EPFL Lecture #4 (Venables)

Notes for EPFL Lecture 4 (Venables)

Lecture notes by John A. Venables. Lecture given 16 Oct 97. Latest version 2 April 09.

The references for this lecture are here.


4. Surface Processes in Epitaxial Growth

4.1 Introduction

  • Why are we studying epitaxial growth?
  • Epitaxial growth is a subject with considerable practical application, most obviously in relation to the production of semiconductor devices, but also to a whole range of other items. For example, magnetic devices such as recording heads have been produced using metallic multilayers, in which alternating layers of magnetic and non-magnetic materials to produces high sensitivity to magnetic fields; another magnetic example is bistable switches where the alignment of the magnetic moments can be parallel or antiparallel in the neighboring layers. Many of these films are produced via epitaxial growth processes, epitaxy meaning the growth of one layer in a particular orientation relationship to the underlying, or substrate layer.

    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.

  • Simple models- how far can we go?
  • In this situation, it seems a good idea to study a relatively simple approach in some depth. This enables one to say clearly what one does, and does not understand. Although it may help to offer advice on what is or is not a good recipe for producing better films or devices, this is certainly not straightforward. Personally, I find this dichotomy an interesting example of the relationship between science and technology. It means that one can use the understanding so gained as a background to appreciate the next technological advance, but that trying to advance the science and the technology are rather different endeavors.

    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.

  • Literature and reference sources
  • This subject has a long history, and a full literature citation is neither possible nor sensible. I would like to know which references you find helpful. The main reviews which I have either used or written include: Frankl and Venables (1970); Matthews (1975); Lewis and Anderson (1978); Kern et al.(1979); Venables et al. (1984); Venables (1994). Growth and properties of ultrathin epitaxial layers (King and Woodruff, 1997) is a research compilation with chapters by myself, by Bauer, by Brune and Kern, and by several other authors. I am certainly not intending to repeat all this material here, but will sketch the arguments leading to the simplest models.

    4.2 Growth Modes and Nucleation Barriers

  • Growth Modes and Adsorption Isotherms
  • The classification of three growth modes (diagram E1) dates from 1958, when Ernst Bauer wrote a much quoted (review) paper in Zeitschrift fur Kristallographie. The layer-by-layer, or Frank- van der Merwe, growth mode arises because the atoms of the deposit material are more strongly attracted to the substrate than they are to themselves. In the opposite case, where the deposit atoms are more strongly bound to each other than they are to the substrate, the island, or Volmer-Weber mode results. An intermediate case, the layer-plus-island, or Stranski-Krastanov growth mode is much more common than one might think. In this case, layers form first, but then for some reason or other the system gets tired of this, and switches to islands.

    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.

  • Nucleation Barriers in Classical and Atomistic Models
  • The same phenomena look a lot more complex when one considers what is going on at the atomic level (diagram E3), and in general only a few of these processes can be put into quantitative models at the same time. It may be useful to refresh your ideas about crystal growth in general now, by rereading Lecture 1, and attempting the problems, since we will be looking further into these atomic processes in more detail in this lecture and the next. In particular, the nucleation barrier concept can be explored in both classical (macroscopic surface energy) or in atomistic terms. The classical nucleation theory proceeds roughly as follows.

    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.

    4.3 Atomistic Models and Rate Equations

  • Rate Equations, Controlling Energies, and Simulations
  • We have considered simple rate equations for adatom concentrations in Lecture 1, and in problem 1.4.1, adding a diffusion gradient in problem 1.4.2. Now we need to add non-linear terms to describe clustering and nucleation of 2D or 3D islands. These equations are governed primarily by energies, which appear in exponentials, and also by frequency and entropic preexponential factors.

    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.

  • Elements of Rate Equation Models
  • But with the above provisos, here goes with the atomistic models. We consider rate equations for the various sized clusters and then try to simplify them. If only isolated adatoms are mobile on the surface,

    nucleus size. In its simplest form, this means that a) we can consider all clusters of size > i to be ‘stable’, in that another adatom usually arrives before the clusters (on average) decay; the reverse is true for clusters of below critical size; b) these subcritical clusters are in local equilibrium with the adatom population.

    The capture numbers are related to the size, stability and spatial distribution of islands, and solving this problem has caused a lot of words to be spilled; it isnít over yet. The simplest mean field model, which I and others worked on long ago, and which several people are working on now, considers a typical cluster of size k immersed in the average density of islands of all sizes. Then one can set up a diffusion equation for the adatom concentration in the vicinity of the k-cluster (size specific), or x-cluster (the average size cluster), which has a Bessel Function solution.

    Using these expressions one can compute the evolution of the nucleation density with time, or more readily with the coverage of the substrate by islands, Z, as the independent variable, as first proposed by Stowell et al. in the early 1970ís. It is also possible to compute other observable quantitities such as the condensation coefficient in the same manner (Venables et al. 1984, p 409-413).

    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 Chrzanís (1994) paper was to show, for the particular case of i = 1 in complete condensation, that uniform depletion solution, with tauc as the argument, is the correct expression in the absence of spatial correlations, and that it agreed with their KMC simulations.

  • Regimes of Condensation
  • Comments on individual treatments
  • My 1987 paper (JAV, Phys. Rev. B36 (1987) 4153-62) concentrated on including vibrations in a self-consistent way, within the mean field framework outlined above. It is very easy to construct a model which is inconsistent with the equilibrium vapor pressure of the deposit unless the vibrations are treated reasonably carefully. This paper builds on the Einstein model calculations which we have done here in lecture 1, and as problems, and is the basis for the model calculations which I have done since.

    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, nx; 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.