Band gap renormalization in diamond using one-shot method

Overview > Band gap renormalization in diamond using one-shot method > List of tutorials

Task

Calculating the zero-point renormalization (ZPR) and the temperature dependence of the indirect band gap in diamond using a one-shot method.

Input

POSCAR

Primitive cell (POSCAR.prim):

C_2_fcc
   1.00000000000000
     0.0000000000000000   1.783493 1.783493
     1.783493   0.0000000000000000   1.783493
     1.783493 1.783493    0.0000000000000000
   C
     2
Direct
  0.0000000000000000  0.0000000000000000  0.0000000000000000
  0.7500000000000000  0.7500000000000000  0.7500000000000000

4x4x4 super cell used in this calculation (POSCAR.4x4x4):

C_128_fcc
   1.00000000000000
    0.00000000   7.13397200   7.13397200
    7.13397200   0.00000000   7.13397200
    7.13397200   7.13397200   0.00000000
   C
  128
Direct
   0.00000000   0.00000000   0.00000000
   0.25000000   0.00000000   0.00000000
   0.50000000   0.00000000   0.00000000
   0.75000000   0.00000000   0.00000000
   0.00000000   0.25000000   0.00000000
   0.25000000   0.25000000   0.00000000
   0.50000000   0.25000000   0.00000000
   0.75000000   0.25000000   0.00000000
   0.00000000   0.50000000   0.00000000
   0.25000000   0.50000000   0.00000000
   0.50000000   0.50000000   0.00000000
   0.75000000   0.50000000   0.00000000
   0.00000000   0.75000000   0.00000000
   0.25000000   0.75000000   0.00000000
   0.50000000   0.75000000   0.00000000
   0.75000000   0.75000000   0.00000000
   0.00000000   0.00000000   0.25000000
   0.25000000   0.00000000   0.25000000
   0.50000000   0.00000000   0.25000000
   0.75000000   0.00000000   0.25000000
   0.00000000   0.25000000   0.25000000
   0.25000000   0.25000000   0.25000000
   0.50000000   0.25000000   0.25000000
   0.75000000   0.25000000   0.25000000
   0.00000000   0.50000000   0.25000000
   0.25000000   0.50000000   0.25000000
   0.50000000   0.50000000   0.25000000
   0.75000000   0.50000000   0.25000000
   0.00000000   0.75000000   0.25000000
   0.25000000   0.75000000   0.25000000
   0.50000000   0.75000000   0.25000000
   0.75000000   0.75000000   0.25000000
   0.00000000   0.00000000   0.50000000
   0.25000000   0.00000000   0.50000000
   0.50000000   0.00000000   0.50000000
   0.75000000   0.00000000   0.50000000
   0.00000000   0.25000000   0.50000000
   0.25000000   0.25000000   0.50000000
   0.50000000   0.25000000   0.50000000
   0.75000000   0.25000000   0.50000000
   0.00000000   0.50000000   0.50000000
   0.25000000   0.50000000   0.50000000
   0.50000000   0.50000000   0.50000000
   0.75000000   0.50000000   0.50000000
   0.00000000   0.75000000   0.50000000
   0.25000000   0.75000000   0.50000000
   0.50000000   0.75000000   0.50000000
   0.75000000   0.75000000   0.50000000
   0.00000000   0.00000000   0.75000000
   0.25000000   0.00000000   0.75000000
   0.50000000   0.00000000   0.75000000
   0.75000000   0.00000000   0.75000000
   0.00000000   0.25000000   0.75000000
   0.25000000   0.25000000   0.75000000
   0.50000000   0.25000000   0.75000000
   0.75000000   0.25000000   0.75000000
   0.00000000   0.50000000   0.75000000
   0.25000000   0.50000000   0.75000000
   0.50000000   0.50000000   0.75000000
   0.75000000   0.50000000   0.75000000
   0.00000000   0.75000000   0.75000000
   0.25000000   0.75000000   0.75000000
   0.50000000   0.75000000   0.75000000
   0.75000000   0.75000000   0.75000000
   0.18750000   0.18750000   0.18750000
   0.43750000   0.18750000   0.18750000
   0.68750000   0.18750000   0.18750000
   0.93750000   0.18750000   0.18750000
   0.18750000   0.43750000   0.18750000
   0.43750000   0.43750000   0.18750000
   0.68750000   0.43750000   0.18750000
   0.93750000   0.43750000   0.18750000
   0.18750000   0.68750000   0.18750000
   0.43750000   0.68750000   0.18750000
   0.68750000   0.68750000   0.18750000
   0.93750000   0.68750000   0.18750000
   0.18750000   0.93750000   0.18750000
   0.43750000   0.93750000   0.18750000
   0.68750000   0.93750000   0.18750000
   0.93750000   0.93750000   0.18750000
   0.18750000   0.18750000   0.43750000
   0.43750000   0.18750000   0.43750000
   0.68750000   0.18750000   0.43750000
   0.93750000   0.18750000   0.43750000
   0.18750000   0.43750000   0.43750000
   0.43750000   0.43750000   0.43750000
   0.68750000   0.43750000   0.43750000
   0.93750000   0.43750000   0.43750000
   0.18750000   0.68750000   0.43750000
   0.43750000   0.68750000   0.43750000
   0.68750000   0.68750000   0.43750000
   0.93750000   0.68750000   0.43750000
   0.18750000   0.93750000   0.43750000
   0.43750000   0.93750000   0.43750000
   0.68750000   0.93750000   0.43750000
   0.93750000   0.93750000   0.43750000
   0.18750000   0.18750000   0.68750000
   0.43750000   0.18750000   0.68750000
   0.68750000   0.18750000   0.68750000
   0.93750000   0.18750000   0.68750000
   0.18750000   0.43750000   0.68750000
   0.43750000   0.43750000   0.68750000
   0.68750000   0.43750000   0.68750000
   0.93750000   0.43750000   0.68750000
   0.18750000   0.68750000   0.68750000
   0.43750000   0.68750000   0.68750000
   0.68750000   0.68750000   0.68750000
   0.93750000   0.68750000   0.68750000
   0.18750000   0.93750000   0.68750000
   0.43750000   0.93750000   0.68750000
   0.68750000   0.93750000   0.68750000
   0.93750000   0.93750000   0.68750000
   0.18750000   0.18750000   0.93750000
   0.43750000   0.18750000   0.93750000
   0.68750000   0.18750000   0.93750000
   0.93750000   0.18750000   0.93750000
   0.18750000   0.43750000   0.93750000
   0.43750000   0.43750000   0.93750000
   0.68750000   0.43750000   0.93750000
   0.93750000   0.43750000   0.93750000
   0.18750000   0.68750000   0.93750000
   0.43750000   0.68750000   0.93750000
   0.68750000   0.68750000   0.93750000
   0.93750000   0.68750000   0.93750000
   0.18750000   0.93750000   0.93750000
   0.43750000   0.93750000   0.93750000
   0.68750000   0.93750000   0.93750000
   0.93750000   0.93750000   0.93750000

