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[101] reports a vacancy formation energy of 5.0eV but with an improved basis set the formation energy reduces to 3.8eV[102]. Using a Green's function method and a different basis set the same authors report a value of 4.4eV[95] 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[103]. 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.