南京航空航天大学学报  2017, Vol. 49 Issue (4): 518-523 PDF

Effect of Line Width on Intergranular Microcrack Evolution in Interconnects
DU Jiefeng, HUANG Peizhen
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics & Astronautics, Nanjing, 210016, China
Abstract: Based on the classical theory of surface diffusion induced by stress migration, a finite element program is developed for simulating the evolution of intergranular microcracks in copper interconnects. The numerical results show that there exists a critical value of the line width under the biaxial tensile stress state. When the line width is equal or less than the critical value, the intergranular microcrack will grow and split into three small microcracks along the grain boundary. When the line width is greater than the critical value, the microcrack will directly evolve into a cylinder. The splitting time of the intergranular microcrack reduces with the line width decreasing, which means that the decrease of the line width will accelerate the splitting process. Both the critical value of the stress and that of the aspect ratio decrease when the line width decreases, that is, the decrease of the line width is beneficial to the microcrack splitting. The critical values of the stress and the aspect ratio decrease when the ratio of the grain-boundary energy to the surface energy increases. And it is easier for the intergranular microcrack to split than the intragranular microcrack.
Key words: intergranular microcrack evolution     interconnects     stress migration     line width     finite element method

 图 1 内连导线中沿晶微裂纹模型 Figure 1 Simplified model of an intergranular microcrack in a conductor

1 有限元方法 1.1 微结构演化的弱解描述

 $\int_{{\rm{surface}}} {\boldsymbol{F} \cdot \delta \boldsymbol{I}} {\rm{d}}s + {\rm{ }}\int_{{\rm{surface}}} {\boldsymbol{p} \cdot \delta i} {\rm{d}}s = - \delta G$ (1)

 $\boldsymbol{J} = {\rm{ }}M\boldsymbol{F}$ (2)
 $j = m\boldsymbol{p}$ (3)

 $\int \{\frac{{\boldsymbol{J}\delta \boldsymbol{I}}}{M} + \frac{{\left( {{\boldsymbol{v}_n} + {\rm{ }}\partial \boldsymbol{J}\partial s{\rm{ }}} \right)\left[ {\delta {\boldsymbol{r}_n} + {\rm{ }}\partial \left( {\delta \boldsymbol{I}} \right)/\partial s} \right]{\rm{ }}}}{m}\} {\rm{d}}s = - \delta G$ (4)

1.2 系统自由能

 $G = \int_{{\rm{surface}}} {{\gamma _{\rm{s}}}} {\rm{d}}A + {\rm{ }}\int_{{\rm{grain}}\;{\rm{boundary}}} {{\gamma _{\rm{b}}}} {\rm{d}}A + \int_{{\rm{volume}}} w {\rm{d}}v$ (5)

1.3 有限单元法

 $\delta G = - \sum {{f_i}\delta {q_i}}$ (6)

 $\sum\limits_j {{H_{ij}} \cdot {{\dot q}_j}} = {f_i}$ (7)

1.4 边界条件

 $\cos \theta = \frac{{{\gamma _{\rm{b}}}}}{{2{\gamma _{\rm{s}}}}}$ (8)

 ${J_1} = {J_{NG}} = 0$ (9)
 ${{\dot y}_1} = {{\dot y}_{NG}} = 0$ (10)
2 计算结果与分析

 图 2 $\hat \sigma$=10，β=10时沿晶微裂纹的演化 Figure 2 Evolution of intergranular microcrack when $\hat \sigma$=10 and β=10

 图 3 分节时间$\hat t$f与线宽$\hat h$的关系 Figure 3 Splitting time $\hat t$f as a function of $\hat h$

 图 5 临界外载$\hat \sigma$c与线宽$\hat h$的关系 Figure 5 Critical value of stress $\hat \sigma$c as a function of $\hat h$

 图 6 临界外载$\hat \sigma$c随γb/γs的变化 Figure 6 Critical value of stress $\hat \sigma$c as a function of γb/γs

 图 7 临界形态比βc随γb/γs的变化 Figure 7 Critical value of aspect ratio βc as a function of γb/γs

3 结论

(1) 对于给定形态比与外载下的沿晶微裂纹，存在一个临界线宽$\hat h$c。当$\hat h$>$\hat h$c时，微裂纹不分节；当$\hat h$$\hat h$c时，微裂纹沿晶界方向扩展并最终分节为3个小裂腔。

