Symmetry breaking in shape transitions of epitaxial quantum dots

Abstract

During heteroepitaxial growth, coherently strained islands form. These "self-assembled quantum dots" then undergo a series of shape transitions with increasing size. The best-known examples are the transitions of Ge on Si(001) and InAs on GaAs(001) from pyramidal islands to multi-facetted domes. Here we examine the transition pathway, using a simple two-dimensional model. We find that the transition occurs via sequential nucleation of individual facets. While the stable states are symmetrical, the transition states are highly asymmetrical. The calculated transition path can pass through a metastable half-dome island shape, consistent with experimental observations. The broken symmetry of the transition state can be "locked in" by intermixing with substrate material, leading to asymmetrical islands.1 Nanoscale islands form spontaneously during the growth of strained heteroepitaxial semiconductor films. This process has attracted intense study as a route to self-assembly of quantum dots. 1 These islands are typically symmetrical in shape, but asymmetrical islands are also observed in experiment 2-5 and in simulations. 6,7 The presence of asymmetry has important implications for both the growth process and the island properties. Asymmetry in quantum dots is particularly significant for optical applications, as it can split degeneracies of quantum dot states and provide a source of optical anisotropy.The complexity of the growth process has made it difficult to identify precisely how asymmetry arises. Both experiments 2 and calculations 8 showed that when the substrate has a modest miscut, islands can have qualitatively asymmetric shapes even in equilibrium.However, asymmetric shapes are also seen on nominally singular surfaces, 3-5 i.e. with the surface oriented in a high-symmetry direction.We therefore study the energetics of heteroepitaxial islands on singular substrates, using a fully faceted two-dimensional (2D) model. 8,9 We focus on the transition from pyramids to domes, the two most-studied shapes for Ge and InAs islands. 1 For any given volume, we calculate the energies of all possible shapes. In this way we determine the entire energy surface, and the transition path and barrier heights along this surface.As expected, with increasing island size there is a first-order transition from symmetric pyramid shapes to symmetric domes. However, we find that shape transitions occur via highly asymmetric transition states, typically involving nucleation of a steeper facet on only one side of the island. The activation barrier for this transition is much smaller than for a symmetrical transition pathway.In addition, we find that the reaction pathway can pass through a metastable asymmetrical "half-dome" shape. If sufficiently long-lived, such a state could appear stable in experiment. Indeed, Ross et al. 3 using in situ microscopy clearly observed the shape transition to occur via such asymmetrical shapes by sequential nucleation of facets. Evidence of metastable half-dome states has also been observed in sim...

Asymmetric shape transitions of epitaxial quantum dots

“…The symmetry-breaking in the pyramid-dome transition is also demonstrated by Spencer & Tersoff [32]. They used the small-slope approximation up to second-order for the elastic energy and obtained a similar topography for the energy surface as shown in figure 2.…”

“…Since our 2D islands only have facets in several specific orientations, for simplicity (as in [32]), we assume that these facets have the same surface energy density, i.e. γ (θ i ) = γ for i = 1, 2, .…”

“…Recently, in [32], a 2D continuum model with a second-order elastic energy approximation has been used to determine the energy of a more general class of faceted island shapes which include both symmetric and asymmetric island shapes. These results show that the transition from a pyramid to a dome (which involves the creation of a new steep facet on each side of the island) generally occurs through a metastable asymmetric half-dome state which has a steep facet on one side only, and there is a critical facet nucleus and associated energy barrier for passing into the metastable state that can be found as a saddle point in a parametrization of the energy surface in terms of the island shape.…”

“…In this article, we use a 2D model similar to [32] but use a third-order approximation to the elastic energy to study the faceted island shape transition. In particular, the higher order elastic energy approximation permits us to determine not only the characteristics of the transition from pyramids to domes, but also the transformation from domes to multifaceted domes.…”

Self-assembly of InAs quantum dots on GaAs(001) by molecular beam epitaxy

“…As its size increases, the 3D precursor somehow becomes a small faceted 3D island with a regular geometric shape, such as a pyramid bound by the {105} facets in a Ge/Si(001) system or by the {137} face facets in InAs/GaAs(001). Once the faceted sidewalls are fully formed around the 3D island, classical 2D layerby-layer nucleation and growth occur on these sidewall facets [214][215][216][217][218][219][220][221][222]. As the 3D island increases in size by facet growth, at some critical point, steeper facets successively appear on its sidewalls.…”