Vacancy Formation Energy Calculation

The necessary tools to perform the basic calculations to investigate the vacancy formation energy is shown in this page.

Note: The page is under a continuous developing stage.

Figure 1. Remove one atom from the centermost part of the fcc Al system.

In order to perform the vacancy formation energy calculation, one needs to enlarge the simulation box size three dimensionally big enough, such that the vacancy formation energy value should be converged. As a matter of fact one needs to realize that typically DFT calculations are performed within the periodic boundary condition (PBC) such that the simulation cell in hand is actually repeated in all dimensions. Hence for the vacancy formation energy calculation, if one uses a tiny simulation cell, because of periodicity, that vacancy is repetitive in all direction, and then it is no longer a single vacancy, but multiple vacancies, and then one needs to consider the interactions between those vacancies as well. We typically put a vacancy in the center of the mesh. A typical vacancy formation energy calculation consists of a 3X3X3 duplicated simulation cell consisting of more than 50 atoms as shown in the adjacent figure.

Initially as you have already calculated the lowest energy structure while doing the Energy-Volume curves, you already have the per atom energy $\epsilon$ of the equilibrium structure. Now you need to remove one atom from the center of the simulation cell, and relax the structure to get the relaxed energy (E_tot) of the system. Vacancy formation energy E_vac can be found by E_tot - N $\epsilon$

Note: For the case of a binary AB alloy, you need to remove both the elements one by one to get the vacancy formation energy for each element A and B individually to get the understanding of what elemental deficit in the system could be more probabilistic in nature (that which has less vacancy formation energy).

Note: The INCAR (ISIF=2) and POTCAR file remain same as given for Al P1-DFT example.

Relaxation

For relaxation (geometric or ionic optimizations) always use ISMEAR = 1 and SIGMA = 0.1 or ISMEAR = 2 and SIGMA = 0.2. It is always recommended to do ionic relaxations at fixed volumes and plot the energy vs volume graph to determine the equilibrium volume. To get accurate energies obtain the structure after relaxation and run a static calculation (no relaxations NSW = 0) with ISMEAR = -5.

Proposed method to relax and get accurate energies
- The system should be relaxed (ISIF=2) with ISMEAR = 1 (Methfessel-Paxton). At the end of the relaxation run VASP will generate new positions in the CONTCAR file. Copy the CONTCAR as POSCAR. Then run a static calculation (no relaxations, NSW = 0) with the tetrahedron method (ISMEAR = -5).

INCAR file for relaxation

LWAVE = .FALSE.
LCHARG = .FALSE.
LREAL = Auto
ISMEAR = 1
ENCUT = 240.3
EDIFF = 1e-6
NSW=100
ISIF=2
IBRION=2

INCAR file for static calculation

LWAVE = .FALSE.
LCHARG = .FALSE.
LREAL = Auto
ISMEAR = -5
ENCUT = 240.3
EDIFF = 1e-6
#NSW=100
ISIF=2
IBRION=2

Running the calculation

To find the energy of the system using VASP

Copy the following INCAR and KPOINTS file to a directory.
Copy the POTCAR file to the same directory
Copy the POSCAR file and remove one atom and modify the total no. of atoms to create a vacancy.
and execute VASP by
```
mpirun -np <no. of processors> vasp_std # for parallel calculation
```
```
vasp_std # for serial calculation 
```

Post-processing

After running the calculation use the equation to calculate vacancy formation energy. E_tot is the Energy you get from VASP. This value can be extracted at the end of the calculation by executing
```
tail -n1 OSZICAR|awk '{print $5}'
```

k-point convergence

The vacancy formation energy should be converged in terms of k-point grid. Therefore try different k-point grids. Remember that the ratio of k-points should be inversely proportional to the lengths of the lattice vectors (in this case the edges of the simulation box).

