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An integrated circuit technology using complementary MOSFETs. CMOS names a circuit technology; active-pixel sensor names a detector architecture.

An ideal insulating gate blocks particle exchange between the gate and substrate of a MOSFET. Their electrochemical potentials, represented by their band-diagram Fermi levels, therefore need not coincide. An applied gate voltage fixes their difference; with $e>0$ and voltage defined by an electron reservoir, ${\tilde{\mu }}_g-{\tilde{\mu }}_s=-e{V}_g$. Contact work-function differences also affect band bending, so equal Fermi levels do not imply flat bands. Each conducting region can separately have a spatially constant equilibrium electrochemical potential.

For an ideal MOSFET with a nondegenerate p-type semiconductor of acceptor density $N_A$, define ${\varphi }_F=(k_{B}T/e)\ln (N_A/n_i)>0$, where $n_i$ is the intrinsic carrier concentration. A positive surface electric potential ${\psi }_s$ lowers the local conduction band and valence band by $e{\psi }_s$. The surface Electron density is $n_s=(n_i^2/N_A)\exp (e{\psi }_s/k_{B}T)$. Equality $n_s=N_A$ gives the conventional onset of strong inversion at ${\psi }_s=2{\varphi }_F$. The earlier crossover $n_s=p_s=n_i$ occurs at ${\psi }_s={\varphi }_F$. Neither condition requires the Fermi level to enter the conduction band.

A positive gate voltage attracts Electrons towards the oxide interface of the p-type semiconductor and repels mobile electron holes. The uncovered ionized acceptors form a negatively charged depletion region. Increasing the voltage further creates a thin electron-rich semiconductor inversion layer at the surface. The conduction band and valence band both bend downwards towards the positively biased gate, because an electron's electrostatic energy is $-e\psi$.

In an ideal nondegenerate substrate with acceptor density $N_A$, write ${\varphi }_F=(k_{B}T/e)\ln (N_A/n_i)>0$ and surface potential ${\psi }_s$ relative to the neutral bulk. The electron and hole surface densities areThe first change of majority-carrier type occurs at ${\psi }_s={\varphi }_F$, where $n_s=p_s=n_i$. The conventional MOS inversion threshold for appreciable strong inversion isAt threshold the inversion sheet is just developing, while a much thicker depletion region lies between it and the neutral p-type bulk. Inversion does not require the conduction band to cross the Fermi level: the surface can remain nondegenerate.

The metal and semiconductor Fermi levels need not be equal. The ideal oxide blocks particle exchange, so the gate and substrate can maintain different electrochemical potentials. In a biased MOSFET, the gate-to-substrate Fermi-level difference is ${\tilde{\mu }}_g-{\tilde{\mu }}_s=-e{V}_g$, when the applied voltage is defined in the electron-reservoir convention. Each region can separately be in internal equilibrium. If an external zero-voltage connection instead equilibrates the reservoirs, their Fermi levels coincide, but different work functions can still cause band bending. Here the chemical potential shown in a band diagram is the electron electrochemical potential; the purely chemical contribution alone is not the equilibrium criterion for charged particles.

Klaus von Klitzing, G. Dorda and M. Pepper reported a silicon MOSFET measurement in 1980. Its gate controlled an semiconductor inversion layer; transverse contacts measured the Hall voltage and separate longitudinal contacts measured dissipation. One sweep held $B=18\mathrm{T}$, $T=1.5\mathrm{K}$ and $I=1\mu \mathrm{A}$ fixed while varying gate voltage. Hall plateaus accompanied small longitudinal voltages. Device-geometry comparisons and high-input-impedance voltage measurements helped test whether contact placement or electrical loading produced the result.^[ref]

This original functional diagram separates transverse and longitudinal readouts; it does not reproduce the historical device's dimensions. The gate inset explains how a field-effect transistor provides an adjustable two-dimensional electron gas. A longitudinal voltage close to zero is a useful check because accidental longitudinal separation between Hall contacts can otherwise contaminate the transverse measurement.