KPOINTS

K-Points
 0
Gamma
 1  1  1
 0  0  0

We only need a single k-point, since the convergence is done via the super-cell size.

INCAR

general:
 System = cd-C
 PREC = Accurate
 ALGO = FAST
 ISMEAR = 0 
 SIGMA = 0.1;
 IBRION = 6
 PHON_LMC = .TRUE.
 PHON_NSTRUCT = 0
 PHON_NTLIST = 1
 PHON_TLIST = 0.0

The tags with "PHON_" control the electron-phonon couplings (EPC). PHON_LMC enables electron-phonon coupling in the calculations using Monte Carlo sampling. By selecting PHON_NSTRUCT=0 a one-shot configuration (ZG configuration) is obtained. The tag PHON_NTLIST selects the number of temperatures for which the structure with the one shot calculation is obtained. This requires also the list of temperatures given by PHON_TLIST which have exact PHON_NTLIST number of elements.
IBRION=6 is selected to obtain the eigenvectors and eigenvalues of the dynamical matrix at the Gamma point.

Calculation

This example will use a one-shot method, where only a single structure that controls the EPC information is required for a given temperature. The "Gamma" version of VASP is used throughout this example since only a single k point is used in the calculations.

The calculation consists of two steps:

Obtain new "distorted" POSCAR file which contains the EPC. This calculation also contains the band gap of the original structure.
Execute simple DFT calculation for the structure that contains the EPC information to obtain the band gap.
Extract ZPR as the difference between the band gaps from the two calculations.

Obtain structure containing EPC

To run the calculation POSCAR.4x4x4 needs to be copied to POSCAR and INCAR.init to INCAR.

Execute VASP.

Copy the OUTCAR file to OUTCAR.init. It will be later used for the band gap of the "undistorted" structure.