KPOINTS

Auto	        #header file
 0		
Monkhorst	#Style of Kpoints
 3  3  3	#Numbers
 0  0  0

POSCAR file for 3X3X3 orthogonal supercell

Al fcc (111)
1.0
    12.1499999999999986      0.0000000000000000      0.0000000000000000
     0.0000000000000000     12.1499999999999986      0.0000000000000000
     0.0000000000000000      0.0000000000000000     12.1499999999999986
108 <----Change to 107 after removing atom
Cartsian
    0.0000000000000000     0.0000000000000000     0.0000000000000000
    0.0000000000000000     2.0249999999999999     2.0249999999999999
    2.0249999999999999     0.0000000000000000     2.0249999999999999
    2.0249999999999999     2.0249999999999999     0.0000000000000000
    0.0000000000000000     0.0000000000000000     4.0499999999999998
    0.0000000000000000     2.0249999999999999     6.0749999999999993
    2.0249999999999999     0.0000000000000000     6.0749999999999993
    2.0249999999999999     2.0249999999999999     4.0499999999999998
    0.0000000000000000     0.0000000000000000     8.0999999999999996
    0.0000000000000000     2.0249999999999999    10.1250000000000000
    2.0249999999999999     0.0000000000000000    10.1250000000000000
    2.0249999999999999     2.0249999999999999     8.0999999999999996
    0.0000000000000000     4.0499999999999998     0.0000000000000000
    0.0000000000000000     6.0749999999999993     2.0249999999999999
    2.0249999999999999     4.0499999999999998     2.0249999999999999
    2.0249999999999999     6.0749999999999993     0.0000000000000000
    0.0000000000000000     4.0499999999999998     4.0499999999999998
    0.0000000000000000     6.0749999999999993     6.0749999999999993
    2.0249999999999999     4.0499999999999998     6.0749999999999993
    2.0249999999999999     6.0749999999999993     4.0499999999999998
    0.0000000000000000     4.0499999999999998     8.0999999999999996
    0.0000000000000000     6.0749999999999993    10.1250000000000000
    2.0249999999999999     4.0499999999999998    10.1250000000000000
    2.0249999999999999     6.0749999999999993     8.0999999999999996
    0.0000000000000000     8.0999999999999996     0.0000000000000000
    0.0000000000000000    10.1250000000000000     2.0249999999999999
    2.0249999999999999     8.0999999999999996     2.0249999999999999
    2.0249999999999999    10.1250000000000000     0.0000000000000000
    0.0000000000000000     8.0999999999999996     4.0499999999999998
    0.0000000000000000    10.1250000000000000     6.0749999999999993
    2.0249999999999999     8.0999999999999996     6.0749999999999993
    2.0249999999999999    10.1250000000000000     4.0499999999999998
    0.0000000000000000     8.0999999999999996     8.0999999999999996
    0.0000000000000000    10.1250000000000000    10.1250000000000000
    2.0249999999999999     8.0999999999999996    10.1250000000000000
    2.0249999999999999    10.1250000000000000     8.0999999999999996
    4.0499999999999998     0.0000000000000000     0.0000000000000000
    4.0499999999999998     2.0249999999999999     2.0249999999999999
    6.0749999999999993     0.0000000000000000     2.0249999999999999
    6.0749999999999993     2.0249999999999999     0.0000000000000000
    4.0499999999999998     0.0000000000000000     4.0499999999999998
    4.0499999999999998     2.0249999999999999     6.0749999999999993
    6.0749999999999993     0.0000000000000000     6.0749999999999993
    6.0749999999999993     2.0249999999999999     4.0499999999999998
    4.0499999999999998     0.0000000000000000     8.0999999999999996
    4.0499999999999998     2.0249999999999999    10.1250000000000000
    6.0749999999999993     0.0000000000000000    10.1250000000000000
    6.0749999999999993     2.0249999999999999     8.0999999999999996
    4.0499999999999998     4.0499999999999998     0.0000000000000000
    4.0499999999999998     6.0749999999999993     2.0249999999999999
    6.0749999999999993     4.0499999999999998     2.0249999999999999
