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

形态和组成的不稳定性在纳米晶体生长的研究中是至关重要的。特别是，在自组织薄膜的生长过程中，控制步聚、蜿蜒和面聚的发生和演化，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，各种二维纳米尺度的图案等。这个项目的总体目标有四个方面。它的第一部分涉及单种外延过程中的一种新的不稳定性，这种不稳定性是由阶跃进化方程中阶跃正则势的非标准项的存在引起的。这里的目标是在一维和二维上描述这种不稳定性，并确定它是否可以被各向异性台阶和台阶动力学抵消。在第二部分中，重点是在二元化合物生长过程中引发的不稳定性，其中表面化学起着真正的作用。这种不稳定性不同于杂质的存在所导致的不稳定性，并且不是由于两种沉积物质之一的有效反希-施沃贝尔屏障。因此，需要更好地了解其潜在机制，并通过相图识别参数空间中的不稳定状态。第三部分是基于台阶铺装的实验证据。我们的目标是推导出，在耗散设置下，一个热力学一致的正则化模型，该模型捕获了生长和升华过程中这种表面不稳定性的特征。接下来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，以解决中尺度上的接触和粗化问题。最后讨论了多组分薄膜在阶梯流动生长过程中的混合、相分离和畴粗化，重点讨论了二元取代合金。与现有理论相比，该理论明确地解释了邻近表面的微观结构。此外，在进化的步骤中，新的边界条件被仔细地推导出来，并用于补充卡恩-希利亚德偏微分方程，该偏微分方程控制原子体扩散。所提出的模型捕捉了生长背后的多方面物理（表面动力学、体弹性和原子扩散、相分离等），并且是多尺度的，因为薄膜被建模为分层结构，这种观点允许在横向和纵向上分辨不同的长度尺度。最后，它的有限元实现对步进流和合金化/偏析/有序之间的相互作用产生了非常需要的洞察力。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，在纳米尺度上制造设备已经变得可行。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，理论是实验不可缺少的指南，它为实验观察提供了坚实的基础，更重要的是，它可以预测在实验未知条件下材料系统的行为。最重要的是薄膜形态和组成的不稳定性，因为它们会导致量子线和量子点等的自组装。因此，控制这些不稳定性对各种纳米结构的生产至关重要。这种转变需要对不稳定性发生和进化背后的物理和化学机制有一个数学上的理解。研究者开发并分析了显示材料薄膜中纳米结构演变的数学模型。他的努力结合了数学建模、分析和计算，并依赖于基础物理学和热力学的知识。这项工作有可能更好地理解生长的不稳定性。最后，该项目还包括对博士生的培训，对他们来说，这一经历将成为物理、应用数学、力学和材料数学的入门。

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.