*Introduction*

The mobile electrons in a semiconductor, carrying electrical charge
under any applied voltage, are the electrons occupying the energy states
of conduction band (E > E_{c}). The counting statistics of conduction
band electrons is based on

number of electrons = (number of states) * (probability of occupancy by an electron).

The number of electron energy states per unit volume over the energy interval, E ~ E+dE, is defined as

number of states = g(E)dE.

Here g(E) is called the density of states (DOS).

Then, an energy state at energy E has a definite probability to be occupied by an electron given by

probability of occupancy = f(E).

This probability is the main conclusion of the *Fermi-Dirac statistics*
and f(E) is called the *Fermi distribution function*.

Therefore, the number of electrons over the energy interval, E ~ E+dE, is found as

number of electrons over E ~ E+dE = (number of states) * (probability
of occupancy)

= g(E)dE * f(E)

Total number of conduction band electrons is found by integrating this:

n = g(E)f(E)dE
over E = E_{c} ~ +infinity.

For holes in the valence band (i.e., the unoccupied states in VB),

number of holes over E ~ E+dE = (number of states) * (probability of
vacancy)

= g(E)dE * [1-f(E)]

And the total number of holes,

p = g(E)[1-f(E)]dE
over E = -infinity ~ E_{v}.

Read the *Mathematical Analysis* for further
details.