(49) 
It is a property of the FermiDirac distribution that at all finite temperatures
Exercise 6.6
Verify this.
Figure 16 illustrates this for pure (intrinsic) insulators and semi conductors.

If the multiplicity of energy states from to
is given by
, then the number of electrons
with energies from to
is
(51) 
Remember that in the case of free electrons in a three dimensional box (a Fermi gas,
for example, a metal) we found that
(53)  
The Fermi energy, , could be found by allocating the electrons sequentially
to the available states at from energies to
.
(54)  
(55) 
What happens in a semiconductor ?
Intrinsic Carrier Concentration
Semiconductor behaviour is defined by the conductivity due to the electrons
crossing the (narrow) band gap due to thermal excitations.
Our aim is to adapt the expression for in a free electron system to be appropriate for electrons (holes) in a semiconductor.
We can then find the number of electrons in the conduction band and holes in the valence band, due to thermal excitation of electrons from the valence band to the conduction band. These "charge carriers" account for the conductivity of the material.
We know that in a semiconductor, the electrons are nearly free, but
not actually free. There is a periodic potential due to the lattice cores, as in figure
3.
We have seen that the electrons acquire an effective mass
in the semi conducting material , which can be more or less than their free mass
at rest . Part of the effect of the periodic lattice potential can therefore be
modelled by simply replacing the mass of the electron with its effective mass in the
equation 52 for the multiplicity of states
,
(56) 
(57) 
Figure 18 shows how Fermi distribution maps to the band gap.
Why place the Fermi energy in the middle of the gap ?
We will prove shortly this actually where it is , in the intrinsic case.
However, for now, consider that the position of the Fermi energy is defined by equation 50, . Then as is inbetween the filled valence band states and empty conduction band states (at K), we note this is an intuitive place for the Fermi level.
When we look at electrons in the conduction band, we note that here electrons are free and we can use the multiplicity of states as for a free electron system, provided we measure the density of states from the conduction band edge upwards.
For electrons in the conduction band
(58) 
When we look at holes in the valence band, we note that here holes are free and we can use the multiplicity of states as for a free electron system, provided we measure the density of states from the valence band edge downwards.
For holes in the valence band
(59) 
In the conduction band,
, so we may approximate
(60) 
The number of electrons in energy interval is now
(61) 
The number of electrons is therefore
(62) 
(63)  
(64)  
In an intrinsic semiconductor, the probability of a hole in the valence band
is the complimentary probability for an electron in the valence band, at finite temperature.
(66) 
Exercise 6.7
Verify this.
As before,
,
so
(67) 

The number of holes in the valence band will be given by the multiplicity of states
for holes in the valence band, multiplied by the probability distribution
of a hole in the valence band.
(68) 
(69) 
(70) 
Exercise 6.8
Compare the equations 65 and 71, for the number of electrons and holes
respectively. Are there forms symmetrical, as expected ?
Consider the product of the number of holes and electrons.
(72) 
Exercise 6.9
What are the consequences of this ?
From now on, we will write
negative charge carriers  
positive charge carriers 
Intrinsic semiconductors
In an intrinsic semiconductor, all the electrons in the conduction band are thermally
excited from the valence band.
Clearly, , that is, . From equations 65 and 71,
(73) 
(74) 
(75) 
Example : Gallium Arsenide
=  0.072  
=  0.4  
=  1.0 eV 
In the intrinsic case
(77) 
(78) 
Example : Silicon and Germanium
cm  =  for Ge 
cm  =  for Si 
Extrinsic semiconductors  Substitutional dopants
Intrinsic semiconductors contain only the elemental material of their pure substance.
Extrinsic semiconductors have been specially modified so that some of the atoms of
the pure substance are substituted by carefully chosen foreign atoms called dopants.
The concentration of dopants ranges from /cm to /cm.
Since the number of pure atoms is usually about /cm, it is clear that
the dopant concentrations are about 1 ppm to 0.1 ppb. This is exceedingly low. Clearly,
the concentrations of unwanted impurities must be even lower than this. It is
therefore understandable that microchips have to be fabricated from the purest material
under carefully controlled clean conditions.
The presence of substitutional impurities changes the electronic structure of the semiconductor locally around the impurity. Usually, the doping is intended to produce a hole state just above the valence band, or an electron state just under the conduction band.

