直拉硅单晶非均匀相变温度场最优控制
Optimal control for heterogeneous phase transition temperature field of Czochralski monocrystalline silicon
摘要点击 188  全文点击 47  投稿时间:2020-01-16  修订日期:2020-08-13
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DOI编号  10.7641/CTA.2020.00040
  2021,38(1):44-52
中文关键词  直拉硅单晶  相变温度场  生长直径  抛物型PDE  分布参数系统
英文关键词  Czochralski monocrystalline silicon  phase transition temperature field  growth diameter  parabolic PDE  distributed parameter system
基金项目  国家自然科学基金项目(61533014), 陕西省自然科学基金(2019JQ–734).
作者单位E-mail
张晶 西安理工大学 zhjing@xaut.edu.cn 
刘丁 西安理工大学 liud@xaut.edu.cn 
杜燕军 西安理工大学  
中文摘要
      针对直拉硅单晶固液界面相变温度场的非均匀性导致晶体直径不均匀问题, 提出一种基于偏微分方程 (PDE)模型的温度场最优控制策略. 考虑生长速率波动的影响, 建立了一种改进的提拉动力学模型, 确定了域边界 演化动力学关系. 研究基于抛物型PDE的时变空间域对流扩散过程的温度模型, 描述了域运动在对流扩散系统上的 单向耦合. 针对无限维分布参数系统建模控制难问题, 采用谱方法进行系统近似, 选取整个空间域的全局和正交的 空间基函数, 通过Galerkin方法对无限维系统进行降维, 获得了该系统的近似模型. 采用线性二次型方法控制晶体 生长温度, 通过仿真实验对相变温度场模型进行验证. 结果表明, 优化后的模型能够获得较为平稳的晶体生长速率, 减小了生长直径的波动, 使得固液界面径向温度分布更加均匀, 验证了该方法的有效性.
英文摘要
      The study proposed an optimal strategy for controlling temperature field based on PDE model (partial differential equation, PDE) in order to resolve non-uniformed diameters during crystal growth resulted by heterogeneity of phase-transition temperature field at solid-liquid interface for Czochralski monocrystalline silicon. In considering the influences brought by growth rate fluctuations, an improved lift-pull kinetic model was set up with dynamic relation been determined for the evolving domain boundary. Since the study was on the basis of the parabolic PDE temperature model of convection- diffusion process in time-varying spatial domain, it described one-way coupling of domain movement in convection-diffusion system. As for controlling difficulties occurred when modelling infinite distributed parameter system, the study adopted spectral method for system approximation. Within which, spatial primary functions that are global and orthogonal in the entire spatial domain were selected; thus, dimension reduction was performed to infinite-dimension system by taking Galerkin method so as to obtain approximation model of the system. Then, linear quadratic type method was applied to control crystal growth temperature, and simulation experiment was conducted for verifying phase-transition temperature field model. The results not only verify the effectiveness of the methods, but also show that the optimized model could stabilize crystal growth rate and reduce the problems of non-uniformed diameters for crystal growth; furthermore, temperature in radial direction at solid-liquid interface is more well-distributed.