    6.0749999999999993     6.0749999999999993     0.0000000000000000
    4.0499999999999998     4.0499999999999998     4.0499999999999998
    4.0499999999999998     6.0749999999999993     6.0749999999999993
    6.0749999999999993     4.0499999999999998     6.0749999999999993 <------------------Remove this atom to make vacancy
    6.0749999999999993     6.0749999999999993     4.0499999999999998
    4.0499999999999998     4.0499999999999998     8.0999999999999996
    4.0499999999999998     6.0749999999999993    10.1250000000000000
    6.0749999999999993     4.0499999999999998    10.1250000000000000
    6.0749999999999993     6.0749999999999993     8.0999999999999996
    4.0499999999999998     8.0999999999999996     0.0000000000000000
    4.0499999999999998    10.1250000000000000     2.0249999999999999
    6.0749999999999993     8.0999999999999996     2.0249999999999999
    6.0749999999999993    10.1250000000000000     0.0000000000000000
    4.0499999999999998     8.0999999999999996     4.0499999999999998
    4.0499999999999998    10.1250000000000000     6.0749999999999993
    6.0749999999999993     8.0999999999999996     6.0749999999999993
    6.0749999999999993    10.1250000000000000     4.0499999999999998
    4.0499999999999998     8.0999999999999996     8.0999999999999996
    4.0499999999999998    10.1250000000000000    10.1250000000000000
    6.0749999999999993     8.0999999999999996    10.1250000000000000
    6.0749999999999993    10.1250000000000000     8.0999999999999996
    8.0999999999999996     0.0000000000000000     0.0000000000000000
    8.0999999999999996     2.0249999999999999     2.0249999999999999
   10.1250000000000000     0.0000000000000000     2.0249999999999999
   10.1250000000000000     2.0249999999999999     0.0000000000000000
    8.0999999999999996     0.0000000000000000     4.0499999999999998
    8.0999999999999996     2.0249999999999999     6.0749999999999993
   10.1250000000000000     0.0000000000000000     6.0749999999999993
   10.1250000000000000     2.0249999999999999     4.0499999999999998
    8.0999999999999996     0.0000000000000000     8.0999999999999996
    8.0999999999999996     2.0249999999999999    10.1250000000000000
   10.1250000000000000     0.0000000000000000    10.1250000000000000
   10.1250000000000000     2.0249999999999999     8.0999999999999996
    8.0999999999999996     4.0499999999999998     0.0000000000000000
    8.0999999999999996     6.0749999999999993     2.0249999999999999
   10.1250000000000000     4.0499999999999998     2.0249999999999999
   10.1250000000000000     6.0749999999999993     0.0000000000000000
    8.0999999999999996     4.0499999999999998     4.0499999999999998
    8.0999999999999996     6.0749999999999993     6.0749999999999993
   10.1250000000000000     4.0499999999999998     6.0749999999999993
   10.1250000000000000     6.0749999999999993     4.0499999999999998
    8.0999999999999996     4.0499999999999998     8.0999999999999996
    8.0999999999999996     6.0749999999999993    10.1250000000000000
   10.1250000000000000     4.0499999999999998    10.1250000000000000
   10.1250000000000000     6.0749999999999993     8.0999999999999996
    8.0999999999999996     8.0999999999999996     0.0000000000000000
    8.0999999999999996    10.1250000000000000     2.0249999999999999
   10.1250000000000000     8.0999999999999996     2.0249999999999999
   10.1250000000000000    10.1250000000000000     0.0000000000000000
    8.0999999999999996     8.0999999999999996     4.0499999999999998
    8.0999999999999996    10.1250000000000000     6.0749999999999993
   10.1250000000000000     8.0999999999999996     6.0749999999999993
   10.1250000000000000    10.1250000000000000     4.0499999999999998
    8.0999999999999996     8.0999999999999996     8.0999999999999996
    8.0999999999999996    10.1250000000000000    10.1250000000000000
   10.1250000000000000     8.0999999999999996    10.1250000000000000
   10.1250000000000000    10.1250000000000000     8.0999999999999996