Hillock formation by surface drift-diffusion driven by the gradient of elastic dipole interaction energy under compressive stresses in bi-crystalline thin films

Year 2015, , 65 - 76, 01.07.2015

Tarik Omer Ogurtani Oncu Akyildiz

https://doi.org/10.17350/HJSE19030000010

Cited By: 1

Abstract

We investigated surface drift diffusion induced grain boundary GB grooving and ridge hillock formation and growth, under the combined actions of the capillary forces and applied uniaxial compressive stresses, in bi-crystal thin films with dynamical computer simulations. In the present theory, the generalized driving force for the stress induced surface drift diffusion includes not only the usual gradient of the elastic strain energy density, but also the elastic dipole tensor interaction energy. During the morphological evolution of GB ridge formation and growth, triple junction TJ displacement and its velocity are continuously tracked down in order to resolve precisely the crossover time and depth at which velocity sign inversion takes place. An incubation time for the onset of the ridge growth stage coupled to the GB-TJ displacement velocity inversion is defined and its dependence on the stress is investigated. This analysis implies that the ridge growth stage is not controlled by Ziegler’s ‘maximum entropy production principle’ but rather Prigogine’s ‘minimum entropy production hypothesis’ for the stationary non-equilibrium states in complex systems, which are exposed to external applied body forces and surface tractions

Keywords

Grain Boundary Grooving, Non-Equilibrium Thermodynamics, Surface/Grain Boundary Diffusion, Compressive Stresses, Thin Films.

