Determining k-points or citical points of NaCl

[subasa]

Hi every one

I have a question regarding the k-points and I was wondering if anyone can guide me in the right path.

At present I am writing a software in which I am trying to design a photonic crystal structure. I want this photonic structure to produce the same DISPERSION DIAGRAM as a natural NaCl crystal. So simply put I am trying to produce an artificial crystal which produces the same results as a natural crystal. I've looked into the concepts of Brillouin zone and k-points but I can not get the big picture in my head to combine it with photonics.

As we know NaCl is a FCC crystal. So my questions are:
1: How would you determine its critical points? Would it be e.g. GKLX or GMKG as I've attached the images with this question? (P.s. the examples that I've mentioned are not for NaCl and are just examples).

2: Once you determine the critical points, what would be the shape of the k-points? As you can see in this picture the first Brillouin zone is the red hexagonal but the k-points are GMK in the green triangle. How was this triangle shape determined? Because if I have a much more complex structure what would I expect the k-points to be look like? What would also NaCl look like?

That's all. Thank you in advance and any help would be appreciated.

 

[kanato]

Are you trying to determine what lines in k-space to walk to make a band plot, or are you trying to find the k-points that fall into the irreducible Brillouin zone (IBZ) for doing integrals in the Brillouin zone? Your second question is not well worded (k-points are points, they don't have a shape) and it sounds like you're talking about the shape of the IBZ so that needs to be clarified.

Making a band plot is simple. Choose a set of high symmetry k-points and calculate bands on paths that connect them. There is no unique way to do it, and different people do it in different ways, but usually you want your lines to go either from Gamma to a zone edge point, or to travel along the zone edge, so in an fcc structure you wouldn't normally plot along X-L since those points do not fall on the zone edge.

Calculating the IBZ is more difficult. The shape of the IBZ is determined by the symmetries in your crystal, which you wouldn't generally write code to determine. Instead you'd generate a grid of k-points in the full BZ by doing $$\vec{k} = n_1/N_1 \vec{g_1} + n_2/N_2 \vec{g_2} + n_3/N_3 \vec{g_3}$$ where $$n_i = 0.5 + 0,1,...,(N_i-1)$$ and $$N_i$$ are integers that determine the total number of k-points. (The 0.5+ part is optional, but can reduce the number of irreducible k-points without reducing the accuracy of the calculation.) Once you have this grid, you go through the symmetries of the crystal to find which k-points are mapped onto others, throw out all but one of them and record the multiplicity of that k-point. The hard part of this method is determining which symmetries your crystal has. Many open source codes such as abinit and pwscf have code to do this (they are GPL so be sure you understand the implications of the license before you take their source code).

 

[subasa]

Hi

Thank you Kanato for your reply and sorry for my late reply. At present I'm overseas and there is no internet. To access the internet I have to find a cafeteria which usually is very busy. But any way regarding my second question (sorry for not clarifying) you are right I am talking about IBZ. You've explained how the shape of IBZ is determined. But its a bit difficult to get it. Are you aware of any books or websites or journals that have done an example? It will be much easier and quicker to get it instead of wasting everyone's time to go back and forth with questions.

Also regarding your comment about the bandplot I have 3 questions:

1: You said that for making a bandplot "Choose a set of high symmetry k-points" and "usually you want your lines to go either from Gamma to a zone edge point, or to travel along the zone edge", does this means that any combinations such as GKWUG or GKWG or GWUG would be fine?

2: Which one of the above combination would be the best and why?

3: Would it be any different if I swap the K-points with each other e.g. GKWG make it as GWKG or GWGK etc? Would they effect the result in any way?

Thank you again.

 

[kanato]

You've explained how the shape of IBZ is determined. But its a bit difficult to get it. Are you aware of any books or websites or journals that have done an example? It will be much easier and quicker to get it instead of wasting everyone's time to go back and forth with questions.

Yes, the correct detection of symmetries is a challenging programming problem. I do not know of books or websites, which is why I pointed to open source codes which have an implementation of this, so you could look through their source code to get an idea of how it is done.

1: You said that for making a bandplot "Choose a set of high symmetry k-points" and "usually you want your lines to go either from Gamma to a zone edge point, or to travel along the zone edge", does this means that any combinations such as GKWUG or GKWG or GWUG would be fine?

Yes, any of those paths would be fine. If you look in the literature you will see various publications that make different choices.

2: Which one of the above combination would be the best and why?

The best combination might be one which travels along each line connecting G to a zone edge point and each line between zone edge points that lies on the zone edge. But it's really not that important to include every possible path [1]. So I would just choose a longer path like GXWKGLW.

[1] Note that sometimes interesting stuff can happen only along some particular part of the path, like having a Dirac point in the band structure. So it is a good idea to inspect these paths, and make sure that, say, the plotted band structure is consistent with the density of states in a band structure calculation to make sure something interesting wasn't missed.

3: Would it be any different if I swap the K-points with each other e.g. GKWG make it as GWKG or GWGK etc? Would they effect the result in any way?

GKWG and GWKG would be identical, just mirrored. GWGK would show redundant information because you go from G to W and then back to G, and you also would lose information about the path from K to W.

 

[subasa]

Thank you Kanato for the quick reply. I'll look into the IBZ to see if I can do any thing. Would it be ok if in the future I'll send you a message regard it?