An analytical model for the energetics of quantum dots: beyond the small slope assumption

Abstract

Analytical models for strained heteroepitaxial quantum dot systems have invariably assumed that the dots have a low-aspect ratio (small slopes) and that the elastic properties of the dot and the substrate are identical. In this paper, a three-dimensional analytical model for the energetics of an array of axisymmetric quantum dots is developed from physical principles. This is valid for high-aspect ratio dots (such as GeSi and InGaAs) and allows the dot and substrate to have different elastic properties. It is shown that these features are very important in determining the strain energy of both isolated dots and arrays of interacting dots. Both the elastic relaxation energy (per unit volume) of a single dot and the elastic interaction energy (per unit volume) between multiple dots are found to be greatest for tall, steep dots and for dots which are stiffer than the substrate. The equilibrium of two-facet dots is investigated and shape transition phase diagrams for small slope monoelastic theory, GeSi and InGaAs are compared. Different features of the bimodal dot size distributions in these systems are explained.

Asymmetric shape transitions of epitaxial quantum dots

“…Daruka and colleagues [20,21] used a 2D model analogous to that in [19] to study the energy of 2D faceted islands as a function of the shape, facet angles and facet energies to obtain a phase diagram for the island type as a function of the volume and surface energy parameters. A similar but more general analysis was completed by Gill & Cocks [25,26], who constructed a 3D model for the energetics of axisymmetric faceted islands, which is valid for high-aspect ratio islands and allows different elastic properties of the film and substrate. They determined similar equilibrium phase diagrams as in [20] but for 3D faceted axisymmetric cone islands, and interpret the results in terms of 3D faceted islands in [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].…”

“…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]. The results we detail here develop the third-order elastic approximation for faceted 2D island geometries.…”

Equilibrium Distributions and the Nanostructure Diagram for Epitaxial Quantum Dots

“…12 Recently, the small slope assumption has been relaxed in the analytic calculations of Gill and Cocks. 13 Daruka and Barabási 14 used the Shchukin expression to determine a map of the surface growth mode as a function of lattice mismatch and the amount of material deposited at zero temperature, and we note that the results of several experimental studies of the surface growth mode have been summarized in maps as a function of temperature and the amount of material deposited. 15,16 The analysis of the island energetics was extended to a thermodynamic equilibrium model by MedeirosRibeiro et al 17 .…”

On modeling the mechanical behavior of heterostructures with quantum dots

“…For example, in the Ge/Si system, the stresses are about 6 GPa. High stresses occurring under heteroepitaxy significantly affect both kinetics [3,4] and thermodynamics [5] of the quantum dot formation and further evolution and the electron energy spectrum inside an individual quantum dot [6]. For this reason, there is a need in the development and realization of approaches to modeling stressed-strained states in quantum-dot heterostructures to work out techniques for developing quantum-dot arrays of a specified morphology and density and to predict their functional characteristics.…”