# 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
EnEF
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
EnEF
kT
ö
ø
+ 1
If we allow also excited states of the dopant, we obtain the fully generalized Fermi distribution
f(Er, EF, T)  =

1
Sgr · exp æ
è
ErEF
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.

2.2.2 Doping and Carrier Concentration

© H. Föll (Semiconductor - Script)