Chapter 2 · Visualization 1 of 6 · Sections 2.4 to 2.6

Band Diagram Explorer

Sweep the gate voltage and watch the energy bands bend at the oxide-silicon interface, from hole accumulation through depletion to an inversion layer of electrons.

Energy bands versus depth (exact Poisson solution). Gold lines are Fermi levels.

Carrier densities at the surface (log scale)

What to look for

VGB = VFB + ψox + ψS

  • The bulk Fermi level (gold) stays flat; the gate Fermi level moves down by qVGB as the gate goes positive.
  • Positive ψS bends the bands down. When the intrinsic level Ei at the surface drops below EF, electrons outnumber holes: inversion starts at ψS = ΦF.
  • At ψS = 2ΦF the surface electron density reaches NA (see the bar chart marker).
  • The depletion width grows like √ψS and then almost stops growing once inversion electrons supply the charge.

Try this

  1. Move VGB − VFB from −1.5 V to +2.5 V and note where holes and electrons cross in the bar chart.
  2. Raise NA from 10¹⁶ to 10¹⁸ cm⁻³. How does the depletion width at the same ψS change?
  3. Increase the oxide thickness to 10 nm. How much more gate voltage do you need to reach the same band bending?
  4. Switch the gate to p⁺ polysilicon. Which numbers move: the bands at a given VGB − VFB, or the absolute VGB?