Question about the phonon dispersion in the tutorial (part 1)

[reynaldo.putra]

Hello,

I have a question about the phonon dispersion section of the phonon tutorial (part 1) https://vasp.at/tutorials/latest/phonon ... phonon-e03.

In here, we plot the phonon dispersion/band structures of the different graphene supercells sized 3x3x1, 4x4x1, 5x5x1 and 6x6x1, as seen in the attached file.

In the discussion of the results, there is a particular note about how the supercell size affects the appearance of slight imaginary modes/'soft' modes, about how these 'soft' modes might appear again for larger supercell sizes but tend to disappear when the supercell size is large enough.

In the plots, only the 5x5x1 supercell has no soft modes. But there is also an extra note on how the k-point mesh could not be kept uniform with 5x5x1 as it is the only structure that requires a non-integer k-point mesh if it were to keep the same mesh coarseness.

My question is, does this indicate that the 5x5x1 supercell is the right size for the proper phonon modes, or is this also affected by the different mesh coarseness?

Thank you for your attention!

[marie-therese.huebsch]

Hello,

In short, it is accidental and not a converged result.

First let me make two points:

Why do we increase the supercell?
We can increase the accuracy of force-constants (i.e considering more distant interacting ions). Force-constants computed via finite difference approach require the displacement of ions. Only ions included in the supercell can be replaced. As the magnitude of these force-constants decreases for pairs of atoms that are further apart we know that the phonon dispersion will eventually converge with reasonably sized supercells. A few notable exceptions were discussed in the tutorial.

Why do we increase the k mesh?
To get more accurate forces via the Hellman-Feynman theorem by accounting for electronic interaction with neighboring unit cells. The calculation of forces does not require a displacement of ions, so long-range interactions can be accounted for by sampling the 1 BZ in reciprocal space.

I think one key comment is in section 2.1:

Mind: Converge all phonon calculations with respect to the size of the supercell. When increasing the supercell size by a factor, reduce the k-point mesh by the same factor so that the sampling density remains invariant.

In the tutorial we run calculations for:

cell | k mesh
3x3x1 | 4x4x1
4x4x1 | 3x3x1
5x5x1 | 2x2x1 (cannot use 2.4x2.4x1)
6x6x1 | 2x2x1

So, how can we interpret that the phonon dispersion for 5x5x1 supercell with k mesh 2x2x1 has no imaginary modes?

The calculation with the largest supercell (6x6x1) shows soft modes. So, 5x5x1 is not converged with respect to supercell size! Based on our calculations, we could not say if 6x6x1 is converged. To check that, you need to perform additional calculations for larger supercells.

The k mesh for the 5x5x1 supercell is 2x2x1 and hence the sampling density is coarser than for all other calculations. This means the forces are of lesser accuracy. It would be better to perform a convergence study of the forces with respect to k-mesh density. You can do that for a small unit cell and then apply the appropriate reduction of k-mesh density for the larger unit cells when computing the force constants using finite differences.

In the end, the fully converged result can (like for graphene) resemble a calculation with less accurate settings. But for an unknown system and novel results, you cannot know if the soft modes (or the lack of them) are a true physical (in)stability or a sign of unconverged settings unless you do a proper convergence study.

Does this clarify what you see in the tutorial?
Best regards,
Marie-Therese

[reynaldo.putra]

Hello,

Yes, I think so, thank you! So if I am understanding this correctly, this is more likely due to the coarser k-mesh density of the 5x5x1 calculation, rather than the 5x5x1 being the proper size of the supercell.

Now I have some follow-up questions.

• If I were to calculate larger supercells, how do I ensure the k-mesh density uniformity? Let's say to move to from a 2x2x1 k-point mesh to a 1x1x1 mesh.

• What is considered a proper convergence study? What are the general steps we need to take to do a proper convergence study? Do we start from the ionic relaxation stage? Or does it have more to do with the phonon calculations? (I apologize if this question sounds vague/naive, I am more or less fumbling around in order to understand...)

Also, I have a separate question, still related to the tutorial.
The section after the explanation of why the 5x5 supercell has a different dispersion pattern is the following:

As we have mentioned before, at the q points commensurate with the supercell the phonon frequencies are computed exactly. All the other points are obtained through interpolation. To illustrate this better, we show how increasing the size of the supercell corresponds to sampling the Brillouin zone with a Γ centered k point mesh. Shown below are the plots of the reciprocal Brillouin zone with the k points generated from a regular sampling of the Brillouin zone corresponding to a certain supercell size.

