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. 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.