Follow the conduction band edge along the surface, from the n⁺ region into the region under the gate. Electrons hop thermally; how many reach the surface depends only on the height of the barrier.
electron (height = its energy)- - -electron quasi-Fermi level EFn— conduction band edge Ec
What to look for
Electrons are drawn at their energy. Most sit near the band edge; a few have extra thermal energy (the exponential tail).
Every electron that reaches the surface came from the n⁺ region through terminal C. Watch the wire when you change VCB or VGB.
The density ratio printed under the plot follows e−barrier/kT: the animation is a Boltzmann equilibrium.
The physics behind it
Barrier height = φbi + VCB − ψS: the n⁺ region sits at φbi + VCB above the neutral body, the surface at ψS.
Hence nsurface = nn⁺e−(φbi+VCB−ψS)/φt = n0e(ψS−VCB)/φt (3.2.3), the same result as with quasi-Fermi levels.
ψS comes from the exact three-terminal equation (3.2.21) for NA = 10¹⁷ cm⁻³, tox = 10 nm, n⁺ polysilicon gate.