This work was carried out using the CASTEP code using first principles molecular dynamics as discussed in Chapter 2 and in reference . The electronic wavefunctions are expanded in a plane wave basis set with periodic boundary conditions, the basis set being expanded to a cut-off of 250 eV for silicon and germanium and 408 eV for carbon. This higher value is required to ensure energy difference convergence for carbon because of the greater depth of the pseudopotential. This in turn arises because of the 2p character of the valence orbitals are not kept away from the nucleus by orthogonalisation to other p orbitals as in silicon or germanium.
The electronic degrees of freedom are relaxed using the conjugate gradients routine and orthogonalisation of the wavefunctions is restored at the end of each iteration. This electronic relaxation is continued until the Hellmann-Feynman forces have converged to better than three significant figures, and then the ions are moved also using a conjugate gradients routine. The ions are allowed to relax until the forces are below 0.001eV/Å. The unit cell for each calculation is not allowed to relax during the run due to the difficulty in dealing with enlarging or reducing the basis set. As the plane wave basis set is not localised there are no Pulay forces arising from changes in the basis set while the ions are relaxing. This was discussed briefly in Chapter 1.
For silicon and carbon in the diamond, ST12 and BC8 structures 4 special k-points at which the band structure is sampled were used. The same special k-point sets were found to be insufficient to converge the very small ST12-diamond energy difference in germanium. Several sets of k-points were tested and a set of 10 was found to be sufficient. This will be discussed further in Section 4.7. It is also possible that the frozen core approximation for the germanium 3d electrons may not be entirely justified, but a full potential treatment would be required in order to explore this. The combination of cut-off and k-point sampling gives total energies which are converged to approximately 0.1eV, however the physically significant quantities are total energy differences between total energy-volume curves which are converged to better that 0.01eV, of which a significant amount can be attributed to curve fitting schemes (Murnaghan vs. polynomial). In germanium the better k-point sampling gives still better convergence.