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Analytical study of elastic relaxation and plastic deformation in nanostructures on lattice mismatched substrates

机译：晶格失配基底上纳米结构的弹性弛豫和塑性变形的分析研究

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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.
机译：鉴于对异质半导体材料以纳米结构形式进行单片集成的兴趣不断增长，我们对在晶格不匹配的基板上生长的不同各向同性几何形状的纳米结构中的弹性应变能进行了分析研究。得出了弹性应力场的解析解，不受小纵横比或特定几何形状的限制。结果表明，在足够大的纵横比下，具有垂直坐标的位移场的弛豫是指数的。这使我们能够找到应变能密度的解析表达式。同时考虑刚性和弹性基板的情况。通过最小化总能量，对于给定的纳米结构几何形状，我们获得了弹性能量密度与纵横比的函数关系。我们的数据表明，弹性能高度依赖于岛的形状，随着接触角的增加，松弛变得更快。我们还给出了弹性松弛的简单表达式，其中拟合系数取决于几何形状。然后考虑位错模型来分析弹性能和位错能之间的竞争。我们计算出临界半径，在该临界半径以下可塑性地抑制塑性变形。我们的结果证明与在硅基板上生长的III-V半导体纳米线的可用实验数据具有良好的定量相关性。

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