Equilibrium Shapes of Small Strained Islands
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ABSTRACT We calculate the equilibrium morphology of a strained layer, for the case where it wets the substrate (Stranski-Krastonow growth). Assuming isotropic surface energy and equal elastic constants in the film and substrate, we are able to calculate two-dimensional equilibrium shapes as a function of the island size and spacing. We present asymptotic results for the equilibrium shape of a thin island where the island height is much smaller than the island width. We also present numerical results of the full equations to describe the island shape when the islands are widely separated. From these solutions we are able to determine the chemical potential of the island as a function of island volume and the strain energy density along the surface of the island for small to medium-sized islands. INTRODUCTION Continuum models have been reasonably successful in explaining many features of the stress-driven morphology changes in epitaxially strained films. A strained planar film is unstable to the formation of corrugations [1], and these corrugations can be suppressed by kinetic considerations [2]. From calculations of the nonlinear evolution of the instability, it has been demonstrated that, for the case of very thick films where the substrate does not influence the motion of the film surface, the instability results in the formation of cusp singularities in the solid [3, 4], which act as nucleation sites for dislocations [5].
In
thinner films the substrate can have a significant impact on the morphology of the film. In particular, the presence of a wetting layer on the substrate means that for thin enough films the morphology is prohibited from forming cusplike features. Instead, strain relief is achieved by the formation of islands lying atop the wetting layer (see [6]), the StranskiKrastonow growth mode. An important issue in modeling island morphologies is how one treats the physics of the wetting layer. Chiu and Gao [7], in two-dimensional, time-dependent calculations, have imposed a wetting layer condition in terms of a boundary layer over which the surface energy of the film changes from the film/vapor and film/substrate surface energies to the substrate/vapor surface energy. In their model, they need to specify a functional form for the transition region along with a boundary layer thickness parameter which determines the scale over which the transition occurs. In-our work, we propose a different wetting layer model which does not rely on a boundary layer thickness parameter. We shall use this model in the calculation of equilibrium island shapes. In addition to being a simple, zero-parameter model for the wetting layer, this model also has the particular feature that it results in well-defined island widths.
283 Mat. Res. Soc. Symp. Proc. Vol. 399 01996 Materials Research Society
S--......y. = h(x) film
x substrate Figure 1: Schematic of Epitaxially Strained Film. We consider a two-dimensional system in which the film and substrate are isotropic materials. THEORY Continuum Model We consider a two-dimens
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