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Analytical Study of Elastic Relaxation and Plastic Deformation in Nanostructures on Lattice Mismatched Substrates

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journal contribution
posted on 07.12.2011, 00:00 by Xu Zhang, Vladimir G. Dubrovskii, Nickolay V. Sibirev, Xiaomin Ren
In view of a continuously growing interest in monolithic integration of dissimilar semiconductor materials in a nanostructure form, we present an analytical study of elastic strain energy in nanostructures of different isotropic geometries grown on lattice mismatched substrates. An analytical solution for the elastic stress field is derived, which is not restricted by the small aspect ratio or particular geometry. It is shown that, at a large enough aspect ratio, the relaxation of displacement fields with the vertical coordinate is exponential. This allows us to find an analytical expression for the strain energy density. The cases of rigid and elastic substrates are considered simultaneously. By minimizing the total energy, we obtain the elastic energy density as a function of aspect ratio for given nanostructure geometry. Our data indicate that the elastic energy is highly dependent on the island shape, with the relaxation becoming faster as the contact angle increases. We also present a simple expression for the elastic relaxation where the fitting coefficient depends on the geometry. A dislocation model is then considered to analyze the competition between the elastic and dislocation energies. We calculate the critical radius below which the plastic deformation is energetically suppressed. Our results demonstrate a good quantitative correlation with the available experimental data on III–V semiconductor nanowires grown on silicon substrates.