The references for this lecture are here.

In most of these applications, the end-point interest is almost always
electrical, magnetic or optical, and there may also be an interest in the
mechanical properties. However, it is not enough to be interested just
in the end-point, since we need to know how to get there, and what
influences the final properties. It is here that the science behind the
essentially *atomic* processes in epitaxial growth can find a good
part of its (societal) justification. However, in delving deeply into
this topic for its own sake (as we are about to do), we should realise
that the technological ends may be better served in other ways. For example,
many multilayer films are produced by sputtering, and are polycrystalline,
albeit with a preferred orientation; another current example is that it
may be better to produce films by depositing clusters rather than single
atoms.

What I am attempting to do here, is to see how far one can go with simple
models involving adsorption and diffusion of atoms, and the new element,
*binding* between atoms on surfaces. Binding introduces cooperative
features into the description, which are non-linear in the adatom
concentration. This opens the way to a discussion of the kinetics of crystal
*nucleation* and growth, as constrasted with the thermodynamics of
adsorption. For both experiment and models, we can discuss these topics
in atomistic terms; indeed the behavior of single atoms actually influences
the end products in many cases.

Additionally, as I hope to show by selected examples in later lectures, the nucleation and growth patterns observed experimentally reflect directly the different types of bonding in solids. This then leads to a better understanding of what to expect in the growth of metals on metals, metals on ionic crystals or on semiconductors, and semiconductors on semiconductors. Such examples reflect back on our knowledge of interatomic forces, vibrations, etc in the context of processes which occur at surfaces.

For each of these growth modes, there is a corresponding adsorption
isotherm (diagram E2), as discussed in Lecture 2.
In the island growth mode, the adatom concentration on the surface is
small at the equilibrium vapor pressure of the deposit; no deposit would
occur at all unless one has a large *supersaturation*. In layer growth,
the equilibrium vapor pressure is approached from below, so that all the
processes occur at *undersaturation*. In the S-K mode, there are a
finite number of layers on the surface in equilibrium. The new element
here is the idea of a *nucleation barrier*, dashed on diagram E2. The
existence of such a barrier means that a finite supersaturation is required
to nucleate the deposit.

This way of looking at the problem is less than 100% realistic, perhaps not surprisingly. It is rather artificial to think about surface energies of monolayers and very small clusters in terms of macroscopic concepts like surface energy. Numerically, the critical nucleus size, i, can be quite small, sometimes even one atom; this is the justification for developing an atomistic model, as discussed in the next section. However, an atomistic model should be consistent with the macroscopic thermodynamic viewpoint in the large-i limit. To ensure this is not trivial; most models don’t even try; if I harp on about this, it is because I am attempting to do this in my research papers. In other words, there are (at least) two traditions in the literature; it would be nice to unify them.

This is the main advantage of such ‘mean field’ models. They are known not to describe fluctuations very well, so various quantities, such as size distributions of clusters, are not described accurately. In current research, using fast computational techniques such as ‘Kinetic Monte Carlo’ (KMC), the early stages can be simulated on moderate size lattices. These KMC ‘experiments’ using the same assumptions can then be used to check whether mean field treatments work for a particular quantity.

The emergence of computer simulation as a third way between experiment and theory is clearly a growth area of our time. To make progress in this area, one has to start with the simplest models, and stick with them until they are really understood. You need to beware generating more heat than light, and in particular of generating special cases which may or may not be of real interest. Simulations can however be very illuminating, and may suggest inputs for simple models that one hadn’t thought of. Animations are immediately appealing, and if Spielberg can do it, why shouldn’t we? The problem lies only in the subsequent claims for correspondence with reality; then a measure of self-discipline is needed, both from the lecturer/writer and the listener/reader.

There are also other approximations for the various sigma’s, such as the
lattice approximation. However, the appropriateness of any of these
depends on the spatial correlations between the islands which develop as
nucleation proceeds. The key point of Bales and Chrzan’s (1994) paper was
to show, for the particular case of i = 1 in complete condensation, that
uniform depletion solution, with tau_{c} as the argument, is the
correct expression in the absence of spatial correlations, and that it
agreed with their KMC simulations.

There have been many related treatments by several groups, mostly in response to the new UHV STM-based experimental results, some of which are described in the next sections. In particular, several groups have been studying these models using a combination of rate equations and KMC techniques. This work has been summarised in various places: in particular, volume 8 of the King-Woodruff series (1997) contains chapters by myself, and by Harald Brune /Klaus Kern which provide references.

Principal recent papers include a detailed comparison of rate equations
and KMC simulations for i = 1 in the complete condensation limit
(G.S. Bales and D.C. Chrzan, Phys. Rev B50 (1994) 6057); Bales has published
on other topics since, including the role of steps and nucleation on
defects. The KMC work is important for checking that the rate equation
treatment works well for average quantities, such as the nucleation density,
n_{x}; but it also shows that the treatment doesn't do a good job
on quantities such as size distributions, which are dependent on the local
environment, i.e. on the spatial correlations which develop during
nucleation and growth.

Several authors (M.C. Bartelt and J.W. Evans, J.G. Amar and F. Family,
P.A. Mulheran and J.A. Blackman, D.D. Vvedensky, A. Zangwill *et al*
in several papers) have characterised size distributions during deposition,
and shown that they are characteristic both of the critical cluster size, i,
and of the spatial correlations. This is a problem which has so far eluded
a rigorous analytical treatment, even though the qualitative features have
been appreciated for almost 30 years.

There are many other related topics (steps, defects, ripening, alloying
plus the current buzz topic *nanostructures*) but I think it will
be better to discuss them in the context of specific experimental examples,
which we will do in the next lecture. If there are any examples you would
like included, let me know by email before
Lecture 5.