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Mathematical Analysis

Fermi statistics relate the carrier concentration (n and p) in the energy band to the Fermi level position EF. In the general case (an arbitrary doping-level, or an arbitrary EF-position)  the accurate relation is, for example, n = Nc (2/p1/2) F1/2(h). [ For more details, see this page or another applet.]

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

n » ni exp( (EF - Ei)/kT ),  if EF < Ec - 3kT [or, Nd < 0.05 Nc] ----- (1)

p » ni exp( -(EF - Ei)/kT ).  if EF > Ev + 3kT [or, Na < 0.05 Nv]

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

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

np = ni2.  -------- [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 EF in the band gap relative to Ei. In an extrinsic n-type semiconductor with doping levels Nd >> Na,

n = Nd - Na, ---- (3)

p = ni2 / n.

The higher the doping level, the closer the EF 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

Ev + 3kT < EF < Ec - 3kT.   (4)

For a very heavily doped material where EF is within 3kT from Ec or Ev, 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the full Fermi function must be used in the derivation of n and p.  Therefore the simple exponential relations (1) no longer valid.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 EF can end up within the band: That is, EF > Ec or EF < Ev 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]