Morphological Development in Strained Alloy Films

应变固体薄膜是量子阱半导体、双极异质结晶体管等新型电子器件的重要组成部分。薄膜中的应变对这些器件是必不可少的，因为它改变了器件中的电子能带，以产生所需的电子性质。然而，薄膜中的晶格也是在生长过程中产生不稳定性的原因，导致非平面薄膜具有不均匀的微结构，如“量子点”和“量子线”。由于与非均匀微结构相关的应变局域化，所产生的电子性质可以由于量子电子效应而得到增强。因此，人们对生长过程中量子点和量子线形态的自然“自组装”产生了浓厚的兴趣。PI的研究将集中于分析应变合金薄膜中这种不均匀微结构的发展的数学模型。应变薄膜中形态发展的理论描述是困难的，因为弹性应变在形态发展中起主要作用；一般情况是自由边界或移动边界弹性问题。虽然在单组分应变薄膜生长的数学模型方面已经取得了很大的进展，但合金应变薄膜的生长模型仍处于初级阶段。这项拟议的研究将采用分析和数值技术相结合的方法，考察由新的合金膜模型产生的微结构作为移动边界弹性问题的非线性解。这项研究将集中在与应变合金薄膜中微结构形成有关的三个方面。首先，通过对非线性自由边界问题的分叉分析，研究在厚膜中产生基于自组装成分调制的新型微结构的可能性。其次，我们将用渐近和数值相结合的方法描述非均匀合金量子点的生长过程，并将理论预测与合作者平行进行的实验进行比较。最后，通过在薄膜生长的宏观模型中引入原子尺度行为的模型，来检验应变合金晶体中小平面拐角的适当的数学建模。这项研究的总体目标是发展一种数学方法，能够对应变合金薄膜中纳米结构的形成进行全面的理论描述。在纳米级的电子器件中，应变固体层由于应变材料的电子性质的改善而发挥着重要的作用。虽然平坦的平面应变薄膜已经被成功地使用，但它们在生长过程中很容易受到不稳定性的影响，并产生纳米级的凸起(“点”)。由于量子物理的电子效应，这些纳米级的“量子点”被发现具有优越的电子行为。因此，有兴趣生长的量子点具有可控的大小和间距，以提供具有特定或最佳电子性质的非材料。这笔赠款支持的研究重点是通过对生长过程的物理模型进行详细建模，开发描述量子点和其他纳米结构形成的数学模型。特别是，这项工作将集中在合金薄膜的生长上，这方面的理论了解还不够深入。理论工作的目标将是评估不同生长参数的影响，以指导开发“最佳”的量子点结构。为了解决数学问题，将发展和应用先进的数学技术来确定解的特征以及它们如何依赖于材料的性质和生长条件。此外，这项研究还将解决如何将原子水平的行为纳入应变薄膜生长的大规模模型中的难题。理论预测将与布法罗大学的一位合作者进行的平行实验进行比较。这项工作的结果将是双重的。对如何处理好应变合金膜生长的数学问题的理解将得到提高。此外，还将开发一种有用的“参数图”，描述影响纳米结构发展的重要物理过程。该参数图可用于指导具有特定或最佳性能的应变合金纳米结构的设计。

Strained solid films are an important component of newly-developedelectronic devices such as the quantum-well semiconductor and thebipolar heterojunction transistor. The strain in the film isessential to these devices because it modifies the electronic band gapin the device to generate the desired electronic properties. Thestrain in the film, however, is also responsible for the generation ofinstabilities during growth, resulting in nonplanar films withinhomogeneous microstructure, such as "quantum dots" and "quantumwires". Because of the strain localization associated with theinhomogeneous microstructure, the resulting electronic properties canbe enhanced due to quantum electronic effects. Thus, there has beenintense interest in natural "self-assembly" of quantum dot and quantumwire morphologies during the growth process. The research of the PIwill focus on the analysis of mathematical models for the developmentof such inhomogeneous microstructures in strained alloy films. Thetheoretical description of morphological development in strained filmsis difficult because of the major role that elastic strain plays inthe development of the morphology; the generic case is that of a freeboundary or moving boundary elasticity problem. While there has beena significant amount of progress made on the mathematical modeling ofthe growth of single-component strained films, models for the growthof alloy strained films are still in their infancy. The proposedresearch will examine the microstructure generated from the new alloyfilm models as nonlinear solutions to the moving boundary elasticityproblem using a combination of both analytical and numericaltechniques. The research will focus on three areas relating to theformation of microstructure in strained alloy films. First, thepossibility of generating a new type of microstructure based onself-assembled compositional modulations in thick films will beinvestigated by a bifurcation analysis of the nonlinear free boundaryproblem. Second the growth of inhomogeneous alloy quantum dots willbe described using a hybrid asymptotic and numerical approach, and thetheoretical predictions will be compared to experiments carried out inparallel by a collaborator. Finally the appropriate mathematicalmodeling of facet corners in strained alloy crystals will be examinedthrough a model which incorporates atomic-scale behavior in amacroscopic model for film growth. The overall goal of the researchis to develop mathematical approaches that enable a comprehensivetheoretical description of nanostructure formation in strained alloyfilms.In nanoscale electronic devices, strained solid layers play animportant role because of the improved electronic properties of thestrained material. While flat, planar strained films have been usedsuccessfully, they can be susceptible to instabilities during thegrowth process and develop nanoscale bumps ("dots"). These nanoscale"quantum dots" have been found to give superior electronic behaviorbecause of quantum-physics electronic effects. There is thus interestin growing quantum dots with a controlled size and spacing to give amaterial with specified or optimum electronic properties. Theresearch supported by this grant focuses on the development ofmathematical models for describing the formation of quantum dots andother nanostructures from detailed modeling of the physics of thegrowth process. In particular, the work will focus on the growth ofalloy films, for which theoretical understanding not well developed.The goal of the theoretical work will be to evaluate the effect ofdifferent growth parameters to guide the development of "optimum"quantum dot structures. To solve the mathematical problem, advancedmathematical techniques will be developed and applied to determine thecharacteristics of the solutions and how they depend on the materialproperties and growth conditions. In addition, the research will alsoaddress the difficult question of how to incorporate atomic-levelbehavior into a large scale model for strained film growth. Thetheoretical predictions will be compared to experiments conducted inparallel by a collaborator here at the University at Buffalo. Theresults of the work will be twofold. The understanding of how totreat the mathematical issues of strained alloy film growth well will be improved. Also, a useful "parameter map" describing the important physical processes that influence the development of nanostructures will be developed. This parameter map can be used as aguide to the design of strained alloy nanostructures with specified oroptimum properties.