In this section, the results of several calculation will be presented. Firstly, the 0K configurations of the hexagonal and tetrahedral interstitials and the vacancy are determined by the use of the empirical potential given in Chapter 4. The Parrinello-Rahman Lagrangian is used to allow the size of the unit cell to change so that no external stresses act upon the supercell. This was not done in the ab initio calculation due to the difficulties in dealing with a changing basis set for the wavefunctions. The relaxed positions of the atoms and the force constants calculated directly from the potential are then used to construct a dynamical matrix for the entire supercell. Subsequent diagonalisation gives the vibrational frequencies and their corresponding eigenvectors which will enable a search for localised modes caused by the defects.
The covalent bond model discussed in Chapter 4 required the bonding
configuration of the defects to be known. The bonding
configuration for the tetrahedral interstitial is well known (for
example, see [103]) and it is this one which is used. The
bonding topologies found from the ab initio calculations
presented above are used for the hexagonal interstitial and vacancy. It
is not possible to use the empirical potential to calculate the
vibrational properties of the 
interstitial formation found in the
ab initio calculations presented above. This is because the
potential models simple four-fold covalent bonds, whereas the
defect was found to contain a three-fold bonding orbital which will not
be described correctly with a simple covalent bond model.
For a given bonding arrangement, the energy minimum is located using a conjugate gradients routine starting from a configuration in which all atoms are given a small random displacement to break all symmetries. The supercells used for the perfect crystal contain 64 and 216 atoms, with plus and minus one atom for the interstitials and vacancy respectively. It is found that the defect formation energies and formation volumes differ slightly between the two sizes of simulation, therefore a larger cell based on 512 and 1000 atoms for the perfect crystal was also used to check that the defects were sufficiently well isolated in the 216 atom cell. This was found to be so.
These defect formation energies and volumes are shown in Table 7.1. It was found that the formation energies and volumes for the two larger simulations are identical showing that the defects are isolated for the simulations based in the 216 atom supercell.
Table: The defect formation energies in eV and formation volumes in
Å 
is shown as the system size changes. Systems I, II, and III are
for supercells based on 64, 216 and 512 atoms respectively. The first
three columns show the formation energies, the next three show formation
volumes.
For this reason, the vibrational simulations are carried out on the 216 atom supercells only. The results of the defect formation energies are in agreement with other empirical results[108] in that the tetrahedral interstitial is energetically more favourable than the hexagonal interstitial. It is also in agreement with the above ab initio calculations although the formation energies are slightly lower than that of other results[95]. This is believed to be due to finite size effects, and in this context, note that there is a strong dependence on system size exhibited in Table 7.1. Also note that the normal correlation between low formation volume and low formation energy is observed in all cases.
The relaxed atomic positions and supercell sizes can now be used to calculate the vibrational modes of the entire system. The theory of lattice vibrations is presented in detail in Chapter 4. Continuing from that point, the method which is used here is presented.
Defects in the crystal remove the periodicity from the structure.
Therefore the entire supercell can be described by l=(0,0,0) only, and
k=1,...,N for N atoms in the simulation. To make a comparison, the
perfect diamond crystal will be treated in the same way. The perfect
lattice is treated as one large unit cell which therefore has a very small
Brillouin zone. This allows sampling at 
to suffice resulting
in 
being a real symmetric
matrix. This ranges over the same reciprocal space points allowed by
that of a fully symmeterised calculation with 
unit cells. Thus, in effect, the phonon frequencies of higher valued
wavevectors relative to the standard diamond cell are being calculated
but without the need for diagonalisation of complex matrices. The phonon
density of states and the displacements associated with each normal mode
can be found. In the case of lattices containing defects, the
periodicity of supercells is unwanted hence sampling of wave vectors
other that is unnecessary.
The l subscripts in the dynamical matrix can be dropped, and setting
, the dynamical matrix of the supercell is now given by
where the matrix of force constants is now
and
By removing the sum over unit cells and carrying out the calculation as if there were no symmetry within the supercell (which is true in the case of crystals containing defects), the phonon frequencies which have higher order wave vectors in the standard FCC-diamond Brillouin zone will be mapped onto the zero wave vector relative to the supercell.
