Exciton band structure

Band structure of a typical strongly correlated system - Ni

Exciton band dispersion serves as a powerful tool for characterizing excitations in a system and can be directly linked to the measured dynamical structure factor. The exciton band structure is a band structure plot where the exciton energies are presented for different q-points. To find the exciton energies at the required $q$-points, the Bethe-Salpeter equation (BSE) (or the Casida equation in case of TDDFT calculation) has to be solved for every such $q$-point.

Excitons at finite q

To solve the BSE (ALGO=BSE or TDHF) at a finite momentum q, the index of the corresponding $q$-point has to be provided via the KPOINT_BSE tag. To calculate the dispersion along a certain path, the k-points along this path can be identified in the OUTCAR file from a preceding GW step or a BSE calculation itself.

For example, bulk hBN with 24x24x2 k-points grid has the following k-points in the irreducible Brillouin zone:

   Subroutine IBZKPT returns following result:
 ===========================================

 Found    122 irreducible k-points:

 Following reciprocal coordinates:
            Coordinates               Weight
->0.000000  0.000000  0.000000      1.000000<--
  0.041667  0.000000  0.000000      6.000000
  0.083333  0.000000  0.000000      6.000000
  0.125000  0.000000  0.000000      6.000000
  0.166667  0.000000  0.000000      6.000000
  0.208333  0.000000  0.000000      6.000000
  0.250000  0.000000  0.000000      6.000000
  0.291667  0.000000  0.000000      6.000000
  0.333333  0.000000  0.000000      6.000000
  0.375000  0.000000  0.000000      6.000000
  0.416667  0.000000  0.000000      6.000000
  0.458333  0.000000  0.000000      6.000000
  0.500000  0.000000  0.000000      3.000000
->0.041667  0.041667  0.000000      6.000000<--
  0.083333  0.041667  0.000000     12.000000
  0.125000  0.041667  0.000000     12.000000
...

In GW and BSE calculations one can find two blocks for Subroutine IBZKPT returns following result:. The first one corresponds to the k-point with symmetries enabled and the second one when the symmetries are turned off. The index in KPOINT_BSE should be provided according to the first block, i.e., with the symmetries. If the symmetries are disabled, the blocks should be identical. The selected q-point with index and coordinates in reciprocal coordinates is written in the BSE calculation to the OUTCAR file:

NQ=  61    0.3333    0.3333    0.0000

KPOINT_BSE = 1 corresponds to the q=0 case, and KPOINT_BSE > 1 would result in the BSE calculation for the specific q-point. The optional three integers in KPOINT_BSE can be used to specify a point outside of the first Brillouin zone.

Warning: With VASP, finite momentum calculations at TDDFT or BSE level, i.e. ALGO=TDHF or BSE, must always use ANTIRES=2 regardless of the solver, functional, or approximation used for the electron-hole interaction. Otherwise, the results will be unphysical!

After the BSE is solved for all the required q-points, the eigenvalues can befound in the corresponding vasprun.xml files:

<varray name="opticaltransitions" > 

The dispersion along XK direction for the two lowest excitons in bulk hBN is shown in the figure.