# Band Structure¶

BAND can calculate the band structure for the standard k-path in the Brillouin zone [1] and saves the corresponding data to the binary file RUNKF.

The band structure is best examined with the GUI module BandStructure see:

BandStructure
Enabled Yes/No
Automatic Yes/No
DeltaK float
FatBands Yes/No
UseSymmetry Yes/No
EnergyAboveFermi float
EnergyBelowFermi float
End

BandStructure
Type: Block Options for the calculation of the band structure.
Enabled
Type: Bool No Calculate band structure If True, Band will calculate the band structure and save it to file for visualization.
Automatic
Type: Bool Yes Automatic generate path If True, BAND will automatically generate the standard path through the Brillouin zone. If False BAND will use the user-defined path in BZPath.
DeltaK
Type: Float 0.1 1/Bohr Interpolation delta-K Step (in reciprocal space) for band structure interpolation. Using a smaller number (e.g. 0.03) will result in smoother band curves at the cost of an increased computation time.
FatBands
Type: Bool Yes Calculate fatbands If True, BAND will compute the fat bands (only if BandStructure%Enabled is True). The Fat Bands are the periodic equivalent of the Mulliken population analysis.
UseSymmetry
Type: Bool Yes Use symmetry If True, only the irreducible wedge of the Wigner-Seitz cell is sampled. If False, the whole (inversion-unique) Wigner-Seitz cell is sampled. Note: The Symmetry key does not influence the symmetry of the band structure sampling.
EnergyAboveFermi
Type: Float 0.75 Hartree Energy above Fermi level Bands with minimum energy larger then FermiEnergy + EnergyAboveFermi are not saved to file. Increasing the value of EnergyAboveFermi will result in more unoccupied bands to be saved to file for visualization.
EnergyBelowFermi
Type: Float 10.0 Hartree Energy below Fermi level Bands with maximum energy smaller then FermiEnergy - EnergyBelowFermi are not saved to file. Increasing the value of EnergyBelowFermi will result in more occupied core bands to be saved to file for visualization. Note: EnergyBelowFermi should be a positive number!

Information on the k-path used for band structure plotting (including the fractional coordinates of high-symmetry k-points) can be found in the section KPath of the output file.

## User-defined path in the Brillouin zone¶

If BZStruct%Automatic is False, BAND will compute the band structure for the user-defined path in the BZPath block.

BZPath
path # Non-standard block. See details.
...
End
End

BZPath
Type: Block Definition of the user-defined path in the Brillouin zone for band structure plotting.
path
Type: Non-standard block True Definition of the k-points in a path. The vertices of your path should be defined in fractional coordinates (wrt the reciprocal lattice vectors)

You should define the vertices of your path in fractional coordinates (wrt the reciprocal lattice vectors) in the Path sub-block. If you want to make a jump in your path, you need to specify a new Path sub-block.

In the following example we define the path Gamma-X-W-K|U-X for a FCC lattice:

BZPath
Path
0.000   0.000   0.000
0.500   0.000   0.500
0.500   0.250   0.750
0.375   0.375   0.750
End
Path
0.625   0.250   0.625
0.500   0.000   0.500
End
End


## Definition of the Fat Bands¶

The fat bands $$F_{i,n,\sigma,\vec{k}}$$ are the periodic equivalent of the Mulliken population. They are defined as:

$F_{i,n,\sigma,\vec{k}} = \sum_j C_{i,n,\sigma,\vec{k}} C_{j,n,\sigma,\vec{k}} S_{i,j,\vec{k}}$

where $$C_{i,n,\sigma,\vec{k}}$$ and $$S_{i,j,\vec{k}}$$ are the orbital coefficients and the overlap matrix elements respectively. The indices $$i$$ and $$j$$ denote basis functions, $$n$$ is the band index, $$\sigma$$ is the spin index and $$\vec{k}$$ is a reciprocal vector in the Brillouin zone.

## Band Gap¶

The band gap (if any) is printed in the output. Here is an example for the NaCl crystal:

----------------------------------------
Band gap information
----------------------------------------
Number of valence electrons                    16
Valence Band index                              8
Top of valence Band (a.u.)                 -0.192
Bottom of conduction Band (a.u.)           -0.039
Band gap (a.u.)                             0.153
Band gap (eV)                               4.173
Band gap (kcal)                            96.235


References

 [1] W. Setyawan and S. Curtarolo, High-throughput electronic band structure calculations: Challenges and tools, Computational Materials Science 49 (2010) 299–312.