This book is one of two companion books on the science of crystallization and both are meant to be teaching vehicles rather than scientific treatises although some parts get close to the line. They are intended to develop a student's understanding of all the interwoven processes that are involved in either the natural geological formation of crystals or in the tailor-making of crystals in the laboratory for a specific use. Although the specific techniques used in the laboratory are many and extremely varied, the basic atomic processes involved are quite similar. Thus, it is felt that a basic description of the processes involved in any one specific technique allows ready extrapolation to the understanding of other techniques.

In this book, many of the experimental examples will relate to crystallization from the melt; however, examples and applications are also given to show how these same principles may be used to understand crystallization from all other nutrient media. In some cases, we will be more interested in the corollaries “Under what conditions does amorphous deposition occur?” and “What is the dissolution rate?”. We shall see that these same principles and ways of thinking allow us to consider the more general case of “phase transformations,” whether the path is to a crystalline or to an amorphous product (e.g., the formation of an SiO2 film on Si during thermal oxidation).

This chapter is dedicated to developing the understanding of interface energetics and molecular attachment kinetics and this can be carried out most effectively by considering a completely pure material (all ΔGc effects are eliminated). Both crystal growth by condensation from a pure vapor and crystal growth from a pure melt are suitable vehicles for pedagogical description. However, since the latter has more technological history than the former at the moment (relatively pure Si boules), it will be used as the major example. Since this application utilizes a seed crystal, we will not deal here with the initial stage of nucleation of a crystal. (Nucleation is dealt with in Chapter 8 (Background Science).)

Crystal growth of bulk Si is largely via the CZ technique illustrated in Fig. 2.1. The melt is generally contained in a pure SiO2 crucible and either induction heated via a graphite susceptor (as seen here) or resistance heated via a graphite heater. The crystal is rotated at some angular frequency to produce axial thermal symmetry and forced convection in the melt.

To initiate the crystal, the melt is stabilized at a temperature slightly above the melting point of Si while a rotating seed crystal, held in a chuck, is lowered into intimate physical contact with the melt and a portion of the seed melted back.

In this chapter we shall report on developments in liquid-metal theory in which, instead of dealing with ions coupled by effective interactions mediated by the conduction electrons, the liquid metal is treated as a two-component system, conduction electrons and positive ions (Cowan and Kirkwood, 1958; Watabe and Hasegawa, 1973; Chihara, 1973; March and Tosi, 1973). Earlier, evidence has been presented for well-defined collective modes of the positive ions in liquid Rb near its melting point. There is, of course, no doubt that, in any dense assembly of conduction electrons, there are well-defined plasma modes. Therefore, a workable model to keep in mind for such a two-component system begins with the assumption that there are two types of well-defined collective excitations; density fluctuations (analogous to phonons in a crystal) and plasmons (Tosi and March, 1973a; for details see Appendix 14.1).

Following the approach of March and Tosi (1973), the two-component theory will be developed in terms of three partial structure factors, Sii = S, the nuclear-nuclear structure factor as observed by neutron scattering, the electron-electron structure factor See, and the “cross” correlations represented by Sie(k).

Electron-ion Hamiltonian and density fluctuation operators

To implement the programme just outlined, let us set up a description of the liquid metal in terms of the local number densities ρe(r, t) and ρi(r, t) of electrons and ions, respectively, at position r and time t.