Group theory and crystal structure

[gdumont]

Hi,

I have the following problem :

I generate GaAs (zinc blende structure) supercells, and then I want to replace some As atoms by N atoms. Let's say I have fcc conventional cell repeated twice in the x, y and z direction so that I have a total of 64 atoms, 32 of Ga and 32 of As. 8 atoms per conventional cell times 2x2x2 = 64. Then I replace 2 of the As atoms by N atoms so that there are
$$\frac{64!}{2! 62!} = \frac{64 \times 63}{2} = 2016$$
possibilities. Of course since the supercell is repeated to infinity there will be a lot of equivalent configurations.

My question is: Is there any way using group theory to determine which of the 2016 possible configurations are equivalent?

If no one knows the answer can anyone suggest a good book about group theory applied to crystal structures?

Thanks!

 

[nbo10]

There are several good books. Liboff has a good introduction book. Tinham is another good book.

 

[AKG]

It seems that since you're replacing 2 of the As atoms, and since there are only 32 As atoms, there are

$$\frac{32!}{2!30!} = \frac{32 \times 31}{2} = 496$$

possibilities. That is, there are 496 possible ways to change the original supercell. The number of ways to build a supercell with 32 Ga, 30 As, and 2 N "from scratch" would be:

$${{64}\choose{32}}{{32}\choose{2}} = \frac{64!}{32!32!}\times 496$$

which is very big. The number of equivalent possibilities depends on the shape of your supercell. In fact, it also depends on the precise placement of your atoms. I'm guessing it looks like something like three mutually perpendicular sticks, and at the end of each stick, you have an arrangement of a cell consisting of 8 atoms. How are the atoms arranged in these cells? Is it something like this:

 

[Dr Transport]

You cannot predict before hand the locations of the N atoms in the lattice. You will end up with a compound $$GaAs_{1-x}N_{x}$$. Remenber that GaAs is a direct band gap material whereas GaN is indirect. To calculate the band gap, do a linear interpolation between the two direct gaps and you should be close.

As for using symmetry arguments, Tinkham is good, but in this case I'd suggest Evarestov and Smirnov, Site Symmetry in Crystals, Theory and Applications it would be much more useful. Group theory should be able to tell you the equivalence positions, it really isn't a difficult problem to calculate provided you have the correct symmetry groups.

 

[Gokul43201]

I'm guessing it looks like something like three mutually perpendicular sticks, and at the end of each stick, you have an arrangement of a cell consisting of 8 atoms. How are the atoms arranged in these cells? Is it something like this:

No AKG, it's something like this : http://www-ncce.ceg.uiuc.edu/tutorials/crystal_structures/gaas.gif

The GaAs structure can be built by first constructing an FCC (face centered cubic) lattice with say, Ga atoms, and then putting As atoms in the tetrahedral interstices between nearest neighbor Ga atoms. Alternatively, you can think of it as an FCC lattice with a basis of 2 atoms - one at each FCC site (the Ga atoms in the picture) and the other displaced from each site by a constant displacement vector (the As atoms in the above pic).

The OP's supercell, I think, consists of 8 of the unit cells from the picture, 2 along each direction.

Note : This is a little deceptive from the picture, but there are as many Ga atoms as there are As atoms (over any large volume).