How to calculate the properties of an isolated neutral vacancy

We present the vacancy isolated created in the FCC lattice of silicon Si.


The original silicon unit cell consists of two atoms in FCC structure. The value of the lattice parameters used is 5.29 Å. The lattice parameters used in the initial unit cell are within 3% of the experimentally measured 5.44 Å. 

 The crystal structure is as follows:

 Si-Diamond                                                 
F   LATTICE,NONEQUIV.ATOMS:  2                              
MODE OF CALC=RELA unit=ang
  9.996655  9.996655  9.996655 90.000000 90.000000 90.000000
ATOM  -1: X=0.00000000 Y=0.00000000 Z=0.00000000
          MULT= 1          ISPLIT= 8
Si1        NPT=  781  R0=0.00010000 RMT=    2.1500   Z: 14.0
LOCAL ROT MATRIX:    1.0000000 0.0000000 0.0000000
                     0.0000000 1.0000000 0.0000000
                     0.0000000 0.0000000 1.0000000
ATOM   2: X=0.25000000 Y=0.25000000 Z=0.25000000
          MULT= 1          ISPLIT= 8
Si2        NPT=  781  R0=0.00010000 RMT=    2.1500   Z: 14.0
LOCAL ROT MATRIX:    0.0000000 0.0000000 0.0000000
                     0.0000000 0.0000000 0.0000000
                     0.0000000 0.0000000 0.0000000
   0      NUMBER OF SYMMETRY OPERATIONS

This unit cell was used to create a supercell using the “supercell” command in Wien2k. A 4x4x4 128-atom supercell was constructed with an FCC structure. 

The crystal structure of the supercell is as follows:

After the supercell was complete, an atom was removed to create the Si vacancy. 



With the help of SGROUP, the number of atoms needed to represent the FCC 127/128-atom supercell was modified to the final supercell model of the vacancy. 


The resultant supercell was adjusted such that there was one vacancy per supercell. The supercell used for the calculation had a BCC structure, space group #44, with 49/50-atoms. This model is equivalent to the FCC 127/128-atom cell through multiplicity and symmetry.

Wien2k saves vital computational time through useful symmetry operations implemented in the code. 

The kmesh used consists of 100 k-points in the first brillouin zone. This kmesh is suitably dense to describe the properties of the vacancy. The supercell is 4 times the size of the initial silicon unit cell, therefore needing approximately 1/4 of the number of k-points to describe the brillouin zone. Typically a silicon unit cell needs only 250-500 k-points to describe the DOS. 

The cell is partitioned into silicon atomic spheres that include one vacancy per supercell. The RMT = 1.11 Å for each silicon sphere; the value corresponding to the periodic table is 1.17 Å. In this calculation a slightly smaller sphere size was chosen so that the atomic spheres do not overlap. The PORT program must efficiently move the spheres during the relaxation process (minimization of forces). In the construction of the supercell the vacancy image was separated by 7.48 Å.

The periodicity of the vacancy in the supercell calculation is large enough to separate the vacancy image with a worthy amount of bulk. This way the vacancy is isolated and the interaction between vacancy image is negligible. Since the vacancy has such localized properties, as mentioned previously, it will be deduced in the results section that the interaction is in fact negligible for this distance separating the periodic vacancy. This calculation serves as a model to bring insight into the associated properties of the vacancy during relaxation and the effect of vacancy-induced states in the bandgap. 

The vacancy geometry and characterization of the local electronic properties will be the focus of the vacancy calculation results section. Although it would be difficult to measure such geometries and electronic states using the present experimental techniques, these characteristics can be modeled using theoretical methods such as Wien2k. There are various dependencies that introduce shortcomings to the calculation. These are not limited to but include many of the initial conditions of the calculation, such as supercell size, symmetry, kmesh, and methods of calculated forces and relaxation. The results will be presented, accomplished by portraying some of the most fundamental properties of the vacancy.


Reference: http://www.physics.niu.edu/physics/_pdf/academic/grad/theses/Fabella.pdf