Semiconductor thermal statistics

(Fermi-Dirac statistics)

Fermi-Dirac Distribution function

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

Density of electrons in conduction band due to thermal probability distribution

Density of holes in valence band due to thermal probability distribution

intrinsic (= pure) semiconductor

∴ n = p = ni ≡ intrinsic carrier density

For (thermal) creation of each electron a hole is also created.

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.

law of mass-action n0p0 = ni2

Due to:

‘mass-action’ law, since the increase of one quantity results in decrease of the other.

p0 = ni2/no

Thermal behavior of intrinsic carrier density

(For silicon)

Extrinsic semiconductors nomenclature

ND = donor impurity density (#/cm3)

NA = acceptor impurity density (#/cm3)

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

Extrinsic: Donor (pentavalent) impurity added to silicon(tetravalent)

With law of mass-action resolves to:

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

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

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.

Counter-doping

for ND > NA

n0 = ND – NA with p0 = ni2/n0

for NA > ND

p0 = NA – ND with n0 = ni2/p0

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

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