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:

   {Enabled     [True|False]}
   {Automatic   [True|False]}
   {FatBands    [True|False]}
   {UseSymmetry [True|False]}
   {DeltaK      value}
(Default: False) Whether or not to calculate the band structure.
(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.
(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.
(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.
(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:

      0.000   0.000   0.000
      0.500   0.000   0.500
      0.500   0.250   0.750
      0.375   0.375   0.750
      0.625   0.250   0.625
      0.500   0.000   0.500

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