Non-Symmetry-Broken Band Structure & Spectral Function

[kavanase]

[Copying in with permission from emailed question]:

Dear Dr. Kavanagh,

Greetings!

I am Sumit Kukreti a research fellow in University of Warwick. I have been working on the transport properties of alloy system. I was following Easyunfold for unfolding the bandstructure. Really It is nice tool.
In the start, I just computed the band unfolding on the simple 222 supercell of the NbFeSb material without doing any atomic relaxation. In principle, the bandstructure should resemble to the primitive one. Although the band geometry of the unfolded one is also matching with the primitive one, I have few question for understanding;

1. As this is just unfolding of the simple supercell (without any relax or symmetry broken), the spectral weight should be 1 and the unfolded band should be of same colour. What are the reason of the different shade of colours (dark and bright) in bandstructure.
2. The bandstructure is not smooth means there is discreetness so what are possibility to make this smooth. Is this related to k-points so how we can increase that one in easyunfold calculations. Or is there any option of interpolation while plotting ?
3. How the spectral weight and spectral function are used in the plot, I am not clear in that.
   I hope you will help me understanding these.

Thank You for giving your precious time

Regards
Dr. Sumit Kukreti
Research Fellow
University of Warwick

[kavanase]

Hi Sumit!

Everything you say is correct from what I can see.

1. The reason for the slightly different shades of colours in your easyunfold spectral plot is likely due to numerical/normalisation effects in generating the spectral function. You should be able to control these with the --intensity option for easyunfold unfold plot (see CLI docs) or with the python API if needed. No other lines appear as expected due to the lack of symmetry breaking.

2. The apparent discreteness I think is just due to the finite grid used in the plot. I think you can control this with the --npoints and --dpi options (CLI docs here) – can you try this and report back?

3. How the spectral weight and spectral function are used in the plot,

   For this, I would advise looking at our JOSS paper / docs page which describe the theory of band unfolding, and references therein for the original theory papers by Popescu & Zunger (copied below), and the underlying code e.g. here.
   Once the spectral function is obtained (i.e. spectral intensity as a function of energy and k-point), it is plotted as a bandstructure 'heatmap' as in the easyunfold plot you show here.

@zhubonan may have more thoughts.

[1] Popescu, V.; Zunger, A. Effective Band Structure of Random Alloys. Phys. Rev. Lett. 2010, 104 (23), 236403. https://doi.org/10.1103/PhysRevLett.104.236403.

[2] Popescu, V.; Zunger, A. Extracting E vs k Effective Band Structure from Supercell Calculations on Alloys and Impurities. Phys. Rev. B 2012, 85 (8), 085201. https://doi.org/10.1103/PhysRevB.85.085201.

[KukretiS]

Dear Dr. Kavanagh,

Thank you very much for clarification and sharing the references. Yes discreteness issue is resolved with --nkpoints option. Additionally I found few coloured states on W high symmetry point. Could you help me understanding the reason of getting these states.

[kavanase]

Great that npoints solves that issue. I have also added some changes in #56 to help with this further.

The issue with the W kpoints looks like a rare bug at specific high-symmetry/degenerate points. @zhubonan is the expert on this part of the code, could you please have a look at this Bonan?

@KukretiS can you share your easyunfold.json file to help with debugging this? Thanks!

[kavanase]

@zhubonan I checked and this issue also appears with the Si222 example:

But seems not to have been an issue before (as it doesn't appear in the older version of the image files in that directory). Should help narrow down the issue!

[KukretiS]

Here, I have shared the easyunfold.json file.
However I would like to know if there possibility to see the colour-bar on the side of the unfolded bandstructure. Please suggest me the way to get that.
easyunfold.json

[kavanase]

You could generate the colourbar using the python API, but I don't think it will be particularly insightful as the units/values have been renormalised.
You can seen the available cmap choices here: https://matplotlib.org/stable/users/explain/colors/colormaps.html#sequential
(PuRd is the default)