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Next: Complex Structures: Empirical Treatment Up: Complex Phases: Ab Initio Previous: Electronic Band Structure


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 tex2html_wrap_inline5370 -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 Lonsdalitegif 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 tex2html_wrap_inline5370 -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, tex2html_wrap_inline5370 -Sn and BC8[39]. On consideration of the difference in pressures in which the III-V semiconductors transform to the tex2html_wrap_inline5370 -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 tex2html_wrap_inline5370 -Sn tex2html_wrap_inline6048 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 tex2html_wrap_inline5370 -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 tex2html_wrap_inline5398 =91.296 tex2html_wrap_inline6586 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 tex2html_wrap_inline5370 -Sn tex2html_wrap_inline6048 BC8 transition. The rhombohedral tex2html_wrap_inline6048 BC8 transition route is rather simple - the A bonds are broken along the (111) direction and rebonded to the tex2html_wrap_inline5268 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.

next up previous
Next: Complex Structures: Empirical Treatment Up: Complex Phases: Ab Initio Previous: Electronic Band Structure

Stewart Clark
Thu Oct 31 19:32:00 GMT 1996