Distribution of electrons and holes among energy states is governed by the Fermi-Dirac statistics which is represented by the distribution function:

where

E = energy of the state,

E_Fn = Fermi energy,

T = absolute temperature, k = Boltzman constant,

f_n(E) = probability that an energy state at E is occupied by an electron.

This formula says that the occupation probability is determined by the relative energy spacing between E and E_Fn, and the temperature. This was demonstrated in a previous applet.

The concentration of conduction band electrons depends on

The accurate formula is

where

Here, F_1/2 is the Fermi integral of order 1/2, defined as:

and N_c is the effective density of states of conduction band given by:

An approximation can be made if E_c - E_F > 3 kT. In which case,

This formula is applicable if n < 0.05 N_c. That is, if the donor impurity concentration, N_d, is smaller than about 5% of Nc. If N_d > 0.05 N_c, then the accurate formula (2) should be used.

At room temperature, the effective density of states is as follows:

Similarly, for a moderately-doped p-type where N_a < 0.05 N_v, hole concentration is:

For heavily-doped p-type where Na > 0.05 Nv,

where,