应力诱发界面迁移下晶内孔洞的演化
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摘要
随着微电子技术的迅猛发展,集成电路中内连导线的失效问题引起广泛关注.内连导线内部孔洞萌生、长大、漂移和失稳变形成狭长裂纹,从而导致电路的开路失效.这是内连导线失效的常见形式.而界面迁移是导致微结构形态演化的主要机制之一.本文基于界面迁移下微结构演化的经典理论和弱解描述,建立了应力诱发界面迁移下微结构演化的有限单元法,并验证了算法的可靠性.对铜内连导线中晶内孔洞的演化进行了数值模拟,详细分析了应力、线宽及形态比对晶内孔洞演化的影响.研究结果表明,椭圆形晶内孔洞存在生长和收缩两种演化分叉趋势.通过大量数值分析得到了晶内孔洞演化的临界应力?σ_c、临界线宽?h_c和临界形态比β_c.当?σ≥?σ_c,?h≤?h_c或β≥β_c时,晶内孔洞会沿长轴长大;反之,晶内孔洞会收缩甚至愈合.此外,应力?σ越大、线宽?h越小或形态比β越大,晶内孔洞越易发生长大,且孔洞面积增大速度越快;?σ越小、?h越大或β越小,晶内孔洞越易发生收缩,且孔洞面积减小速度越快.
With the rapid development of microelectronics technology, the failure of interconnects in the integrated circuit raises wide attention. The interconnects inevitably exist some drawbacks, such as voids and cracks. If the drawbacks nucleate, grow and change their shape to form crack-like slits oriented perpendicular to an interconnect line, an open circuit could result. This is a common form of interconnects failure. And interface migration is one of the main mechanisms leading to the evolution of microstructure. Based on the classic theory and weak statement of interface migration, a finiteelement method is developed to simulate the evolution of intragranular voids in copper interconnects caused by interface migration induced by stress migration. The validity of the method is confirmed by the agreement of the numerically simulated the undulating surface with that predicted theoretically. Through a large number of numerical simulations, we find that the evolution of the intragranular voids has two trends, namely, void growth and void shrinkage. And the shape of the void is governed by the stress, σ, the linewidth, h, and the initial aspect ratio of the intragranular void, β, and there exist critical values for these parameters( σ_c,h_c and β_c). When σ≥σ_c,h≤h_c or β≥β_c, the intragranular void will grow along the major axis; otherwise, the intragranular void will shrink into a cylinder. The increase of the stress, or the aspect ratio, or the decrease of the linewidth is beneficial to void growth. And the area of void growth will increase faster with bigger σ, smaller h or bigger β. But, the decrease of the stress or the aspect ratio, or increase the linewidth accelerates void shrinkage and the shrinkage area will decrease faster with smaller σ, bigger h or smaller β.
With the rapid development of microelectronics technology, the failure of interconnects in the integrated circuit raises wide attention. The interconnects inevitably exist some drawbacks, such as voids and cracks. If the drawbacks nucleate, grow and change their shape to form crack-like slits oriented perpendicular to an interconnect line, an open circuit could result. This is a common form of interconnects failure. And interface migration is one of the main mechanisms leading to the evolution of microstructure. Based on the classic theory and weak statement of interface migration, a finiteelement method is developed to simulate the evolution of intragranular voids in copper interconnects caused by interface migration induced by stress migration. The validity of the method is confirmed by the agreement of the numerically simulated the undulating surface with that predicted theoretically. Through a large number of numerical simulations, we find that the evolution of the intragranular voids has two trends, namely, void growth and void shrinkage. And the shape of the void is governed by the stress, σ, the linewidth, h, and the initial aspect ratio of the intragranular void, β, and there exist critical values for these parameters( σ_c,h_c and β_c). When σ≥σ_c,h≤h_c or β≥β_c, the intragranular void will grow along the major axis; otherwise, the intragranular void will shrink into a cylinder. The increase of the stress, or the aspect ratio, or the decrease of the linewidth is beneficial to void growth. And the area of void growth will increase faster with bigger σ, smaller h or bigger β. But, the decrease of the stress or the aspect ratio, or increase the linewidth accelerates void shrinkage and the shrinkage area will decrease faster with smaller σ, bigger h or smaller β.
