[Pw_forum] Reciprocity of Supercells & BZ Sampling

[yavar pour azar]

Dear Professor Cantele.

Thanks for your time,consideration and detailed description.

Best regards.

Yavar



On Tue, Jul 17, 2012 at 1:35 PM, Giovanni Cantele <
giovanni.cantele at spin.cnr.it> wrote:

>
> 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
>
> --
>
> **** PLEASE NOTICE THE NEW E-MAIL ADDRESS: giovanni.cantele at spin.cnr.it
>
> Giovanni Cantele, PhD
> CNR-SPIN
> c/o Dipartimento di Scienze Fisiche
> Universita' di Napoli "Federico II"
> Complesso Universitario M. S. Angelo - Ed. 6
> Via Cintia, I-80126, Napoli, Italy
> Phone: +39 081 676910
> Skype contact: giocan74
>
> ResearcherID: http://www.researcherid.com/rid/A-1951-2009
> Web page: http://people.na.infn.it/~cantele
>                      http://www.nanomat.unina.it
>
>
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