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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. |
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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
Oi (oxygen interstitials) and Ci(Carbon
substitutionals). |
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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 non-equilibrium 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? |
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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. |
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However, using the word "impurity" instead of "alloy" implies that we are talking about
small amounts of B in crystal
A. |
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The decisive parameter is the solubility of the impurity atom as a function of
temperature. |
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In a first approximation, the equilibrium concentration of
impurity atoms is given by the usual
Arrhenius
representation, akin to the case of vacancies or self-interstitials.
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. |
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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. |
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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. |
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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: |
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A 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). |
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| cC |
= |
z · cF · cV(T)
1 – z · cF |
· exp |
ΔSC
k |
· exp |
HC
kT |
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With cC = concentration of
vacancy-impurity atom complexes, cF = concentration of
impurity atoms, cV = equilibrium concentration of
(single) vacancies, and ΔSC
or HC = binding entropy or enthalpy, resp., of the
pair. z, again, is
the coordination number. |
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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
· cF must be some correction factor, obviously
accounting for the possible case of rather large impurity concentrations
cF. Why? - Well, for small
cF, this term is just about 1! |
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Note also that as far as equilibrium goes, we have a kind of
mixed case here. The impurity atoms have some concentration
cF that is not an
equilibrium concentration. But if we redefine equilibrium as the state of
crystal plus impurities (essentially we simply change the
G0 = Gibbs energy of the "perfect" crystal
in one of our first equations),
than the concentration concentration cC of
vacancy-impurity atom complexes is an equilibrium concentration. |
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The equation above
for cC is quite similar to
what we had for the divacancy
concentration. |
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If you forget the "correction factor" for a moment,
we have identical exponential terms describing the binding enthalpy, and
pre-exponential factors of z · cV ·
cF for divacancies and z ·
cV · cV for the vacancy -
impurity complexes. |
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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
vacancy-impurity complexes - I sincerely hope that you can see why! |
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The total vacancy concentration
c1V(total) (= concentration of isolated vacancies +
concentration of vacancies in the complexes) as opposed to
the equilibrium concentration without
impurities c1V(eq) is given by |
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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. |
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cC 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 cV >>
cC under normal circumstances. |
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We will find out if this is true and
more about vacancy - impurity complexes in an exercise. |
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You don't have to do it all yourself; but at
least look at it - it's worth it. |
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© H. Föll
