The equation for f(E) is the Fermi-Dirac probability function and is

Fermi Level:

The probability that an electron occupies an energy level E is given by the Fermi-Dirac probability function and It can be expressed by

\({{\rm{f}}_{\rm{}}}\left( {\rm{e}} \right) = \frac{1}{{1 + \exp \left[ {\frac{{E - {E_F}}}{{KT}}} \right]}}\)

The probability function of a hole existing below Fermi level = 1 - f (e)

\(=1- \frac{1}{{1 + \exp \left[ {\frac{{E - {E_F}}}{{KT}}} \right]}}\)

\(1-{{\rm{f}}_{\rm{}}}\left( {\rm{e}} \right) = \frac{1}{{1 + \exp \left[ {\frac{{-(E - {E_F)}}}{{KT}}} \right]}}\)

Fermi level for n-type:

The density of electrons in a semiconductor is related to the density of available states and the probability that each of these states is occupied, i.e.

n(E) = gc(E) f(E)

gc(E) = Density of states in the conduction band

f(E) = Probability of electron at Energy Level, E

\(f\left( E \right) = \frac{1}{{1 + \exp \left( {\frac{{E - {E_F}}}{{{K_B}T}}} \right)}}\)

The above can be approximated as:

\(f\left( E \right) \cong \exp \left( { - \frac{{\left( {{E} - E_F} \right)}}{{{K}T}}} \right)\)   ---(1)

The majority carrier density in a semiconductor is obtained by integrating the product of the density of states gc(E) and the probability density function, over all possible states.

For electrons in the conduction band, the integral is taken from the conduction band, labeled EC, to the top, i.e.

\(n = \mathop \smallint \limits_{ - \infty }^{{E_v}} {g_c}\left( E \right)\left[ {f\left( E \right)} \right]dE\)

Using Equation (1), the above equation becomes:

\(n = \mathop \smallint \limits_{ - \infty }^{{E_v}} {g_c}\left( E \right) \exp \left( { - \frac{{\left( {{E} - E_F} \right)}}{{{K}T}}} \right)\)

Assuming all the density of states to be concentrated at the top of the conduction band with effective density given by Nc, the above Equation becomes:

\(n = {N_c}\exp \left[ {\frac{{ - \left( {{E_c} - {E_F}} \right)}}{{KT}}} \right]\)

Since for an n-type semiconductor, ND is the majority carriers, we can write:

\(N_D = {N_c}\exp \left[ {\frac{{ - \left( {{E_c} - {E_F}} \right)}}{{KT}}} \right]\)

\(\frac{N_D}{N_C} = \exp \left[ {\frac{{ - \left( {{E_c} - {E_F}} \right)}}{{KT}}} \right]\)

Taking natural log on both the sides, we get:

\(ln(\frac{N_D}{N_C}) = -{\frac{{\left( {{E_c} - {E_F}} \right)}}{{KT}}}\)

\(KT~ln(\frac{N_C}{N_D}) = E_C-E_F\)

\(E_F=E_C-KT~ln(\frac{N_C}{N_D})\)

Similarly for p-type:

\(E_F=E_V+KT~ln(\frac{N_V}{N_A})\)