It is my understanding that this may be done by comparing unit cells between the substrate and the material to be grown, but I would want to delve into the details as to how this lattice matching is actually done..
Which parameter should I use IF I want to calculate the lattice mismatch factor between a certain crystal with a substrate? (and whI had parameter should I choose I find more than one parameter i.e. a b and c for a certain crystal) From what I’ve read, the lattice mismatch between two interfaces “f” is calculated by the below formula:
where aepi is related to the crystal lattices of the layer and asub, to the substrate
http://fy.chalmers.se/OLDUSERS/hyonju/research/Epitaxial%20Growth/index.htm
If all you're seeking is the lattice constant, I'm not sure why you'd need an algorithm, you'd only need a list. Typically we're interested in other (functional) properties as well. The most classic example is the famous bandgap versus lattice parameter chart(s) for III-V (and other) semiconductors. An example taken from here: https://woodall.ece.ucdavis.edu/resources/ is attached.
I'm studying oxides and there are several review papers on strain engineering that I find useful, e.g. Schlom et al., MRS Bull. 2014 DOI: 10.1557/mrs.2014.1
Lior thanks.Ok... let me re-word it:
How do I get this “lattice constant” ? I’ve surveyed the litterature for these constants and for some crystals I get 3 i.e. lattice constants a b c, and not just one...
which one of these should I use IF I want to calculate the lattice mismatch factor between a certain crystal with a substrate? From what I’ve read, the lattice mismatch between two interfaces “f” is calculated by the below formula:
where aepi is related to the crystal lattices of the layer and asub, to the substrate
http://fy.chalmers.se/OLDUSERS/hyonju/research/Epitaxial%20Growth/index.htm
Ah, now your question is more clear to me. I recommend reading briefly on Goldschmidt's tolerance factor (there's a decent introduction in Wikipedia). When a,b,c of the perovskite are not the same (but all angles are still 90 deg) it's referred to as an orthorhombic distortion (a=b, not equal to c is called tetragonal distortion), and its one consequence of a tolerance factor lower than 1 (at a certain range). This is common in many of the rare-earth titanates and several other systems (note, this can also be the result of a Jahn-Teller distortion rather than a tolerance factor, e.g.: LaMnO3)
The zero-order (and incorrect) approach/approximation is to consider the crystal as a pseudo-cubic system, which is: what would be apc if a cubic structure would have the same unit cell volume. In other words: apc3 = a*b*c
However, a typical (orthorhombic) nomenclature uses a larger primitive cell (a consequence a symmetry reduction, typically due to octahedral rotations), and you'd need to divide a and b by sqrt(2) and c by 2 (sometimes it's b/2 and c/sqrt(2) ) to get the correct apc. Your sanity check for the perovskite oxides would be getting a value in the neighborhood of 3.7 ~ 4.1 A, otherwise you're missing a factor such as those above.
This approximation would give a very rough estimate and it is strictly incorrect. If the difference between a/sqrt(2), b/sqrt(2) and c/2 is very small, then the approximation becomes more reasonable. Note that some substrates have the same issue, e.g. DyScO3 , NdGaO3 and others.
Sometimes, such as the case of the ferroelectric tetragonal phase of BaTiO3, it is of high importance to use the actual lattice constants because the direction of the long axis is the ferroelectric easy axis, and people specifically tailor the epitaxial strain to get it in the right direction.
Thank you Lior, this is helpful. Is this concept excluisve to perovskites or would it also apply to other crystal types?
Andrew, I only know this in perovskites (doesn't mean it doesn't exist elsewhere...)
Feel free to follow up with questions if my answer wasn't clear. Making sure that you're around ~4A is the best way to circumvent calculation errors early on.
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