引文
1付云伟,张龙,倪新华等.考虑夹杂相互作用的复合陶瓷夹杂界面的断裂分析.力学学报,2016,48(1):154-162(Fu Yunwei,Zhang Long,Ni Xinhua,et al.Interface cracking analysis with inclusion interaction in composite ceramic.Chinese Journal of Theoretical and Applied Mechanics,2016,48(1):154-162(in Chinese))
2 Liu Z,Yu H.A numerical study on the effect of mobilities and initial profile in thin film morphology evolution.Thin Solid Films,2006,513(1):391-8
3 耿亚南,蔡宗熙.内压载荷作用下薄膜椭球的稳定性分析.力学学报,2016,48(5):1343-1352(Geng Yanan,Cai Zongxi.Stability of a pressurized ellipsoidal balloon.Chinese Journal of Theoretical and Applied Mechanics,2016,48(5):1343-1352(in Chinese))
4 Xia L,Bower AF,Suo Z,et al.A finite element analysis of the motion and evolution of voids due to strain and electromigration induced surface diffusion.Journal of the Mechanics&Physics of Solids,1997,45(9):1473-1493
5 熊骏,李振环,朱亚新等.基于微结构动态演化机制的单晶镍基高温合金晶体塑性本构及其有限元模拟.力学学报,2017,49(4): 763 -781(Xiong Jun,Li Zhenhuan,Zhu Yaxin,et al.Microstructure evolution mechanism based crystal-plasticity constitutive modelfor nickel-based superalloy and its finite element simulation.Chinese Journal of Theoretical and Applied Mechanics,2017,49(4):763- 781 (in Chinese))
6 吴丰顺,张金松,吴懿平等.集成电路互连引线电迁移的研究进展.半导体技术,2004,29(9):15-21(Wu Fengshun,Zhang Jinsong,Wu Yiping,et al.Progress of electromigration in IC interconnect metallic line.Semiconductor Technology,2004,29(9):15-21(in Chinese))
7 Suzuki T,Nakamura T,Mizushima Y,et al.Stress migration phenomenon in narrow copper interconnects.Journal of Applied Physics,2007,101(4):753
8 Ho PS.Motion of inclusion induced by a direct current and a temperature gradient.Journal of Applied Physics,1970,41(1):64-68
9 Ho PS,Kwok T.Electromigration in metals.Reports on Progress in Physics,1989,52(3):301
10 张金松,吴懿平,王永国等.集成电路微互连结构中的热迁移.物理学报,2010,59(6):4395-4402(Zhang Jinsong,Wu Yiping,Wang Yongguo,et al.Thermomigration in micro interconnects in integrated circuits.Acta Physica Sinica,2010,59(6):4395-4402(in Chinese))
11 郭佳惠,祝六花.关于超大规模集成电路制造中的应力迁移问题.电子器件,2000,23(4):262-266(Guo Jiahui,Zhu Liuhua.The problem about stress migration in the VLSI manufacture.Chinese Journal of Electron Devices,2000,23(4):262-266(in Chinese))
12 Gan ZH,Shao W,Mhaisalkar SG,et al.Experimental and numerical studies of stress migration in Cu interconnects embedded in different dielectrics.AIP Conference Proceedings,2006,817(817):269-274
13 刘晴,徐凯宇.考虑应变能密度的铜互连导线电迁移应力分析.力学季刊,2017(2):359-368(Liu Qing,Xu Kaiyu.Electromigration stress analysis of copper.Chinese Quarterly of Mechanics,2017(2):359-368(in Chinese))
14 Dong X,Li Z.An analytical solution for motion of an elliptical void under gradient stress field.Applied Physics Letters,2009,94,071909
15 Wang Y,Yao Y.A theoretical analysis of the electromigrationinduced void morphological evolution under high current density.Acta Mechanica Sinica,2017,33:868-878
16 Huang Q,Lilley CM,Divan R.An in situ investigation of electromigration in Cu nanowires.Nanotechnology,2009,20(7):075706
17 Maniatty AM,Ni J,Liu Y,et al.Effect of microstructure on electromigration-induced stress.Journal of Applied Mechanics,2016,83(1):13-13
18 Lin SK,Liu YC,Chiu SJ,et al.The electromigration effect revisited:Non-uniform local tensile stress-driven diffusion.Scientific Reports,2017,7(1):3082
19 Wang H,Li ZH,Sun J.Effects of stress and temperature gradients on the evolution of void in metal interconnects driven by electric current and mechanical stress.Modelling&Simulation in Materials Science&Engineering,2006,14(4):607
20 He DN,Huang PZ.A finite-element analysis of intragranular microcracks in metal interconnects due to surface diffusion induced by stress migration.Computational Materials Science,2014,87:65-71
