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*Mathematical Analysis*

Fermi statistics relate the carrier concentration (n and p) in the energy
band to the Fermi level position E_{F}. In the general case (an
arbitrary doping-level, or an arbitrary E_{F}-position) the
accurate relation is, for example, n = N_{c} (2/p^{1/2})
F_{1/2}(h). [ For more details, see
this page or another
applet.]

When the doping level is not too high [or, E_{F} is well within
the bandgap], the concentration is approximately given as follows: [Boltzman
approximation]

n » n_{i} exp( (E_{F}
- E_{i})/kT ), if E_{F} < E_{c} - 3kT
[or, N_{d} < 0.05 N_{c}] ----- (1)

p » n_{i} exp( -(E_{F}
- E_{i})/kT ). if E_{F} > E_{v} + 3kT
[or, N_{a} < 0.05 N_{v}]

where n_{i} is the intrinsic carrier concentration and E_{i}
is the intrinsic Fermi level. (In a pure semiconductor, free of any chemical
impurities or structural defects, n = p = n_{i }and E_{F}
= E_{i}). For Si at room temperature 300K, n_{i }=
1.00E10 cm^{-3 }[1]. [*Thanks
to Michael Godfrey of Stanford Univ. for pointing out the correct n _{i}-value!*]

Equation (1) shows that as E_{F} goes above E_{i} [n-type],
electron density n *increases *exponentially and hole density p *decreases
*exponentially. As E_{F} goes below E_{i} [p-type],
the opposite holds for n and p. Note that in Boltzman approximation, the
*np *product is constant, independent of E_{F}:

np = n_{i}^{2}. -------- [mass-action
law] (2)

This is called the *mass-action-law*. The *np *product is
independent of dopant concentration as shown in the following. In semiconductor
processing, the carrier concentration is controlled by the introduction
of dopant chemical impurities. This in turn sets the position of E_{F}
in the band gap relative to E_{i}. In an extrinsic n-type semiconductor
with doping levels N_{d} >> N_{a},

n = N_{d} - N_{a}, ---- (3)

p = n_{i}^{2 }/ n.

The higher the doping level, the closer the E_{F} to the band
edge, Ec or Ev. Note that the above relations (1) hold only if a Boltzman
approximation is valid for f(E). That is, if

E_{v} + 3kT < E_{F} < E_{c} -
3kT. (4)

For a very heavily doped material where E_{F} is within 3kT
from E_{c} or E_{v}, or even inside a band (not bandgap),
the full Fermi function must be used in the derivation of n and p. Therefore
the simple exponential relations (1) no longer valid. These very heavily
doped semiconductors are often called *degenerately *doped because
E_{F} can end up within the band: That is, E_{F }> E_{c}
or E_{F} < E_{v} where Ef is degenerate with one of
the band states. For more discussion, see another
applet and the accompanying Math
Analysis section.

[Apology for the applet: for Si, the applet does not give you an accurate concentration. We will fix it soon. 11/30/98]