 Supercells as a way to study solid state solutions

Hi all,

We would like to be able to calculate the vibrational properties of solid solutions
for example, A_(1-x)B_xO. We assumed that this would involve creating a supercell in the *.gin file and then inputting the appropriate concentration of A,B atoms.

However we foresee a problem that is illustrated by example 5 - MgO. Our approach assumes that doubling the unit cell size of the MgO (hence cubing the number of atoms in the unit cell) would result in identical dispersion curves to MgO itself. However the larger unit cell gives different dispersion! In addition both example 5 which assumes a=4.212 with an 8 eight atom basis and example 5 assuming unit cell size 2a=8.414 angstroms with 32 atom basis give the incorrect number of dispersion branches (which should be 6).

Any ideas as to what we are doing incorrectly? Was our original idea wrong?

David

misprint - if the unit cell side length is doubled, the number of atoms in unit cell would increase by a factor of 8.

Hi David,

I think the issue here is that your idea isn’t quite right. The dispersion curves for larger cells are perfectly valid, but would be a convolution of multiple dispersion curves for a smaller cell, since dispersion is specific to a given unit cell representation. I’m not sure it really makes much sense to think about dispersion curves for a solid solution since dispersion is only really measured for quasi infinite ordered crystals. You could focus on the phonon density of states which can be converged for any cell size and would show a peaks for defect modes etc. If you wanted to do phonon dispersion you would have to do it in a mean field approximation (which is OK for cell averages and mechanical properties, but not so good for thermodynamics and local effects). GULP allows you to use partial occupancies on sites and so you could do it this way. Of course accurate modelling of solid solutions is better approached by a multi scale way to compute the phase diagram. See the papers of Victor Vinograd and co-workers for more information.
Regards,
Julian

Hi Julian,

I agree with your point about the the meaningfulness of dispersion in solid solutions. Let’s put this aside for a minute and discuss the relation between the unit cell and the dispersion relations as I think I may not be understanding things correctly.

I am thinking very naively about the number of gamma point modes. The number of modes at the gamma point determined by neutron, IR and Raman spectroscopy, is a definite number. In the case of MgO there is one LO mode and one doubly degenerate TO mode (one restrahlen band in reflectance for example). These properties cannot depend on the choice of unit cell. However in the calculation the number of gamma point modes depends on choice of unit cell ! Using the two-atom basis cell there are two non-zero frequency modes modes, the same as measured by various spectroscopies. Using a simple cubic cell with 8 atom basis there are 6 different frequency optical modes at gamma. Using a simple cubic cell with 32 atom basis, I count 22 different frequency modes at the gamma point . How can this be?

I probably do not understand what you mean by a convolution of dispersion curves. Are you saying that non-zero k vectors in the Brillouin zone are mapped onto gamma when you change size of unit cell? If so, I will have to give that concept a think.

Regards,
David

Hi David,
I agree it can be confusing when thinking about k points and cells. Here are the key points to make clear:

1. There will always be 3N modes at any k point and at gamma 3 modes will be zero due to translational invariance.
2. How many unique frequencies there are will depend on the symmetry of the k point leading to varying degeneracies.
3. If you increase the size of the cell this corresponds to sampling an equivalent number of k points for the smaller cell. For example if you double a cell in one direction (x) then this is equivalent to calculating at (0,0,0) and (0.5,0,0) for the original cell. So the F centred cell of MgO samples 4 k points from the primitive cell.
4. A more subtle point is that the LO/TO splitting depends on the direction of approach to gamma and so the frequencies in the dispersion curves aren’t always the same as those from a single k point calculation at gamma.
Hope that clarifies things a bit.
Julian