The new POSCAR file containing the configurations pertaining to the EPC is given as POSCAR.T=0..

Calculate electronic levels of structure containing EPC

Copy the file POSCAR.T=0. to POSCAR.

Delete (or comment out with #) all the lines in the INCAR file related to EPC so that it looks like the following:

System = cd-C
PREC = Accurate
ALGO = FAST
ISMEAR = 0 
SIGMA = 0.1;

Execute VASP.

Copy OUTCAR to OUTCAR.T=0..

Extract ZPR

We extract the band gap renormalization as

$\Delta E_{\mathrm{rm}} = E_{\mathrm{EPC}}-E_{\mathrm{no}}$

where $E_{\mathrm{EPC}}$ and $E_{\mathrm{no}}$ are the band gaps with and without EPC, respectively.

Since the 4x4x4 cell of cubic diamond has 3 degenerate bands at the valence band maximum and 6 degenerate bands at the conduction band minimum, they are averaged in the calculation of the band gap. This is necessary since the convergence with respect to cell size is drastically improved this way. Also all perturbation theory calculations in literature evaluate the band gap the same way, which ensures compatibility of the different calculational methods.

The band gaps are extracted from the previously saved files OUTCAR.init and OUTCAR.T=0. using the following script:

Click to show/extract_zpr.sh

Extract zpr

To use the script please type:

bash extract_zpr.sh

The output of the script should look like the following:

The band gap (in eV) without zero-point vibrations is:
4.4049
The band gap (in eV) including zero-point vibrations is:
4.05102
The zero-point renormalization of the band gap (in eV) is:
-0.353883

Better accuracy

The accuracy of the band gap renormalization depends dominantly on the size of the super cell, so this is the quantity that has to be usually converged in this type of calculation.

This example contains a POSCAR file for a 5x5x5 cell (POSCAR.5x5x5). After repeating all above steps with this POSCAR file the following results should be obtained:

The band gap (in eV) without zero-point vibrations is:
4.1421
The band gap (in eV) including zero-point vibrations is:
3.83717
The zero-point renormalization of the band gap (in eV) is:
-0.304933

The interested user can try to further increase the cell size, by making a super cell from the primitive cell (POSCAR.prim) provided by this tutorial (how to build a super cell is for example covered here).

Temperature dependence of the band gap

Here the temperature dependence of the band gap due to EPC is calculated. The input files are located in the directory TEMP_DEPENDENCE. Switch to this directory. Copy POSCAR.4x4x4 to POSCAR and INCAR.init to INCAR. The INCAR file contains following lines which are different from the calculation of the ZPR:

PHON_NTLIST = 8
PHON_TLIST = 0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0

After running the calculations several new POSCAR files are created.

Before running the calculations copy INCAR.run_temp to INCAR.

Run a standard VASP calculation for each of them to obtain the band gap or use the script run_temperature.sh provided with this calculation:

Click to show/run_temperature.sh

Run temperature cd carbon

To run the script please edit the file and set your VASP executable path (vasp_exec) and the number of processors you are going to use (np). To run the calculation type:

bash ./run_temperature.sh

This step produces several OUTCAR files which can be analyzed using the script extract_temp.sh:

Click to show/extract_temp.sh

Extract temp cd carbon

To run the script please type the following:

bash ./extract_temp.sh

This script calculates the relative change of the band with respect to temperature (that means the offset is shifted to zero). The result is written to gap_vs_temp.dat. To plot the data please type the following:

gnuplot -e "set terminal jpeg;set xlabel 'T (K)'; set ylabel 'band gap'; set style data lines; plot 'gap_vs_temp.dat', 'C_exp_points_offset0.dat' w circles, 'C_exp_fit_offset0.dat'" > gap_vs_temp.jpg

The resulting curve should look like the following:

The experimental data (blue lines and circles) are taken from reference ^[1].

Improving temperature dependence by adding volume effects

In the previous step we see that the experimental slope of the temperature dependence of the band gap is underestimated. To improve on that we will add also the volume dependence. The volume dependence is calculated from quasi harmonic calculations ^[2].

First save your obtained band gap vs. temperature curve, since it will be overwritten otherwise. Type the following:

mv gap_vs_temp.dat gap_vs_temp_novol.dat

Go out of the directory ./TEMP_DEPENDENCE and go to the directory ./QUASI_HARMONIC. For the purpose of this tutorial the quasi harmonic calculations will be performed for the 4x4x4 cell but for exact calculations one needs to go to larger cell sizes until the results are converged.

