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I understand that some codes allow for or have external tools to perform band unfolding in super cell calculations. Normally these codes give band structures that are somewhat grainy and not nearly as clean as we normally see in a non-supercell band structure.

What benefits exist to band unfolding over leaving the original band structure when reporting supercell band structures? Presumably you could directly compare the supercell with the dopant (or other change) to the unmodified supercell. Is there an analogous method to density of states in supercell calculations?

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  • $\begingroup$ Nope you were totally right, thanks for proof reading it. $\endgroup$ – Tristan Maxson Aug 29 '20 at 2:40
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Some reasons why you may want to do band unfolding have been explained by Jack. What I would like to add here concerns your second question about doing band structure calculations in the supercell: you don't want to do that.

Supercells are typically needed when studying non-periodic systems using calculations with periodic boundary conditions. These could include the study of point defects (non-periodic in all 3 dimensions), line defects (non-periodic in 2 directions), surfaces or interfaces (non-periodic in 1 direction), etc. In all these cases, you build a supercell that is long along the non-periodic direction (or directions) so that you attempt to approach the non-periodic limit. In reality, what you have is still a periodic system in that direction, but the period is large enough that it is indistinguishable from a truly non-periodic system (of course this is the ideal, in practice you may not be able to use a large-enough supercell to reach this desired limit).

So what does this all mean for a band dispersion? A band dispersion is a relation between the energy and momentum of an electron, $E(\mathbf{k})$, where the momentum $\mathbf{k}$ takes some allowed value in the first Brillouin zone. We need to distinguish two scenarios:

  1. Real non-periodic system. In a real non-periodic system, the size of the Brillouin zone along the non-periodic direction(s) is zero, that is, there is no Brillouin zone along that direction. This means it makes no sense to talk about a band dispersion along the non-periodic direction. The correct way to think about this is in terms of densities of states, which are well-defined even for a non-periodic system.
  2. Simulating a non-periodic system with a supercell. In this case, you will have a very long supercell along the non-periodic direction, which means that you will have a very short Brillouin zone along that direction, but crucially it will not be zero. So in principle you could plot/calculate a dispersion along this short Brillouin zone direction as you suggest. However, this dispersion has no physical meaning. You will get band folding, so a large number of overlaping bands. The larger the supercell, the more bands you will get, until they form a sort of continuum. This is actually the band folding slowly building up the density of states of the truly non-periodic system, which is what is really meaningful in a non-periodic system. Therefore, my advice would be to focus on the physically meaningful density of states when attempting to simulate non-periodic systems.
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There are many applications/advantages for supercell band unfolding. Take the band unfolding program KPROJ as an example:

A k-projection technique (supercell band unfolding) that includes the $k_\perp$-dependence of the surface bands is used to separate the contributions arising from the silicene and the substrate, allowing a consistent comparison between the calculations and the angle-resolved photoemission experiments.

Studying the effects of doping and interfacing.

Understanding STS experiments.

The most important point is that you can figure out which states are coming from which atoms with band unfolding technique.

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