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Distribution of electrons and holes among energy states is governed by the Fermi-Dirac statistics which is represented by the distribution function:
E = energy of the state,
EFn = Fermi energy,
T = absolute temperature, k = Boltzman constant,
fn(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 EFn, and the temperature. This was demonstrated in a previous applet.
The concentration of conduction band electrons depends on
(Ec - EF) / kT
The accurate formula is
Here, F1/2 is the Fermi integral of order 1/2, defined as:
and Nc is the effective density of states of conduction band given by:
An approximation can be made if Ec - EF > 3 kT. In which case,
This formula is applicable if n < 0.05 Nc. That is, if the donor impurity concentration, Nd, is smaller than about 5% of Nc. If Nd > 0.05 Nc, 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 Na < 0.05 Nv, hole concentration is:
For heavily-doped p-type where Na > 0.05 Nv,