[Pw_forum] Reciprocity of Supercells & BZ Sampling

[Giovanni Cantele]

On Jul 14, 2012, at 7:34 AM, yavar pour azar wrote:

> Dear QE users,
> 
> I have a conceptual problem with k-point sampling in supercells.
> 
> Suppose we have a simple cubic unit cell for the bulk, which it's electronic structure, total energy and etc... is calculated by n*n*n BZ Sampling.
> for example I can see band width and band curvature along k axis, and i will have max and min points for every band in different k_points such gamma, X, L,......
> 
> Now, if we want to make a n*n*n supercell from this unit cell, it is the  known fact that just gamma point calculations is enough.
> 
> Anyone can see following sentences in most QE tutorials; 
> "Increase supercell in real space by a factor N along a ; EXACTLY same results obtained by reducing  divisions in k mesh (in the new smaller BZ) by factor N"
> 
> This is the problem that I cant understand : when we substitute the unit cell by n*n*n  supercell ( k=0), the calculated bands with new sampling has no curvature and extension along k axis, because we have just a K point. Total Energy, probably is same as former unit cell (by integer factor), but if we will lose information about other aspects of electronic structure by this sampling?
> I would appreciate it if anyone can help me to understand this, or refer me to a source.
> 
> 
> 
> PS:  suppose supercell sustained no relaxation or similar changes, and we have just the two different scales from one infinite, periodic lattice.
> 
> 
> 
> Thanks in advance.
> 
> Yavar Taghipour Azar 
> PhD student
> Physics Group, AEOI, Tehran, Iran _______________________________________________
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If you build your supercell correctly, of course no information will be lost. 
Reverting to the example you did --namely, the same cubic structure simulated by using either a 1x1x1 unit cell with an NxNxN sampling
or an NxNxN supercell with a 1x1x1 sampling of the BZ-- you should consider that the band structure you plot in both cases is along a path
in the BZ that has NOTHING to do, in principle, with the sampling of the BZ itself.

The BZ sampling controls the convergence of the calculated properties (the ground-state charge density, for example) with respect to the BZ
integration.
The path along the BZ you use for band structure calculations is instead just a collection of k-points inside the BZ which you choose in order
to plot the band structure of a system. Of course, for the NxNxN supercell you expect to get a number of bands which is NxNxN times that
of the 1x1x1 calculation: this is the so called band structure folding. That means that your band structure, even though apparently
different, is just the same but described within a BZ which is NxNxN times smaller than for the 1x1x1 unit cell.
In conclusion, even for the NxNxN supercell, if you select a path in the BZ, you'll find the same electronic bands, gaps, curvature, even though
described within a different BZ.

Giovanni

--

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Giovanni Cantele, PhD
CNR-SPIN
c/o Dipartimento di Scienze Fisiche
Universita' di Napoli "Federico II"
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Skype contact: giocan74

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                     http://www.nanomat.unina.it

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