力及力—电耦合作用下微结构中扩散、应力和变形分析
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摘要
化学和力学交互作用下的结构失效与原子或者离子的扩散密切相关。一方面,固体结构中物质扩散会引起局部应力变化；另一方面,在制造和使用过程中产生的应力和外力会对扩散产生影响。因此,正确理解应力作用下的物质扩散和迁移过程,对认识与控制结构的长寿命服役老化和可靠性衰退具有重要意义。本文围绕应力及力-电耦合作用下结构中的物质扩散、应力和变形问题,进行了相关的理论分析和实验研究。主要研究工作和结论如下：
以热力学理论和菲克定律为基础,建立了包含扩散和应力耦合效应的通用理论模型。利用有限差分法求解,分析了承受单向应力梯度的平板中,扩散物质的浓度和扩散致应力随时间的变化规律及扩散和应力集中的交互作用。研究表明：在扩散致应力、外力和扩散的耦合作用下,当外应力梯度和扩散方向相同时,外力会促进物质扩散,同时降低扩散本身导致的应力；在几何结构不连续或者材料固有缺陷部位,物质扩散导致的应力会降低结构的应力集中。
根据力与力矩的平衡原理,推导出膜/基结构中扩散致弯曲曲率和扩散致应力的闭合解,建立了包含热残余应力和扩散致应力影响的自洽扩散模型。该理论模型可用于分析在扩散和应力耦合作用下,膜/基结构中物质的扩散行为、应力分布和弯曲变形。研究发现,结构中的残余应力可以延长或者缩短扩散致弯曲达到稳定状态的时间,但不会改变达到稳定状态后弯曲曲率的大小,弹性模量比、厚度比和扩散物质在结构中的扩散率对膜/基悬臂梁结构的弯曲变形和应力分布有显著影响,这一结果对膜/基悬臂梁型微器件的结构优化有一定意义。
考虑纳米尺度下的表面粗糙现象和表面能效应,根据能量最小原理,建立了扩散导致的膜/基微悬臂梁结构应力分布和弯曲变形的解析解,定量分析了表面粗糙度和表面残余应力的影响。结果表明,纳米尺度下薄膜的表面粗糙度和上、下表面残余应力可显著改变膜/基微悬臂梁的弯曲变形和应力分布。
针对电子封装和互联结构的力-电耦合失效问题,进行了力-电耦合作用下铜的拉伸蠕变实验,分析了不同应力水平和电流下的蠕变变形行为。以晶界扩散理论为基础,建立了稳态蠕变应变率和电流密度及应力之间的关系。理论计算和实验结果对比分析表明,铜的蠕变应变和稳态蠕变应变率随应力和电流增加而增大；在较低应力水平下,电场不会改变蠕变变形的控制机理；但随着应力水平提高,蠕变变形会由受晶界扩散控制转变为受位错滑移或者攀移控制。在低应力和电场作用下,稳态蠕变应变率的理论预测值与实验结果较为相符。
The structure damage caused by the interaction of chemical and mechanical effects is in-timately associated with the diffusion of atoms or ions. Diffusion of atoms in structure can lead to the evolution of local stresses. On the other hand, the stresses produced during prac-tical manufacture and application can also influence the diffusion process. Therefore, under-standing of mass diffusion and migration is significant for analyzing and controlling the age-ing and reliability deterioration of long life structure. Studies on the diffusion behavior, stress and deformation under stress and electro-mechanical coupling effects have been systemically investigated in this contribution. The main contents and conclusions are as follows:
A self-consistent diffusion model including the coupling effects of diffusion and stress is developed under the frame of the thermodynamic theory and Fick's law. With the help of fi-nite difference method, the evolutions of concentration and diffusion-induced stress during diffusion process and the interaction between diffusion and stress concentration are analyzed. Results show that the external stress field will accelerate the diffusion process, and thus pro-mote the value of concentration while reduce the magnitude of stress induced by diffusion when the direction of diffusion is identical to that of external stress gradient with the coupling effects. The stress concentration around geometrical discontinuities and defects could be re-lieved by diffusion of solute.
According to the equilibrium conditions of force and moment, the exact closed-form so-lutions of curvature and diffusion-induced stress in the film/substrate structure are derived. And then, the self-consistent diffusion model involving the coupling effects of thermal misfit stress and diffusion-induced stress are developed. The theoretical model is available for ana-lyzing the diffusion behavior, diffusion-induced bending and stress with the coupling effects of diffusion and stress. Results indicate that thermal misfit stress can extend or shorten the time for achieving the saturated state, but won't change the magnitude of curvature at saturated state. The diffusivity and other structure parameters have great influence on the diffu-sion-induced curvature and stress distribution, which are meaningful for structural optimiza-tion of film/substrate cantilever-based device.
Based on energy minimization principle, the analytic solutions of diffusion-induced bending curvature and stress in film/substrate system are derived with coupling effects of rough surface and surface energy. The effects of surface roughness and residual surface stress have been analyzed quantitatively. Results show that the diffusion-induced stress and bending of film/substrate structure will dramatically changes with the variation of surface roughness and residual surface stress of the top and bottom surface.
Aiming at the failure of electronic packing and interconnection structure under the coupling effects of stress and electric field, tensile creep experiments on creep behavior of copper have been carried out. The creep behavior is analyzed under different stress and elec-tric current. Based on the grain boundary diffusion theory, a simple expression has been de-rived for the prediction of the steady state creep rate. According to the comparison of experi-ment results and theoretical predictions, the steady creep strain rate and creep strain of copper increase with increasing stress and electric current. The creep deformation is controlled by grain boundary diffusion at lower stress level, while it's controlled by dislocation glide or climb at higher stress level. The theoretical predictions of steady creep strain rate agree with the experiment results under low stress and electric field.
