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 ELPH_RUN=.TRUE. and 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 ELPH_SELFEN_GAPS=.TRUE.

Basic usage

The first step of an electron-phonon calculation is necessary the 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 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 KPOINTS_DENSE grid. The format is the same as KPOINTS. Note that NBANDS governs the number of bands used in the self-consistent field calculation, while 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

 PREC = Accurate
 ENCUT = 500
 EDIFF = 1e-8
 ISMEAR = 0; SIGMA = 0.01
 LREAL = .FALSE.
 LWAVE = .FALSE.
 LCHARG = .FALSE.
 
 #run electron-phonon calculation
 ELPH_RUN = .TRUE.
 ELPH_DRIVER = EL
 
 # use exact diagonalization and compute all the bands
 ELPH_NBANDS = -2
 KPOINTS_OPT_MODE = 2
 
 # compute gap renormalization
 ELPH_SELFEN_DELTA = 0.01
 ELPH_SELFEN_FAN = .TRUE.
 ELPH_SELFEN_DW = .TRUE.
 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 (NBANDS) and plane-waves (ENCUT). To avoid a more cumbersome double-convergence with ENCUT and ELPH_NBANDS we recommend setting ELPH_NBANDS to be equal to the maximum number of plane-waves. This can be done automatically by setting ELPH_NBANDS=-2

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

In a practical example, assume the electron-phonon potential was generated for Silicon with an ENCUT=500 which yields NGX=NGY=NGZ=28 in the primitive cell. The values of NGX,NGY and NGZ chosen by vasp can be monitored in the OUTCAR file. A convergence study can be performed by running multiple calculations with ENCUT=200, then 300, then 400 and finally 500 with NGX=28; NGY=28; NGZ=28 explicitly set in the INCAR file. You can verify that the calculations with a lower cutoff run faster, hence the choice of the 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 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.

Together with the k-point convergence, the broadening parameter ELPH_SELFEN_DELTA should be monitored as well. The usual approach is to also extrapolate the result to zero broadening. This is made easier by the fact that multiple values can be passed on to ELPH_SELFEN_DELTA with vasp computing the self-energy for each of these broadening and reporting the results in the OUTCAR file and to vaspout.h5. For each value of the broadening, a new electron self-energy accumulator is created with the corresponding settings and values reported in the OUTCAR.

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 $q\rightarrow0$. 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 treatment is done for the inter-atomic force constants. The behavior, of this treatment, is governed by the following INCAR tags

 ENCUTLR = 150
 ELPH_LR = 1
 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 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.