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Mathematical Analysis

The Fermi function is

f(E) = 1 / [ 1 + exp( (E-EF)/kT ) ]       (1)

whose physical meaning is the pobability of electron occupancy for an energy state at energy E. It is plotted as a function of Energy in the left hand side of this applet.

For E - EF >> kT (say, for E > EF + 4 kT)

f(E) ~ exp( -(E-EF)/kT ).

This means that the probability of occupancy approaches zero exponentially with increasing E for states at energy E > EF + 4kT. This is the so-called Boltzman approaximation (that is, f(E) approximated by a simple exponential term).

For E - EF << - kT (say, for E < EF - 4kT),

f(E) ~ 1 - exp( (E-EF)/kT ).

This means that the probability of occupancy approaches unity with decreasing energy for E < EF - 4kT.

Note that the maximum value of f(E) is unity.

For EF - 4kT < E < EF + 4kT, no simple exponential function approaximation is possible and the full expression, formula (1), must be used for f(E).

Note that f(E) = 1/2 at E = EF.