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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,

EFn = Fermi energy,

T = absolute temperature, k = Boltzman constant,

fn(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 EFn, and the temperature. This was demonstrated in a previous applet.

The concentration of conduction band electrons depends on

(Ec - EF) / kT

The accurate formula is

(2)

where

Here, F1/2 is the Fermi integral of order 1/2, defined as:

and Nc is the effective density of states of conduction band given by:

.

An approximation can be made if Ec - EF > 3 kT. In which case,

.    (3)

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

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

 Si GaAs Ge Eg(eV) 1.12 1.42 0.66 Nc [cm-3] 2.8x1019 4.7x1017 1.04x1019 Nv [cm-3] 1.04x1019 7.0x1018 6x1019

Similarly, for a moderately-doped p-type where Na < 0.05 Nv, hole concentration is:

.    (4)

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

.   (5)

where,