以热力学理论和菲克定律为基础,建立了包含扩散和应力耦合效应的通用理论模型。利用有限差分法求解,分析了承受单向应力梯度的平板中,扩散物质的浓度和扩散致应力随时间的变化规律及扩散和应力集中的交互作用。研究表明：在扩散致应力、外力和扩散的耦合作用下,当外应力梯度和扩散方向相同时,外力会促进物质扩散,同时降低扩散本身导致的应力；在几何结构不连续或者材料固有缺陷部位,物质扩散导致的应力会降低结构的应力集中。
根据力与力矩的平衡原理,推导出膜/基结构中扩散致弯曲曲率和扩散致应力的闭合解,建立了包含热残余应力和扩散致应力影响的自洽扩散模型。该理论模型可用于分析在扩散和应力耦合作用下,膜/基结构中物质的扩散行为、应力分布和弯曲变形。研究发现,结构中的残余应力可以延长或者缩短扩散致弯曲达到稳定状态的时间,但不会改变达到稳定状态后弯曲曲率的大小,弹性模量比、厚度比和扩散物质在结构中的扩散率对膜/基悬臂梁结构的弯曲变形和应力分布有显著影响,这一结果对膜/基悬臂梁型微器件的结构优化有一定意义。
考虑纳米尺度下的表面粗糙现象和表面能效应,根据能量最小原理,建立了扩散导致的膜/基微悬臂梁结构应力分布和弯曲变形的解析解,定量分析了表面粗糙度和表面残余应力的影响。结果表明,纳米尺度下薄膜的表面粗糙度和上、下表面残余应力可显著改变膜/基微悬臂梁的弯曲变形和应力分布。
针对电子封装和互联结构的力-电耦合失效问题,进行了力-电耦合作用下铜的拉伸蠕变实验,分析了不同应力水平和电流下的蠕变变形行为。以晶界扩散理论为基础,建立了稳态蠕变应变率和电流密度及应力之间的关系。理论计算和实验结果对比分析表明,铜的蠕变应变和稳态蠕变应变率随应力和电流增加而增大；在较低应力水平下,电场不会改变蠕变变形的控制机理；但随着应力水平提高,蠕变变形会由受晶界扩散控制转变为受位错滑移或者攀移控制。在低应力和电场作用下,稳态蠕变应变率的理论预测值与实验结果较为相符。
The structure damage caused by the interaction of chemical and mechanical effects is in-timately associated with the diffusion of atoms or ions. Diffusion of atoms in structure can lead to the evolution of local stresses. On the other hand, the stresses produced during prac-tical manufacture and application can also influence the diffusion process. Therefore, under-standing of mass diffusion and migration is significant for analyzing and controlling the age-ing and reliability deterioration of long life structure. Studies on the diffusion behavior, stress and deformation under stress and electro-mechanical coupling effects have been systemically investigated in this contribution. The main contents and conclusions are as follows:
A self-consistent diffusion model including the coupling effects of diffusion and stress is developed under the frame of the thermodynamic theory and Fick's law. With the help of fi-nite difference method, the evolutions of concentration and diffusion-induced stress during diffusion process and the interaction between diffusion and stress concentration are analyzed. Results show that the external stress field will accelerate the diffusion process, and thus pro-mote the value of concentration while reduce the magnitude of stress induced by diffusion when the direction of diffusion is identical to that of external stress gradient with the coupling effects. The stress concentration around geometrical discontinuities and defects could be re-lieved by diffusion of solute.
According to the equilibrium conditions of force and moment, the exact closed-form so-lutions of curvature and diffusion-induced stress in the film/substrate structure are derived. And then, the self-consistent diffusion model involving the coupling effects of thermal misfit stress and diffusion-induced stress are developed. The theoretical model is available for ana-lyzing the diffusion behavior, diffusion-induced bending and stress with the coupling effects of diffusion and stress. Results indicate that thermal misfit stress can extend or shorten the time for achieving the saturated state, but won't change the magnitude of curvature at saturated state. The diffusivity and other structure parameters have great influence on the diffu-sion-induced curvature and stress distribution, which are meaningful for structural optimiza-tion of film/substrate cantilever-based device.
Based on energy minimization principle, the analytic solutions of diffusion-induced bending curvature and stress in film/substrate system are derived with coupling effects of rough surface and surface energy. The effects of surface roughness and residual surface stress have been analyzed quantitatively. Results show that the diffusion-induced stress and bending of film/substrate structure will dramatically changes with the variation of surface roughness and residual surface stress of the top and bottom surface.
Aiming at the failure of electronic packing and interconnection structure under the coupling effects of stress and electric field, tensile creep experiments on creep behavior of copper have been carried out. The creep behavior is analyzed under different stress and elec-tric current. Based on the grain boundary diffusion theory, a simple expression has been de-rived for the prediction of the steady state creep rate. According to the comparison of experi-ment results and theoretical predictions, the steady creep strain rate and creep strain of copper increase with increasing stress and electric current. The creep deformation is controlled by grain boundary diffusion at lower stress level, while it's controlled by dislocation glide or climb at higher stress level. The theoretical predictions of steady creep strain rate agree with the experiment results under low stress and electric field.
引文
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