导线焊点界面扩散相变过程模拟和分析

The intermediate-phase growth and void growth at the contacting interfaces in a solder joint under the influence of electromigration, thermomigration and elastic stresses are critical issues for the failure analysis in integrated circuits. The mechanisms of electromigration and thermomigration were originally developed during 1950s. Nowadays, due to the trend in miniaturization in integrated circuits, electromigration and themomigration begin to play much more important roles, which made further understanding of both mechanisms highly necessary. Research of this project is two-fold: on one hand, both the intermediate-phase growth and the void growth at the contacting interfaces of solder joints will be simulated using a continuum diffuse interface model to investigate their growth behaviors under the influence of electromigration, thermomigration and elastic stresses; on the other hand, built upon our work of the inventive analysis of the mechanism of electromigration from the perspective of electromagnetism, further analysis of the temperature-dependence and the stressing effect of electromigration will be performed, moreover, analysis will also be extended to the mechanism of thermomigration analogously. Our simulation work will determine the growth behaviors of intermediate phases and viods, which are important application problems related to solder joint technology. Our work of analysis will not only help to build the theoretical ground for understanding the physical effects at the interfaces in solder joints, but also contribute to the further understanding of both mechanisms of electromigration and thermomigration. In addition, electromigration and thermomigration are diffusion problems under the influence of external fields, which belongs to the category of irreversible thermodynamics. Thus, our analysis will also help to contribute to the development of irreversible thermodynamics. Simulations will be performed using a software package developed jointly by group members in the research group of the Department of Mathematics in the University of California, Irvine. This package, which adopts an adaptive nonlinear multigrid finite difference method, excels in solving high-order multivariable coupled differential equations with field-dependent coefficients.

导线焊点界面上受电、热迁移和弹性应力影响的中间相生长和空穴生长，是集成电路失效问题中的关键。同时，对电、热迁移机制的理解均来源于上世纪五六十年代，而集成电路微型化的发展非常需要进一步增进对于电、热迁移机制的理解。本项目的工作包含模拟和分析两方面：首先,利用连续体扩散界面模型来模拟中间相和空穴的生长，发现其在电、热和弹共同影响下的生长规律；其次，在申请人已有的从电磁学的新角度出发对电迁移进行分析的工作之上扩展，增进对于电迁移温度依赖和应力效应等新问题的理解，并把分析扩展到热迁移的机制。此模拟工作会直接解决焊点工业界关心的界面相和空穴生长规律问题。而分析工作不仅会给电、热迁移机制带来新的理解，而且电、热迁移属于不可逆热力学科中的外场下扩散问题，也会促进这一学科的进展。模拟的软件将采用本人曾在国外数学系参与开发的一个多网格自适配有限差分软件包，能有效求解模型中的高阶多变量耦合变系数微分方程。

导线焊点界面上受电致迁移、热致迁移和弹性应力影响的中间相和孔洞生长，是集成电路失效问题中的关键。同时，集成电路微型化的发展也非常需要进一步增进对于电致、热致迁移机制的理解。电致、热致迁移问题本身属于热力学中的不可逆过程，对其理解同时也会促进不可逆热力学学科的进展。从另一个角度，电致、热致迁移属于多场耦合过程。多场耦合过程的本质实质反应了能量在不同形式之间的转化。多场耦合过程不仅包含不可逆过程，也包含可逆过程之间的耦合，比如电弹耦合效应。电弹耦合效应在声纳和感应器上都有着广泛的应用。本项目的工作包含模拟和分析两方面：首先,利用连续体扩散界面模型来模拟导线焊点界面上中间相和孔洞的生长，发现其在电、热和弹共同影响下的生长规律；其次，展开对不可逆和可逆耦合过程，比如电致、热致、热电等不可逆过程和电弹耦合等可逆耦合过程，的机制的一般性分析。本项目取得的重要结果如下：(1) 在有效电荷数依赖于电场和相场参数时，相界面的迁移速率主要取决于界面两侧合金相中的有效电荷数之差；在弹性应力的影响下，拉应力会加速孔穴的合并过程，而剪应力或复杂应力则会阻碍空洞的合并并最终到达稳态；（2）在申请人已有的从电磁学的新角度出发对电迁移进行分析的工作之上扩展，提出了在高温下，麦克斯韦尔应力会导致一定程度的晶格畸变，从而使得原子的扩散速率会大为增加，形成一种新的电致迁移机制，同时为失效分析中广泛应用的Black方程提供了热力学证明；利用相场变分方法，完成了对热电现象和热致迁移机制的分别讨论以及热力学不可逆过程的一般性分析，获得了重要物理参数的表达式，而且赋予了这些物理参数明确又简洁的物理涵义，同时并对Onsager关系的宏观证明进行了尝试；从能量守恒的角度出发对电弹可逆耦合过程进行了分析，发现了它们的控制或本构方程均满足能量守恒定律，从而能便捷的讨论耦合过程中能量的转化，同时能更深入的了解这些耦合过程的机制。此项目的模拟工作解决了几个焊点工业界关心的界面相和空穴生长规律问题。而分析工作则能增进对不可逆和可逆过程，如电致、热致迁移以及电弹耦合效应机制的理解，从而促进不可逆热力学的进展。

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