Computation of the Semiclassical Limit of Schroedinger's Equation, Anisotropic Grain Growth, and Epitaxial Growth Using Kinetic Monte Carlo

使用动力学蒙特卡罗计算薛定谔方程的半经典极限、各向异性晶粒生长和外延生长

• 批准号: 1115252
• 负责人: Peter Smereka
• 金额: $ 15万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-15 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1115252&HistoricalAwards=false
• 关键词: Computation; Semiclassical; Limit; Schroedinger; Equation

This proposal involves three projects where each one builds from prior NSF support. This first project involves the simulation of grain boundary motion in two and three dimensions using our recently developed multiphase variational level set framework. It is planned to extend this formulation to allow for the possibility of arbitrary surface tension between grains. This will be accomplished by combining this formulation with minimizing movements. The second project concerns the efficient computation the semi-classical limit of the Schroedinger equation. The proposed algorithm is based on the observation that if one transforms the Schroedinger equation using a transformation inspired by Gaussian wave packets, one arrives at another Schroedinger equation that is much more amenable to computation in the semiclassical limit project. The third project concerns modeling and efficient simulation of epitaxial growth using kinetic Monte Carlo (KMC). We plan to model and simulate both liquid drop epitaxy and heteroepitaxial growth. In each of these cases we aim to develop highly efficient KMC algorithms to allow simulation on time and lengths not previously possible, thereby facilitating model development and allowing comparison with experiments.Each of the proposed projects has the potential to have a significant impact on problems that are both fundamental and technologically important. Epitaxial growth is scientifically interesting since it has effects on both nanoscales and mesoscales. It is technologically relevant since quantum dot materials are made in this way. Our proposed techniques will greatly increase the simulation speed thereby facilitating model development. The study of grain boundary motion using curvature flow is a classic problem in applied and computational mathematics which has importance in material science since various properties of polycrystalline substances can crucially depend on the details of the grain patterns. The fast simulation of the semi-classical limit of the Schroedinger equation could provide deeper insight into chemical reaction dynamics, molecular-surface scattering, and photodissociation, for example.

该提案涉及三个项目，每个项目都是在先前NSF支持的基础上构建的。这第一个项目涉及到模拟晶界运动在二维和三维使用我们最近开发的多相变分水平集框架。计划扩展该配方以允许颗粒之间任意表面张力的可能性。这将通过将该配方与最小化运动相结合来实现。第二个项目是关于薛定谔方程半经典极限的有效计算。所提出的算法是基于这样的观察，如果一个转换的薛定谔方程的启发高斯波包的变换，一个到达另一个薛定谔方程，这是更适合于计算的半经典极限项目。第三个项目涉及使用动力学蒙特卡罗（KMC）的外延生长建模和有效的模拟。 我们计划对液滴外延和异质外延生长进行建模和模拟。在每一种情况下，我们的目标是开发高效的KMC算法，使模拟的时间和长度上以前不可能的，从而促进模型的开发，并允许与experiments.Each提出的项目有可能产生重大影响的问题，是根本性的和技术上的重要性。外延生长在科学上是有趣的，因为它对纳米尺度和中尺度都有影响。 这在技术上是相关的，因为量子点材料是以这种方式制成的。 我们提出的技术将大大提高模拟速度，从而促进模型的开发。利用曲率流研究晶界运动是应用数学和计算数学中的经典问题，它在材料科学中具有重要意义，因为多晶物质的各种性质可以关键地依赖于晶粒图案的细节。 薛定谔方程的半经典极限的快速模拟可以为化学反应动力学、分子表面散射和光解离等提供更深入的了解。