Mathematical Sciences: Mathematical Modeling of Island Formation in Strained Semiconductor Films

9622930斯宾塞这项研究的目标是开发数学模型来预测和控制应变固体薄膜中的形态发展，这在半导体器件应用中具有重要的技术意义。该研究项目集中在薄膜生长过程中发生的应力驱动的形态不稳定性的后果。特别地，将用形态不稳定性的数学模型来解释“岛”形态的形成。该研究计划包括两个项目，一个专注于单组分薄膜，第二个专注于合金薄膜。在第一个项目中，将开发一个形成三维岛屿的“最先进”模型。该模型将包括对薄膜和其底层之间的润湿层的关键处理。在第二个项目中，我们将为合金薄膜中更复杂的形貌发展问题建立一个基本模型，其中成分变化和应力变化是耦合的。在这两个项目中，数学模型都是关于薄膜形状的非线性自由边界问题。这些问题将使用包括分析、渐近和数值方法在内的应用数学技术进行分析。具体地说，将得到岛屿形状的渐近描述，其利用了岛屿通常具有比宽度小得多的高度的事实。这项工作的结果将与从合作研究项目和已发表的实验结果中观察到的张紧薄膜系统中的岛屿进行比较。将对这些模型进行评估，以确定它们可用于预测和控制形貌的程度，以及确定作为长期研究计划的一部分，改进应变薄膜生长数学模型的未来方向。这项研究的目标是建立数学模型来预测和控制应变固体薄膜中的形貌发展。应变固体薄膜在半导体器件应用中具有重要的技术意义。通过将固体薄膜沉积到不同材料的底层衬底上，从蒸汽中生长应变薄膜。由于薄膜与衬底的结合，薄膜生长在应力状态下。在这些薄膜的生长过程中，薄膜中的应力可能会导致凹凸不平的形成，也就是“岛”的形成。这些岛的存在对薄膜器件的电子性能有至关重要的影响，因此了解是什么控制了岛的形成能够更好地控制应变薄膜器件的电子性能。这项研究的目的是从薄膜生长过程的物理推导的数学模型中描述岛屿的形成。这一数学模型是对传统应变薄膜实验研究的补充。开发这种模型的主要好处是，它允许人们快速而容易地确定不同的材料参数和工艺参数对薄膜生长的影响。因此，该模型在应变固体薄膜的开发和生产中有三个主要应用。首先，通过改变模型中的参数，该模型可以作为一种低成本的方式来对不同的材料和生长结构进行“实验”。其次，通过确定获得具有特定物理和电子特性的应变固体薄膜所需的必要工艺输入，该数学模型可用于帮助设计材料。最后，该数学模型可以帮助确定在工业上制造应变薄膜的最佳和/或可接受的工艺条件。***

9622930 Spencer The objective of this research is to develop mathematical models to predict and control morphology development in strained solid films, which are of great technological importance in semiconductor device applications. The research project focuses on the consequences of the stress-driven morphological instability which occurs during film growth. In particular, the formation of the "island" morphology will be explained in terms of mathematical models for the morphological instability. The research program consists of two projects, one focusing on single-component films, the second focusing on alloy films. In the first project, a "state of the art" model for the formation of three-dimensional islands will be developed. This model will include a crucial treatment of the wetting layer between the film and its underlying substrate. In the second project, a basic model will be developed for the more complicated problem of morphology development in alloy films, where composition variations and stress variations are coupled. In both projects the mathematical models are nonlinear free boundary problems for the shape of the film. These problems will be analyzed using applied mathematical techniques which include analytical, asymptotic, and numerical methods. In particular, an asymptotic description for the island shape will be derived which takes advantage of the fact that islands generally have a much smaller height than width. The results of the work will be compared to observations of islands in strained film systems from both collaborative research projects and published experimental results. The models will be evaluated to determine the extent to which they can be used to predict and control morphologies, as well as to determine future directions for improving mathematical models of strained film growth as part of a long-term research program. %%% The objective of this research is to develop mathematical models to predict and control mo rphology development in strained solid films. Strained solid films are of great technological importance in semiconductor device applications. The strained films are grown from a vapor through the deposition of the solid film onto an underlying substrate of a different material. Because of the bonding of the film to the substrate, the film is grown in a state of stress. During the growth of these films, the stresses in the film can lead to the formation of bumps, or "islands." The presence of these islands has a crucial effect on the electronic properties of the thin film device, so a knowledge of what controls island formation enables a better control over the electronic properties of the strained film device. The objective of this research is to describe island formation from physically-derived mathematical models of the film growth process. This mathematical model represents a complementary alternative to traditional experiment-based research on strained films. The primary benefit of developing such a model is that it allows one to quickly and easily determine how the growth of the film is affected by the different material parameters and process parameters. Thus, the model has three main applications in the development and production of strained solid films. Firstly, by changing the parameters in the model, the model can be used as a low-cost way to "experiment" with different materials and growth configurations. Secondly, the mathematical model can be used to help engineer materials by determining the necessary process inputs required to achieve a strained solid film with specified physical and electronic properties. Finally, the mathematical model can assist in the determination of optimum and/or acceptable processing conditions for the manufacture of strained films in industry. ***