Chapter 2 · Animation 3 of 4 · Sections 2.6 and 2.7
Where the Inversion Electrons Come From
Step the gate of a MOS capacitor to a high voltage. With no source of electrons nearby, the silicon goes into deep depletion and the inversion layer builds up only as fast as thermal generation supplies it.
electronholethermal generation of a pair⊖ acceptor ion
Build-up after the step, in real time (Zerbst picture)
What to look for
Right after the step the depletion region is much wider than its equilibrium maximum: the gate charge has nothing else to pull in.
Each generated pair is separated by the field: the electron joins the inversion layer, the hole leaves through the body. The depletion region shrinks as electrons arrive.
The time scale is set by thermal generation, seconds to minutes in good silicon. This is the slowness behind the low-frequency/high-frequency difference in C–V curves.
The physics behind it
Equilibrium needs np = ni² everywhere. In deep depletion np ≪ ni², so there is net thermal generation, roughly ni/τg per unit volume.
Zerbst picture: dQ′I/dt ≈ q ni (xd − xd,final)/τg, generation in the excess depletion width; with Q′I fixed, VGB = VFB + ψs + (γC′ox√ψs + |Q′I|)/C′ox gives ψs.
In a transistor the source and drain supply electrons in picoseconds, so this delay does not limit MOSFETs; it is a property of the isolated two-terminal structure.