# Fermi Distribution for Dopant Levels

| The correct Fermi distribution for most dopant levels, i.e. the probability that an electron is occupying an energy level belonging to a dopant atom is
f(E, EF, T) = | 1
--- | ---
½ · exp | æ è | E n – EF kT | ö ø | + 1
|  | The reason for the factor 1/2 instead of the usual 1 is that there is a spin degeneracy, i.e. the energy is the same for different spins.
|  | fDop (E,T) is thus the probability that the level is occupied by an electron of either spin. This applies to group III acceptors, or group V donors as doping elements for group IV semiconductors.
| There also might be cases were dopants can accommodate two electrons (which then must have paired spin). The Fermi distribution formulated for acceptors in this case is
f( E, EF , T) = | 1
--- | ---
2 · exp | æ è | En – EF kT | ö ø | + 1
| If we allow also excited states of the dopant, we obtain the fully generalized Fermi distribution
f(E r, EF , T) = | 1
--- | ---
S gr · exp | æ è | E r – EF kT | ö ø | + 1
|  | With Er = energy of the r-th state; gr = degeneracy/spin factor.
| Interesting, but rather irrelevant as long as we simply assume completely ionized donors and acceptors.

With frame

2.2.2 Doping and Carrier Density

© H. Föll (Semiconductors - Script)