
In all semiconductors, lattice
defects change the electronic properties of the material locally, and this may
result in electronic energy states in the band gap of the semiconductor and
this is true for all kinds of lattice defects 


Semiconductor technology actually depends
completely on this fact. Doping a
semiconductor, after all, mostly means the incorporation of (usually)
substitutional extrinsic point
defects in defined concentrations in defined regions of the crystal 
we have B, As and P for Si. 


Our extrinsic point defects now exist in two
states: we have some concentration [P]^{0} of e.g. a neutral
donor like P and some concentration [P]^{+} of ionized
donors; and [P]^{0} + [P]^{+} = [P_{0}], the
total concentration of P holds at all conditions. 


The concentration [P]^{+} is
simply given by the the total concentration times the probability that the
electronic state associated with the P impurity atom is not occupied by an electron. 


If this electrons state is at an energy
E_{D} in the band gap,
basic
semiconductor physics tells us that for a given
E_{F} and temperature T the concentration
of ionized impurity atoms is given by 


[P^{+}]_{ } 
= [P^{0}]
·_{ } 
{1 –
f(E_{D}, E_{F}; T)} 




»
[P^{0}] _{ }· 
exp 
E_{F} –
E_{D}
kT_{ } 



There is no reason whatsoever that a
vacancy (or any other point defect you care to come up with) should not have a
energy level (or even more than one) in the band gap of its host semiconductor.
This level then will be occupied or not occupied by electrons exactly like the
extrinsic point defect. 


If the vacancy is mobile at the temperature
considered, it will diffuse around  exactly like an extrinsic mobile
defect. 


If the temperature changes, the intrinsic point
defects concentration changes to the extent that it can establish equilibrium 
in pronounced contrast to the extrinsic point
defects. 

It should be clear form this, that
intrinsic point defects in semiconductors are not all that simple. Charge
states must be considered that depend on primary doping with extrinsic point
defects and temperature. If things get really messy, the intrinsic point
defects change the actual doping and their mobility (or diffusion coefficient)
depends on their charge state. 

Looking at jus at few topics in the
case of Si, we obtain a bunch of complex relations, which shall only be
touched upon: 


Once again, the equilibrium concentration of
charged point defects depends on the Fermi
energy E_{F} (which is the chemical potential of
the electrons). As an example, for a negatively charged vacancy we obtain 


c(V ^{–}_{ }) 
= c(V) · exp 
E_{F} –
E_{A}
kT_{ } 




With E_{F} = Fermi energy,
and E_{A} = acceptor level of
the vacancy in the band gap. 


This tells us that besides the formation energies
and entropies, we now also must know the
energy levels
of the defects in the band gap! 


The dependence of the concentration of
arbitrarily charged point defects on the carrier concentration (i.e. on doping)
is given by 


c_{Vx}(n)
c_{Vx}(n_{i}) 
= 
æ
ç
è 
n
n_{i} 
ö
÷
ø 
–x 




With n_{i}, n
= (intrinsic) carrier density, x = charge state of point
defect. 

As a Si special, we also must consider
selfinterstitials (which, if
you remember, we always can safely neglect for just about any other elemental
crystal) 


Local equilibrium between vacancies and
interstitials follows this relation: 


c_{V}(loc) · c_{i}(loc) 
» 
c_{V}(equ) · c_{i}(equ) 



Considering that carrier densities
and the Fermi energy depend on the temperature, too, things obviously get
complicated! 

It thus should not be a big surprise
that the scientific community still has not come up with reliable, or least
undisputed numbers for the basic properties of intrinsic point defects in
Si, not to mention the more complicated semiconductors. 


But do not let yourself be deceived by this:
While you might have problems coming up
with numbers for e.g. the vacancy concentration in Si at some
temperature and so on, the Si
crystal has no problems whatsoever to "produce" the
concentration that is just right for this condition. 