1. Semiconductor thermal statistics
(Fermi-Dirac statistics)

2. Fermi-Dirac Distribution function

3. Fermi-Dirac distribution overlaid on energy band-gap of an intrinsic semiconductor

4. Density of electrons in conduction band due to thermal probability distribution

5. Density of holes in valence band due to thermal probability distribution

6. intrinsic (= pure) semiconductor
For (thermal) creation of each electron a hole is also created.
∴ n = p = ni ≡ intrinsic carrier density

7. extrinsic semiconductor
extrinsic semiconductor. (Additional) electrons and holes caused by (select) impurities
Donor impurity (pentavalent) = extra electron
Acceptor impurity (trivalent) = shortage of electron = extra hole
Carrier densities n and p , however they arise, must still obey thermal distribution statistics.

8. law of mass-action n0p0 = ni2 p0 = ni2/no Due to:
‘mass-action’ law, since the increase of one quantity results in decrease of the other.
p0 = ni2/no

9. Thermal behavior of intrinsic carrier density
(For silicon)

10. Extrinsic semiconductors nomenclature
ND = donor impurity density (#/cm3)
NA = acceptor impurity density (#/cm3)

11. Typical: (donor) impurity of 1 part in 107
(for silicon): NSi = 5 × 1022 atoms/cm3
∴ ND = 5 × 1015 #/cm3
Compare to intrinsic: ni0 = 1.5 × 1010 #/cm3

12. Extrinsic: Donor (pentavalent) impurity added to silicon(tetravalent)
With law of mass-action resolves to:

13. Extrinsic semiconductors: Donor impurities
(for ND >> ni,)
n0 ≅ ND
with p0 =ni2/ND
by law of mass-action
Example: ND = 1 x 1016 #/cm3 = n0
p0 = 2.25 x 104 #/cm3

14. Extrinsic semiconductors: Acceptor impurities
p0 ≅ NA
(for NA >> ni)
with n0 =ni2/NA
by law of mass-action
Example: NA = 5 x 1014 #/cm3 = p0
∴ n0 = 0.45 x 106 #/cm3

15. Counter-doping. Both donor and acceptor impurities
For counter-doping of ND > NA. Donor electrons fill the acceptor sites first and what is left over is then n ≅ ND – NA.

16. Counter-doping n0 = ND – NA p0 = NA – ND for ND > NA for NA > ND
with p0 = ni2/n0
for NA > ND
p0 = NA – ND
with n0 = ni2/p0

17. Thermal equilibrium: Fermi Level
n = NC exp [(EF – EC)/kT]
p = NV exp [-(EF – EV)/kT]
For intrinsic semiconductor n = p = ni
∴ ni = NC exp [(Ei – EC)/kT]
& ni = NV exp [-(Ei – EV)/kT]
Defines a specific Fermi level Ei = intrinsic Fermi level

18. Intrinsic and extrinsic Fermi levels
n = ni exp [(EF – Ei)/kT]
p = ni exp [-(EF – Ei)/kT]
Displacement of EF from Ei identifies density and type of charge-carrier