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*Mathematical Analysis*

Distribution of electrons and holes among energy states is governed by the Fermi-Dirac statistics which is represented by the distribution function:

(1)

where

E = energy of the state,

E_{Fn} = Fermi energy,

T = absolute temperature, k = Boltzman constant,

f_{n}(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 E_{Fn}, and the temperature.
This was demonstrated
in a previous applet.

The concentration of conduction band electrons depends on

(E_{c} - E_{F}) / kT

The accurate formula is

(2)

where

Here, F_{1/2} is the Fermi integral of order 1/2, defined as:

and N_{c} is the effective density of states of conduction band
given by:

.

An approximation can be made if E_{c} - E_{F} > 3
kT. In which case,

. (3)

This formula is applicable if n < 0.05 N_{c}. That is, if
the donor impurity concentration, N_{d}, is smaller than about
5% of Nc. If N_{d} > 0.05 N_{c}, then the accurate formula
(2) should be used.

At room temperature, the effective density of states is as follows:

Si | GaAs | Ge | |

E_{g}(eV) |
1.12 | 1.42 | 0.66 |

N_{c} [cm^{-3}] |
2.8x10^{19} |
4.7x10^{17} |
1.04x10^{19} |

N_{v} [cm^{-3}] |
1.04x10^{19} |
7.0x10^{18} |
6x10^{19} |

Similarly, for a moderately-doped p-type where N_{a} < 0.05
N_{v}, hole concentration is:

. (4)

For heavily-doped p-type where Na > 0.05 Nv,

. (5)

where,