A step-flow model for the heteroepitaxial growth of strained, substitutional, binary alloy films with phase segragation: I. Theory

We develop a step-flow model for the heteroepitaxy of a generic, strained, substitutional, binary alloy. The underlying theory is based on the fundamental principles of modern continuum thermodynamics. In order to resolve the inherent disparity in the spatial scales-continuous in the lateral directions vs. atomistically discrete along the epitaxial axis-we represent the film as a layered structure, with the layer height equal to the lattice parameter along the growth direction, thus extending the classical BCF framework [W. K. Burton, N. Cabrera, and F. C. Frank, Philos. Trans. Roy. Soc. London Ser. A, 243 (1951), pp. 299-358] to growth situations in which the bulk behavior impacts the surface evolution. Our discrete-continuum model takes the form of a free-boundary problem for the evolution of monoatomic steps on a vicinal surface, in which interfacial effects on the terraces and along the step edges couple to their bulk counterparts (i.e., within both film and, indirectly, substrate). In particular, the proposed constitutive theory is such that the film layers are endowed with (generalized) Ginzburg-Landau free energies that account for phase segregation and, concomitantly, competition between gradient-driven coarsening and elastic refining of the separated domains. Importantly, the bulk and terrace effects are intertwined with the step dynamics via novel boundary conditions at the step edges derived from separate balance laws for configurational and microforces. Specifically, the former forces are associated with the evolution of defects (in the present setting, the steps), whereas the latter forces accompany micro- and nanoscopic changes in an order parameter (for a binary alloy subject to diffusion-mediated phase separation, the atomic density of one of its components or, equivalently, the relative atomic density), and the postulated balances should be viewed as generalizations to a dynamic, dissipative setting-such as epitaxial growth, a far-from-equilibrium process-of more standard variational calculations.