View source for Bandgap renormalization due to electron-phonon coupling

The band-structure renormalization within the nonadiabatic Allen Heine Cardona (AHC) theory is computed from the real part of the electron self-energy evaluated at the Kohn-Sham (KS) eigenvalue.
This calculation is activated by default when {{TAGO|ELPH_RUN|True}} and {{TAGO|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 {{TAGO|ELPH_SELFEN_GAPS|True}}.

==Basic usage==
The first step of an electron-phonon calculation is the [[Electron-phonon_potential_from_supercells|computation of the electron-phonon potential]], which corresponds to the derivatives of the KS potential with ionic displacement.
This calculation produces the {{FILE|phelel_params.hdf5}} file which is a required input for the subsequent calculation.
The electron-phonon matrix elements are computed using the KS states obtained from a non-self-consistent-field calculation on a dense k-point mesh.
This k-point mesh is specified in the {{FILE|KPOINTS_DENSE}} file, which has the same format as the regular {{FILE|KPOINTS}} file.
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 via {{FILE|KPOINTS_DENSE}}.

The computation of the electronic bandgap renormalization can be done using the following {{TAG|INCAR}} file:

  {{TAGBL|PREC}} = Accurate
  {{TAGBL|ENCUT}} = 500
  {{TAGBL|EDIFF}} = 1e-8
  {{TAGBL|ISMEAR}} = 0; SIGMA = 0.01
  {{TAGBL|LREAL}} = .FALSE.
  {{TAGBL|LWAVE}} = .FALSE.
  {{TAGBL|LCHARG}} = .FALSE.
  
  #run electron-phonon calculation
  {{TAGBL|ELPH_RUN}} = .TRUE.
  {{TAGBL|ELPH_DRIVER}} = EL
  
  # use exact diagonalization and compute all the bands
  {{TAGBL|ELPH_NBANDS}} = -2
  {{TAGBL|KPOINTS_OPT_MODE}} = 2
  
  # compute gap renormalization
  {{TAGBL|ELPH_SELFEN_DELTA}} = 0.01
  {{TAGBL|ELPH_SELFEN_FAN}} = .TRUE.
  {{TAGBL|ELPH_SELFEN_DW}} = .TRUE.
  {{TAGBL|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 ensures that the result is precise, provides an error estimate and reveals the computationally most favorable settings.

==Basis set convergence==
First, we will deal with convergence of the bandgap renormalization with respect to 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 {{TAGO|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 an {{TAG|ENCUT}} that would yield a set of {{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.

Let us consider an example calculation where the electron-phonon potential was generated for Silicon with an {{TAGO|ENCUT|500}} which yields {{TAG|NGX}}={{TAG|NGY}}={{TAG|NGZ}}=28 in the primitive cell. 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.
{{TAGO|NGX|28}}, {{TAGO|NGY|28}} and {{TAGO|NGZ|28}} are set explicitly in the {{FILE|INCAR}} file.
You can verify that the calculations with a lower cutoff run faster, highlighting the importance of convergence tests for reaching a good compromise between accuracy and computational time.

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

Apart from the convergence with respect to the basis set, one should perform a convergence with respect to the k-point sampling.
This step implies running the calculation for increasingly dense k-point meshes specified in the {{TAG|KPOINTS_DENSE}} file.
The convergence behavior with respect to k-points is mostly independent from the convergence behavior with respect to the plane-wave basis.
Therefore, it is recommended to study both types of convergence separately to save on time.

Furthermore, we recommend extrapolating the bandgap renormalization to infinite k-point density.
This is most easily accomplished by plotting the value of the renormalization as a function of the inverse k-point density.
In addition to the k-point convergence, the broadening parameter {{TAG|ELPH_SELFEN_DELTA}} should be monitored as well.
The usual approach is to also extrapolate the result to zero broadening.
{{NB|tip|Multiple values can be specified for {{TAG|ELPH_SELFEN_DELTA}}. VASP then computes the self-energy for each of these broadenings and reports the results in the {{FILE|OUTCAR}} and {{FILE|vaspout.h5}} files. For each value of the broadening, a new electron self-energy accumulator is created with the corresponding settings and values reported in the {{FILE|OUTCAR}} file. This should help simplify the convergence study.}}

==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 same [[Phonons:_Theory#Long-range_interatomic_force_constants_(LO-TO_splitting)|treatment is done for the inter-atomic force constants]].
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.