Chapter 1 · Visualization 2 of 6 · Sections 1.2.4 to 1.2.5

Potential and Carrier Density

Raise the electrostatic potential of one point relative to another in equilibrium: the bands bend, the Fermi level stays flat, and the carrier densities change by a factor of ten for every 60 mV.

Energy bands between two flat regions (the bands bend by −qψ₂₁)

Electrostatic potential ψ(x)

Carrier densities versus potential (log scale): a straight line, one decade per φ_t ln 10

What to look for

n₂/n₁ = exp(ψ₂₁/φ_t), p₂/p₁ = exp(−ψ₂₁/φ_t), n p = n_i² exp((E_Fn − E_Fp)/kT)

Positive ψ₂₁ lowers all electron energies at point 2 by qψ₂₁. The Fermi level does not move in equilibrium, so it ends up closer to E_c: more electrons, fewer holes.
In the log plot both curves are straight lines. Each φ_t ln 10 ≈ 60 mV (at 300 K) changes n and p by one decade, in opposite directions, so n p stays n_i².
Switch on the quasi-Fermi split: E_Fn and E_Fp separate by qV_split and the product n p becomes n_i² exp(V_split/φ_t). This is the situation inside a forward-biased junction.

Try this

Set ψ₂₁ = +0.12 V at 300 K. Predict n₂/n₁ and compare with the readout (about 100).
Raise T to 450 K. Does the number of mV per decade grow or shrink? What does that do to the steepness of the curves?
Turn on the non-equilibrium option for p-type with V_split = 0.3 V. By how many decades does n p exceed n_i²? Check against V_split/(φ_t ln 10).
Make ψ₂₁ negative. Which carrier is depleted at point 2?