
Consider a real crystal  take even a hyperpure single
crystalline Si crystal if you like. It's not perfect! It just is not. It
will always contain some impurities. If the impurity concentration is below the
ppm level, then you will have ppb, or ppt or ppqt
(figure that
out!), or...  it's just never going to be zero. 


The highest vacancy concentration
your are going to have in simple metals close to the melting point is around
10^{–4} = 100 ppm; in Si it will be far lower. On
the other hand, even in the best Si you will have some ppm of
O_{i} (oxygen interstitials) and C_{i}(Carbon
substitutionals). 


In other words  it is quite likely
that besides your intrinsic
equilibrium point defects (usually vacancies) squirming around in equilibrium
concentration, you also have comparable concentrations of various
extrinsic nonequilibrium point
defects. So the question obviously is: what is going to happen between the
vacancies and the "dirt"? How do intrinsic and extrinsic point
defects interact? 

Let's look at the impurities first.
Essentially, we are talking
phase diagrams
here. If you know the phase diagram, you know what happens if you put
increasing amounts of an impurity atom in
your crystal. Turned around: If you know what your impurity "does",
you actually can construct a phase diagram. 


However, using the word "impurity" instead of "alloy" implies that we are talking about
small amounts of B in crystal
A. 

The decisive parameter is the solubility of the impurity atom as a function of
temperature. 


In a first approximation, the equilibrium
concentration of impurity atoms is given by the usual
Arrhenius
representation, akin to the case of vacancies or selfinterstitials.
This is often only a good approximation below the eutectic temperature (if
there is one). Instead of the formation energies and entropies, you resort to
solubility energies and
entropies. 


There is a big difference
with intrinsic point defects, however. The concentration of impurity
atoms in a given crystal is pretty much constant and not a quantity that can
find its equilibrium value. After all, you can neither easily form nor destroy
impurity atoms contained in a crystal. 


That means that thermal equilibrium is only
obtained at one specific temperature, if at
all. For all other temperatures, impurity atoms are either
undersaturated or oversaturated. 

Now the obvious: Vacancies,
divacancies, interstitials etc. may interact with impurity atoms to form complexes  provided that there is some attractive
interaction. Interactions may be elastic (e.g. the lattice deformation of a big
impurity interstitial will attract vacancies) or electrostatic if the point
defects are charged. Schematically it may look like this: 





An impurity  vacancy complex
(also known as Johnson complex)
is similar to a divacancy, just one of the partners is now an impurity atom.
The calculation of the
equilibrium
concentration of impurity  vacancy complexes thus proceeds in analogy
to the calculations for double
vacancies, but it is somewhat more involved. We obtain (for
details use the link). 


c_{C} 
= 
z · c_{F} · c_{V}(T)
1 – z · c_{F} 
· exp 
DS_{C}
k 
· exp 
H_{C}
kT 








» 
z · c_{F} · c_{V}(T) 
· exp 
DS_{C}
k 
· exp 
H_{C}
kT 




With c_{C} = concentration
of vacancyimpurity atom complexes, c_{F} = concentration
of impurity atoms, c_{V} = equilibrium concentration of
(single) vacancies, and DS_{C}
or H_{C} = binding entropy or enthalpy, resp., of the
pair. z, again, is
the coordination number. 


That the coordination number z
appears in the equation above is not surprising  after all there are always
z possibilities to form one complex. Note that the term 1
– z · c_{F} must be some correction factor,
obviously accounting for the possible case of rather large impurity
concentrations c_{F}. Why?  Well, for small
c_{F}, this term is just about 1 and we get the
approximation from above. 


Note also that as far as equilibrium goes, we have
a kind of mixed case here. The impurity atoms have some concentration
c_{F} that is not an
equilibrium concentration. But if we redefine equilibrium as the state of
crystal plus impurities (essentially we simply change the
G_{0} = Gibbs energy of the "perfect" crystal
in one of our first equations),
than the concentration c_{C} of vacancyimpurity atom
complexes is an equilibrium concentration.


The equation above
for c_{C} is quite similar to
what we had for the divacancy
concentration. 


If you forget the "correction factor"
for a moment, we have identical exponential terms describing the binding
enthalpy, and preexponential factors of z ·
c_{V} · c_{F} for divacancies and
z · c_{V} ·
c_{V} for the vacancy  impurity complexes. 


In both cases the concentrations decreases
exponentially with temperature. However, assuming identical binding enthalpies
for the sake of the argument, in an Arrhenius plot the slope for divacancies
would be twice that of vacancyimpurity complexes  I sincerely hope that you
can see why! 

The total vacancy concentration
c_{1V}(total) (= concentration of isolated vacancies +
concentration of vacancies in the complexes) as opposed to
the equilibrium concentration without
impurities c_{1V}(eq) is given by 


c_{V}(total) 
= 
c_{V}(eq) + c_{C} 




That's what equilbrium means! If impurity atoms
snatch away some vacancies that the crystal "made" in order to be in
equilibrium, it just will make some more until equilibrium is restored. 


c_{C} thus can be seen as a
correction term to the case of the perfect (impurity free) crystal which
describes the perturbation by impurities. This implies that
c_{V} >> c_{C} under normal
circumstances. 

We will find out if this is true and
more about vacancy  impurity complexes in an exercise. 


You don't have to do it all yourself;
but at least look at it  it's worth it. 






