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Let's say I want to find the electronic energy bands from DFT calculations in VASP along a chosen path in the Brillouin zone.

Now, I want to make sure I choose a path that captures the conduction band minimum (and valence band maximum.) How can I find such a path?

In principle, I could calculate the band structure very densely over the full Brillouin zone. This, however, sounds computationally demanding, and I still don't know how dense my grid must be.

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  • $\begingroup$ What do you mean by "the full Brillouin zone"? The idea is using the path between the high symmetry points. This should be enough (as it include the full system symmetry) to detect (visually or computationally) the minimum/maxima of the bands. $\endgroup$
    – Camps ♦
    Aug 31, 2021 at 18:30
  • $\begingroup$ Related: physics.stackexchange.com/q/193420/49107 $\endgroup$
    – Anyon
    Aug 31, 2021 at 20:44
  • $\begingroup$ @Camps, What if the minima/maxima is not at a high symmetry point? Then I guess I would need to consider "all" points". $\endgroup$
    – B. Brekke
    Sep 1, 2021 at 8:07
  • $\begingroup$ @Anyon Thank you. The question makes it clear that there are two possibilities, either the minima/maxima are at a high symmetry point, or it is not. The first case is most "probable". However, I am not sure what is the case for my crystal. So the question still stands, how can I be sure? $\endgroup$
    – B. Brekke
    Sep 1, 2021 at 8:09

2 Answers 2

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The band extrema may be at an arbitrary point in the Brillouin zone, so determining their position can become computationally demanding. The following strategy is something that is sometimes used to locate Weyl points (band crossings) in a band structure, which I think should also be useful for locating band extrema. The strategy would be:

  1. Sample the full Brillouin zone with a uniform $\mathbf{k}$-point grid.
  2. Locate the band extrema in your $\mathbf{k}$-point grid.
  3. Sample a patch of the Brillouin zone around the band extrema of your previous grid.
  4. Iterate until convergence of the location of the band extrema.

There are still some choices to be made (density of sampling grids, size of patches), and I imagine this will require some trial-and-error as ideal values will depend on the specific band structure (effective masses, bandwidth).

The approach does sound computationally demanding, but you should be able to diagonalize the Hamiltonian over the required $\mathbf{k}$-point grids in a non-self-consistent manner, which should accelerate the calculations.

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  • $\begingroup$ I suppose a derivative-free optimization method, such as the simplex method, may be more efficient than sampling on uniform grids? $\endgroup$
    – wzkchem5
    Sep 2, 2021 at 8:52
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    $\begingroup$ @wzkchem5 I guess there are many possibilities to adapt the above strategy, I have never tried any. $\endgroup$
    – ProfM
    Sep 2, 2021 at 8:56
  • $\begingroup$ +1 and welcome back! I didn't see any answers from you since June! When you say "non-self-consistent" do you just mean something like "with a method that doesn't use SCF iterations"? It came across to me like you were talking about diagonalizing the Hamiltonian such that the solution is not self-consistent. Either way, great answer! $\endgroup$
    – Nike Dattani
    Sep 2, 2021 at 13:14
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    $\begingroup$ @NikeDattani thanks! Been very busy indeed, but hopefully will be able to participate more moving forward. By "non-self-consistent" I mean that one can perform the usual SCF calculation on a usual grid, and then once the DFT Hamiltonian is known, there is no need for further self-consistency when evaluating bands at specific $\mathbf{k}$-points. Essentially the same approach used to calculate band structures along high symmetry lines. $\endgroup$
    – ProfM
    Sep 2, 2021 at 18:58
  • $\begingroup$ @ProfM Thanks for the clarification about "non-self-consistent" ! Good to hear you've been keeping busy, feel free to come and go and participate as you see fit --- you're still in our top 10 users list! $\endgroup$
    – Nike Dattani
    Sep 2, 2021 at 19:21
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In addition to ProfM's answer, for many materials it's typical for the band extrema to be on high symmetry points or lines. Sampling these lines can often help you find the extrema or a point close to it.

However, there are several schemes for determining what lines in $\mathbf{k}$-space to sample, including:

These schemes are accessible via the Python code pymatgen and the HighSymmKpath class.

Note that this method will not be robust for all materials! Sometimes very fine uniform sampling of the Brillouin zone is required as ProfM notes. In this case, tools which help to interpolate between $\mathbf{k}$-points like BoltTraP2 may help you refine your search.

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  • $\begingroup$ is there any material of which the vbm and cbm are not on the high symmetry point? $\endgroup$
    – Jack
    Dec 22, 2021 at 5:52
  • $\begingroup$ Yes, many! I at least know of materials where the VBM/CBM is along a high-symmetry line rather than on a point. $\endgroup$
    – Matt Horton
    Jan 5, 2022 at 0:08

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