In writing a preface, an author is faced with the question: what is this book of mine? Of course, in the end only the reader will decide what it really is. The scope of this preface, as of all prefaces, is to say what it was intended to be.

This book tries to offer a reasonably complete description of the physical phenomena which make solid materials grow in a certain way, homogeneous or not, rough or smooth. These phenomena belong to chemistry, quantum physics, mechanics, statistical mechanics. However, chemistry, mechanics and quantum physics are essentially the same during growth as they are at equilibrium. The statistical aspects are quite different. For this reason, the authors have insisted on statistical mechanics.

Another reason to emphasize the statistical mechanical concepts is that they will probably survive. The concepts developed many years ago by Frank, or more recently by Kardar, Parisi and Zhang are still valid while, for instance, quantum mechanical calculations of the relevant energy parameters will certainly evolve a lot in the next few years. We have not considered it useful to devote too many pages to them, but we have tried to present the frame in which the data can be inserted, as soon as they are known.

However, although emphasis is on statistical mechanics, other aspects are not ignored, even though they may have been treated somewhat superficially. The reader will find more detailed information in an extensive bibliography, where all titles are given in extenso, thus making its use much easier.

This book is mainly devoted to growth, and therefore to non-equilibrium processes.

A system composed of particles, whose quantum mechanical zero point energy is large compared to their interaction energy, does not solidify even at absolute zero temperature and remains a so-called quantum liquid. Typical examples of such systems are conduction electrons in metals and liquid helium. Many conductors undergo a phase transition at their critical temperature Tc and become superconducting below it. Similarly, liquid 4He under its vapour pressure becomes a superfluid at Tc = 2.17 K (Tλ is often used instead of Tc), and liquid 3He at Tc = 0.9 mK. Although the phenomena are called superconductivity for a charged system like conduction electrons, and superfluidity for a neutral system like liquid helium, they are characterised by the same basic property that the wave nature of the particles manifests itself on a macroscopic scale, as we shall see below.

Phenomena

The most important factor in determining the properties of a quantum liquid is the statistics of the constituent particles. Liquid 3He becomes a typical Fermi liquid below 0.1 K and a superfluid below a few mK, in contrast with the bosonic liquid 4He mentioned above. The two superfluids are quite different both phenomenologically and microscopically. Table 1.1 shows systems which exhibit, or are predicted to exhibit (in brackets), superconductivity or superfluidity. In the case of fermion systems pair formation is responsible for the phenomena so that their properties differ according to the pairing type.

The Ginzburg–Landau (GL) theory has been employed for general analysis of the physical properties of an ordered phase. This theory is especially useful when the system under consideration has a spatial variation, due to an applied external field for example. It should also be noted that this theory serves as a basis for studying the effects of fluctuations, that are neglected in the mean-field approximation. The GL theory was proposed before the microscopic theory of superconductivity by BCS and has played an important rôle in the analysis of superconducting phenomena even after the emergence of BCS theory. In the present chapter, the time-independent GL theory and its applications are first discussed; its extensions and the effect of fluctuations are then considered.

GL theory of superconductivity

The free energy Fs of the system is not only a function of the temperature T and other ordinary thermodynamic variables but a function (in fact, a functional in general since we are dealing with a system with spatial variations) of the order parameter. The order parameter of a spin-singlet s-wave superconductor is the pair wave function Ψ(x) = 〈ψ↑(x)ψ↓(x)〉, which is equivalent to the gap parameter Δ(x) = gΨ(x) in the BCS model. Here the order parameter is written Ψ(x) including a multiplicative constant. Let us write the free energy as Fs = Fs({Ψ(x)}), where such variables as T and V are and will be dropped to simplify the notation.

Brief history

Cellular automata (often termed CA) are an idealization of a physical system in which space and time are discrete, and the physical quantities take only a finite set of values.

Although cellular automata have been reinvented several times (often under different names), the concept of a cellular automaton dates back from the late 1940s. During the following fifty years of existence, cellular automata have been developed and used in many different fields. A vast body of literature is related to these topics. Many conference proceedings), special journal issues and articles are available.

In this section, our purpose is not to present a detailed history of the developments of the cellular automata approach but, rather, to emphasize some of the important steps.

Self-reproducing systems

The reasons that have led to the elaboration of cellular automata are very ambitious and still very present. The pioneer is certainly John von Neumann who, at the end of the 1940s, was involved in the design of the first digital computers. Although von Neumann's name is definitely associated with the architecture of today's sequential computers, his concept of cellular automata constitutes also the first applicable model of massively parallel computation.

Von Neumann was thinking of imitating the behavior of a human brain in order to build a machine able to solve very complex problems. However, his motivation was more ambitious than just a performance increase of the computers of that time.