Directed self-assembly (DSA) of block-copolymers (BCPs) is one of the most promising developments in the cost-effective production of nanoscale devices. The process makes use of the natural tendency for BCP mixtures to form nanoscale structures upon phase separation. The phase separation can be directed through the use of chemically patterned substrates to promote the formation of morphologies that are essential to the production of semiconductor devices. Moreover, the design of substrate pattern can formulated as an optimization problem for which we seek optimal substrate designs that effectively produce given target morphologies. In this paper, we adopt a phase field model given by a nonlocal Cahn--Hilliard partial differential equation (PDE) based on the minimization of the Ohta--Kawasaki free energy, and present an efficient PDE-constrained optimization framework for the optimal design problem. The design variables are the locations of circular- or strip-shaped guiding posts that are used to model the substrate chemical pattern. To solve the ensuing optimization problem, we propose a variant of an inexact Newton conjugate gradient algorithm tailored to this problem. We demonstrate the effectiveness of our computational strategy on numerical examples that span a range of target morphologies. Owing to our second-order optimizer and fast state solver, the numerical results demonstrate five orders of magnitude reduction in computational cost over previous work. The efficiency of our framework and the fast convergence of our optimization algorithm enable us to rapidly solve the optimal design problem in not only two, but also three spatial dimensions.
翻译:集成聚合物(BCPs)自发自组装(DSA)是纳米级设备成本-效益生产中最有希望的进展之一。这一过程利用了BCP混合物在分期分离时形成纳米结构的自然趋势。阶段分离可以通过使用化学型式基质来引导,以促进形成对生产半导体装置至关重要的形态。此外,设计基质模式可以作为一种优化问题,我们为此寻求最优化的亚值设计,以有效产生目标形态。在本文件中,我们采用了由非本地卡赫-赫利亚尔部分差异方程式(PDE)提供的阶段性设计模型,该模型以尽量减少Ohta-Kawasaki自由能源为基础,为最佳设计问题提供一个高效的PDE限制优化框架。设计变量是用于模拟基质化学型模式的圆形或条形指导员额的位置。为了解决随后产生的优化问题,我们建议采用一个阶段性亚精度亚精度亚精度设计模型,由非本地卡-希勒(PDDE)的精度设计模型设计模型设计模型模型模型,以快速计算出我们之前的精确度成本-卡度缩缩缩缩缩缩缩缩图,以显示我们之前的数值计算结果的缩缩略度的缩略图。