This chapter continues the study of high density metastable phases found in the tetravalent group IV semiconductors. Presented here is a treatment based on empirical potentials which allows intuitive insights into the stability and pressure response of these materials, along with estimates of free energy to describe finite temperature behaviour.

*Ab initio* calculations were presented in Chapter 3 which are
based on density functional theory in the local density approximation.
These calculations reproduced accurately the properties known
experimentally, and furthermore gave some predictions for the exact
behaviour of the internal structure of BC8 and ST12 as a function of
pressure. Despite the success that density functional total energy
calculations have had in describing the relative phase stability in
crystals there remain limitations which render the treatment of several
important physical properties beyond that which can be achieved by
these methods. This is particularly true in the case of free energy
calculation at finite temperature and relaxation of unit cell
dimensions under hydrostatic pressure. The explicit treatment of finite
temperature effects by first principles methods is extremely difficult
to incorporate in practice, although a rigorous extension of density
functional theory to finite temperature does exist[52]. This
is because of the long simulations required to obtain good
thermodynamic averages and the difficulties associated with changing
the box volume with thermal expansion and fluctuations. Current
computational power limited the non-zero temperature calculations
presented in Chapter 3 to single unit cells and did not allow for
changes in the size of the unit cell as described below.

Regarding structural relaxation, the plane wave pseudopotential total energy method has been shown to substantially alleviate complications which arise in atomic force calculations. These difficulties include basis set corrections (Pulay forces) which must be considered when localised basis sets are used and corrections which arise from non-self-consistency in the solution of the Kohn-Sham equations which are particularly problematic in full potential treatments. The advantages of the pseudopotential plane wave methods, however, do not apply in the case of unit cell relaxation which remains a formidable task to implement for first principles calculations. Schemes which have been proposed to accomplish this show extremely unfavourable scaling with system size.

These two complications associated with density functional methods
coupled with the urge to explore the P-T diagram of dense metastable
phases of Group-IV semiconductors have provided the motivation for the
present work. Chapter 3 reported *ab initio* pseudopotential
density functional calculations which accurately reproduced known
structural properties and allowed for predictions as to the internal
relaxation of the BC8 and ST12 structures of Group-IV elements under
pressure. To make the thermodynamic problem more tractable, the *ab
initio* force calculation is now replaced with an empirical potential
which agrees well with the structural results.

With a view towards a full theoretical investigation of the P-T diagram
of these phases, including hydrostatic effects, the structural
analysis in Chapter 3 was first repeated using a
previously published empirical model for covalent
bonding[74, 75]. This potential was parameterised to model
quite different situations such as the diamond structure and small
clusters. It has, however, been shown to be successful in other regimes
very different from that for which it was parameterised. It has
previously been shown to provide an accurate description of defects,
vibrational properties, rebonding effects in surface reconstructions
and in the formation of clusters. It is now also parameterised to give
a good description of the optic phonon frequencies similar to that of
silicon. It has not been applied previously in the present context of
dense phases of covalent semiconductors, so comparisons with the *ab
initio* structural results are essential.

The compute-intensive nature of density functional calculations means
that finite temperature studies at many points in the phase diagram
are currently impractical, especially in view of the pressure and
temperature dependence of the *c*/*a* ratio although finite
temperature calculations at the relaxed unit cell size are now able to
be done and are performed in Chapter 3. To investigate the stability
of the phases it is, however, necessary to evaluate free energies
throughout the pressure-temperature phase space. To acheive this, in
the next section we outline the method of lattice dynamics, which is
included here for completeness but essentially taken from the famous
"Dynamical Theory of Crystal Lattices" by Born and Huang. From that,
the empirical potential enables calculations to be carried out for a
model covalent material, whose properties are fitted as closely as
possible to those of silicon in the diamond structure.

Thu Oct 31 19:32:00 GMT 1996