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Global thermal equilibrium at arbitrary
temperatures, i.e. the absolute minimum of
the free enthalpy, can only be achieved if there are mechanisms for the
generation and
annihilation of point
defects. |
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There must be sources and
sinks for vacancies and
(intrinsic) interstitials that operate with small activation energies -
otherwise it will take a long time before global equilibrium will be achieved.
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At this point it is essential to
appreciate that an ideal perfect (= infinitely large) crystal has no sources and sinks - it can never be in thermal
equilibrium. An atom, for example, cannot simply disappear leaving a vacancy
behind and then miraculously appear at the surface, as we assumed in
equilibrium thermodynamics, where it does not matter how a state is reached. |
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On the other hand, infinitely large perfect
crystals do not exist - but semiconductor-grade dislocation-free single Si
crystals with diameters of 300 mm and beyond, and lengths of up to 1
m are coming reasonably close. They form a special case as far as point
defects are concerned. |
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Otherwise we need other
defects - grain boundaries, dislocations, precipitates and so on as
sources and sinks for point defects. And that is what we almost always have in
regular metals or ceramics. |
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How a grain boundary can act as
source or sink for vacancies is schematically shown in the pictures below. |
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It is clear form these drawings that the
activation energy (which is not the
formation energy of a vacancy!!) needed to emit (not to form from scratch!) a vacancy from a
grain boundary is small. |
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| Grain boundary absorbs 1 vacancy, i.e. acts as
sink for one more jump of the proper
atom. |
Grain boundary emits 3 vacancies, i.e. acts as
source for one more jump of the 3
proper atoms. |
| The red arrows indicate the jumps of individual
atoms. The flux of the vacancies is always opposite to the flux of diffusing
atoms. |
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We thus may expect that at
sufficiently high temperatures (meaning temperatures large enough to allow
diffusion), we will be able to establish global point
defect equilibrium in a real (= non-ideal) crystal, but not really
global crystal equilibrium, because a
crystal with dislocations and grain boundaries is never at equilibrium. |
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Sources and sinks are a thus a
necessary, but not a sufficient ingredient for point defect
equilibrium. We also must require that the point defects are able to move,
there must be some diffusion - or you must resign yourself to waiting for a
long time. |
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At low
temperatures, when all diffusion effectively stops, nothing goes
anymore. Equilibrium is unreachable. For many practical cases however, this is
of no consequence. At temperatures where diffusion gets
sluggish, the equilibrium
concentration ceq is so low, that you cannot measure
it. For all practical purposes it surely doesn't matter if you really achieve,
for example, ceq = 10–14, or if you
have non-equilibrium with the actual concentration c a thousand
times larger than ceq (i.e. c =
10–11). For all practical purposes we have simply
c = 0. |
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At high
temperatures, when diffusion is fast, point defect equilibrium will
be established very quickly in all real crystals with enough sources and
sinks. |
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The intermediate temperatures thus are of interest. The
mobility is not high enough to allow many point defects to reach convenient
sinks, but not yet too small to find other point defects. |
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In other words, the average diffusion length or mean distance covered by a
randomly diffusing point defect in the time interval considered, is smaller
than the average distance between sinks, but larger than the average distance
between point defects. |
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Global point defect equilibrium as
the best solution is thus unattainable at medium
temperatures, but local
equilibrium is now the second best choice and far preferable to a huge
supersaturation of single point defects slowly moving through the crystal in
search of sinks. |
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Local equilibrium then simply refers to the state
with the smallest free enthalpy taking into account
the restraints of the system. The most simple restraint is that the
total number of vacancies in vacancy clusters of all sizes (from single vacancy
to large "voids") is constant. This acknowledges that vacancies
cannot be annihilated at sinks under these conditions, but still are able to
cluster. |
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Let us illustrate this with a
relevant example. Consider vacancies in a metal crystal that is cooled down
after it has been formed by casting. |
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As the temperature decreases, global equilibrium
demands that the vacancy concentration decreases exponentially. As long as the
vacancies are very mobile, this is possible by annihilation at internal
sinks. |
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However, at somewhat lower temperatures, the