21 He DN,Huang PZ.A finite-element analysis of in-grain microcracks caused by surface diffusion induced by electromigration.International Journal of Solids&Structures,2015,62:248-255
22 杜杰锋,黄佩珍.电迁移诱发表面扩散下沿晶微裂纹的演化.固体力学学报,2017,38(1):39-46(Du Jiefeng,Huang Peizhen.The evolution of intergranular microcracks due to surface diffusion induced by electromigration.Chinese Journal of Solid Mechanics,2017,38(1):39-46(in Chinese))
23 Suo Z.Motions of microscopic surfaces in materials.Advances in Applied Mechanics,1997,33(8):193-294
24 Sun B,Suo Z,Yang W.A finite element method for simulating interface motion—I.Migration of phase and grain boundaries.Acta Materialia,1997,45(5):1907-1915
25 Herring C.Surface tension as a motivation for sintering//Fundamental Contributions to the Continuum Theory of Evolving Phase Interfaces in Solids.Berlin,Heidelberg:Springer1999:33-69
26 Mullins WW.Theory of thermal grooving.Journal of Applied Physics,1957,28(3):333-339
27 Prevost JH,Baker TJ,Liang J,et al.A finite element method for stress-assisted surface reaction and delayed fracture.International Journal of Solids&Structures,2001,38(30-31):5185-203
28 Huang JM,Yang W.Three-dimensional evolution of interfaces under evaporation-condensation kinetics:A finite-element simulation.Modelling&Simulation in Materials Science&Engineering,1999,7(1):87
29 Yu H.Crack nucleation from a single notch caused by stressdependent surface reactions.International Journal of Solids&Structures,2005,42(13):3852-3866
30 Huang PZ Sun J,Li ZH.Evolution of penny-shaped microcracks by interface migration.International Journal of Solids and Structures,2003,40(8):1959-1972
31 Huang PZ,Sun J,Li ZH.Axisymmetric finite-element simulation of grain growth behaviour.Modelling&Simulation in Materials Science&Engineering,2002,11(1):41
32 Srolovitz DJ.On the stability of surfaces of stressed solids.Acta Metallurgica,1989,37(2):621-625
2 Liu Z,Yu H.A numerical study on the effect of mobilities and initial profile in thin film morphology evolution.Thin Solid Films,2006,513(1):391-8
3 耿亚南,蔡宗熙.内压载荷作用下薄膜椭球的稳定性分析.力学学报,2016,48(5):1343-1352(Geng Yanan,Cai Zongxi.Stability of a pressurized ellipsoidal balloon.Chinese Journal of Theoretical and Applied Mechanics,2016,48(5):1343-1352(in Chinese))
4 Xia L,Bower AF,Suo Z,et al.A finite element analysis of the motion and evolution of voids due to strain and electromigration induced surface diffusion.Journal of the Mechanics&Physics of Solids,1997,45(9):1473-1493
5 熊骏,李振环,朱亚新等.基于微结构动态演化机制的单晶镍基高温合金晶体塑性本构及其有限元模拟.力学学报,2017,49(4): 763 -781(Xiong Jun,Li Zhenhuan,Zhu Yaxin,et al.Microstructure evolution mechanism based crystal-plasticity constitutive modelfor nickel-based superalloy and its finite element simulation.Chinese Journal of Theoretical and Applied Mechanics,2017,49(4):763- 781 (in Chinese))
6 吴丰顺,张金松,吴懿平等.集成电路互连引线电迁移的研究进展.半导体技术,2004,29(9):15-21(Wu Fengshun,Zhang Jinsong,Wu Yiping,et al.Progress of electromigration in IC interconnect metallic line.Semiconductor Technology,2004,29(9):15-21(in Chinese))
7 Suzuki T,Nakamura T,Mizushima Y,et al.Stress migration phenomenon in narrow copper interconnects.Journal of Applied Physics,2007,101(4):753
8 Ho PS.Motion of inclusion induced by a direct current and a temperature gradient.Journal of Applied Physics,1970,41(1):64-68
9 Ho PS,Kwok T.Electromigration in metals.Reports on Progress in Physics,1989,52(3):301
10 张金松,吴懿平,王永国等.集成电路微互连结构中的热迁移.物理学报,2010,59(6):4395-4402(Zhang Jinsong,Wu Yiping,Wang Yongguo,et al.Thermomigration in micro interconnects in integrated circuits.Acta Physica Sinica,2010,59(6):4395-4402(in Chinese))
