a b s t r a c tBy introducing a homogeneous piezoelectric material and its Green's function, we present a new semianalytical three-dimensional perturbation method for general inhomogeneity problems in anisotropic and piezoelectric solids. This method removes the limitations associated with previous analytical methods, which often ignore the anisotropic properties or the difference between the material properties of the inhomogeneity and its surrounding matrix. As an important application, the proposed theory is employed to calculate the elastic and electric fields in a truncated pyramidal InAs/GaAs quantum-dot (QD) nanostructure. Numerical results demonstrate that the anisotropy of the materials and the difference between the material constants of the QD and the matrix have a significant influence on the strain and electric fields. The relative differences of the strain and electric field inside the QD between the simplified isotropic and homogeneous model and the real anisotropic and heterogeneous one may reach 22% and 53%, respectively. The accuracy of the calculated elastic strain and electric fields is improved greatly by a second order approximate solution (OAS). Since the third OAS nearly coincides with the second one, good convergence of the iteration procedure is demonstrated. Moreover, contours of the hydrostatic strain and electric potential within and around the QD are also presented and analyzed.
The maximum and minimum principal strains of an InAs quantum wire (QWR) buried in a GaAs matrix are computed using the boundary element method (BEM), the inclusion method, and molecular statics, and the results from each method are compared with each other. The first two methods are based on continuum mechanics and linear elasticity, while the third is atomistic. The maximum principal strains are largely in agreement among the different methods, especially outside the QWR, though in the centre of the QWR, the discrepancy between the continuum and atomistic methods can be as large as 11.9%. The gradients of the strain tensor are in agreement among the methods. The inclusion method is faster than the BEM, and both continuum methods are an order of magnitude faster than molecular statics. Although the inclusion method, unlike the BEM, ignores the difference in material properties between the QWR and its surrounding matrix, its results are in better agreement with the molecular statics results than the results from the BEM. The rough quantitative and qualitative agreements indicate the utility of classical continuum methods for estimating strain profiles in nanoscale structures.
A model has been developed to simulate the growth of arrays consisting of a substrate on which alternating layers of quantum dots (QDs) and spacer layers are epitaxially grown. The substrate and spacer layers are modeled as an anisotropic elastic half-space, and the QDs are modeled as point inclusions buried within the half-space. In this model, the strain at the free surface of this half-space due to the buried point QDs is calculated, and a scalar measure of the strain at the surface is subsequently determined. New point QDs are placed on the surface where the previously calculated scalar strain measure is a minimum. Following available DFT results, this scalar strain measure is a weighted average of the in-plane strains. This model is constructed under the assumption that diffusional anisotropy can be neglected, and thus, the results are more in agreement with results from experiments of growth of SiGe QDs than experiments involving QDs of (In,Ga)As.
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