2.4 Point Defects in Ionic Crystals

2.4.1 Motivation and Basics

Point defects in ionic crystals (e.g. NaCl or AgCl2) and oxides (e.g. SnO2 or ZrO2) are quite important and put to technical uses.
Unfortunately (from the metal oriented persons point of view) the scientific community working with those materials has its own way for dealing with point defects, which differs in some respects from the viewpoint of the metal and semiconductor community.
There are historical and "cultural" reasons for this, but there are also good reasons. Essentially, in dealing with more complicated crystals - and ionic materials or oxides are always more complicated than metals or simple semiconductors - a more chemical point of view is traditional and useful. Let us look at some important points that have to be considered in this context.
First, we look at the stoichiometry of these crystals.
Ionic crystals must consist of at least two different kinds of ions. They may then contain point defects in concentrations far above thermal equilibrium (as defined relative to a perfect crystal), if the real material is non-stoichiometric. If you imagine a single crystal of, lets say, NaCl with the composition Na1 - dCl and d << 1, i.e close to, but not exactly at stoichiometry (which is what you always would expect in reality), your only way of forming a crystal seems to be to use some point defects as integral part of the crystal.
You might consider, e.g., to introduce a concentration of d vacancies on the Na lattice sites, or to put a concentration of d Cl ions in interstitial positions, or to mix both defect types in a ratio where the sum of the concentrations somehow equals d.
But now lets think again. If you consider a crystal of Na1 - dCl, you are really talking about a crystal with N atoms of negatively charged Cl- ions and N · (1 – d) positively charged Na+ ions, which means that the crystal would carry a net negative charge of e · d · N and thus a dramatically high energy. No such crystal can exist - there must always be equal numbers of Na+ and Cl ions - as long as there are no impurity atoms.
This leads us to the second point, the necessity for charge equilibrium or "zero net charge condition" considered before.
If we stay with the above example of NaCl, we are forced to conclude that a NaCl crystal would be necessarily perfectly stoichiometric - it cannot grow in any other way. However, no crystal exists without some impurities. If, for example, some Ca atoms are to be included into an otherwise perfectly stoichiometric NaCl crystal, they will always be doubly charged Ca++ ions, and we now must remove twice the number of Na+ ions to preserve charge neutrality (or introduce twice the number of additional Cl- ions). Obviously we now must introduce a Na vacancy for every Ca++ ion included in the crystal (or Cl- interstitials and so on).
The concentration of vacancies now could be much higher than the thermal equilibrium concentration. But we still may have equilibrium; namely chemical equilibrium, or, if the defects are charged, electrochemical equilibrium!
We see with this simple example, that there is a linkage between stoichiometry, charge neutrality, impurities and defects, with the added complication that it is not necessarily clear which kinds of point defects must be present in what concentration.
We also see that point defects in concentrations that have nothing to do with the thermal equilibrium concentration in perfect crystals may be an integral part of a real ionic crystal.
The simple example, however, makes also clear that stoichiometry, impurity, and charge neutrality considerations still do not tell us exactly what kinds of point defects are needed in which concentration, but at best will give some integral numbers.
Let us look at a third point. It concerns the surface and its interaction with the surroundings - this is where many applications come in.
Consider a ZrO2 crystal in thermal equilibrium with a gas containing a certain O2 concentration, at a temperature where the oxygen in the crystal is mobile to some extent (maybe because there are O-vacancies?). We must expect some "chemical" reaction to take place. Some additional oxygen may be incorporated into the crystal, or some oxygen may diffuse out of the crystal into the gas. The tendency of whatever is going to happen in this case will be determined by the conditions for chemical equilibrium, or, in other word, by the chemical potentials of the participating species.
But we must expect that point defects are involved in whatever happens across the interface. For the particular example given (which happens to describe the principle of an oxygen sensor) we must expect that some electrical effects take place as well because introducing excess oxygen (always negatively charged) into the crystal or taking some out, will influence the charge distributions and thus electrical potentials in the crystal.
Some electrochemical equilibrium will be reached that contains electrical potential differences - a voltage develops across the interface.
The common denominator in all considerations made so far was: We always had some kind of linkage between "chemistry" as expressed in reactions between atoms or in stoichiometric considerations, and (usually charged) point defects..
We now get the idea of what needs to be done for a general treatment of point defects and ionic crystals:

We want to define point defects in a way
where they can be included
into the familiar concept of chemical reaction equations
We then treat them the same way we treat chemical reactions

In other words, we want to write equations analogous to
n · A  +  m · B    Û    AnBm

[A]n · [B]n
 =  const.
With the option that "A" and "B" may refer not only to atoms, but also to point defects.
We want to do this in a way where we can use the full box of tools developed for chemical reactions, e.g. the mass action law (shown above), chemical potentials and activities instead of concentrations, the concept of kinetics, of chemical equilibrium, etc.
As is turns out, this is possible, but it is not obvious how to do this right. There are several approaches and compromises to achieve the best description. We will look at this in the next subchapter.

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© H. Föll (Defects - Script)