The stability of elastically strained nanorings and the formation of quantum dot molecules

Abstract

Self-assembled nanorings have recently been identified in a number of heteroepitaxially strained material systems. Under some circumstances these rings have been observed to break up into ringshaped quantum dot molecules. A general non-linear model for the elastic strain energy of nonaxisymmetric epitaxially strained nanostructures beyond the small slope assumption is developed. This model is then used to investigate the stability of strained nanorings evolving via surface diffusion subject to perturbations around their circumference. An expression for the fastest growing mode is determined and related to experimental observations. The model predicts a region of stability for rings below a critical radius, and also a region for larger rings which have a proportionally small thickness. The predictions of the model are shown to be consistent with the available results. For the heteroepitaxial InP on In 0.5 Ga 0.5 P system investigated by Jevasuwan et al. (2013), the nanorings are found to be stable below a certain critical size. This is in good quantitative agreement with the model predictions. At larger sizes, the rings are unstable. The number of dots in the resulting quantum dot molecule is similar to the mode number for the fastest growing mode. Second order terms show that the number of dots is expected to reduce as the height of the ring increases in proportion to its thickness. The strained In 0.4 Ga 0.6 As on GaAs nanorings of Hanke et al (2007) are always stable and this is in accordance with the findings of the analysis. The Au nanorings of Ruffino et al. (2011) are stable as well, even as they expand during annealing. This observation is also shown to be consistent with the proposed model, which is expected to be useful in the design and tailoring of heteroepitaxial systems for the self-organisation of quantum dot molecules.

Asymmetric shape transitions of epitaxial quantum dots

“…Our calculations are based on a third-order approximation to the elastic energy. While we do not have a rigorous error bound on this approximation, Gill [29] uses an analogous approximation for a 3D conical island and determines the error in the elastic energy as a function of the surface slope. At slope values corresponding to the maximum slope on our pyramid, dome and multidome these errors are less than 1%, 2% and 13%, respectively, suggesting the third-order approximation is quantitatively accurate for pyramids and domes, and reasonably accurate for multifacet domes.…”

“…For example, the small-volume results in [18] use a small-slope approximation for the island shape to find a two-term asymptotic approximation for the elastic energy as described in [19]. The two-term approximation is also used in [20,21,23,24], while a third-order elastic energy approximation was developed for smooth surfaces in 2D in [31], for axisymmetric nanoring structures in [29], and for smooth surfaces in 3D in [27,28]. Other approaches involve numerical solution of the elasticity problem using a boundary integral method for the 2D geometry [18,22] or an analytic fitting model of the elastic relaxation function for the 3D axisymmetric geometry [25,26].…”

“…an island ridge geometry) with isotropic surface energy is given in [18], showing that small island shapes are governed by a linear integro-differential equation, with numerically determined island shapes at larger volume approaching the shape of a ball sitting atop the substrate. A related body of work in the literature [19][20][21][22][23][24][25][26][27][28][29] has focused on determining how the island shape is affected by other effects such as anisotropy of surface energy, non-ridge (three-dimensional, 3D) geometries, non-uniform film composition and film substrate wetting as discussed below. Rigorous existence and regularity results for the free boundary problem appear in [30].…”