Epitaxial self-assembled quantum dots (SAQDs) result from Stranski-Krastanow growth whereby epitaxial 3D islands
form spontaneously on a planar thin film. Common systems are GexSi1-x/Si and InxGa1-xAs/GaAs. SAQDs are typically
grown on a (001) surface. The formation and evolution of SAQDs is governed in large part by the interaction of surface
energy and elastic strain; however, the surface energy density is quite complicated and not well understood. Many growth
processes take place at high temperature where stress and entropy effects can have a profound effect on the surface free
energy. There are three competing theories of the nature of the planar (001) surface: I. It is a stable crystal facet. II. It is a
stable non-faceted surface. III. It is an unstable crystal antifacet. Each leads to a different theory of the SAQD formation
process. The first theory appears most often in modeling literature, but the second two theories take explicit account of the
discrete nature of a crystal surface. Existing observational and theoretical evidence in support of and against these theories
is reviewed. Then a simple statistical mechanics model is presented that yields a phase-diagram depicting when each of the
three theories is valid. Finally, the Solid-on-Solid model of crystal surfaces is used to validate the proposed phase diagram
and to calculate the orientation and height dependence of the surface free energy that is expressed as a wetting chemical
potential, a wetting modulus and surface tilt moduli.
Semiconductor epitaxial self-assembled quantum dots (SAQDs) have potential for electronic and optoelectronic applications
such as high density logic, quantum computing architectures, laser diodes, and other optoelectronic devices. SAQDs
form during heteroepitaxy of lattice-mismatched films where surface diffusion is driven by an interplay of strain energy and
surface energy. Common systems are GexSi1-x/Si and InxGa1-xAs/GaAs. SAQDs are typically grown on a (001) crystal
surface. Self-assembled nanostructures form due to both random and deterministic effects. As a consequence, order and
controllability of SAQD formation is a technological challenge. Theoretical and numerical models of SAQD formation
can contribute both fundamental understanding and become quantitative design tools for improved SAQD fabrication if
they can accurately capture the competition between deterministic and random effects. In this research, a stochastic model
of SAQD formation is presented. This model adapts previous surface diffusion models to include thermal fluctuations in
surface diffusion, randomness in material deposition and the effects of anisotropic elasticity, anisotropic surface energy
and anisotropic diffusion, all of which are needed to model average SAQD morphology and order. This model is applied
to Ge/Si SAQDs which are group IV semiconductor dots and InAs/GaAs SAQDs which are III-V semiconductor dots.
Abstract Epitaxial self-assembled quantum dots (SAQDs) are of interest for nanostructured optoelectronic and electronic devices such as lasers, photodetectors and nanoscale logic. Spatial order and size order of SAQDs are important to the development of usable devices. It is likely that these two types of order are strongly linked; thus, a study of spatial order will also have strong implications for size order. Here a study of spatial order is undertaken using a linear analysis of a commonly used model of SAQD formation based on surface diffusion. Analytic formulas for film-height correlation functions are found that characterize quantum dot spatial order and corresponding correlation lengths that quantify order. Initial atomic-scale random fluctuations result in relatively small correlation lengths (about two dots) when the effect of a wetting potential is negligible; however, the correlation lengths diverge when SAQDs are allowed to form at a near-critical film height. The present work reinforces previous findings about anisotropy and SAQD order and presents as explicit and transparent mechanism for ordering with corresponding analytic equations. In addition, SAQD formation is by its nature a stochastic process, and various mathematical aspects regarding statistical analysis of SAQD formation and order are presented.
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