References

• 1. Genin FY, Mullins WW, Wynblatt P. The effect of stress on grain boundary grooving. Acta Metall Mater 41 (1993) 3541– 3547. doi: 10.1016/0956-7151(93)90234-J
• 2. Mullins WW. Theory of Thermal Grooving. J Appl Phys 28 (1957) 333. doi: 10.1063/1.1722742
• 3. Philofsky E, Ravi K, Hall E, Black J. Surface Reconstruction of Aluminum Metallization -- a New Potential Wearout Mechanism. 9th Reliab. Phys. Symp. IEEE, pp 120–128, 1971.
• 4. Chang CY, Vook RW, Lee YC, Hoshi I. Isothermal annealing of hillocks in Al-Cu films. Thin Solid Films 181 (1989) 57–63. doi: 10.1016/0040-6090(89)90472-0
• 5. Ericson F, Kristensen N, Schweitz J-A, Smith U. A transmission electron microscopy study of hillocks in thin aluminum films. J Vac Sci Technol B Microelectron Nanom Struct 9 (1991) 58. doi: 10.1116/1.585790
• 6. Genin FY. Effect of stress on grain boundary motion in thin films. J Appl Phys 77 (1995) 5130. doi: 10.1063/1.359324
• 7. 7. Genin FY. The initial stages of the formation of holes and hillocks in thin films under equal biaxial stress. Acta Metall Mater 43 (1995) 4289–4300. doi: 10.1016/0956- 7151(95)00132-F
• 8. Genin FY, Siekhaus W. Experimental study to validate a model of hillock’s formation in aluminum thin films. J. Appl. Phys. 79 (1996) 3560–3566.
• 9. Kim D, Nix WD, Vinci RP, Deal MD, Plummer JD. Study of the effect of grain boundary migration on hillock formation in Al thin films. J Appl Phys 90 (2001) 781. doi: 10.1063/1.1381045
• 10. Chaudhari P. Hillock growth in thin films. J Appl Phys 45 (1974) 4339. doi: 10.1063/1.1663054
• 11. Presland AEB, Price GL, Trimm DL. Hillock formation by surface diffusion on thin silver films. Surf Sci 29 (1972) 424–434. doi: 10.1016/0039-6028(72)90229-4
• 12. Hull D, Rimmer DE. The growth of grain-boundary voids under stress. Philos Mag 4 (1959) 673–687. doi: 10.1080/14786435908243264
• 13. Gao H, Zhang L, Nix WD, Thompson CV, Arzt E. Cracklike grain-boundary diffusion wedges in thin metal films. Acta Mater 47 (1999) 2865–2878. doi: 10.1016/S1359- 6454(99)00178-0
• 14. Zhang L, Gao H. Coupled grain boundary and surface diffusion in a polycrystalline thin film constrained by substrate. Zeitschrift für Met 93 (2002) 417.
• 15. Barvosa-Carter W, Aziz MJ, Gray LJ, Kaplan T. Kinetically driven growth instability in stressed solids. Phys Rev Lett 81 (1998) 1445–1448. doi: 10.1103/PhysRevLett.81.1445
• 16. Lahiri SK. Stress Relief and Hillock Formation in Thin Lead Films. J Appl Phys 41 (1970) 3172. doi: 10.1063/1.1659383
• 17. Ogurtani TO. Unified theory of linear instability of anisotropic surfaces and interfaces under capillary, electrostatic, and elastostatic forces: The regrowth of epitaxial amorphous silicon. Phys Rev B 74 (2006) 1–25. doi: 10.1103/ PhysRevB.74.155422
• 18. Kirchheim R. Stress and electromigration in Al-lines of integrated circuits. Acta Metall Mater 40 (1992) 309–323. doi: 10.1016/0956-7151(92)90305-X
• 19. Basaran C, Lin M, Ye H. A thermodynamic model for electrical current induced damage. Int J Solids Struct 40 (2003) 7315–7327. doi: 10.1016/j.ijsolstr.2003.08.018
• 20. Sukharev V, Nix WD, Zschech E. A model for electromigrationinduced degradation mechanisms in dual-inlaid copper interconnects: Effect of microstructure. J Appl Phys 102 (2007) 053505. doi: 10.1063/1.2775538
• 21. Akyildiz O, Oren EE, Ogurtani TO. Grain boundary grooving in bi-crystal thin films induced by surface driftdiffusion driven by capillary forces and applied uniaxial tensile stresses. Philos Mag 92 (2012) 804–829. doi: 10.1080/14786435.2011.634850
• 22. Ogurtani TO. Mesoscopic nonequilibrium thermodynamics of solid surfaces and interfaces with triple junction singularities under the capillary and electromigration forces in anisotropic three-dimensional space. J Chem Phys 124 (2006) 144706. doi: 10.1063/1.2185625
• 23. Ogurtani TO, Oren EE. Irreversible thermodynamics of triple junctions during the intergranular void motion under the electromigration forces. Int J Solids Struct 42 (2005) 3918– 3952. doi: 10.1016/j.ijsolstr.2004.11.013
• 24. Ogurtani TO, Akyildiz O. Morphological evolution of voids by surface drift diffusion driven by capillary, electromigration, and thermal-stress gradients induced by steady-state heat flow in passivated metallic thin films and flip chip solder joints. I. Theory. J Appl Phys 104 (2008) 023521. doi: 10.1063/1.2958088
• 25. Ogurtani TO, Akyildiz O. Morphological evolution of voids by surface drift diffusion driven by the capillary, electromigration, and thermal-stress gradient induced by the steady state heat flow in passivated metallic thin films and flip-chip solder joints. II. Applications. J Appl Phys 104 (2008) 023522. doi: 10.1063/1.2958303
• 26. Yeremin EN. The Foundations of Chemical Kinetics. MIR Publishers, Moscow, Russia , 1979.
• 27. Beer G, Watson JO. Introduction To Finite And Boundary Element Methods For Engineers. John Wiley & Sons, New York, USA, 1992.
• 28. Gear CW. Numerical initial value problems in ordinary differential equations. Prentice-Hall, Englewood Cliffs, N. J., 1971.
• 29. Pan J, Cocks ACF. A numerical technique for the analysis of coupled surface and grain-boundary diffusion. Acta Metall Mater 43 (1995) 1395–1406. doi: 10.1016/0956- 7151(94)00365-O
• 30. Friesen C. Reversible stress changes at all stages of Volmer– Weber film growth. J Appl Phys 95 (2004) 1011. doi: 10.1063/1.1637728
• 31. Prigogine I. Introduction to Thermodynamics of Irreversible Processes. Interscience Publishers, New York, USA, 1961.
• 32. Ogurtani TO, Celik A, Oren EE. Morphological evolution in a strained-heteroepitaxial solid droplet on a rigid substrate: Dynamical simulations. J Appl Phys 108 (2010) 063527. doi: 10.1063/1.3483937
• 33. Sokolnikoff I. Mathematical Theory of Elasticity. McGrawHill Book Co., New York, 1956.
• 34. Martyushev LM, Seleznev VD. Maximum entropy production principle in physics, chemistry and biology. Phys Rep 426 (2006) 1–45. doi: 10.1016/j.physrep.2005.12.001
• 35. Aharonov Y, Bohm D. Significance of electromagnetic potentials in the quantum theory. Phys Rev 115 (1959) 485– 491. doi: 10.1103/PhysRev.115.485
• 36. Kröner E. Kontinuums theorie der Versetzungen und Eigenspannungen. Springer, Berlin, 1958.
• 37. Hong QZ, Zhu JG, Mayer JW, Xia W, Lau SS. Solid phase epitaxy of stressed and stress-relaxed Ge-Si alloys. J Appl Phys 71 (1992) 1768. doi: 10.1063/1.351212
• 38. Smithells CJ. Metals Reference Book Vol. 3. Butterworths, London, 1967.