Each energy band in a crystal accommodates a large number of electron energy levels. According to Pauli’s exclusion principle any energy level can be occupied by two electrons only, one spin up and down . however, all the available energy states are not filled in an energy band. The separation between the consecutive energy level is very small around 10^{(-27)} eV due to which the energy states are not filled in an energy band.

FERMI DIRAC DISTRIBUTION FUNCTION.

The carrier occupancy of the energy states is represented by a continuous distribution function known as the Fermi-Dirac distribution function, given by

$f(E)= 1/(1+e^{((E-E_F)/kT)} )$

This indicates the probability that a particular quantum state at the energy level E is occupied by an electron. Here k is Boltzmann’s constant and T is absolute temperature of the semiconductor. The energy $E_F$ is called Fermi energy that corresponds to a reference level called Fermi level.

IN n-TYPE SEMICONDUCTOR

At 0K the fermi level $E_{Fn}$ lies between the conduction band and the donor level.

As temperature increases more and more electrons shift to the conduction band leaving behind equal number of holes in the valence band. These electron hole pairs are intrinsic carriers.

With the increase in temperature the intrinsic carriers dominate the donors.

To maintain the balance of the carrier density on both sides the fermi level $E_{Fn}$ gradually shifts downwards.

Finally at high temperature when the donor density is almost negligible $E_{Fn}$ is very close to $E_{Fi}$.