Observations on the Mechanics of Strained Epitaxial Island Growth
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ABSTRACT An epitaxial material island which has a lattice parameter differing by a small amount for that of its substrate is considered within the framework of continuum mechanics. The strain distribution in the island is determined for a range of aspect ratio, taking into account the compliance of the substrate. It is demonstrated that the total free energy of a strained island is minimum for some value of aspect ratio, and that this value depends on the volume of the island. To consider strain relaxation, the nucleation of a dislocation at the edge of a strained island is examined and the equilibrium aspect ratio of a dislocated island is computed. In particular, it is shown that an island can reduce its free energy by reducing its aspect ratio and, simultaneously, forming an interface misfit dislocation. The simulations are based on the numerical finite element method. INTRODUCTION The physical system discussed here is an isolated crystalline island growing epitaxially on a relatively large substrate crystal with similar elastic properties but slightly different lattice parameter. The mismatch strain is defined in terms of the lattice parameter d, for the island and ds for the substrate as c, = (ds - di)/di. The system is modeled within the framework of continuum mechanics under the assumption that the configuration is two dimensional. The materials are taken to be isotropic elastic solids, with the island and substrate materials each having elastic shear modulus t and Poisson ratio v. The common biaxial modulus is M
=
2p(l + v)/(1 - v). The shape of the cross sectional profile of the
island surface is specified a priori to be a circular arc. However, the absolute size of the island and its height-to-width aspect ratio are allowed to vary arbitrarily. The influence of the material volume and aspect ratio on the mechanical characteristics of the island are of central importance. The very difficult question of how islandsbegin to form is beyond the scope of continuum mechanics, and is not addressed here. Due to its finite lateral extent and the nature of the constraint imposed by the substrate, the strain distribution in the island is necessarily nonuniform. Thus, as the first feature examined, this strain distribution is determined for various aspect ratios. This question has been addressed previously, but neglecting the compliance of the substrate, through lattice statics calcuations. 1 The total free energy of the island is assumed to consist of the elastic strain energy, the free surface energy, and the energy of the shared interface. For a given volume of material, both the surface energy and the strain energy vary significantly with the aspect ratio. A question of fundamental interest for the process of island growth to coalescence as a film, or for the process of forming island-like patches to achieve quantum sized dot structures of electronic significance 2 , is whether or not the structure becomes unstable against formation of dislocations. The issues of nucleation of a dislocation at the edge of an is
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