## Abstract

We derive, in the form of coupled partial differential equations, the evolution equations for the epitaxial growth, via step flow, of a multispecies crystal on a stepped surface. Both adsorption–desorption on the terraces and attachment–detachment along the step edges are accompanied by chemical reactions and adatom diffusion. Moreover, we account for deposition from either a vacuum, e.g., in molecular beam epitaxy, or a gas, e.g., during vapour phase epitaxy (chemical or physical). Our theory (i) endows the steps with a *thermodynamic* structure whose main ingredients are a free-energy density and species edge chemical potentials, (ii) incorporates *anisotropy* into the terrace species diffusion as well as into the edge free energy, species mobilities, attachment–detachment and reaction-rate coefficients, (iii) allows for large departures from local equilibrium along the steps, and (iv) ensures the consistency of the constitutive relations for the terrace and edge chemical rates with the second law. In particular, a configurational force balance at each step yields a generalization of the classical Gibbs–Thomson relation. Finally, the special case of steady-state growth of a *binary compound* is discussed.

## Footnotes

↵Epitaxy designates growth during which the film inherits, at least in the early stages, the crystalline structure of the substrate. If the film and substrate are made of distinct materials, the mismatch between their lattice parameters generates a misfit stress in the film. Here the resulting strain is ignored as the emphasis is on the interplay between diffusion and chemistry. During step flow, adsorption–desorption occurs mainly on the terraces, followed by the diffusion of adatoms, i.e. adsorbed atoms or molecules, until the steps are reached where attachment–detachment and edge diffusion take place. Additionally, for multicomponent systems, chemical reactions are present, both on the terraces and along the steps, with the latter leading to the incorporation of particles into the bulk, by which the step edges evolve laterally.

↵Notable exceptions include Jabbour & Bhattacharya (1999), Pimpinelli & Videcoq (2000), Vladimirova

*et al*. (2000) and Pimpinelli*et al*. (2003).↵As pointed out by these authors, the possibility of step bunching in binary materials offers an alternative mechanism for the development of instabilities that does not recourse to the controversial assumption of an

*inverse*Ehrlich–Schwoebel barrier (cf. Vladimirova*et al*. 2000; Krug in press). See also Jabbour & Bhattacharya (1999), who show that, in the presence of*competition*between the vapour-phase precursors for open adsorption sites, even slight deviations from vapour stochiometry can lead to a drastic drop in the growth rate during chemical vapour deposition.↵Within continuum physics, configurational forces are associated with the evolution of grain boundaries, phase interfaces, film surfaces and disclinations in liquid crystals, etc. (cf. Gurtin 2000; Cermelli & Fried 2002; Fried & Gurtin 2003, 2004; Jabbour & Bhattacharya 2003). In the context of epitaxy on vicinal surfaces, these forces can be viewed as

*driving the lateral motion of steps*.↵The steady-state approximation is valid for growth situations in which the time-scale associated with the motion of steps is negligible in comparison with the time-scale for diffusion of adatoms on terraces.

↵Note that, in accordance with the above continuum description, the length-scale considered herein, although submicroscopic, is large enough that a distinction between kinks, corners and straight segments along a step is unwarranted. Instead, it is implicit that attachment of adatoms to the steps occurs preferentially at kinks.

↵The classical BCF model is based on the assumption of

*infinite*attachment–detachment kinetics, i.e.*K*^{±}→∞, so that the step acts as a perfect sink:The step velocity is then determined by (3.5) which now reduces towith*ϱ*^{b}assumed constant and the adatom flux given by a*linearized*Fick's law . Finally, note that*j*^{±}are now constitutively indeterminate, and are specified a posteriori by (3.4) once the adatom densities and step velocity have been resolved.↵External supplies can be assigned

*arbitrarily*and, although this may seem artificial, the availability of these supplies provides us with a useful conceptual tool to investigate the constitutive dependence of*j*^{±}on the variables of the theory.↵See, for example, Heidug & Lehner (1985), Truskinowsky (1987) and Abeyaratne & Knowles (1990).

↵The terrace and edge configurational forces

and**l**do not expend power on (**g***t*) as they act*internally*to the migrating control surface.↵This convexity condition is automatically satisfied for single- and multi-component ideal lattice gases (see §§4 and 5).

↵To illustrate the anisotropy of the step, assume that lies on the [001]-plane of a

*cubic*crystal. Hence, the step free-energy density has square symmetry, i.e. , and , where , and are smooth periodic functions with period 2*π*. Additionally, we require that , and have minima at the high-symmetry directions*ϑ*=0,*π*/2,*π*and 3*π*/4 (cf. Van der Eerden 1993). Analogous constitutive relations can be postulated for*α*^{±},*β*and*D*^{s}.↵We still allow for species diffusion along the step which, as before, is endowed with an anisotropic free-energy density.

↵The positive-definiteness of the matrix of matrices

**D**_{ik}signifies that for all . (An analogous definition holds for the matrix of scalar coefficients*α*_{ik}.) Note that (5.9)_{1}has the classical Fickean form for multicomponent systems, while (5.9)_{2}generalizes the standard expression for the deposition rate to the case of multispecies crystals. If the diffusion of adatoms is*isotropic*for all species, then the mobility matrices have the form**D**_{ik}=*D*_{ik}**1**. The explicit form of (5.9)_{3}will be discussed more explicitly below in the context of ideal lattice gas.↵In the context of MBE, and are assumed constant. This is consistent with the second law (5.8). Indeed, when adsorption of

*i*-particles occurs,*F*_{i}>0 and so that .- Received October 20, 2004.
- Accepted April 5, 2005.

- © 2005 The Royal Society

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