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

JabbourDMS-0605039 形态和成分的不稳定性在晶体生长的研究中是至关重要的。 特别是，控制自组织膜生长过程中台阶聚束、曲折和刻面的发生和演变，以及相分离和化学有序化，为纳米结构的系统生产铺平了道路，量子线和量子点，各种二维纳米图案等。本项目的总体目标是四个方面。 第一部分讨论了单物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演化方程中阶跃巨正则势的非标准项的存在而引起的。 这里的目标是在一个和两个维度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阶地动力学抵消。 在第二部分中，重点讨论了在二元化合物生长过程中，表面化学起着关键作用的不稳定性。 这种不稳定性不同于由杂质的存在而产生的不稳定性，并且不是由于两种沉积物种之一的有效的inverseEhrlich-Schwoebel势垒。因此，需要更好地了解其潜在的机制，并确定，通过相图，不稳定的制度在参数空间。 第三部分是台阶刻面的实验依据。 我们的目标是在耗散环境中，得到一个在生长和升华过程中都能捕捉到这种小面不稳定性特征的物理一致的正则化模型。 其次是通过最近开发的算法，以解决问题ofaceting和粗化在中尺度的自由边界problemvia的数值研究。 最后一部分讨论了多元薄膜分步流动生长过程中的混合、相分离和畴粗化，重点是二元替代合金。 与现有的理论相比，该理论明确地解释了相邻表面的微观结构。 此外，新的边界条件在thevolving步骤仔细推导，并用于补充theCahn-Hilliard偏微分方程的原子体扩散。 所提出的模型捕捉了多方面的物理（表面张力，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，因为膜被建模为分层结构，这是一种允许在横向和纵向上分辨不同长度尺度的视图。 最后，它的有限元implementationyield急需的洞察力之间的相互作用的阶梯流和合金化/偏析/有序。 随着纳米技术的出现，从量子计算机到用于生物医学应用的纳米机电系统，制造纳米级设备已经成为可能。这产生了丰富的实验和理论工作，其中，重要的是，是跨学科的性质，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，理论比在宏观尺度上更是实验不可缺少的指南，它为实验观察提供了坚实的基础，更雄心勃勃的是，它预测了材料系统在实验未知条件下的行为。 最重要的是膜形态和组成的不稳定性，因为它们导致例如，量子线和量子点。 因此，控制这些不稳定性对各种纳米结构的生产至关重要。 这反过来又需要对不稳定性的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。 这项工作有可能使人们更好地理解生长不稳定性。 最后，该项目涉及到对博士生的培训，对他们来说，这一经验是对物理应用数学、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.