Graphs of Murnaghan equation fits to energy against volume for silicon and germanium in the ST12, BC8 and diamond phases are shown in Figures 4.3 and 4.4. The initial unit cell dimensions and internal structural parameters for ST12 were obtained from the empirical calculations described in Chapter 4. These graphs show that in each case the relaxed structures are unstable with respect to diamond at low pressures.
Figure 4.3: Graph of energy against volume for fully relaxed diamond, BC8
and ST12 structures in silicon.
Figure 4.4: Graph of energy against volume for fully relaxed diamond, BC8
and ST12 structures in germanium.
To obtain the curve for ST12 under hydrostatic pressure it was necessary to perform several series of calculations at several different c/a ratios, and to construct Figure 4.5 which is a contour plot of the energy against volume and c/a. The hydrostatic curve is then the lowest value of the projection of this surface onto the volume axis. Figure 4.6 shows the variation of c/a with volume under hydrostatic pressure.
Figure 4.5: Contour plot of energy against c/a ratio and atomic volume
for ST12 silicon.
Figure 4.6: Graph showing the variation of c/a ratio in ST12 silicon as a
function of volume.
In silicon there is no intercept between the diamond- 
-Sn common
tangent (Figure 4.3) and BC8 or ST12 curves, from which it
is deduced that these phases are, at best, metastable. The common
tangent construction can also be used to predict the -Sn
BC8 transition pressure using the metal calculation
described in another section. From these graphs it is
predicted that, for silicon, a pressure of only 11GPa will transform
the diamond structure to BC8. This is lower than the observed pressure
for the diamond -Sn transition at
12.5GPa[12]. This experimental value is above the theoretical
value because of hysteresis effects resulting from the absence of an
easy transition mechanism. The theoretical diamond BC8
transition pressure is greater than the calculated diamond
-Sn pressure (9GPa) found here. The calculated
pressure between -Sn and BC8 is 8GPa in good agreement with
depressurisation experiments and suggestive of a relatively easy
transition path. This will be discussed further when the III-V
semiconductors are discussed which show no metastable high density
phases on depressurisation from their high pressure metallic phases.
In germanium the broad picture is similar but with ST12 being stable at
intermediate pressure. The contour plot of energy against c/a and
volume and the variation of c/a with volume for germanium ST12 are
similar to that of silicon shown on Figs. 4.5 and
4.6. Unlike silicon, the common tangent between -Sn and
diamond cuts the ST12 curve (not shown), implying that for a range of pressures
ST12 is the stable structure. This result appears to contradict the
observation of the first transition being to -Sn. However,
given the long experimental lifetime of the ST12 phase at ambient
pressure[18, 22], it is clear that no easy
diamond ST12 transformation path exists and is therefore
likely that before the kinetic barrier can be overcome, the
experimental pressure has been raised to that at which -Sn is
stable. Thus the only way to make ST12 germanium is via the -Sn
phase. It may even be that the retransformation from -Sn to BC8
is easier than to ST12 because both structure have evenfold rings. The
BC8 structure will also be favoured at low temperatures (this will be
discussed in Chapter 4) although at room temperature, BC8 germanium has
been observed to exist for several hours[18] when the sample
was rapidly depressurised from the high pressure metallic phase.
For all phases, the calculated results for the equilibrium structural parameters are in good agreement with those reported in previous studies. The lattice parameters and bulk moduli as determined by fits to a Murnaghan equation of states,
where 
is the bulk modulus and B' is its
pressure derivative. These are given in Table
4.1. As is
usual in LDA calculations, the lattice parameters are underestimated
by about 2% (except for diamond germanium where the underestimate is
only 0.5%). This means it is expected that
the bulk moduli will be overestimated; where experimental data exist this
is indeed the case, B 
is overestimated by up to 8%.
Table 4.1: Calculated structural properties of carbon, silicon and
germanium in the diamond, BC8 and ST12 structures. The units are in eV
and Å.
The values are taken from fits to the Murnaghan equation
of states.
Determination of the equilibrium axial ratio in both ST12-Si and ST12-Ge is crucial for accurate determination of energy differences in these phases. This is because it is found that the ST12 total energy for both materials varies by several tens of meVs over a c/a ratio range of 1.1 to 1.3. For the case of germanium the equilibrium axial ratio occurs at 1.24 which represents a 6% overestimate of the experimental value[18, 22]. The calculated values for the relaxed atomic positional parameters, which are also sensitive to the c/a ratio, are discussed in more detail in the following section.
The Murnaghan equation of state assumes a linear pressure dependence of
the bulk modulus. The pressure derivative of the bulk modulus,
, is a dimensionless quantity describing the third volume
differential of the energy in units normalised to the volume and bulk
modulus. In all cases it is found that this is in the region 4-6, typical of
experimental values. The fitting errors for this quantity are of order
25%, being especially unstable with respect to data points taken at high
compression.
It is also observed that there is some debate as to whether a Murnaghan fit is appropriate in these structures[59]. Some of the calculated points are at unphysical compressions of up to 30%. Since compression is taken up by a combination of internal distortion and bond compression, the approximation that dB/dP is a constant throughout this huge pressure range is extremely doubtful.
The calculated valence charge densities for germanium in the BC8 structure is shown in figure 4.7 and ST12 structure is shown in Figure 4.8. The figures for the BC8 structure show the valence charge density in a plane containing both A and B bonds. Figure 4.9 shows a schematic representation of the atomic positions and bonding configurations for both structures.
Figure 4.7: Valence charge density of germanium in the BC8 structure showing
both A and B bonds. The charge density for BC8 Si on the same plane
is similar.
Figure 4.8: Valence charge density of germanium in the ST12 structure. The
charge density for ST12 Si on the same plane is similar.
Figure 4.9: Schematic diagram illustrating the atomic positions and bonding
configurations for planes through the BC8 and ST12 structures.