[Wien] Band Identity in spaghetti

[Andrei Postnikov]

On Thu, 7 Apr 2005, Dr. Sharat  Chandra wrote:

| Dear Wien experts
| 
| In the band structure plot generated by spaghetti, when any two bands are 
| crossing at some point in the plot, the resulting degeneracy is lifted at 
| the point of crossing and the two bands are separated. This results in a 
| change of band identity. Suppose, at first band 1 is lower in energy, then 
| after the point of crossing band 2 will be lower in energy. But after the 
| point of band bending, band 2 will be higher in energy throughout the 
| range. I want to know in what way the identity of the bands are given. Are 
| they set after all the crossings are taken into account, or they are 
| labelled starting from the band which is lowest in energy?

Dear Sharat,

the bands are numbered in the order of increasing energy
(not only in WIEN but probably in any bandstructure code).
There reason is, there is no other systematic way of numbering the
bands. Moreover the crossings usually occur at high-symmetry points
(along which you'd usually plot your spaghetties),
but if you want to trace a band, and go around a high-symmetry
point, you can end up with different numbering, depending on
your path. Moreover the exact numbering does not affect any
properties (physically, I mean). Certainly there can be
a numerical inaccuracy if you use a finitely sparse k-mesh and 
interpolation. This can be a problem especially in calculations
involving matrix elements (e.g. optics, of Green's function),
yet the error can be controllably suppressed by using finer k-mesh.

| This becomes a problem, if the identies of the bands change after the 
| crossing is taken into account. For example, I am interested in 
| calculating the contribution to DOS from each band separately using the 
| joint program. But this can be done correctly, only if the band identities 
| do not change.

The following argument has been offered to me long ago by Igor Mazin,
in a similar discussion: you start to trace a band from any (symmetric)
point in the BZ, you go around the BZ along a non-trivial path, and
good chance that you'll return to the same k-point at different energy
hence in another band. (Try it!) Hence there is no unambiguous
band numbering other than by increasig energy.

Andrei Postnikov

 +-- Dr. habil. Andrei Postnikov ----- Tel. +49-541-969.2377 -- Fax .2351 ---+
 | Institut fuer Festkoerperforschung - FZ Juelich, D-52425 Juelich, Germany |
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