Band Structure

BAND can calculate the band structure for the standard k-path in the Brillouin zone [65] and saves the corresponding data to the binary file RUNKF.

The band structure is best examined with the GUI module BandStructure (see BAND-GUI tutorial Getting Started with BAND).

BandStructure
   Enabled [True | False]
   Automatic [True | False]
   DeltaK float
   FatBands [True | False]
   UseSymmetry [True | False]
   EnergyAboveFermi float
   EnergyBelowFermi float
End

BandStructure

Type:Block Description:Options for the calculation of the band structure.

Enabled

Type:Bool Default value:False Description:If True, Band will calculate the band structure and save it to file for visualization.

Automatic

Type:Bool Default value:True Description: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 Default value:0.1 Unit:1/Bohr Description: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 Default value:True Description: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 Default value:True Description: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 Default value:0.75 Unit:Hartree Description: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 Default value:10.0 Unit:Hartree Description: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 Description:Definition of the user-defined path in the Brillouin zone for band structure plotting.

path

Type:Non-standard block Recurring:True Description: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
   SubEnd
   Path
      0.625   0.250   0.625
      0.500   0.000   0.500
   SubEnd
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