The energy vs. volume graph for carbon is shown in Figure 4.10. It is immediately clear that the atomic volumes of all phases are rather similar, and therefore that BC8 is unlikely to be even a metastable phase. The energy difference is also very much greater than for silicon. These results for BC8 are in agreement with previous calculations, where the rather surprising conclusion that BC8 may exist was drawn (from extrapolation of the murnaghan curves to 12Mbar. In Figure 4.11 the value of the BC8 internal parameter x is shown. It is clearly very different from Si and Ge, and consideration of the bondlengths shows that the short A bond is much shorter than the B bonds.
Figure 4.10: Graph of energy against volume for fully relaxed diamond, BC8 and ST12 structures in carbon.
Figure 4.11: Graph showing the variation of the BC8 internal parameter, x, as a function of normalised atomic volume in carbon, silicon and germanium.
All these results suggest that carbon is behaving very differently to the other materials, and the difference in bonding can be clearly seen in Figures 4.12, 4.13 and 4.14. Far from being fourfold coordinated, the valence electrons are concentrated in one bond only, with the three B bonds much reduced.
Figure 4.12: Valence charge density of carbon in the BC8 structure showing both A and B bonds.
Figure 4.13: Valence charge density of carbon in the ST12 structure.
Figure 4.14: Three dimensional representation of a valence charge density isosurface of carbon in the BC8 structure.
To examine this more closely, the amount of charge in each bond was estimated by integration of the valence charge density. There is no unique way of doing this: the method used here consists of defining the bonds by the line joining the two atoms, and then associating with that bond the region of space closer to the bond's line than to any other bond's line. This construction produces space-filling polyhedra and ensures that the total integrated density is equal to the number of electrons. It has the drawback that purely spherical charge densities give the same number of electrons per bond as purely covalent ones, but in the current case covalency is assumed, and studying whether the bonds are of similar order is all that is required. In Figure 4.15, the charge densities in slices of these polyhedra perpendicular to the bond direction along C-C bonds in the BC8 structure are compared to the corresponding bonds in silicon. The electron density in the Si bonds is localized between the atoms with a total charge of 2.0 electrons in each type of bond. This situation corresponds to completely covalent bonding in BC8 silicon. In carbon however, it is evident that the B bonds are different in nature from those in BC8 silicon. The charge density has a minimum between the two B-bonded atoms, suggesting a much weaker bond. Integration over the charge density gives a total charge of only 1.8 electrons in the carbon B bonds. For the BC8 carbon A bond, the integrated charge is approximately 2.6 electrons. This suggests that BC8 carbon A bonds are stronger than single bonded C-C. This is attributed to the formation of an unstable molecular crystal phase of carbon, with the fourfold coordinated BC8 phase being still less stable.
Figure 4.15: Valence charge density contour plot for BC8 Si (top left) and C (bottom right). The electron charge density along the A and B bonds are shown suggesting that significantly different bonding occurs in C in the BC8 structure. Integration of the charge densities along bond directions reveal that there are 2 electrons in both the A and B bonds in BC8 Si. For C there are approximately 2.6 electrons per A bond suggesting a stronger than single C-C bond exists. Only 1.8 electrons per B bond are found for C in the BC8 structure.
The apparent metastable BC8 state in carbon arises from constraining the simulation to have certain symmetries. The Hellmann-Feynman forces must have the same symmetries as the ions and the current method of minimisation does not break these symmetries. Consequently, although the BC8 phase is extremely unfavourable in carbon, the minimisation routine finds the best compromise structure subject to those assumed symmetries, which turns out to be the molecular crystal.
There have been a number of previous calculations of BC8 carbon, limited by computing resources to much smaller cutoff energies. The results of these are rather similar to Figure 4.10 but have been interpreted as generating a phase transition between diamond and BC8 at extremely high pressures (Biswas et al quote the massive pressure of 1200GPa). This crossover is not observed here, and moreover the evidence of the charge density plot leads one to suspect that BC8 may not even be metastable.
To test this latter hypothesis, a calculation was performed in which all the atoms were given a small random displacement from their equilibrium sites. From there, free relaxation under the Hellmann-Feynman forces restored the atoms to their BC8 symmetric positions. Thus the BC8 phase is at least metastable against any small atomic displacements.
For the ST12 structure of carbon, the equilibrium c/a ratio is 1.30 and the equilibrium atomic volume is slightly larger than that calculated for diamond. This suggests that, for carbon, unlike the other group IV elements considered, the ST12 structure does not constitute a dense phase.
The calculated difference between diamond and BC8 silicon was found to be 0.11eV per atom. This is in good agreement with the value reported by Biswas et al. We find the difference in energy of the diamond and BC8 structure in carbon to be 0.7 eV/atom which is also in good agreement with Biswas et al. In germanium the differences are very small, about 0.01eV/atom to ST12-Ge and 0.03eV/atom to BC8-Ge. This distinct trend down the group can be understood as an increasing ease of distortion of hybrids with principle quantum number. Our results for the differences in energy between diamond and the BC8 and ST12 structures are summarised in Table 4.2.
Table 4.2: Differences in energies, E, between the diamond and BC8 and ST12 structures in carbon, silicon and germanium. The units are in eV/atom. Note that the difference in energy between the diamond and ST12 structures changes by nearly two orders of magnitude for the group IV elements considered here.