Size dependency of strain in arbitrary shaped anisotropic embedded quantum dots due to nonlocal dispersive effects

Both quantitative and qualitative knowledge of strain and strain distributions in quantum dots are essential for the determination and tailoring of their optoelectronic properties. Typically strain is estimated using classical elasticity and then coupled to a suitable band structure calculation approach. However, classical elasticity is intrinsically size independent. This is in contradiction to the physical fact that at the size scale of a few nanometers, the elastic relaxation is size dependent and a departure from classical mechanics is expected. First, in the isotropic case, based on the physical mechanisms of nonlocal interactions, we herein derive (closed-form) scaling formulas for strain in embedded lattice-mismatched spherical quantum dots. In addition to a size dependency, we find marked differences in both spatial distribution of strain as well as in quantitative estimates especially in cases of extremely small quantum dots. Fully recognizing that typical quantum dots are neither of idealized spherical shape nor isotropic, we finally extend our results to cubic anisotropy and arbitrary shape. In particular, an exceptionally simple expression is derived for the dilation in an arbitrary shaped quantum dot. For the more general case (incorporating anisotropy), closed-form results are derived in the Fourier space while numerical results are provided to illustrate the various physical insights. Apart from qualitative and quantitative differences in strain states due to nonlocal effects, an aesthetic by-product for the technologically important polyhedral shaped quantum dots is that strain singularities at corners and vertices (which plague the classical elasticity formulation) are absent. Choosing GaAs as an example material, our results indicate that errors as large as hundreds of meV may be incurred upon neglect of nonlocal effects in sub-10-nm quantum dots.