Novel Instabilities During the Epitaxy of Single- and Multi-Species Films: A Multiscale Approach

JABBURE DMS-0605039的形态和成分不稳定性在纳米晶体生长研究中具有重要意义。特别是，在自组织薄膜的生长过程中，控制阶跃聚束、弯曲和小面的开始和演化，以及相分离和化学有序，为系统地生产纳米结构铺平了道路，例如量子线和量子点，各种二维纳米图案等。它的第一部分涉及到单种外延过程中的一种新的不稳定性，这种不稳定性是由于阶跃卷积方程中存在一个非标准项--阶地巨正则势的跃迁所导致的。这里的目标是在一维和二维上表征这种不稳定性，并确定它是否可以被各向异性的阶跃和阶地动力学所抵消。在第二部分，重点是在二元化合物的生长过程中引发的不稳定性，其中表面化学起着重要的作用。这种不稳定性不同于杂质的存在，并且不是由于两个沉积物种中的一个物种的有效逆Ehrlich-Schwoebel势垒所致。因此，需要更好地了解其潜在机制，并通过相图确定参数空间中的不稳定区域。第三部分是基于阶跃刻面的实验证据。我们的目标是在耗散的背景下，推导出一个热力学上一致的正则化模型，该模型能够捕捉到生长和升华过程中这种刻面不稳定性的特征。其次是通过最近开发的算法对由此产生的自由边界问题进行数值研究，以解决中尺度上的粗化和粗化问题。最后一部分讨论了多组元薄膜阶跃流生长过程中的混合、相分离和磁区粗化，重点讨论了二元置换合金。与已有的理论相比，对邻近表面的微观结构进行了详细的解释。此外，在演化步骤中，新的边界条件被仔细地推导出来，并用来补充控制原子体扩散的Cahn-Hilliard偏微分方程组。该模型考虑了多方面的物理问题(表面动力学、体弹性和原子扩散、相分离等)。这是生长的基础，并且是多尺度的，因为薄膜被建模为分层结构，这一视图允许在横向和轴向上解决不同的长度尺度。最后，它的有限元实现产生了对阶跃流动和合金化/偏析/有序之间的相互作用的迫切需要的洞察。随着纳米技术的出现，从量子计算机到生物医学应用的纳米机电系统，制造纳米级的设备已经成为可能。这产生了大量的实验和理论工作，重要的是，这些工作本质上是跨学科的，涉及材料工程师、凝聚态物理学家和应用数学家。在纳米尺度上，比在宏观尺度上更是如此，理论是实验不可或缺的指南，它为实验观察提供了可靠的基础，并更雄心勃勃地预测了材料系统在实验未知条件下的行为。最重要的是薄膜形态和成分的不稳定性，因为它们导致例如量子线和点的自组装。因此，控制这些不稳定性对各种纳米结构的生产至关重要。这反过来需要对不稳定性发生和演变的物理和化学机制有一个数学上的理解。研究人员开发并分析了显示材料薄膜中纳米结构演变的数学模型。他的工作结合了数学建模、分析和计算，并依赖于基础物理和热力学的知识。这项工作有可能对生长稳定性有更好的理解。最后，该项目涉及对博士生的培训，对他们来说，这一经历是对物理应用数学、力学和材料数学的介绍。

JabbourDMS-0605039 Morphological and compositional instabilities are of centralimportance 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 orderingduring the growth of self-organizing films, paves the way to thesystematic production of nanostructures, e.g., quantum wires anddots, various two-dimensional nanoscale patterns, etc. Theoverall objective of this project is four-fold. Its first partis concerned with a novel instability during single-speciesepitaxy that results from the presence of a nonstandard term, thejump in the terrace grand canonical potential, in the stepevolution equations. The goal here is to characterize thisinstability both in one and two dimensions and to determine if itcan be offset by anisotropic step and terrace kinetics. In thesecond part, the focus is on an instability triggered during thegrowth of binary compounds where surface chemistry plays agenuine role. This instability differs from that resulting fromthe presence of impurities and is not due to an effective inverseEhrlich--Schwoebel barrier for one of the two deposited species. Hence the need to better understand its underlying mechanisms andto identify, via phase diagrams, the unstable regimes inparameter-space. The third part is based on experimentalevidence of step faceting. The goal there is to derive, in adissipative setting, a thermodynamically consistent regularizedmodel that captures the features of this faceting instabilityboth during growth and sublimation. This is followed by thenumerical investigation of the resulting free-boundary problemvia algorithms recently developed to tackle the problems offaceting and coarsening at the mesoscale. The last part dealswith intermixing, phase separation, and domain coarsening duringthe step-flow growth of multicomponent films, with emphasis onbinary substitutional alloys. In contrast with existingtheories, the microstructure of the vicinal surface is explicitlyaccounted for. Moreover, novel boundary conditions at theevolving steps are carefully derived and used to complement theCahn--Hilliard PDE's that govern atomic bulk diffusion. Theproposed model captures the multifaceted physics (surfacekinetics, bulk elasticity and atomic diffusion, phase separation,etc.) that underlies growth and is multiscale in that the film ismodeled as a layered structure, a view that permits theresolution of the disparate length scales in the lateral andepitaxial directions. Finally, its finite-element implementationyields much needed insight into the interplay between step flowand alloying/segregation/ordering. With the advent of nanotechnologies, it has become feasibleto manufacture devices at the nanoscale, from quantum computersto nano-electro-mechanical systems for biomedical applications. This has generated a wealth of experimental and theoretical workwhich, importantly, is interdisciplinary in nature, involvingmaterials engineers, condensed-matter physicists, and appliedmathematicians. At the nanoscale, much more so than at themacroscopic one, theory is an indispensable guide to experimentby providing a sound basis for experimental observations and,more ambitiously, by predicting the behavior of material systemsunder experimentally uncharted conditions. Of centralimportance are instabilities in the film morphology andcomposition, as they lead to the self-assembly of, e.g., quantumwires and dots. Controlling these instabilities is thereforecrucial to the production of various nanostructures. This inturn requires a mathematical understanding of the physical andchemical mechanisms underlying the onset and evolution ofinstabilities. The investigator develops and analyzesmathematical models showing the evolution of nanostructures infilms of materials. His effort combines mathematical modeling,analysis, and computation, and relies on knowledge of theunderlying physics and thermodynamics. The work has thepotential of yielding a better understanding of growthinstabilities. Finally, the project involves the training of adoctoral student for whom this experience serves as anintroduction to physical applied mathematics, mechanics, andmathematics of materials.