The empirical potential has been used to calculate the phonon spectra
for diamond, BC8 and ST12 phases at various pressures up to 10GPa.
Atomistic relaxations, as described in Chapter 1, have been performed
under constant pressure allowing the atoms and box size to relax into
their equilibrium state. These atomic positions and cell parameters are
used in conjunction with the potential to calculate the vibrational
frequencies allowed in the crystals by the methods described in Section
5.2. The phonon dispersion curves are calculated along
several lines of high symmetry, but to calculate thermodynamic
properties, only the phonon frequency density of states are required
rather the dispersion curves. For this reason it is necessary to
calculate the allowed phonon frequencies at points throughout the
Brillouin zone rather than along lines of high symmetry. A good
description of the vibrational density of state is required as
illustrated in Chapter 3 where the vibrational spectrum of only the
point modes of the ST12 structure and and X modes
of the diamond and BC8 structures could be calculated due to the large
compute time of ab initio calculations. It was found that the BC8
structure had lower vibrational free energy than ST12 at all
temperatures and therefore would always be more stable that the ST12
structure. A similar result is obtained with the empirical potential on
calculation of the point modes only. In order to obtain a good
density of states the Brillouin zone is sampled in a regular grid of
k-points for the diamond structure and 
in the BC8 and
ST12 structures. It will be shown in Section 5.6 that this
full description of the vibrational spectra means that the ST12
structure always has a lower vibrational free energy than the BC8
structure and hence the possibility of ST12 becoming the more stable
structure is found.
The phonon dispersion curves along several lines of high symmetry for the diamond, BC8 and ST12 structures at zero pressure are shown in Figures 5.5-5.10. The phonon density of states used in the free energy calculations are also given in these figures.
Figure 5.5: Phonon dispersion curves for the diamond structure along
several lines through the first Brillouin zone of the the primitive unit
cell.
Figure 5.6: Phonon density of states for the diamond structure calculated
on a regular grid of k-points.
Figure 5.7: Phonon dispersion curves for the BC8 structure along
several lines through the first Brillouin zone of the the primitive unit
cell.
Figure 5.8: Phonon density of states for the BC8 structure calculated
on a regular grid of k-points.
Figure 5.9: Phonon dispersion curves for the ST12 structure along
several lines through the first Brillouin zone of the the primitive unit
cell.
Figure 5.10: Phonon density of states for the ST12 structure calculated
on a regular grid of k-points.
There is no experimental data on phonon frequencies for the ST12
structure and very little on BC8 due to the difficulty in making
reasonable samples. The phonon density of states for the diamond
structure is in excellent agreement to that of experiment[84].
The peaks from 3 to 5 THz correspond to the low frequency acoustic
modes associated with bond bending and can easily be fitted to
experiment by adjustment of the C parameter in the potential. The
high frequency TO( ) modes can also be fitted by setting the second
derivative of the A and B functions in the potential to give a
frequency of 15.5THz. Fitting only these two values produces a
dispersion curve for modes that are not fitted which is also in
reasonable agreement with experiment[84]. This gives
confidence that the results for the BC8 and ST12 structures will also
be reliable since the potential is made to describe the tetrahedral
nature of such structures. A recent light scattering
experiment[85] obtained frequencies of Raman active modes at
ambient pressure in the range of 10.5THz to 13THz although the authors do not
identify which modes they are. The BC8 structure phonon dispersion
curve (Figure 5.7) and density of states (Figure
5.8) show a high density of modes at 12.8THz at the
point (and could therefore be Raman active) and also along the
and 
lines at approximately
12.5THz to 13.0THz respectively and at the H point at 10.3THz
(although these modes cannot be seen by Raman experiments). This
good agreement of the -modes to a Raman experiment also
indicates that the potential is transferable to these complex phases to
which the parameters are not fitted.
The general trend in the density of states as the structure becomes
more complex (diamond 
BC8 ST12) with
increasing numbers of different bond lengths and bond angles shows a
reduction in the height of the peaks of the high frequency (TO) modes
in favour of the lower frequency bond bending modes. This behaviour is
also observed as the pressure is increased. It is also interesting to
note that there is a range of forbidden frequencies in the higher range
of the spectrum for the ST12 structure. As pressure increases, the gap
increases from 0.8THz at ambient pressure to almost 1.6THz at 10GPa.
In all cases the effect of pressure on the structures is to increase the frequencies of all the modes although this increase varies - in general the high frequency (mostly bond stretching) modes shows a large change in frequency with pressure while the acoustic modes remain relatively unchanged. The frequencies of the zone centre phonons have been picked out for comparison with Raman spectroscopy. The variation in frequency with pressure of zone centre phonons in BC8 is remarkably similar to that measured experimentally[86, 85] in silicon BC8. The low frequency modes are almost unchanged with pressure, consistent with the TA phonons in most tetrahedral semiconductors. These may even be anomalous in that their frequency is slightly reduced with pressure. This implies that the effective force constants for these modes are unchanged or even slightly weakened by pressure. The lack of anomalous modes in the simulation can be associated to the lack of explicit bondlength dependence in the term which describes the low frequency modes. The effect of such a change in pressure on the phonon modes can be characterised by the Grüneisen parameter defined by the ratio of the logarithmic change in frequency to volume, thus:
Therefore a softening of a mode with pressure results in a negative Grüneisen parameter. The Grüneisen parameters have been calculated for the zone centre modes and are shown in Table 5.2.
Table 5.2: Grüneisen parameters for the non-degenerate zone centre modes
for the BC8 and ST12 structures modelled by an empirical
potential.
The variation in frequency of the zone centre modes for the BC8 structure with increasing pressure is shown in Figure 5.11.
Figure 5.11: Change in zone-centre phonon frequencies with increase in
pressure for the ST12 (left hand plot) and BC8 (right hand plot)
structures.
Comparison to experimental results on BC8 silicon[85] show remarkably good agreement. The equivalent plot for the ST12 structure is also shown although there exists no experimental data for comparison.
The predicted Grüneisen parameter for the TA(X) phonon is small,
0.28, an order of magnitude smaller than the bond stretching modes, but
non-negative. This mode is dominated by the bond bending
term[77, 80, 82], and the Grüneisen parameter is
always positive for models with this type of bond bending term, because
the third derivative of 
is
which is negative for all angles
occurring in the BC8, ST12 and diamond structures. Since the entropy
calculations performed below requires a sum of (logarithms of)
frequencies and not the product, the absolute error in this quantity,
regardless of sign, is the important feature. This is small.