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The mobile electrons in a semiconductor, carrying electrical charge under any applied voltage, are the electrons occupying the energy states of conduction band (E > Ec). 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 = Ec ~ +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 ~ Ev.

Read the Mathematical Analysis for further details.