(followed by the attached image)

Does this explanation about interpolation of points outside of the supercell has something to do with why the dispersion patterns can have soft modes if the cell size is not large enough? I'd like to ensure I am understanding this correctly.

Thank you!

[marie-therese.huebsch]

If I were to calculate larger supercells, how do I ensure the k-mesh density uniformity? Let's say to move to from a 2x2x1 k-point mesh to a 1x1x1 mesh.

Let's consider the example of graphene with

cell | k mesh
3x3x1 | 4x4x1
4x4x1 | 3x3x1
6x6x1 | 2x2x1
12x12x1 | 1x1x1

If you multiply the number of unit cells along a direction with the number of k points along the direction the value should be constant. In the example above that is 12.

What is considered a proper convergence study? What are the general steps we need to take to do a proper convergence study? Do we start from the ionic relaxation stage? Or does it have more to do with the phonon calculations? (I apologize if this question sounds vague/naive, I am more or less fumbling around in order to understand...)

This is a very general concept, but of course, it can be translated to specific steps for the example at hand. Generally, a convergence study employs increasingly accurate settings and checks the behaviour of a specific quantity of interest. You stop increasing the settings to higher accuracy once the quantity of interest is converged within the desired accuracy.

Now, more specifically for phonon calculations typical settings to converge are ENCUT, k-mesh density, and supercell size. For ENCUT and k mesh the quantity of interest are the forces in the smallest unit cell. For the supercell size it is rather the force constant matrix, but that is difficult to judge. It is better to look at the phonon dispersion and check the changes. Your reference is always the quantity of interest for the tightest settings. So for the phonon dispersion it is the calculation for the largest supercell. The degree to which you want to converge a result depends on the conclusions you want to draw. For instance, a comparison to experiment. "Proper" was just my way of saying considering all the relevant facts^^

Does this explanation about interpolation of points outside of the supercell has something to do with why the dispersion patterns can have soft modes if the cell size is not large enough?

No, I would not say so. What do you mean by that?

Marie-Therese

[marie-therese.huebsch]

If I were to calculate larger supercells, how do I ensure the k-mesh density uniformity? Let's say to move to from a 2x2x1 k-point mesh to a 1x1x1 mesh.

Let's consider the example of graphene with

cell | k mesh
3x3x1 | 4x4x1
4x4x1 | 3x3x1
6x6x1 | 2x2x1
12x12x1 | 1x1x1

If you multiply the number of unit cells along a direction with the number of k points along the direction the value should be constant. In the example above that is 12.

What is considered a proper convergence study? What are the general steps we need to take to do a proper convergence study? Do we start from the ionic relaxation stage? Or does it have more to do with the phonon calculations? (I apologize if this question sounds vague/naive, I am more or less fumbling around in order to understand...)

This is a very general concept, but of course, it can be translated to specific steps for the example at hand. Generally, a convergence study employs increasingly accurate settings and checks the behaviour of a specific quantity of interest. You stop increasing the settings to higher accuracy once the quantity of interest is converged within the desired accuracy.

Now, more specifically for phonon calculations typical settings to converge are ENCUT, k-mesh density, and supercell size. For ENCUT and k mesh the quantity of interest are the forces in the smallest unit cell. For the supercell size it is rather the force constant matrix, but that is difficult to judge. It is better to look at the phonon dispersion and check the changes. Your reference is always the quantity of interest for the tightest settings. So for the phonon dispersion it is the calculation for the largest supercell. The degree to which you want to converge a result depends on the conclusions you want to draw. For instance, a comparison to experiment. "Proper" was just my way of saying considering all the relevant facts^^

Does this explanation about interpolation of points outside of the supercell has something to do with why the dispersion patterns can have soft modes if the cell size is not large enough?

No, I would not say so. What do you mean by that?

Marie-Therese

[Fermi1976]

Hi,
Try 6x6x1 supercell and 3x3x1 k-mesh. See (Annals of Nuclear Energy 161 (2021) 108437).

I hope this will help!

Regards,