In order to carry out a theoretical study of complex materials, it is important to model the interactions between the ions in these structures as accurately as possible, but still keeping the calculations computationally feasible. The two methods used to model these interactions are described below.
With the onset of more powerful computers, the molecular dynamics technique has become an effective tool in the physics of condensed matter systems. In the molecular dynamics method, the forces acting on particles in a cell are found and the classical Newtonian equations of motion are solved numerically. The largest part of a molecular dynamics simulation is the evaluation of the forces which are required to be known in order to find the relaxed ionic positions. In general each particle can interact with all the other particles in the simulation, although one method of increasing the computational speed is to limit the range of the potential.
There are several methods by which the forces on particles can be evaluated. One of the simplest and most computationally efficient is by the use of an empirical potential, where the nature of the interactions between particles is fitted to various properties which are found experimentally such as lattice parameter and bulk modulus. In general, an empirical potential is constructed by summing contributions of pairwise interactions, 3-body interactions and so on. Such a potential can be expressed generally as
where is the position of particle i relative to a given origin, is the separation between particles i and j, and are constants which are fitted to data obtained, for example, by experiment, and f and g are chosen functional forms which best represent the interactions. It is then a simple matter to obtain the force on particle i by
Using such a potential can give great deal of insight into microscopic structure of solids and liquids.
Although empirical potentials are of great use, the are limited by the fact that they can only describe a system to the accuracy of the parameterisation of the potential. In general they are good at describing the interactions of the system to which their parameters were fitted but their transferability to other environments can be quite poor. There are many different interactions which are necessary to fit the potential to in order to describe even a small range of physical properties. For example, the Tersoff silicon potential accurately describes properties such as lattice parameter, elastic constants and phonon dispersion curves but requires the complexity of 14 different fitted parameters. But such a complex empirical potential still gets such basic quantities as the melting temperatures wrong because it was not designed to examine such properties. Thus transferability is limited. It may then be asked, why use empirical potentials which only reproduce the experimental results to which they fitted? It is because they will describe details of structures which may not be amenable to experimentation. For example, in Chapter 4, the phonon density of states of complex tetrahedrally bonded structures are found which is a non-trivial experiment to perform. This potential is fitted to properties of covalently bonded diamond-like materials. From the calculations presented in Chapter 3, it is found that the nature of this diamond-like bonding is similar to that of the complex structures found in diamond and germanium. Therefore, although the potential is not fitted to these complex materials, the environment in which the ions are found is very similar to the diamond structures. That is, they are 4-fold coordinated and covalently bonded.
In order to perform molecular dynamics simulations using an empirical potential, one must know beforehand the type of structure (that is, the nature of the bonding in the material) in order to use the correct model in which to construct the potential. In many cases this is not possible. To model interactions in which no a priori bonding information is known, an approach at a more fundamental level is required. One must turn to the formidable task of solving the Schrödinger equation for the electrons (in fact it is the Kohn-Sham equations that are solved where the many electron interactions are approximated by a local potential). It is this method that is described fully in Chapter 2 and then used in later chapters where the bonding topologies of the structures under consideration are unknown. On analysis it is found that some structures are well described by a covalent bond charge model which lends itself to a simple empirical parameterisation. Thus, calculations can be performed which would be computationally prohibitive in a first principles quantum mechanical method.
There are obviously great advantages in using a molecular dynamics method where the only specification of the atomic numbers of the ions present (so called ab initio methods) are required. The drawback is the extremely large compute intensive nature of the calculations. Before an evaluation of the forces on the ions can be performed, a massive minimisation calculation is required in the extremely large phase space of the basis set of the electronic wave functions - the size of this calculation can easily overwhelm any but the smallest simulations. For example, the calculations which are presented in Chapters 6 and 7 are approaching the reasonable limit for the size of ab initio calculations with only 64 atoms, whereas an empirical calculation of a similar scale would be using particles. The empirical potentials allow this because forces are calculated by the evaluation of a simple function.
Although the empirical potential, modelling a system of complicated `springs', is an invaluable tool in determining the microstructure of bulk systems as demonstrated in the calculations in Chapters 4 and 6, which are unfeasibly large for ab initio calculations, they give very little information about the electronic structure of a system. (In fact, they are usually based on some supposition about the general form of the electronic structure). In ab initio calculations the electronic charge density is evaluated directly, and is in fact the fundamental quantity used in the calculations. An example of such a charge density is illustrated in Figure 2.1
Figure 2.1: The figure shows several surfaces of constant charge density of the valence electrons of silicon in a complex metastable configuration known as ST12. It can be seen that the high regions of charge density lie between pairs of atoms showing the covalently bonded nature of the material. Figures such as these lend support that a covalent bond charge model will be a good description for the ST12 structure in silicon.
The covalent bonds are clearly shown. Examination of the charge densities from ab initio calculations on the structures described in later chapters show where an empirical covalent bond charge model will be valid.