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Frenkel defects are, like
Schottky defects, a speciality of ionic crystals. Consult this
illustration
modul for pictures and more details. |
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In fact, the discussion of this defect in
AgCl in 1926 by Frenkel
more or less introduced the concepts of point defects in crystals to science.
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In ionic crystals,
charge neutrality requires (as
we will see) that defects come in pairs with opposite charge, or at
least the sum over the net charge of all charged point defects must be zero.
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"Designer
defects" (defects carrying name tags) are special cases of the general point defect situation
in non-elemental crystals. Since any ionic crystal consists of at least two
different kinds of atoms, at least two kinds of vacancies and interstitials are
possible in principle. |
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Thermodynamic equilibrium always allows all possible kinds of point defects simultaneously
(including charged defects) with arbitrary concentrations, but always requiring
a minimal free enthalpy including the electrostatic
energy components in this case. |
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However, if there is a charge inbalance,
electrostatic energy will quickly override everything else, as we will see. As
a consequence we need charge neutrality in total and in any small volume element of the crystal - we
have a kind of independent boundary condition for equilibrium. |
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Charge neutrality calls for at least two kinds of differently charged point defects. We
could have more than just two kinds, of course, but again as we will see, in
real crystals usually two kinds will suffice. |
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One of two simple
ways of maintaining charge neutrality with two different point defects is to
always have a vacancy - interstitial pair, a combination we will call a
Frenkel pair. |
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The generation of a Frenkel defect is easy to
visualize:
A lattice ion moves to an interstitial site, leaving a vacancy behind. The ion
will always be the positively charged one, i.e. a cation interstitial, because
it is pretty much always smaller than the negatively charged one and thus fits
better into the interstitial sites. In other words; its formation enthalpy will
be smaller than that of a negatively charged interstitial ion. Look at the
pictures to
see this very clearly. |
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It may appear that electrostatic forces keep the
interstitial and the vacancy in close proximity. While there is an attractive
interaction, and close Frenkel pairs do exist (in analogy to
excitons, i.e.
close electron-hole pairs in semiconductors), they will not be stable at high
temperatures. If the defects can diffuse, the interstitial and the vacancy of a
Frenkel pair will go on independent random walks and thus can be anywhere, they
do not have to be close to each other after their generation. |
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Having vacancies and interstitials is
called Frenkel disorder, it consists of
Frenkel pairs or the Frenkel defects. |
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Frenkel disorder is an extreme case of general
disorder; it is prevalent in e.g. Ag - halogen crystals like
AgCl. We thus have |
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This implies, of course, that
vacancies carry a charge; and that is a
bit of a conceptual problems. For ions as interstitials, however, their charge
is obvious. How can we understand a charge "nothing"? |
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Well, vacancies can be seen as charge carriers in
analogy to holes in
semiconductors. There a missing electron - a hole - is carrying the opposite charge of the electron. |
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For a vacancy, the same reasoning applies. If a
Na+ lattice ion is missing, a positive charge is missing in the volume element that contains the
corresponding vacancy. Since "missing" charges are non-entities, we
have to assign a negative charge to the
vacancy in the volume element to get the
charge balance right. |
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Of course, any monoatomic crystal could (and
will) have arbitrary numbers of vacancies and interstitials at the same time as intrinsic
point defects; but only if charge consideration are important
ni = nv holds exactly; otherwise the
two concentrations are uncorrelated and simply given by the formula for the
equilibrium concentrations. |
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Indeed, since the equilibrium concentrations are
never exactly zero, all crystals will have
vacancies and interstitials present at the
same time, but since the formation energy of interstitials is usually much
larger than that of vacancies, they may be safely neglected for most
considerations (with the big exception of Silicon!). |
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Of course, in biatomic ionic
crystals, there could (and will) be two
kinds of Frenkel defects: cation vacancy and cation interstitial; anion vacancy
and anion interstitial; but in any given crystal one kind will always be
prevalent. |
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We will take up all these finer points in modules
to come, but now let's just look at the simple limiting case of pure Frenkel
disorder. |
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With the equilibrium condition
∂G/∂n = 0 we obtain for the
concentration cFP of Frenkel pairs |
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cFP |
= |
nFP
N |
= |
( |
N'
N |
) |
1/2 |
· exp |
SFP
2k |
· exp |
HFP
2kT |
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The factor 1/2 in the exponent comes from equating the
formation energy HFP or entropy, resp., with a
pair of point defects and not with an
individual defect. |
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What is the reality, i.e. what kind
of formation enthalpies are encountered? Surprisingly, it is not particularly
easy to find measured values; the link, however, will give some
numbers. |
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That was rather straight forward, and
we will not discuss Frenkel defects much more at this point. We will, however,
show in the next subchapter from first principles that, indeed, charge
neutrality has to be maintained. |
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© H. Föll