In both dense phases, as the volume is reduced the distortions of the tetrahedra increase. In the BC8 structure these distortions are uniquely described by the variation in the x parameter as described in Section 4.3. For the BC8 structures of Si, Ge and C the structural response to applied pressure is shown is represented in Figure 4.11 where the calculated variation in the free structural parameter, x, is shown. It is clear that both BC8-Si and BC8-Ge behave similarly under pressure as the equilibrium value of x and its slope are nearly equal in the two materials. For Si, the value of x which fully relaxes the structure is found to be x=0.1001 at the equilibrium lattice constant of 6.54Å. The experimental value is reported to be x=0.1003+/-0.0008. For fully relaxed BC8-Ge, x=0.1013. In BC8-carbon, the equilibrium value of x=0.0935 is approximately 6% smaller than it is for Si or Ge.
Although plots of x give a complete representation of the data from the calculations, which are done at constant volume, it is more informative to examine the variation of bondlengths with pressure, and this is plotted in Fig. 4.16. Notice that under pressure x increases, which tends to compensate for decreasing in , but enhances the reduction in . The effect on is small but gives rise to a small additional decrease in with pressure, in addition to the reduction in .
Figure 4.16: Variations in bondlengths and near neighbour distances with volume change for BC8 silicon. The graph for germanium is similar. Note that is most sensitive to isotropic compression. The A bond changes by less than 0.1Å over the range shown. This insensitivity results in a bondlength crossover at a lattice parameter of 6.25Å.
The effect of this compensation against decreasing is that whilst at ambient pressure the A bonds are shorter, as the pressure increases the B bond contracts until at all bonds are the same length. There is no special symmetry associated with this coincidence, and the B bond length continues to contract. A qualitative explanation of this comes from the empirical model in Chapter 4.
Consideration of the differentials of expressions (4.1) to (4.5) makes it clear that decreases with increasing x (pressure) while increases. Thus pressure serves to increase the distortion of the tetrahedra: the price which must be paid for maintaining bondlengths. It is thus clear that the observation of increasing x with pressure means that the bond-stretching forces have a greater bearing on the structure than the bond-bending ones.
In ST12 the structure is only fully defined by four internal parameters. For this reason it is essential to relax the structure under the Hellmann-Feynman forces, and thus study of ST12 is ideal for the plane-wave basis set method. The relationships between these internal parameters and the bondlengths and bond angles are complex, and for clarity only the latter quantities are considered. The evolution of the internal structure under pressure is described by Figure 4.17 which depict the change of the three different bondlengths under pressure.
Figure 4.17: Variation of the three different bondlengths on ST12 germanium with volume. The graph for silicon is similar.
Following the notation of Kasper and Richards for ST12-Ge, the four atomic positional parameters are shown for axial ratios of 1.2, 1.25 and 1.3 in Figure 4.18. It is evident from the figure that the z-parameter for the 8(b) site has the strongest dependence on isotropic compression of the unit cell and that the positional parameters are not greatly affected by changes in axial ratio over the range considered here. The ST12-Ge positional parameters near the equilibrium volume and c/a ratio may be found from Table 4.1. The calculated value of x for the 4(a) sites is 0.0882 compared to 0.0912 found experimentally. The calculated x, y and z values for the 8(b) sites are x=0.1693, y=0.3771 and z=0.2454. The corresponding experimental values at ambient pressure are x=0.1730, y=0.3784 and z=0.2486. The agreement is encouraging and suggests that future experiments using recent advances in angle-dispersive powder diffraction to determine the pressure dependence of this structure would be of value.
Figure 4.18: The relaxed ST12 structural parameters for germanium against volume. The circles, squares and triangles represent axial ratios of 1.25, 1.30 and 1.35 respectively.
As in BC8, the ST12 internal parameters vary with pressure to maintain the bondlengths at the expense of further distorting the bond angles. For a constant c/a ratio the bondlengths are reduced more slowly as a function of volume than the unit cell parameters. In practice the c/a ratio also changes under pressure in such a way as to reduce the changes in bondlengths. The fully relaxed unit cell dimensions and atomic positions in for silicon, germanium and carbon in the BC8 and ST12 structures are summarised in Tables 4.3 and 4.4.
Table 4.3: The full set of positions are given for the BC8 structure in terms of the one free structural parameter x. Also shown is the relaxed lattice parameter a in Å and the corresponding x for carbon, silicon and germanium.
Table 4.4: The full set of positions are given for the ST12 structure in terms of the 4 free positional parameters x, y, z and x which correspond to the (8b)-x, (8b)-y, (8b)-z and (4a)-x sites respectively. Also shown is the relaxed lattice parameters a and c in Å and the corresponding positional parameters for carbon, silicon and germanium.