Morphological and compositional instabilities are of central importance in the study of nanocrystal growth. In particular, controlling the onset and evolution of step bunching, meandering, and faceting, as well as phase segregation and chemical ordering during the growth of self-organizing films, paves the way to the systematic production of nanostructures, e.g., quantum wires and dots, various two-dimensional nanoscale patterns, etc. The overall objective of this project is four-fold. Its first part is concerned with a novel instability during single-species epitaxy that results from the presence of a nonstandard term, the jump in the terrace grand canonical potential, in the step evolution equations. The goal here is to characterize this instability both in one and two dimensions and to determine if it can be offset by anisotropic step and terrace kinetics. In the second part, the focus is on an instability triggered during the growth of binary compounds where surface chemistry plays a genuine role. This instability differs from that resulting from the presence of impurities and is not due to an effective inverse Ehrlich--Schwoebel barrier for one of the two deposited species. Hence the need to better understand its underlying mechanisms and to identify, via phase diagrams, the unstable regimes in parameter-space. The third part is based on experimental evidence of step faceting. The goal there is to derive, in a dissipative setting, a thermodynamically consistent regularized model that captures the features of this faceting instability both during growth and sublimation. This is followed by the numerical investigation of the resulting free-boundary problem via algorithms recently developed to tackle the problems of faceting and coarsening at the mesoscale. The last part deals with intermixing, phase separation, and domain coarsening during the step-flow growth of multicomponent films, with emphasis on binary substitutional alloys. In contrast with existing theories, the microstructure of the vicinal surface is explicitly accounted for. Moreover, novel boundary conditions at the evolving steps are carefully derived and used to complement the Cahn--Hilliard PDE's that govern atomic bulk diffusion. The proposed model captures the multifaceted physics (surface kinetics, bulk elasticity and atomic diffusion, phase separation, etc.) that underlies growth and is multiscale in that the film is modeled as a layered structure, a view that permits the resolution of the disparate length scales in the lateral and epitaxial directions. Finally, its finite-element implementation yields much needed insight into the interplay between step flow and alloying/segregation/ordering.
With the advent of nanotechnologies, it has become feasible to manufacture devices at the nanoscale, from quantum computers to nano-electro-mechanical systems for biomedical applications. This has generated a wealth of experimental and theoretical work which, importantly, is interdisciplinary in nature, involving materials engineers, condensed-matter physicists, and applied mathematicians. At the nanoscale, much more so than at the macroscopic one, theory is an indispensable guide to experiment by providing a sound basis for experimental observations and, more ambitiously, by predicting the behavior of material systems under experimentally uncharted conditions. Of central importance are instabilities in the film morphology and composition, as they lead to the self-assembly of, e.g., quantum wires and dots. Controlling these instabilities is therefore crucial to the production of various nanostructures. This in turn requires a mathematical understanding of the physical and chemical mechanisms underlying the onset and evolution of instabilities. The investigator develops and analyzes mathematical models showing the evolution of nanostructures in films of materials. His effort combines mathematical modeling, analysis, and computation, and relies on knowledge of the underlying physics and thermodynamics. The work has the potential of yielding a better understanding of growth instabilities. Finally, the project involves the training of a doctoral student for whom this experience serves as an introduction to physical applied mathematics, mechanics, and mathematics of materials.
