Three-dimensional simulations of self-assembly of hut-shaped Si–Ge quantum dots

Abstract

Optical properties of multi-stacked InGaAs/GaNAs quantum dot solar cell fabricated on GaAs (311)B substrate J. Appl. Phys. 112, 064314 (2012) Optical polarization properties of a nanowire quantum dot probed along perpendicular orientations Appl. Phys. Lett. 101, 111112 (2012) Electroluminescence from quantum dots fabricated with nanosphere lithography Appl. Phys. Lett. 101, 103105 (2012) Nucleation features and energy levels of type-II InAsSbP quantum dots grown on InAs(100) substrate Appl. Phys. Lett. 101, 093103 (2012) Highly luminescing multi-shell semiconductor nanocrystals InP/ZnSe/ZnS Appl. Phys. Lett. 101, 073107 (2012) Additional information on J. Appl. Phys. This article presents the results of three-dimensional modeling of heteroepitaxial thin film growth with the objective of understanding recent experiments on the early stages of quantum dot formation in SiGe/Si systems. We use a continuum model, based on the underlying physics of crystallographic surface steps, to study the growth of quantum dots, their spatial ordering and coarsening behavior. Using appropriate parameters, obtained from atomistic calculations, the ͑100͒ orientation is found to be unstable under compressive strains. The surface energy now develops a minimum at an orientation that may be interpreted as the ͑105͒ facet observed in SiGe/Si systems. This form of the surface energy allows for the growth of quantum dots without any barrier to nucleation-dots are seen to start off via a surface instability as shallow stepped mounds, which steepen continuously to reach their low energy orientations. During the very initial stages of growth, mounds are seen to grow in a dense array with several of them impinging on each other and subsequently coalescing to form larger mounds. This behavior occurs due to the competition between surface energy which seeks to minimize the free-energy by the formation of islands with side-walls at the strain stabilized orientations and repulsive elastic interactions between such closely spaced islands. Using simple analytical calculations, we show the existence of a critical island size for this coalescence behavior. A key result of our analysis is the inverse scaling of this critical size with the misfit strain in the film. While energetic analyses may be used to obtain useful insights, the growth of quantum dots is essentially a nonequilibrium process and requires a fundamental understanding of the kinetics. Numerical studies show that the growth kinetics has a profound effect on surface morphology: arrays of well-separated islands or, alternatively, intersecting ridges are obtained in different kinetic regimes. We also study an alternative model of a stable but nonfacet ͑100͒ orientation and point out the inconsistencies of this assumption.

Continuum model for the long-range elastic interaction on stepped epitaxial surfaces in2+1dimensions

“…The step line energy and the elastic energy due to force dipole interaction are relatively well known in the continuum framework in the literature. 1,4, [31][32][33][34][35]37,[39][40][41][42]46 A stepped surface changes its morphology by motion, nucleation, and annihilation of steps, instead of changing continuously. Based on the Burton-Cabrera-Frank ͑BCF͒ theory, 17 the evolution equation is given by 30,[32][33][34][35]37,[39][40][41][42]46 …”

“…The self-misfit energy of a step ␥ is thus E misfit ͑␥͒ = 1 2 E misfit ͑␥ , ␥͒. In some continuum theories, the continuum expressions for strained epitaxial surface above roughening transition temperature were directly used for stepped surfaces, 31,33,46 in which the surface profile is a smooth function h͑x , y͒ and the traction on the surface is given by Eq. ͑3͒.…”

“…However, their continuum model for the long-range elastic effect on strained stepped surface was obtained as a simple average of the force monopole interaction between steps, which is the same as the traditional integral expression above roughening transition temperature. Ramasubramaniam and Shenoy 46 generalized the continuum model in 1 + 1 dimensions based on continuum variational principles proposed by Shenoy and Freund 33 to 2 + 1 dimensions. Even though the discrete features of the step line energy and force dipole interaction between steps were included in their models, for the long-range elastic interaction due to misfit, the traditional expression above roughening transition temperature was directly used.…”

Nano-scale equilibrium crystal shapes

“…With decreasing structure size, the issues of finite size and shape effects become non-negligible [4,5] including issues of stability against decay and structural rearrangement [6][7][8]. Small structures are also increasing sensitive to external perturbations, such as the stress caused by the substrate interface [9][10][11][12][13]. In this work we will address the issues of nanoscale equilibrium crystal shape for the surface free energy of the continuum step model [14,15], which allows experimentally determined thermodynamic parameters for step free energies to be connected rigorously to structural predictions.…”

Continuum modelling of semiconductor heteroepitaxy: an applied perspective

“…Once the instability is triggered, growth of faceted islands is observed [127,156,[201][202][203][204], as shown, for example, in Figure 6(b) and (c). The existence of preferential orientations forces the possible island geometries to be bounded by such planes.…”