Chapter 3 · Animation 4 of 4 · Section 3.5
Floor and Ceiling: the Pinch-off Voltage
Keep the gate fixed and raise VCB . The surface potential rides on a rising floor until it meets the ceiling set by the gate; the electrons live in the gap, and the gap closes at the pinch-off voltage.
- - - ceiling ψsa (VGB ) - - - floor φ0 + VCB ━ ψS
Inversion charge versus VCB at this VGB (amber dot: now; amber line: VP )
What to look for The ceiling depends only on the gate; the floor only on VCB . ψS follows the lower of the two. The number of electrons drawn in the gauge is proportional to the exact |Q′I |, and the gap height tracks it. The cross-section shows the same thing in space: the inversion layer thins and the depletion region deepens as VCB rises. The physics behind it Without electrons the gate would set ψS = ψsa (VGB ) = (−γ/2 + √(γ²/4 + VGB − VFB ))² (3.2.24). Charge balance gives |Q′I | ≈ nC′ox (ψsa − ψS ); in strong inversion ψS ≈ φ0 + VCB , so |Q′I | ≈ nC′ox (VP − VCB ). Pinch-off voltage VP = ψsa (VGB ) − φ0 (3.5.7): the VCB at which the strong-inversion formula gives zero charge.
Model: exact three-terminal equations, NA = 10¹⁷ cm⁻³, tox = 10 nm, n⁺ polysilicon gate, φ0 = 2ΦF + 6φt . Equilibrium at every instant (the real response is nanoseconds). Gauge: one electron ≈ 5×10⁻⁹ C/cm²; cross-section: one electron ≈ 4×10⁻⁹ C/cm². Section 3.5 and Figs. 3.13–3.17 of Tsividis and McAndrew, Operation and Modeling of the MOS Transistor .