Chemistry Reference
In-Depth Information
where is a reference energy called the Fermi energy. At meaning that
the probability of a state being occupied by an electron for the energy level at the Fermi
level is 1/2. For not too highly doped
n
-type material the Fermi level is well below the
conduction band such that and the Fermi function reduces to the simpler
Maxwell-Boltzmann distribution function:
The total density of electrons for not too heavily doped
n
-type material can then be
found by the product of the density of allowed states in the conduction band,
g
(
E
), and
the probability that these states are filled and integrating over the conduction band:
Similarly, for moderately doped
p
-type material, the density of holes in the valence
band is given by
where and are the effective densities of energy states at the bottom of the con-
duction band and at the top of the valence band, respectively. For large dopant con-
centrations
or
for silicon, these
equations are no longer
valid as the Fermi-Dirac distribution cannot be approximated by the Maxwell-
Boltzmann distribution function and a different distribution function should be used.
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At very high doping levels or the semiconductor is degenerated
because the Fermi level is within the conduction or the valence band. As a result,
allowed states exist very near the Fermi level, just as in metal. Consequently, the elec-
tronic properties of heavily doped semiconductors become similar to those of metals.
Equations (1.2) and (1.3) essentially depict the chemical potential of the elec-
trons,
and of the holes,
respectively:
At equilibrium,
Eqs. (1.4) and (1.5) become
where is the intrinsic carrier density. Equation (1.6) means that at thermal equilib-
rium the product of the electron and hole densities for a given semiconductor and tem-
perature is a constant. For silicon at room temperature,
which
leads to
