## Carrier concentration in Intrinsic semiconductors:

Let us consider 'dn' number of electrons in the conduction band between Ec and E(c+dE) levels. g(E) is the density of states and f(E) is probability of occupancy in that perticular levels. Then dn=g(E)f(E)dE.

By integrating dn in between Ec and infinity, entire electron concentration will be obtained.

By integrating dn in between Ec and infinity, entire electron concentration will be obtained.

Similarly for holes in valence band,

for intrinsic semiconductor we know n=p=ni so by multiplying both the above equations we get

so the carrier concentration in intrinsic semiconductor is

*Carrier concentration of Extrinsic semiconductor:*

*Carrier concentration of n-type semiconductor:*

Let Nd is the donor concentration and Ed is the donor energy level. At very low temperatures, donor energy levels are filled with electrons. With the increase in temperature, the donor electrons will move to conduction band.

At lower temperatures, electron concentration in conduction band is given by

At lower temperatures, electron concentration in conduction band is given by

Concentration of electrons in donor energy level is given by

Equating both the above equations,

By applying logarithms on both sides and by rearranging the equation, we get

Substituting the above equation in the concentration of electrons in conduction band, we g

The Fermi level shifts downwards when temperature increases and finally reaches to middle of band gap. That is at high temperatures, the n-type semiconductor behaves like intrinsic semiconductor.

*Carrier concentration of p-type semiconductor:*

Let Na is the acceptor concentration, Ea is the acceptor energy level. At low temperatures acceptor levels are filled with holes. With the increase in temperature, the acceptor electrons will move to conduction band.

At lower temperature, the holes concentration in valance band is given by

At lower temperature, the holes concentration in valance band is given by

The concentration of electrons in acceptor level is given by

These two are equal at lower temperatures and hence by equating we get

By applying logarithms and rearranging the equation, we get

Substituting this equation in the concentration of holes in valance band, we get