View source for Bandgap renormalization due to electron-phonon coupling

The band-structure renormalization within the nonadiabatic Allen, Heine and Cardona is computed from the real part of the electron self-energy evaluated at the Kohn-Sham eigenvalue.
This calculation is activated by default when {{TAG|ELPH_RUN}}=.TRUE. and {{TAG|ELPH_DRIVER}}=EL.
For the particular case where we want to determine the bandgap we can compute the self-energy only for the states that form the gap (including all the degenerate states). The selection of these states can be done automatically by VASP using {{TAG|ELPH_SELFEN_GAPS}}=.TRUE.

==Basic usage==
The first step of an electron-phonon calculation is necessary the [[Electron-phonon_potential_from_supercells|computation of the electron-phonon potential]], which corresponds to the derivatives of the Kohn-Sham potential with ionic displacement. The result of this calculation is a {{FILE|phelel_params.hdf5}} which is a required input for the present calculation.
The electron-phonon matrix elements are computed using the Kohn-Sham states obtained from a non-self-consistent field calculation on a {{FILE|KPOINTS_DENSE}} grid. The format is the same as {{FILE|KPOINTS}}.
Note that {{TAG|NBANDS}} governs the number of bands used in the self-consistent field calculation, while {{TAG|ELPH_BANDS}} governs the number of bands that will be used in the electron-phonon calculation and are computed in the grid specified by the {{KPOINTS_DENSE}} grid.

The computation of the renormalization of the electronic bandgap can be done using the following INCAR file

  {{TAG|PREC}} = Accurate
  {{TAG|ENCUT}} = 500
  {{TAG|EDIFF}} = 1e-8
  {{TAG|ISMEAR}} = 0; SIGMA = 0.01
  {{TAG|LREAL}} = .FALSE.
  {{TAG|LWAVE}} = .FALSE.
  {{TAG|LCHARG}} = .FALSE.
  
  #run electron-phonon calculation
  {{TAG|ELPH_RUN}}=.TRUE.
  {{TAG|ELPH_DRIVER}}=EL
  
  # use exact diagonalization and compute all the bands
  {{TAG|ELPH_NBANDS}} = -2
  {{TAG|KPOINTS_OPT_MODE}} = 2
  
  # compute gap renormalization
  {{TAG|ELPH_SELFEN_DELTA}}=0.01
  {{TAG|ELPH_SELFEN_FAN}}=.TRUE.
  {{TAG|ELPH_SELFEN_DW}}=.TRUE.
  {{TAG|ELPH_SELFEN_GAPS}}=.TRUE.

To get an accurate value while using the smallest possible amount of computational resources, we recommend performing a basis set and k-point sampling convergence study. This allows to ensure that the result is precise, an estimation of the error as well as choosing the most computationally favorable settings.

==Basis set convergence==
First, we will deal with convergence of the bandgap renormalization with the number of electronic states ({{TAG|NBANDS}}) and plane-waves ({{TAG|ENCUT}}). To avoid a more cumbersome double-convergence with {{TAG|ENCUT}} and {{TAG|ELPH_NBANDS}} we recommend setting {{TAG|ELPH_NBANDS}} to be equal to the maximum number of plane-waves. This can be done automatically by setting {{TAG|ELPH_NBANDS}}=-2

The derivatives of the electron-phonon potential are contained in the {{FILE|phelel_params.hdf5}} file on a pre-selected grid ({{TAG|NGX}}, {{TAG|NGY}}, {{TAG|NGZ}}).
This means that we should avoid running the electron-phonon calculation with a {{TAG|ENCUT}} that would yield a {{TAG|NGX}}, {{TAG|NGY}} and {{TAG|NGZ}} that is different from the {{FILE|phelel_params.hdf5}} file.
We can, however, choose a smaller {{TAG|ENCUT}} and set {{TAG|NGX}}, {{TAG|NGY}} and {{TAG|NGZ}} manually in the {{FILE|INCAR}} file to be the same as the one in the {{FILE|phelel_params.hdf5}} file.
This allows running the calculation with different values of {{TAG|ENCUT}} and monitoring how that affects the final value of the bandgap renormalization.

In a practical example, assume the electron-phonon potential was generated with an {{TAG|ENCUT}}=500 which yields {{TAG|NGX}}={{TAG|NGY}}={{TAG|NGZ}}=28 in the primitive cell for a particular material. The values of {{TAG|NGX}},{{TAG|NGY}} and {{TAG|NGZ}} chosen by vasp can be monitored in the {{FILE|OUTCAR}} file.
A convergence study can be performed by running multiple calculations with {{TAG|ENCUT}}=200, then 300, then 400 and finally 500 with {{TAG|NGX}}=28; {{TAG|NGY}}=28; {{TAG|NGZ}}=28 explicitly set in the {{FILE|INCAR}} file.
You can verify that the calculations with a lower cutoff run faster, hence the choice of the {{TAG|ENCUT}} value for a production calculation is always a trade-off between accuracy and computational time.

==K-point sampling convergence and extrapolation to infinity==

Apart from the convergence with the basis set, one should perform a convergence with the k-point sampling.
This step simply implies running the calculation for increasingly dense meshes specified in the {{TAG|KPOINTS_DENSE}} file.
The convergence with k-points and basis set are only slightly interdependent, so converging the two parameters separately is a very good starting point.
We recommend users to extrapolate the value to infinite k-point density, this is done by plotting the value of the gap renormalization as a function of the inverse k-point density.

==Special treatment of the dipole interaction for polar materials==

The convergence of the bandgap renormalization for polar materials (i.e. materials with a gap and non-zero born-effective charges) is specially challenging.
This is because the electron-phonon potential diverges as <math>q\rightarrow0</math>. This divergence is due to a long-range electrostatic interaction between the ions that is not screened by the electrons, as it would be in the case of metals. In these cases, VASP can remove this long-range component from the electron-phonon potential in the supercell, Fourier interpolate it and add it back in the primitive cell. The behavior, of this treatment, is governed by the following {{FILE|INCAR}} tags

  {{TAG|ENCUTLR}} = 150
  {{TAG|ELPH_LR}} = 1
  {{TAG|IFC_LR}} = 1

Note that the special treatment is automatically activated when the phelel_params.hdf5 file was generated with the dielectric tensor and born effective charges present in it.
The value of {{TAG|ENCUTLR}} should be the smallest possible for efficiency reasons, but large enough such that the final quantity does not depend on it.
This value also should not be too large, otherwise the results might become unphysical.