Collaborative Research: Computational Problems in Heterogeneous Nanomaterials

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In many applications ranging from energy to biomedicine, nanocrystalline materials, such as quantum dots and nanowires, promise to yield revolutionary new technologies. The realization of this promise is hindered by the challenges inherent in reproducibly fabricating nanocrystalline materials with controlled morphologies and compositions. These nanomaterials are typically heterogeneous and consist of alloys with multiple constituents. While there has been much work on formulating conditions under which spatially ordered nanocrystals with nearly uniform shapes and sizes may be produced, a quantitative description of the mechanisms that determine the spatial distribution of the alloy components, which is crucial to device performance, is still poorly understood. The investigators and their collaborators address this issue in this proposal. They study the nonlinear dynamics of heterogeneous, strained strained nanocrystalline materials by (1) developing and applying state-of-the-art adaptive numerical methods to large-scale computation and (2) performing analytical, numerical and modelling studies of important constituent processes. The investigators focus on the dynamic, nonlinear coupling among shape, elastic stress and composition in the context of (i) the dynamics of thin film alloys and quantum dots under far-from-equilibrium processing conditions where there may be bulk and surface transport of the different constituents, as well as phase decomposition; and (ii) the coarsening dynamics and stability of capped nanocrystals. The cap material is needed in applications to provide the confinement potential for charge carriers as well as passivation against the external environment. These problems are characterized by the presence of multiple constitutive components, bulk-surface interactions, complex pattern formation and/or singularities (i.e. spatial complexity). The mathematical models involve high-order spatial derivatives (e.g. up to sixth-order), evolving free boundaries and highly nonlinear interactions that make analysis and simulation difficult, particularly in 3D. The highly nonlinear nature of these problems makes fast, accurate and robust numerical methods essential to their study. Nanocrystalline alloy materials have physical properties that make them ideally suited for a wide range of potential applications including advanced electronic and magnetic devices as well as biological and chemical sensors. The properties of nanoscale devices are determined both by the spatial composition of the heterogeneous nanocrystal components and the nanocrystal geometry. Recent advances in experimental techniques have enabled the characterization of nanoscale composition variation in nanocrystals. However, a quantitative understanding of these variations remains elusive and yet is critical to device performance. The investigators and their collaborators address this issue by developing new mathematical models, theory and computational methods that make it possible to characterize and quantify the interactions among nanocrystal shape, elastic stress and composition. The investigators also consider capped nanostructures where the cap material provides protection from the environment that is needed in many applications including the use of nanocrystals in silicon-based electronic circuits. The interaction among the nanocrystalline and capping materials introduces additional complexity. These problems are multidisciplinary and progress requires the combined expertise of the investigators in materials science and applied and computational mathematics. Through this study, the investigators provide guidance in the quantitative interpretation of experimental measurements of composition variation in nanocrystals and suggest optimized processing conditions to achieve desired device shape, composition and performance. The project establishes a new collaboration between two institutions and provides interdisciplinary training of two Ph.D students and one postdoctoral researcher. In addition, the investigators build on their recent success and continue to develop and teach a course on crystal and epitaxial growth for gifted high school students as part of the Calif. State Summer School for Mathematics and Science (COSMOS) at UC Irvine. This course also helps to recruit new math and science majors and enhance the participation of high school students in research.