﻿ 硅通孔中电致应力的有限元分析
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Finite element analysis of electromigration induced stress in through-silicon-via
Su Fei, Lu Zixing, Liu Ping, Wang Yuan
School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract:Through-silicon-via (TSV) plays a key role in chip's vertical interconnection in 3D electronic package, so its reliability shows great importance. As the current density through TSV increasing, current induced stress has larger and larger influence on the TSV's reliability. The methodology for evaluation of electromigration induced stress in TSV was developed based on the coupling equation of stress-mass diffusion and the principle of finite element method (FEM), together with the user defined element on the platform of ABAQUS. The numerical simulation of this model was set and its accuracy was verified with analytical solution. The electromigration problem was simulated with a finite element model; The evolution and distribution of electromigration induced stress, strain and vacancy concentration in copper pour hatch were described. So the reliability of 3D electronic package could be assessed someway.
Key words: electromigration     electromigration induced stress     electromigration induced strain     through-silicon-via (TSV)     finite element

TSV的直径通常为数微米到数十微米量级,内壁上沉积有数纳米厚的扩散阻挡层和1~2 μm的SiO2绝缘层,并以电镀方式将孔用铜填充作为电信号通道.

1.2 方程的离散化处理

1.3 残差向量和Jacobian矩阵的定义

2 有限元分析模型的建立及验证

 图 1 无限大有限厚平板[9]Fig. 1 Plate with infinite-aspect and finite-thickness[9]
 图 2 无限大有限厚板的有限元模型及网格(1/2模型)Fig. 2 Finite element model and mesh of the plate shown in fig.1(1/2 model)

 参数 数值 T/K 473 杨氏模量E/GPa 72 泊松比ν 0.33 k/(J/K) 1.38×10-23 Dv/(cm2/s) 2.7×10-16 Z* 4.0 Cv0/cm-3 6.02×1015 ρ/(Ω·cm) 1.139×10-8~2.07×10-7 Ω/cm3 1.66×10-23 τs/s 0.001 8

 图 3 球应力沿厚度方向的分布Fig. 3 Spheric stress distribution along the thickness direction
 图 4 体积应变沿厚度方向的分布Fig. 4 Volumetric strain distribution along the thickness direction

 图 5 空位浓度沿厚度方向的分布Fig. 5 Vacancy concentration distribution along the thickness direction
3 TSV结构的电迁移及电致应力研究

 图 6 TSV结构模型及有限元网格Fig. 6 Model of TSV structure and finite element mesh

 参数 数值 T/K 473 E/GPa 120 ν 0.34 k/(J/K) 1.38×10-23 Dv/(cm2/s) 1.2×10-5 Z* 0.7 Cv0/cm-3 8.413×1015 ρ/(Ω·cm) 1.673×10-6 Ω/cm3 1.18×10-23 τs/s 0.001 8

 图 7 稳态时TSV结构球应力的分布云图Fig. 7 Spheric stress distribution of TSV structure at steady state
 图 8 稳态时TSV结构球应变(无量纲)的分布云图Fig. 8 Spheric strain distribution of TSV structure at steady state

 图 9 不同电流密度时TSV中的最大球应力Fig. 9 Maximum spheric stress at different current densities in TSV

 图 10 不同电流密度时TSV中的最大Mises应力Fig. 10 Maximum Mises stress at different current densities in TSV
4 结 论

1) 本文给出了弹性材料属性下电致应力计算的有限元模型,此模型准确地阐明了电迁移过程中扩散场和位移场的相互作用及其影响,并能计算电致应力场、应变场以及空位浓度场的演化过程.

2) 利用ABAQUS中的用户自定义单元接口,实现了电致应力的有限元计算,并进行了验证性计算.

3) 针对TSV中铜填充的电迁移问题进行了计算,发现TSV结构中的电致球应力在电流密度为1~5 MA/cm2的情况下可达数十到数百兆帕,而Mises应力只有数十兆帕.阳极处过大的球应力可能是造成钝化层的破裂,而阴极处较大的球应力可能对界面的完整性造成不良影响.

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#### 文章信息

Su Fei, Lu Zixing, Liu Ping, Wang Yuan

Finite element analysis of electromigration induced stress in through-silicon-via

Journal of Beijing University of Aeronautics and Astronsutics, 2014, 40(11): 1500-1506.
http://dx.doi.org/10.13700/j.bh.1001-5965.2013.0646