There are three scripts in this directory to perform the quasi harmonic calculations:

quasi_harm_4x4x4_diamond_create_pos_and_run_vasp.sh
quasi_harm_4x4x4_diamond_make_energy_vs_volume_plots.sh
quasi_harm_4x4x4_diamond_obtain_fitting.sh

For a quasi harmonic fit we need free energy vs. volume curves at different temperatures. Starting from the equilibrium volume we need to create POSCAR files at different volumes. The script to do this is given as:

Click to show/quasi_harm_4x4x4_diamond_create_pos_and_run_vasp.sh

Quasi harm 4x4x4 diamond create pos and run vasp

This script creates 15 POSCAR files where the volume is varied in both directions in steps of 2 percent with respect to the starting volume. It also runs the necessary VASP calculations to obtain the dynamical matrix. To run this script please set your executable path and number of processors in the script and type:

bash ./quasi_harm_4x4x4_diamond_create_pos_and_run_vasp.sh

This script renames all the OUTCAR files for each volume which are needed in the next step. In that step the free energy vs. volume curves need to be extracted for each temperature. To do this use the script:

Click to show/quasi_harm_4x4x4_diamond_make_energy_vs_volume_plots.sh

Quasi harm 4x4x4 diamond make energy vs volume plots

To use this script please type:

bash ./quasi_harm_4x4x4_diamond_make_energy_vs_volume_plots.sh

The free energy curves for each volume are save to OUTTEMP_*

Finally to obtain the equilibrium volume at each temperature use the following script:

Click to show/quasi_harm_4x4x4_diamond_obtain_fitting.sh

Quasi harm 4x4x4 diamond obtain fitting

To run that script type:

bash ./quasi_harm_4x4x4_diamond_obtain_fitting.sh

The output should look like the following:

temperature: 0.00000000e+00, a_latt: 7.18012
temperature: 1.00000000e+02, a_latt: 7.18014
temperature: 2.00000000e+02, a_latt: 7.18064
temperature: 3.00000000e+02, a_latt: 7.18267
temperature: 4.00000000e+02, a_latt: 7.18613
temperature: 5.00000000e+02, a_latt: 7.19037
temperature: 6.00000000e+02, a_latt: 7.19493
temperature: 7.00000000e+02, a_latt: 7.19959
temperature: #NOTEMP, a_latt: 7.15218

Now switch back to the folder ./TEMP_DEPENDENCE and replace the lattice parameters at each temperature in the POSCAR.T=* files by the one obtained for the above fit. After that rerun the scripts run_temperature.sh and extract_temp.sh.

To plot the newly obtained curve together with the other curves please type:

gnuplot -e "set terminal jpeg;set xlabel 'T (K)'; set ylabel 'band gap'; set style data lines; plot 'gap_vs_temp_novol.dat', 'C_exp_points_offset0.dat' w circles, 'C_exp_fit_offset0.dat', 'gap_vs_temp.dat'" > gap_vs_temp_volume.jpg

The resulting plot should look like the following:

Now we see in this plot that by adding volume effects the slope is much better recovered. For this tutorial we only used a 4x4x4 cell since larger cells would be already quite time consuming, but for the converged 5x5x5 cell both curves should look slightly worse compared to experiment. A discrepancy between experiment and theory is expected, since the electron exchange and correlation are not sufficiently described within PBE which was used in this example. To get a really excellent agreement one needs to use the GW approximation ^[3].

Strictly the correct way to add volume effects to EPC would be to first change the volume for each temperature and then calculate the EPC for that temperature. In this tutorial and also in reference ^[3], it is done the other way around. Hence the EPC needs to be calculated only once. In reference ^[3] we observed that the two approach give very similar results.

References

Download

EPC cd-C.tgz

Overview > Band gap renormalization in diamond using one-shot method > List of tutorials

[ODonnell:APL:1991-1] K.P. ODonnel and X. Chen, Appl. Phys. Lett. 58, 2924 (1991).

[Baroni:RMG:2010-2] S. Baroni, P. Giannozzi, and E. Isaev, Rev. Min. Geochem. 71, 39 (2010).

[karsai:njp:2018-3] F. Karsai, M. Engel, E. Flage-Larssen, and G. Kresse, New J. of Phys. 20, 123008 (2018).

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[2]

[3]