Calculating High-Symmetry Point Coordinates in Band Structure Computations

[saray1360]

Hi,

In all the tutorials of DFT softwares I have encountered computing of the band structure for different systems by using e.g. L-Gamma-X-(U)Gamma circuits. In the texts I have read I have found about X & Gamma but what are L & (U)-Gamma?

I also enountered that they have separated the k-lengths into 10, 12 and 17 divisions? why is that? and how can we find L Gamma and X coordinates?

Looking forward to hearing from you soon.

Regards,
Sara Y

 

[kanato]

Are you looking at the abinit tutorials?

These points are generally special points of high symmetry in the first Brillouin zone (FBZ). These are almost always on the surfaces of the FBZ, usually in the center of faces, at corners or centers of edges. There's a picture of the first Brillouin zone for an fcc lattice here, with the special point marked:
http://en.wikipedia.org/wiki/Brillouin_zone

X is in the center of the square face, L is in the center of the hexagonal face, and U is in the center of the edge that connects those two faces. These points are different for each reciprocal zone (for instance, a hexagonal crystal does not have an X special point). The one exception is the Gamma point, this is always at the zone center, with coordinate (0,0,0).

The 10, 12, and 17 are the number of points along the line from one point to another that are used to calculate the band structure. These numbers are arbitrary, however it is common to choose numbers proportionate to the distance, for instance X-U is a much shorter distance than L-U, so you would choose to use fewer k-points along the X-U direction.

 

[saray1360]

Thank you very much Kanato.
Yes I am reading abinit tutorial.

 

[aplahemant]

Continuing with the same question, can anybody demonstrate how to calculate the coordinates of these high symmetry points?
Gamma is (0 0 0), X is (0.5 0 0). Now how do you compute for rest of the points? What axis these co-ordinates refer to? and are they 'reduced'? I want to calculate for tetragonal lattice which has symmetry points namely Gamma, X, M, Z, R and A .