(2) 沿晶微裂纹分节时间随着线宽的减小而减小，即减小线宽可以加速微裂纹分节。

(3) 临界外载与临界形态比都随着线宽的减小而减小，即，减小线宽有利于微裂纹分节。且当$\hat h$≤20时，临界外载和临界形态比对线宽的依赖性更强。

(4) 临界外载、临界形态比随着晶界能与表面能比值的增大而减小，且沿晶微裂纹比晶内微裂纹更易发生分节。

 [1] KORHONEN M A, PASZKIET C A, LI C Y. Mechanisms of thermal stress relaxation and stress induced voiding in narrow aluminium-based metallizations[J]. Journal of Applied Physics, 1991, 69(12): 8083–8091. DOI:10.1063/1.347457 [2] ZHAI C J, WALTER Y H, MARATHE A P, et al. Simulation and experiments of stress migration for Cu/low-k BEoL[J]. IEEE Transactions on Device and Materials Reliability, 2004, 4(3): 523–529. DOI:10.1109/TDMR.2004.833225 [3] HULL D, RIMMER D E. The growth of grain-boundary voids under stress[J]. Philosophical Magazine, 1959, 4(42): 673–687. DOI:10.1080/14786435908243264 [4] RAJ R, ASHBY M F. Intergranular fracture at elevated temperature[J]. Acta Metallurgica, 1975, 23(6): 653–666. DOI:10.1016/0001-6160(75)90047-4 [5] CHUANG T E, RICE J R. The shape of intergranular creep cracks growing by surface diffusion[J]. Acta Metallurgica, 1973, 21(12): 1625–1628. DOI:10.1016/0001-6160(73)90105-3 [6] PREVOST J H, BAKER T J, LIANG J, et al. A finite element method for stress-assisted surface reaction and delayed fracture[J]. International Journal of Solids and Structures, 2001, 38(30/31): 5185–5203. [7] LIU Z, YU H. A numerical study on the effect of mobilities and initial profile in thin film morphology evolution[J]. Thin Solid Films, 2006, 513(1/2): 391–398. [8] BOWER A F, SHANKAR S. A finite element model of electromigration induced void nucleation, growth and evolution in interconnects[J]. Modelling and Simulation in Materials Science and Engineering, 2007, 15(8): 923. DOI:10.1088/0965-0393/15/8/008 [9] LIU Z, YU H. Stress relaxation of thin film due to coupled surface and grain boundary diffusion[J]. Thin Solid Films, 2010, 518(20): 5777–5785. DOI:10.1016/j.tsf.2010.05.079 [10] SINGH N, BOWER A F, SHANKAR S. A three-dimensional model of electro migration and stress induced void nucleation in interconnect structures[J]. Modelling and Simulation in Materials Science and Engineering, 2010, 18(6): 65006. DOI:10.1088/0965-0393/18/6/065006 [11] HUANG P Z, ZHANG Z Z, GUO J W, et al. Axisymmetric finite-element analysis for interface migration-controlled shape instabilities of plate-like double-crystal grains[J]. Advanced Materials Research, 2012, 460: 230–235. DOI:10.4028/www.scientific.net/AMR.460 [12] LEVITAS V I, LEE D, PRESTON D L. Interface propagation and microstructure evolution in phase field models of stress-induced martensitic phase transformations[J]. International Journal of Plasticity, 2010, 26(3): 395–422. DOI:10.1016/j.ijplas.2009.08.003 [13] OGURTANI T O, AK YILDIZ O. Cathode edge displacement by voiding coupled with grain boundary grooving in bamboo like metallic interconnects by surface drift-diffusion under the capillary and electromigration forces[J]. International Journal of Solids and Structures, 2008, 45(3/4): 921–942. [14] SHEWMON P G. The movement of small inclusions in solids by a temperature gradient[J]. Transactions of the Metallurgical Society of AIME, 1964, 230(4): 1134–1137. [15] PAN J, COCKS A. A numerical technique for the analysis of coupled surface and grain-boundary diffusion[J]. Acta Metal Mater, 1995, 43(4): 1395–1406. DOI:10.1016/0956-7151(94)00365-O [16] SUN B, SUO Z. A finite element method for simulating interface motion—Ⅱ. Large shape change due to surface diffusion[J]. Acta Metallurgica, 1997, 45(12): 4963–4962. [17] HE D N, HUANG P Z. A finite-element analysis of intragranular microcracks in metal interconnects due to surface diffusion induced by stress migration[J]. Computational Materials Science, 2014, 87: 65–71. DOI:10.1016/j.commatsci.2014.01.063 [18] HE D N, HUANG P Z. A finite-element analysis of in-grain microcracks caused by surface diffusion induced by electromigration[J]. International Journal of Solids and Structures, 2015, 62: 248–255. DOI:10.1016/j.ijsolstr.2015.02.039 [19] HERRING C. Surface tension as a motivation for sintering[C]//The Physics of Powder Metallurgy. New York: McGraw-Hill, 1951:33-69.http://link.springer.com/chapter/10.1007/978-3-642-59938-5_2