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

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.

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

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 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

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