The calculations were carried out on a silicon crystal containing 216 atoms in the perfect lattice. Similar calculations were also done on crystals containing interstitial atoms in the hexagonal and tetrahedral sites and on a crystal containing a vacancy. Point defects are well described by the covalent bond charge potential used here, and the validity of using it to calculate phonon spectra is established here and in Chapter 4.
Similar phonon density of states curves were found for the 64 and 216 atom simulations which gives confidence that the unit cells are large enough for such calculations. Figure 7.9 show plots of the density of states of phonon frequency for the larger simulations.
Figure 7.9: Density of states plots of vibrational frequencies for the
perfect and defected crystals. The density is in arbitrary units. (a) is
for the vacancy, (b) the tetrahedral interstitial, (c) the hexagonal
interstitial, and (d) is the perfect crystal. Note here that sampling at
is equivalent to 256 k-points in the primitive diamond cell
and is therefore much less than the brillouin zone sampling carried out
in Chapter 4. This illustrates the difficulty in performing calculations
where the symmetries of a crystal are lost.
The shapes of the curves are similar to that found in Chapter 4 and to experiment[109] although the density of the low frequency bond bending modes has been enhanced relative to the high frequency. It can be seen that the defects cause all the peaks to broaden, which is most notable at the higher frequencies where several localised modes are located. There are additional modes of higher frequency added to the spectrum which are associated with the defects in the structure. These have frequencies above the cut off frequency for the perfect diamond structure and, as shall be demonstrated below, are localised near the defects.
The eigenvector corresponding to a particular mode gives the directions and relative amplitudes that each of the atoms in the supercell move under excitation of that mode[110]. Localised modes are found by searching through the normalised eigenvectors for large displacements of any individual atom. In the cases of the interstitials, the localised mode with the largest element from the matrix of normalised eigenvectors corresponds to the interstitial atom itself. Figures 7.10(b) and 7.10(c) show the atoms around the interstitial and the directions and relative amplitudes in which they move under excitation of this mode.
Figure 7.10: Line drawings of the silicon crystals showing the localised
modes. The bold lines at each atomic site show the direction and
relative magnitudes of the vibrational mode under consideration. Figure
(a) shows the vacancy where the positions of the atoms surrounding the
defect can be seen to relax to reduce the vacancy formation volume.
There are four atoms vibrating with large amplitude for this localised
mode surrounding the vacant site at the top of the diagram. (b) shows
the tetrahedral interstitial and (c) the hexagonal interstitial where
the single localised vibration of the interstitial atoms are clearly
seen, with very little motion of the surrounding atoms.
It can be seen from Figure 7.9 that modes of higher frequency than the maximum found in the perfect crystal occur. In the vacancy the density of states no longer has a sharp cut-off, but instead tails off with the highest frequency mode found at 15.98THz. The tetrahedral interstitial causes four localised mode higher than the cut-off for the perfect diamond structure. These are at 18.25THz, 17.49THz, 16.92THz and 16.16THz. Similarly, the hexagonal interstitial causes high frequency modes at 17.71THz (doubly degenerate), 16.94THz (double), 16.36THz (single), 16.17THz (double) and 15.98THz (single). The maximum frequency mode that can be supported by the perfect structure is 15.80THz. The perfect crystal is found to have no such localised modes. For the vacancy, Figure 7.10(a), the atoms close to the defect are found to have large vibrational amplitudes in the high frequency modes while the other atoms in the supercell are almost stationary. The localised modes shown in the figure correspond to the eigenvectors with the largest single element. It is found that this is also the eigenvector associated with the highest frequency (18.25THz for the tetrahedral interstitial and 17.71THz for the hexagonal interstitial). Each eigenvector is normalised to unity (that is, the sum of the coefficients of each vector is one). The vibrational energy of the localised mode is concentrated in a single atom. For the hexagonal interstitial the fraction of the vibrational energy localised in this atom is 0.502, while for the tetrahedral interstitial it is 0.505. This is an order of magnitude greater than that of the largest normalised element of any eigenvector for the perfect crystal. The vacancy shows similar characteristics, with the localised modes having the highest frequencies, although the majority of the vibrational energy is associated with the four atoms adjacent to the vacant site.
