Lecture notes by John A. Venables. Lecture given 21 March 96.

Notes updated 12 Dec 96.

E. Surface Processes in Epitaxial Crystal Growth

Refs: Luth, Chap 3.5 and 3.6, pages 94-114; Zangwill, Chap 16, pages 421-432; R. Kern, G. LeLay and J.J Metois, Current Topics in Materials Sci 3 (1979) 139; J.A. Venables, G.D.T. Spiller and M. Hanbuchen, Reports in Progress on Physics 47 (1984) 399; J.A. Venables, Phys. Rev. B36 (1987) 4153; Surface Science 299/300 (1994) 798.

E1. Growth Modes and Nucleation Barriers

a) 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.

Bauer was the first to systematise these growth modes in terms of surface energies. If we deposit material A on B, we get layer growth if A < B + *, where * is the interface energy, and vice-versa for island growth. The S-K mode arises because the interface energy increases as the layer thickness increases; typically this layer is strained to (more or less) fit the substrate. Pseudomorphic growth is the term used when it fits exactly.

For each of these growth modes, there is a corresponding adsorption isotherm (diagram E2), as discussed in section C. 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.

b) 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 section 1.3, and revisiting problems 2 and 3, since we will be looking into these atomic processes in more detail in the next section. 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.

Draw the case where A > B, so that we have 3D islands. Then we can construct a free energy diagram G(j) for islands containing j atoms, which has the form

G(j) = -j + j2/3X, at supersaturation ,

where X is a surface energy term of the form X = kCkk + CAB(* -B), where the first term represents the surface energy of the various faces of the island A, and the second term represents the interfacial energy between A and B; the geometrical constants Ck, CAB depend on the shape of the islands. The form of such curves for different values of  and X (in arbitary units, but think kT) are shown in diagram E4. The nucleation barrier results because the there is a maximum in these curves, where the slope is zero. Differentiating, we can see that this maximum occurs at j = i, and that

i = (2X/3)3; G(i) = 4X3/(272).

The same argument can be followed through for 2D clusters, i.e. monolayer thick islands. In this case, the relevant supersaturation is expressed in relation to the corresponding step in the adsorption isotherm, i.e. ‘ = kT ln (p/p1) for the nucleation of the first condensed monolayer. The form is now G(j) = -j‘ + j1/2X, where the square root expression results from the extra edge energy X. Finding the maximum in the same way leads to

i = (X/2‘)2; G(i) = X2/(4‘), where ‘=  - c

and c = (A + * - B)2/3, with  as the atomic volume of the deposit. Thus, in this formulation, a measurement of the pressure of the steps in the adsorption isotherm directly determines the surface energy difference (A + * - B).

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.