Structural and vibrational calculations have been performed on the diamond, BC8 and ST12 structures using an empirical potential designed specifically for 4-fold covalently bonded structures. The length scales have been fitted to that of diamond-Si. The agreement between experimental, ab initio and empirical results for the structural properties of high-density phases of Si and Ge gives us confidence to proceed with empirical calculations to a scale beyond that which can be achieved by ab initio techniques. As a result of the above calculations it is found that a model whose parameters are fitted to describe the diamond structure, the BC8 structure becomes unstable with respect to ST12 at high temperature and pressure. Note that both phases are metastable with respect to diamond, each having very different topologies. It therefore seems unlikely that BC8 silicon will transform to the ST12 structure simply by heating. In fact it is found that on heating, BC8 silicon transforms into the Lonsdaleite structure, the monoatomic equivalent of wurtzite. However, in view of the kinetic difficulty in transforming from -Sn to diamond, it may be possible to depressurise silicon from the -Sn phase at high temperature to a pressure above 8GPa to form ST12 silicon.
Although the empirical model has not been explicitly parameterised for germanium, it is possible to make some general comments. The parameter C governs the shear moduli and the vibrational frequency of the low-frequency phonons. In germanium the shear moduli are considerably smaller than in silicon, so a smaller value of C would be appropriate. This would lead in turn to a downscaling of the phonon frequencies, and therefore of the entropies and entropy difference. In germanium, therefore, it would expected that the transition temperature should be much lower than in silicon - perhaps below room temperature. Thus it might be possible that BC8 germanium will be produced and remain stable over the ST12 structure by depressurisation below room temperature.
The structural properties of BC8 and ST12 silicon and germanium are well described by a model designed for covalently bonded materials. This suggests that although the electrical properties of BC8 may be dominated by a small Fermi surface, the primary contribution to structural bonding comes from covalent bonds. The same model is then used to determine phonon spectra based on the covalent concepts of bond bending and bond stretching.
Although all structures are based on covalent bonding, the bulk moduli are lower in the denser phases than in diamond. This is because they are able to contract both by bond shortening and bond bending, whereas diamond can contract only by bond shortening.
As noted previously, both silicon and germanium can be described with this model, the main difference being that the bond-bending forces are lower in germanium relative to the bond stretch.
The ST12 structure has a wider spread of bond angles but more closely matched bond lengths. Its many degrees of freedom allow it to take up external pressure with internal relaxations, giving it a large compressibility. These internal modes also give rise to many low frequency phonon modes, making ST12 a high entropy structure and therefore favoured at high temperature and in germanium where the large bond-bending distortions are less unfavourable.
Experimentally, both ST12 and BC8 germanium have been reported, but the conditions in which they were made are not clearly documented and seem to lack reproducibility. ST12 silicon has not yet been found. The calculations presented here suggest that by conducting high pressure experiments at different temperatures, the preferred phase can be altered. In particular, it may be possible to synthesize ST12 silicon by depressurisation from the metallic -tin structure at high temperature.