Total energy pseudopotential calculations have been performed on high density phases of the Group IV elements C, Si and Ge and also on a high density phase of several III-V compound semiconductors. Using a density function molecular dynamics method similar to that proposed by Car and Parrinello[46], the fully relaxed BC8 and ST12 structures of these elements and the compound form, SC16, have been found. This was determined by relaxation under the influence of the Hellmann-Feynman forces. There is excellent agreement with experimental data where it exists. The structural response to compression has also been investigated. Using these results, other properties could also have been calculated such as the response to optical properties by pressure (although other methods exist such as the GW approximation[70] which produces a much better band gap than LDA can, giving better results on optical properties).
Using this molecular dynamics method, properties such as lattice parameters, bulk moduli and behaviour of the internal structure under pressure has been determined. These are all found to be in good agreement with experiment where available data exists, although, as usual within the LDA the structures tend to be overbound slightly. This overbinding means that the lattice parameters are inevitably too small (although in structures which have a lattice parameter which actually increases with pressure, for example Se-I, this lattice parameter can be overestimated[59, 71]). It is found that the lattice parameters are approximately 2% too small with a binding energy 15% too large. It is well established, however, that the energy differences between various structures as used here are well reproduced by LDA.
The calculations presented here predict that BC8 silicon is the low temperature metastable phase. Ab initio calculations of entropy at high temperatures are currently impractical over the full pressure-temperature phase space and so those calculations are carried out using an empirical potential and are presented in Chapter 4.
In germanium the energies of all phases are similar, reflecting the ease with which the germanium bonds can be distorted, and that cohesive energy is dominated by the requirement of fourfold coordination. It was found that the internal parameters in both BC8 and ST12 germanium adjust to maintain all bondlengths to be similar. The energy difference between the ST12 and diamond structures were much smaller than that of carbon or silicon. Convergence of this difference required the use of a denser k-point set for band structure sampling.
Although the germanium energy differences are close to the accuracy of
the code used here, it is predicted that within a range of pressures
ST12 will actually be the stable phase of germanium. It would be
interesting to attempt to verify this experimentally, although the
large kinetic barrier makes the direct transition between diamond and
ST12 impossible. It seems that ST12 may be made via the 
-Sn
phase. BC8 is found to be close but slightly higher in energy which has
now been confirmed experimentally, where at room temperature BC8 Ge is
found for short periods of time before transforming to the more stable
Lonsdalite
structure[18].
In carbon it was found that diamond will be completely stable with respect to BC8 or ST12 at all pressures considered, and that those phases, while not mechanically unstable, have fundamentally different bonding to Si and Ge.
In the case of III-V semiconductors, the SC16 or ST12 phases have not
been found experimentally (although recently SC16 CuCl has been
found[72, 73]). The calculated transition pressures from
zincblende to SC16 structures are about 1.5GPa which are an order of
magnitude smaller than that for zincblende to -Sn. This suggests
that the SC16 structure is thermodynamically stable over a large range
of pressures for the III-V semiconductors considered here. This is in
contrast to silicon where these calculations show it to be only
metastable. The case of the III-V's is somewhat similar to germanium
where the intermediate phase is ST12.
All calculations presented here indicate that the SC16 structure should
be found. This difference between experimental results and theoretical
can be resolved by considering the kinetics of the phase transition
which was not considered in the calculation presented here. It has
recently been shown that there is a simple transition route in silicon
between the high pressure metallic phase, -Sn and
BC8[39]. On consideration of the difference in pressures in
which the III-V semiconductors transform to the -Sn on the
upstroke and back to zincblende on the downstroke, it was concluded
that this transformation is not easy. Considering the same reaction
kinetics as that proposed for -Sn 
BC8 Si it was
found that like bonds would be formed in the III-V SC16 structure which
is extremely unfavourable[67]. As a result the difficult
retransformation to the zincblende structure is favoured in the compound
semiconductors.
The results presented here and in reference[39] show that SC16 should be a stable phase with respect to higher pressure metallic phases. Therefore possible methods of forming it can be suggested, such as cooling the melt of the high pressure metallic phase slowly while allowing the pressure to reduce. If formed, the large kinetic barrier would ensure that SC16 III-V's would be long lived at ambient pressure.
Added Note: Very recent high pressure studies on phases in
silicon formed from depressurisation from the -Sn structure show
that a new rhombohedral phase exists under pressure with 16 atoms per
unit cell (R. O. Piltz, J. McLean, S. J. Clark, P. D. Hatton, J. Crain,
G. J. Ackland and G. S. Pawley, Phys. Rev. B, To be published,
1994). X-ray diffraction results on the structure show it to have a
lattice parameter of a=6.43Å with a rhombohedral angle of
=91.296 
at 8.2GPa. Our preliminary calculations find
that is topologically different from BC8-Si, containing 5-fold rings -
the first crystalline structure of silicon to do so. This has been
confirmed from the experimental data. It seems likely that this is the
intermediate phase in the -Sn BC8 transition. The
rhombohedral BC8 transition route is rather simple - the A
bonds are broken along the (111) direction and rebonded to the 
neighbour. Relaxation of the atoms and unit cell gives a rhombohedral
distortion and forms 4 different bondlengths, all of which are of the
order 2.5Å. If this is the case, then the 5-fold ring structure will
inhibit the formation of SC16 III-V compounds due to the presence of
like-species bonding.