The electron-phonon averaged (EPA) approximation is described in Adv. Energy Mater. 2018, 1800246 and arXiv:1511.08115.
There are two examples, silicon and half-Heusler HfCoSb (from the paper above), containing all the input and output files (output files are gzipped). Each example has two job submission scripts, submit1.sh and submit2.sh, which follow the same computational workflow (see below). There are several python scripts called from submit2.sh, they require python package BRAVE to convert QE output to BoltzTraP input. Alternatively, this conversion can be performed using python script qe2boltz.py included in boltztrap-1.2.5.
- Run pw.x to obtain the SCF solution
- Run ph.x with
fildvscf = 'dvscf'to compute derivatives of the SCF potential - Run ph.x with
electron_phonon = 'epa'to compute the electron-phonon matrix elements and write them to file 'silicon.epa.k' - Run pw.x with
calculation = 'nscf'to obtain the eigenvalues on a fine k-grid - Run epa.x to read the electron-phonon matrix elements from file 'silicon.epa.k', average them over wavevector directions, and write them to file 'silicon.epa.e'
- Run BoltzTraP to read the averaged electron-phonon matrix elements from file 'silicon.epa.e' and compute the transport properties
Format of the input file 'silicon.epa.in' for epa.x:
| Content | Description |
|---|---|
silicon.epa.k |
input data used by epa.x (contains electron-phonon matrix elements in momentum-space, produced by ph.x) |
silicon.epa.e |
output data produced by epa.x (contains electron-phonon matrix elements in energy-space, used by BoltzTraP) |
egrid |
job type, 'egrid' for the standard EPA averaging procedure from momentum to energy space |
6.146000 -0.4 10 0 0 |
valence energy grid edge (VBM energy), grid step (negative = downwards from VBM), number of bins, range of valence bands (0 0 = all valence bands) |
6.602500 0.4 10 0 0 |
conduction energy grid edge (CBM energy), grid step (positive = updards from CBM), number of bins, range of conduction bands (0 0 = all conduction bands) |
0.0 0 0 |
parameters used by job type 'gdist' which constructs the histogram for plotting a distribution of squared matrix elements |
Both valence and conduction energy grids consist of 10 bins of 0.4 eV width (these could be different for the two grids). The valence energy grid extends 4 eV below the VBM (valence band maximum) and the conduction energy grid extends 4 eV above the CBM (conduction band minimum). All energies are in eV.
Transitions between the valence and conduction energy grids are not implemented, there are only valence-to-valence and conduction-to-conduction transitions. This is only valid if the band gap is larger than the highest phonon energy.
In case of a metal or a narrow-gap semiconductor, one can define a single energy grid that spans both valence and conduction bands. For example, if the Fermi level is at 5 eV, the energy grids can be set as follows:
| Content | Description |
|---|---|
2.0 -10.0 1 0 0 |
valence energy grid is 3 eV below the Fermi level and is not functional |
2.0 0.5 12 0 0 |
conduction energy grid spans the range from 2 eV to 8 eV, that is, from 3 eV below the Fermi level to 3 eV above the Fermi level |
The extents of valence and conduction energy grids are determined by the range of Fermi levels for which one needs to compute the transport properties. Let's say the Fermi level spans from 1 eV below the VBM to 0.5 eV above the CBM. Then the valence and conduction energy grids have to cover that range, plus the largest phonon energy (because the electron energy can change by as much as the largest phonon energy during the electron-phonon scattering, so all electrons in that energy range will contribute to the transport properties). Extending the energy grids outside of that range won't have any effect on the transport properties (since it doesn't affect the electronic transitions in the selected range of Fermi energies). Let's say the largest phonon energy is 0.2 eV, then the energy grids have to cover the range from 1.2 eV below the VBM to 0.7 eV above the CBM. If the grid steps are set to 0.1 eV for both grids, this requires 12 bins in the valence energy grid and 7 bins in the conduction energy grid. If the grid steps are increased to 0.2 eV, the numbers of bins can be decreased to 6 and 4, respectively.
Choose some initial values for the grid steps and numbers of bins, then run epa.x, and examine the output. Look at the numbers in countv and countc columns. These are numbers of eigenvalues that fall in each energy bin. If there are any zeros or small numbers in these columns, one has to either increase grid steps or decrease numbers of bins or increase numbers of k-/q-points (the latter requires rerunning pw.x with calculation = 'nscf').
What should be reasonable values for countv and countc? This, of course, is entirely dependent on the material. As a rule of thumb, anything less than 3-5 is too small, something in 5-10 or 20-50 range is reasonable, and anything above 50-100 is too large. Note that since the electron energy dispersion is generally not homogeneous, there will always be some bins with small values (5-10 and below) and other bins with large values (100-200 and above) for the same grid step. Therefore, one has to focus on those bins that have the smallest values of countv and countc, and make sure that it doesn't go below 3-5 while decreasing grid steps.
Add the following line to BoltzTraP input file 'silicon.def' to switch BoltzTraP to the EPA mode:
88, 'silicon.epa.e', 'old', 'formatted', 0
If BoltzTraP is unable to open file 'silicon.epa.e' or read its content, it will automatically fall back to the CRT (constant relaxation time) mode.
Create file 'silicon.ke0j' with content '.TRUE.' to make BoltzTraP compute the electronic part of the thermal conductivity at zero electric current.
EPA patches to Quantum ESPRESSO, BoltzTraP, and BoltzTraP2 are distributed under GPL-2.0, LGPL-3.0+, and GPL-3.0+ licenses, respectively.