vacancies are less mobile and have not enough time to reach sinks like grain
boundaries, but can still cover distances much larger than their average
separation. This means that divacancies, trivacancies and so on can still form
- up to large clusters of vacancies, either in the shape of a small hole or
void, or, in a two-dimensional form, as small dislocation loops. Until they
become completely immobile, the vacancies will be able to cover a distance
given by the diffusion length L (which depends, of course, on how
quickly we cool down). |
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In other words, at intermediate temperatures
small vacancy clusters or agglomerates can be formed. Their maximum size is
given by the number of vacancies within a volume that is more or less given by
L3 - more vacancies are simply not available for any
one cluster. |
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Obviously, what we will get depends very much on
the cooling rate and the mobility or diffusivity of the vacancies. We will
encounter that again; here is a
link looking a bit
ahead to the situation where we cool down as fast as we can. |
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It remains to find out which mix of
single vacancies and vacancy clusters will have the smallest free enthalpy,
assuming that the total number of vacancies - either single or in clusters -
stays constant. This minimum enthalpy for the specific restraint (number of
vacancies = const.) and a given temperature then would be the local
equilibrium configuration of the system. |
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How do we calculate this? The
simplest answer, once more, comes from using the the mass-action law. We
already used it for deriving the
equilibrium concentration of the divacancies. And we did not assume that the vacancy concentration was in
global thermal equilibrium! The mass action law is valid for any starting concentrations of the ingredients - it
simply describes the equilibrium concentrations for the set of reacting
particles present. This corresponds to what we called local equilibrium
here. |
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The reaction equation from sub-chapter
2.2.1 was 1V + 1V
↔ V2 and in this case this is a
valid equation for using the mass action law. The result obtained for the
concentration of divacancies with the single vacancy concentration in global thermal equilibrium was |
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| c2V |
= (c1V)2 · |
z
2 |
· exp |
ΔS2V
k |
· exp |
B2V
kT |
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Don't forget that concentrations here are defined
as n/N, i.e. in relative units (e.g. c = 3,5 ·
10–5) and not in absolute units, e.g. c = 3,5
· 1015 cm–3. |
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For arbitrary clusters with n
vacancies (1V + 1 V + ... + 1V ↔
Vn) we obtain in an analogous way for the concentration
cnV of clusters with n vacancies |
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| cnV |
= (c1V)n · |
z
2 |
· exp |
ΔSnV
k |
· exp |
BnV
kT |
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with BnV = average
binding energy between vacancies in an n-cluster, and
c1V = const. concentration of the vacancies (and no longer the thermal equilibrium concentration
!). |
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The essential point now is to realize
that these equations still work for local equilibrium! They now describe the
local equilibrium of vacancy clusters if a
fixed concentration of vacancies is given.
The situation now is totally different from
global equilibrium. If we consider divacancies for example, we have: |
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Global
equilibrium: |
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c2V(eq) <<
c1V(eq) ; and c2V(eq)
rapidly decreases with decreasing
temperatures since c1V(eq) decreases. |
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Local
equilibrium: |
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c2V is increasing with decreasing T since
c1V stays about
constant, but we still have the exp+BnV/ kT term that
increases with T |
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Whereas the concentration of clusters
may still be small, they now contain most of the vacancies. |
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Generally speaking, it is always energetically
favorable, to put the surplus vacancies in clusters instead of keeping them in
solid solution if there is no possibility to annihilate them completely. It
thus comes as no surprise that in rapidly cooled down crystals with not to many
defects that can act as sinks, we will find some vacancy clusters at room
temperature |
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It also should come as no surprise
that the same is true for impurity atoms - vacancy clusters. The equations
governing this kind of point defect agglomeration are, after all,
quite similar to the equations
discussed here. |
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If you now take the extreme case of a rather
perfect Si single crystal (no sinks for point defects), where we do not
just have vacancies at thermal equilibrium, but also some relevant
concentration of interstitials, interstitial oxygen and substitutional carbon,
you might well wonder what one will find at room temperature. |
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Well - don't wonder! Get to work! It is not all
that clear. And even if that puzzle has been solved before you reach productive
scientisthood, there is always GaAs, or InP, or SiC, or -
well, you will find something left to do, don't worry. |
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© H. Föll
