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*Introduction*

As a function of the Fermi level, E_{F}, the electron
and hole concentrations are given, respectively, by:

__Fermi - Dirac__

**n = N _{c} F_{1/2}(h_{c}),
where h_{c }= (E_{F} - E_{c})/kT,
and**

**p = N _{v} F_{1/2}(h_{v}),
where h_{v }= (E_{v} - E_{F})/kT.**

These are the exact formulae for electron and hole concentrations,
according to the *Fermi-Dirac statistics*. The mysterious function,
*F** _{1/2}*,
is called the

*F _{n}*(h)
= [1 / G(n+1)] x

People do not like to work with a black box function and
thus they try to use an explicit and familiar function when they can. Fortunately,
a simple approximation is available when h_{
}< -3. This is the so-called Maxwell-Boltzman approximation:

*F _{1/2}*(h)
» exp(h) for
h

__Maxwell-Boltzman__

**n »
N _{c} exp[ -(E_{c} - E_{F})/kT ],
**

**p »
N _{v} exp[ -(E_{F} - E_{v})/kT ].**

Applies if E_{v} + 3 kT < E_{F}
< E_{c} - 3 kT.

The following applet demonstrates the applicability of MB approximation vs. the Fermi level position. Here FD is the exact concentration [2] according to the Fermi-Dirac integral [3].

In the laboratory (i.e., *cleanroom *[footnote]),
the electron concentration in the conduction band is determined by the
doping level of Donor impurities and the temperature of the material.

In the classroom, the electron concentration, *n*, is expressed
in terms of the Fermi level position, *E _{F}*, and the temperature,

*n depends on E _{c} - E_{F} and kT.*

The mathematical formula can take an *approximate*, but* *simple
exponential *Maxwell-Boltzman* form if

E_{c} - E_{F} > 3 kT (or, N_{d} <
0.05N_{c}) --- [n-type]

E_{f }- E_{v} > 3 kT (or, N_{a} < 0.05N_{v})
--- [p-type]

The more complicated, accurate* Fermi-Dirac formula* must be used
if

*E _{c} - E_{F} < 3 kT* ---
[heavily-doped n-tyep]

The simple approximation is possible for lightly-to-moderately-doped
semiconductor (called *nondegenerate semiconductor* ). Otherwise,
the full formula must be used in heavily-doped semiconductor (also called,
*degenerate semiconductor* because the fermi energy can be degenerate
with, or 'the same' as, a band state.)

The applet demonstrates the discrepancy in electron (or hole in p-type) concentration between the approximate formula [Maxwell-Boltzman approximation] and the full, accurate formula [Fermi-Dirac formula] when the concentration is high.

footnote-1: In the cleanroom (or fab line), the majority carrier concentration is controlled by the precise doping level of impurity atoms.