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).

Options for the band structure can be specified in the BZStruct block:

BZStruct
   {Enabled     [True|False]}
   {Automatic   [True|False]}
   {FatBands    [True|False]}
   {UseSymmetry [True|False]}
   {DeltaK      value}
End

Enabled

(Default: False) Whether or not to calculate the band structure.

Automatic

(Default: True) If True BAND will automatically generate the standard path through the Brillouin zone. If False BAND will use the user-defined path in BZPath.

FatBands

(Default: True) If True BAND will compute the fat bands (note: only if BZStruct%Enabled=True). The Fat Bands are the periodic equivalent of the Mulliken population analysis.

UseSymmetry

(Default: True) 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.

DeltaK

(Default: 0.1) Step (in reciprocal space, unit: 1/Bohr) 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.

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. 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