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[28] 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.

Thu Oct 31 19:32:00 GMT 1996