The lattice vacancy in diamond structure silicon has probably been the most studied defect in a semiconductor, therefore only a brief account of the work presented here will be given for completeness. Interest in vacancies derives from its importance for understanding the mechanisms responsible for diffusion in silicon. There have been many reports on the formation energy, the result varying tremendously depending on the method used and the charge of the defect. Also, most ab initio methods have not allowed for full relaxation of the surrounding crystal structure of an isolated vacancy. Other errors have included incomplete basis sets used because of the large number of calculations necessary. For example, Car et al reports a vacancy formation energy of 5.0eV but with an improved basis set the formation energy reduces to 3.8eV. Using a Green's function method and a different basis set the same authors report a value of 4.4eV although allowing only nearest neighbour relaxation gives a formation energy of 3.8eV. A first principles calculation carried out by the supercell method (which will be applied to the vacancy in this section) quoted a neutral vacancy formation energy of 3.6eV. In that case the size of the unit cell was very small and therefore could only include nearest neighbour relaxation. In each case there appears to be inaccuracies in the results either due to incomplete basis sets or inadequate system size which will not allow a full relaxation of the system. It will be seen in the following calculation that there is substantial relaxation occurring around the vacancy.
In Section 7.3 the change in the properties of the defect with supercell size is investigated. Due to computational limitations the system size used here is a cubic unit cell consisting of 63 atoms (64 for the perfect crystal with one removed for the vacancy). It is shown later that a 63 atom cell the calculation is not quite converged, but the errors incurred are small.
The initial configuration for the simulation consists of 63 atoms displaced a small amount in a random direction from their perfect diamond lattice sites. It would be possible to relax these positions using the empirical potential and use that final result to start the ab initio calculation, but this would require that bonding topology surrounding the vacancy to be used as input. Anyway, it is preferable that no initial assumptions are used in the calculation, therefore the full relaxation of the system is purely from the ab initio Hellmann-Feynman forces.
It is found that there is substantial relaxation towards the vacant site by each of the surrounding four atoms. This is illustrated in Figure 7.1 summarising the change in atomic distances from the vacant site.
Figure 7.1: Schematic diagram showing the relaxation of the near neighbours around the vacancy defect site calculated by the ab initio method. The distances (in Å) on the outer shell of atoms show the distance from the vacant site. The lengths of the two new bonds are also indicated.
Each atom moves inwards from the initial first neighbour distance from the vacancy of 2.351Å to 2.22Å. In turn, two new covalent bonds are formed between pairs of atoms of length 2.41Å and 2.38Å while the non-bonded pairs are slightly closer together than the bulk second nearest neighbour distance at 3.74Å. This new bonding configuration forms two five fold rings and two seven fold. Unlike in several previous calculations[103, 95], the unit cell is large enough to examine the relaxation undergone by the second shell of atoms around the defect. Of the twelve atoms in the second shell of neighbours, which started at the diamond silicon second neighbour distance of 3.839Å, eight remain almost unmoved, relaxing outwards slightly to 3.85Å. The remaining four atoms in the second shell move inwards to a distance of 3.67Å from the vacant site. The third shell remains almost unchanged. This configuration is found to have a defect formation energy of 3.38eV. This is 0.5eV lower than that of previously published values which do not allow for relaxation on such a long range as presented here.
There is also a significant change in the electronic structure of the crystal on the introduction of the vacancy. The electronic density of states for silicon in the perfect diamond structure calculated by the ab initio methods described in Chapter 2, is given in Figure 7.2.
Figure 7.2: Electronic density of states for silicon in the perfect diamond structures. The top valence band is at 0eV.
A schematic diagram of the region about the Fermi level is shown in Figure 7.3 as found after full relaxation of the structure containing the vacancy.
Figure 7.3: Schematic diagram of the region about the Fermi level of Si-diamond containing a defect as found from the 63 atom supercell calculation. The dotted lines show both the conduction bands and valence bands broadening into the gap. The localised band gap state is also indicated.
The tail of both the conduction and valence bands are broadened into the gap reducing the semiconducting gap. The triply degenerate point state found in the top band is found to split, having energies of , eV and eV, where is the highest energy valence band. There is also a non-degenerate state in the gap close to the conduction band at eV. On reconstruction of the valence electronic charge density using this state only, it is found that it is localised orbital at the site of the vacancy.