11 郭佳惠,祝六花.关于超大规模集成电路制造中的应力迁移问题.电子器件,2000,23(4):262-266(Guo Jiahui,Zhu Liuhua.The problem about stress migration in the VLSI manufacture.Chinese Journal of Electron Devices,2000,23(4):262-266(in Chinese))
12 Gan ZH,Shao W,Mhaisalkar SG,et al.Experimental and numerical studies of stress migration in Cu interconnects embedded in different dielectrics.AIP Conference Proceedings,2006,817(817):269-274
13 刘晴,徐凯宇.考虑应变能密度的铜互连导线电迁移应力分析.力学季刊,2017(2):359-368(Liu Qing,Xu Kaiyu.Electromigration stress analysis of copper.Chinese Quarterly of Mechanics,2017(2):359-368(in Chinese))
14 Dong X,Li Z.An analytical solution for motion of an elliptical void under gradient stress field.Applied Physics Letters,2009,94,071909
15 Wang Y,Yao Y.A theoretical analysis of the electromigrationinduced void morphological evolution under high current density.Acta Mechanica Sinica,2017,33:868-878
16 Huang Q,Lilley CM,Divan R.An in situ investigation of electromigration in Cu nanowires.Nanotechnology,2009,20(7):075706
17 Maniatty AM,Ni J,Liu Y,et al.Effect of microstructure on electromigration-induced stress.Journal of Applied Mechanics,2016,83(1):13-13
18 Lin SK,Liu YC,Chiu SJ,et al.The electromigration effect revisited:Non-uniform local tensile stress-driven diffusion.Scientific Reports,2017,7(1):3082
19 Wang H,Li ZH,Sun J.Effects of stress and temperature gradients on the evolution of void in metal interconnects driven by electric current and mechanical stress.Modelling&Simulation in Materials Science&Engineering,2006,14(4):607
20 He DN,Huang PZ.A finite-element analysis of intragranular microcracks in metal interconnects due to surface diffusion induced by stress migration.Computational Materials Science,2014,87:65-71
21 He DN,Huang PZ.A finite-element analysis of in-grain microcracks caused by surface diffusion induced by electromigration.International Journal of Solids&Structures,2015,62:248-255
22 杜杰锋,黄佩珍.电迁移诱发表面扩散下沿晶微裂纹的演化.固体力学学报,2017,38(1):39-46(Du Jiefeng,Huang Peizhen.The evolution of intergranular microcracks due to surface diffusion induced by electromigration.Chinese Journal of Solid Mechanics,2017,38(1):39-46(in Chinese))
23 Suo Z.Motions of microscopic surfaces in materials.Advances in Applied Mechanics,1997,33(8):193-294
24 Sun B,Suo Z,Yang W.A finite element method for simulating interface motion—I.Migration of phase and grain boundaries.Acta Materialia,1997,45(5):1907-1915
25 Herring C.Surface tension as a motivation for sintering//Fundamental Contributions to the Continuum Theory of Evolving Phase Interfaces in Solids.Berlin,Heidelberg:Springer1999:33-69
26 Mullins WW.Theory of thermal grooving.Journal of Applied Physics,1957,28(3):333-339
27 Prevost JH,Baker TJ,Liang J,et al.A finite element method for stress-assisted surface reaction and delayed fracture.International Journal of Solids&Structures,2001,38(30-31):5185-203
28 Huang JM,Yang W.Three-dimensional evolution of interfaces under evaporation-condensation kinetics:A finite-element simulation.Modelling&Simulation in Materials Science&Engineering,1999,7(1):87
29 Yu H.Crack nucleation from a single notch caused by stressdependent surface reactions.International Journal of Solids&Structures,2005,42(13):3852-3866
30 Huang PZ Sun J,Li ZH.Evolution of penny-shaped microcracks by interface migration.International Journal of Solids and Structures,2003,40(8):1959-1972
31 Huang PZ,Sun J,Li ZH.Axisymmetric finite-element simulation of grain growth behaviour.Modelling&Simulation in Materials Science&Engineering,2002,11(1):41
32 Srolovitz DJ.On the stability of surfaces of stressed solids.Acta Metallurgica,1989,37(2):621-625