For example, in a group IV semiconductor, each atom is tetrahedrally located with respect to its neighbours. Its valence electrons participate in covalent bonds with these neighbours. A substitutional donor impurity would have one extra valence electron. This additional electron would not be able to participate in the covalent bonding to the neighbouring atoms. To a first approximation, it occupies a hydrogenic orbital around its core, but with an extremely large dimension. The wave function of this hydrogenic orbital is typically some thousand angstroms (Figure 21). Usually, the doping is not of a sufficiently high concentration that the wavefunctions of the extra substitutional donor impurity electrons atoms overlap each other. These dopant electron states are therefore localised. As the dopant electron is well screened from its core, it is easily ionised to the conduction band. The position of the dopant state in the band gap is therefore just under the conduction band. There are some devices where the material is deliberately overdoped, so that the substitutional donor impurity electron wavefunctions overlap and form a band. Figure 22 illustrates the foregoing text.

In extrinsic materials at moderate temperatures, the conduction is dominated by the electrons or holes originating from impurity or defect states in the lattice. Even when the temperature is too low for intrinsic electronhole pair generation, these localised states within the band gap can be thermally ionised to generate free electrons or holes as shown in figure 23. This is the extrinsic regime of the semiconductor. Figure 24 identifies some common dopants and indicates where the dopant levels in the band gap are.

Ionisation energy of dopant states
We model a donor state as a positive core with one electron in the hydrogenic analogue
scenario.
(79) 
(80) 
(81) 
(82) 
Example : The Phosphorus donor impurity in Silicon
Relevant data :
=  11.7 for Si  
=  0.19 near the bottom of the conduction band in Si 
(83)  
(84)  
(85) 
So, according to this model, the donor impurity electron requires 0.019 eV to become ionised. The position in the band gap of the donor impurity level is therefore just 0.019 eV below the conduction band edge.
Majority carriers
In general, one impurity type dominates in an extrinsic semiconductor.
The associated carrier is known as the majority carrier.
For an type material, the majority carrier is an electron.
For an type material, the majority carrier is a hole.
Position of the Fermi level in extrinsic semiconductors
We consider first the case of type doping with a substitutional donor impurity.
Just as we did before in the case of the
intrinsic semiconductor, we can calculate the number of electrons in the conduction
band as a result of thermal excitation of electrons from the donor levels
(ie. ionisation of the donors).
As before, the FermiDirac distribution gives the probability of populating the
conduction band. We use the free electron multiplicity of states, which can
describe the multiplicity of states in the case of a semiconductor with a band gap provided
we replace the free electron mass with the effective mass .
Since the electrons are coming from the donor level, not the valence band, we replace
the energy of the the valence band with that of the donor level, .
This situation is represented in figure 25. Compare it to the situation
of the intrinsic semiconductor in figure 18

Using similar arguments and mathematics as in the intrinsic case, we find the Fermi level
in the case of type material :
Similarly we find the Fermi level in the case of type material :
(87) 
It turns out that the second terms in the last two equations are so small that they can be
neglected. For an type semiconductor
(88) 
(89) 
(90) 
(91) 
(93) 
(94) 
(95) 
Law of MassAction for semiconductors
(96) 
Exercise 6.9
Verify this.
Similarly, for a type semiconductor :
(97) 
(98) 
The temperature dependance of the charge carrier concentration, for example, for
electrons in ntype material,
(99) 
(100) 
(101) 
Consider again the diagram (figure 20) showing the plot of versus .
Majority and Minority carriers
Semiconductors often have both donors and acceptors present. Usually one type dominates.
An type semiconductor is one dominated by negative charge carriers.
(102) 
(103) 
(104) 
(105) 