varphi }_F=\frac{k_{B}T}{q}\ln \left(\frac{N_A}{n_i}\right)$	$\begin{array}{r}{\psi }_s:surfacepotential(bandbending\\ atthesemiconductorsurface)\end{array}$ $k_B:Boltzmannconstant$
Maximum depletion width (at threshold)	$W_{max}=\sqrt{\frac{4{\epsilon }_s{\varphi }_F}{q{N}_A}}$	$W({\psi }_s)=\sqrt{\frac{2{\epsilon }_s{\psi }_s}{q{N}_A}}$	$\begin{array}{r}{W}_{max}:maximumdepletionwidth,reached\\ at{\psi }_s=2{\varphi }_{F}andpinnedthereafter\end{array}$
Threshold voltage	$V_T=V_{FB}+2{\varphi }_F+\frac{\sqrt{4q{\epsilon }_s{N}_A{\varphi }_F}}{C_{ox}}$	$V_T=V_{FB}+2{\varphi }_F+\gamma \sqrt{2{\varphi }_F}$	$\begin{array}{r}{V}_T:gatevoltageattheonsetofstrong\\ inversion\end{array}$
Body-effect coefficient	$\gamma =\frac{\sqrt{2q{\epsilon }_s{N}_A}}{C_{ox}}$	$V_T(V_{SB})=V_{FB}+2{\varphi }_F+\gamma \sqrt{2{\varphi }_F+V_{SB}}$	$\gamma :body-effectcoefficient$ $\begin{array}{r}{V}_{SB}:source-to-bodyreverse\\ bias(appliesoncethisstructureis\\ biasedasaMOSFETterminal,notforan\\ isolatedMOScapacitor)\end{array}$
Semiconductor (depletion-layer) capacitance	$C_s({\psi }_s)=\frac{{\epsilon }_s}{W({\psi }_s)}=\sqrt{\frac{q{\epsilon }_s{N}_A}{2{\psi }_s}}(0<{\psi }_s<2{\varphi }_F)$	$W({\psi }_s)=\sqrt{\frac{2{\epsilon }_s{\psi }_s}{q{N}_A}}$	$\begin{array}{r}{C}_s:semiconductor-sidecapacitance\\ perunitarea,validinthedepletion\\ regime\end{array}$
Flat-band semiconductor capacitance	$C_{s,FB}=\frac{{\epsilon }_s}{L_D}$	$L_D=\sqrt{\frac{{\epsilon }_s{k}_{B}T}{q^2{N}_A}}$	$\begin{array}{r}{C}_{s,FB}:semiconductor-sidecapacitance\\ evaluatedat{\psi }_s=0\end{array}$ $\begin{array}{r}{L}_D:extrinsicDebyelength(seePN\\ JunctionDiodepage)\end{array}$ $k_B:Boltzmannconstant$
Total MOS capacitance (per unit area)	$\frac{1}{C}=\frac{1}{C_{ox}}+\frac{1}{C_s({\psi }_s)}$		$\begin{array}{r}C:totalgatecapacitanceperunitarea\\ measuredinahigh-frequency\\ C-Vsweep\end{array}$

Important Questions

Ques: Why does MOS capacitance depend on bias voltage instead of behaving like a fixed parallel-plate capacitor?

An MOS structure is really two capacitors in series: the fixed oxide capacitance $C_{ox}$, and a voltage-dependent semiconductor capacitance $C_s$ set by how far the surface depletion region extends. As the gate voltage sweeps from accumulation through depletion toward inversion, $C_s$ first drops (as $W({\psi }_s)$ grows with ${\psi }_s$) and then, once strong inversion sets in and an inversion layer of minority carriers screens the surface, the depletion width saturates at $W_{max}$ and stops growing further — at low measurement frequencies $C_s$ rises back up as the inversion charge can respond to the AC signal, while at high frequencies the inversion layer cannot generate/recombine fast enough to follow the signal and $C$ stays pinned near its minimum. This bias-dependent, frequency-dependent $C$-$V$ curve is the standard characterization tool used to extract $N_A$, $t_{ox}$, and $V_{FB}$ experimentally. Reference: S.M. Sze and Kwok K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2007; D.A. Neamen, Semiconductor Physics and Devices, 4th ed., McGraw-Hill, 2012.

Ques: Why is $2{\varphi }_F$ specifically chosen as the surface-potential criterion for the onset of strong inversion?

By convention, strong inversion is defined as the point where the minority-carrier (inversion-layer) concentration at the surface equals the bulk majority-carrier concentration $N_A$. Since the bulk Fermi level sits a potential ${\varphi }_F$ below midgap in p-type material, bending the bands at the surface by exactly $2{\varphi }_F$ makes the surface Fermi-level separation from the conduction band mirror the bulk separation from the valence band — producing surface electron and bulk hole concentrations of the same order of magnitude. This is an idealized onset criterion rather than a sharp physical switch (real channel conduction in a MOSFET turns on gradually around this point), but it is the standard convention used to derive a single, well-defined threshold voltage. Reference: D.A. Neamen, Semiconductor Physics and Devices, 4th ed., McGraw-Hill, 2012.