Semiconductor Equilibrium No external forces (voltages, electric fields, temp.gradients) First Consider pure crystal Then Consider addition of dopants

Semiconductor Equilibrium Charge carriers Electrons in conductance n(E) = gc(E)fF(E) n(E) - prob. dens. of electrons gc(E) - conductance density fF(E) - Fermi-Dirac prob. function Holes in valence p(E) = gV(E)(1 - fF(E)) p(E) - prob. dens. of holes gv(E) - valence density

Semiconductor Equilibrium Charge carriers(cont.)

Semiconductor Equilibrium Charge carriers(cont.) Example Find the probability that a state in the conduction band is occupied and calculate the electron concentration in silicon at T = 300K. Assume Fermi energy is .25 eV below the conductance band Note low probability per state but large number of states implies reasonable concentration of electrons

Semiconductor Equilibrium Charge carriers(cont.) For intrinsic semiconductor, concentration of electrons in conductance band is equal to holes in the valence band. Thus,

Semiconductor Equilibrium Dopant Atoms (n-type semiconductor) Phosphorous has 5 valence electrons Energy-band diagram

Semiconductor Equilibrium Dopant Atoms (p-type semiconductor) Boron has 3 valence electrons Energy-band diagram

Semiconductor Equilibrium The Extrinsic Semiconductor n-type p-type

Semiconductor Equilibrium The Extrinsic Semiconductor Example Consider doped silicon at 300K. Assume that the Fermi enery is .25 eV below the conduction band and .87 eV above the valence band. Calculate the thermal equilibrium concentration of e’s and holes

Semiconductor Equilibrium The Extrinsic Semiconductor The n0p0 product That is, the product of n0 and p0 is a constant for a given semiconductor at a given temperature.

Semiconductor Equilibrium Statistics of donors and acceptors Ratio of electrons in donor state total electrons Example Consider phosporous doped silicon at T = 300K and at a concentration of Nd = 1016 cm-3. Find the fraction of electrons in the donor state.

Semiconductor Equilibrium Compensated semiconductors Formed by adding both donor and acceptor impurities in the same region Energy-band diagram

Semiconductor Equilibrium Compensated semiconductors (cont.) With the assumption of charge neutrality, we can derive Example Consider a silicon semiconductor at T = 300K in which Na = 1016 cm-3 and Nd = 3 1015 cm-3. Assume ni = 1.5 1010 cm-3 and find p0 and n0.

Semiconductor Equilibrium Position of Fermi energy level As a function of doping levels As a function of temperature for a given doping level