Figure 5.1 shows the relative stability of the diamond, BC8 and ST12 phases for the potential with the length scale parameters fitted to silicon. From these curves the lattice parameter and cohesive energy for these phases can be deduced and are shown in Table 5.1. The graphs representing ST12 are under hydrostatic pressure, obtained by minimising enthalpy with respect to all internal parameters and the c/a ratio. Including these degrees of freedom makes a significant difference to the curve, softening the bulk modulus considerably.
Figure 5.1: Graphs of energy against volume for the diamond, BC8 and ST12 phases. The range shown for each curve represents the structure under pressures from -3 GPa to 10 GPa.
Table 5.1: Lattice parameters in Å and cohesive energies in eV given by the empirical potential.
The potential can sometimes give rise to ambiguities in choosing the neighbours, especially if considerable relaxation is required after the choice has been made. BC8 has a single internal parameter to relax. It has four nearest neighbours and so there is no difficulty in determining the bonding arrangement . Although the fifth neighbour is relatively close the charge density plots in the previous chapter confirm the assumption implicit in the form of the empirical potential that there is no bond to it. If the fifth neighbour is included as one of the four bonds at the expense of the B bond, it gives a higher energy structure. Including it at the expense of the A bond leads to massive increase in x, achieved by simulated annealing, back to a BC8 structure: the broken A bond now becomes the new fifth neighbour (see added note in Chapter 3).
ST12 has four internal degrees of freedom and, because of its tetragonal symmetry, two independent lattice parameters. In germanium ST12 the bondlengths are on average about 1% longer than in the diamond structure, which leads to a density about 10% greater. All the atoms are fourfold coordinated, and there are two distinct atomic environments a and b. The eight type b atoms per unit cell form spiral chains with a unique helicity, bridged by the four type a atoms. A combination of seven and fivefold rings means that the bond angles are much more diverse than in BC8, but also enables the bondlengths to be much closer to one another than in BC8.
In the ST12 structure the enthalpy is minimised with respect all the internal degrees of freedom and also the lattice parameters. Once again, it is a fourfold coordinated phase and so there is no difficulty in selecting the correct bonding arrangement. As with BC8, the charge density plots in Chapter 3 give a clear indication that four covalent bonds is the correct description of the bonding.
The diamond structure can only respond to pressure by contraction of its bonds, but pressure increase in BC8 and ST12 can be taken up by increased internal distortion and change in bondlengths. The ST12 structure can also relax by changing the c/a ratio. In practice all of these happen simultaneously.
This simultaneous relaxation is non-trivial to implement by ab initio methods. Due to problems with changing basis sets, many ab initio calculations are unable to calculate forces and then relax the relative atomic positions. They are therefore restricted to rescaling all bondlengths. With a plane wave basis set this problem is somewhat alleviated, and for a given unit cell the atomic position can be determined. This still does not fully allow for relaxation of c/a ratio with pressure. Implementation of the unit cell relaxation is now a relatively straight forward procedure. By minimizing the enthalpy using a conjugate gradients procedure with the Parrinello-Rahman Lagrangian it is possible to allow both internal and unit cell parameters to relax simultaneously, as they would in a real material.
Figure 5.2 shows the importance of including all degrees of freedom in the case of ST12. The graph shows the change in length of the three distinct bonds with volume under full relaxation and under conditions of constant c/a. For reference, the variation of a typical bondlength when no relaxation is allowed is also shown (it simply remains proportional to the cube root of the atomic volume).
Figure 5.2: Comparison of empirical bond lengths in ST12 with full relaxation (full lines), fixed c/a ratio (dashed lines) and no relaxation (dotted lines).
It is clear that the change in c/a with pressure is such as to reduce the necessary change in bondlengths. In both ST12 and BC8 the material is able to reduce its volume without commensurate reduction in bondlengths by adjusting its internal degrees of freedom. This has a very significant effect on the bulk modulus, which is lower in the denser phases than in diamond, in spite of a much greater number of bonds per unit volume. It can be seen from the energy-volume curves how the effect of increasingly large numbers of free structural parameters rapidly decrease the bulk modulus. The curvature of each graph quickly decreases as the number of free parameters change from in the diamond BC8 ST12 structures. Note that in polycrystalline ST12, large changes in the c/a ratio may be inhibited.
Figure 5.3 shows the variation of the BC8 internal parameter, x, with change in volume as determined by empirical methods. In figure 5.4 it is shown that the variation in neighbour distances with decreasing volume which is found to be in reasonable agreement with the ab initio results. Note that LDA invariably underestimates the lattice parameter of solids by about 2%, while the empirical model has its length scale set to reproduce the diamond structure exactly. Allowing for this, comparison with the ab initio simulation shows that the pressure evolution of the structure is well described by the model.
Figure 5.3: Variation in the BC8 internal parameter x with pressure in the empirical model.
The empirical model predicts that x is slightly larger found from the ab initio calculations, but the trend of increasing x with pressure is still found.
Figure 5.4: Nearest neighbour distances in the BC8 structure using the empirical model. These are in reasonable agreement with the ab initio calculations performed in Chapter 3.
The cross-over in BC8 bond-lengths occurs with both empirical and ab initio methods, although at different pressures. The empirical model gives an intuitive explanation for this. The compression of the type A bond is restricted by repulsion arising from the orthonormality requirement with the type B bonds. Since the angle is smaller than , the AB overlap is greater than the BB overlap, and so this term dominates the differential short-ranged repulsion. There are three times as many AB angles to the A bond as to the B bond, so we expect its compression to be three times more difficult. Indeed, this result is found.