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Finite-element simulation of a phase-field model for inclusion electromigration in {110}-oriented single crystal metal interconnects due to interface diffusion anisotropy

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Abstract

Microdefects that exist in interconnect lines, such as inclusions, undergo a complex morphological evolution due to electromigration, which poses a challenge to the reliability of the integrated circuit. The investigation of the morphological evolution for inclusions driven by electromigration can be beneficial to improve the performance of the integrated circuit, and the applicability of the nanopattern. In this paper, a phase-field model based on the Cahn–Hilliard equations with anisotropic interface diffusion is established and the corresponding finite-element program is developed to study the evolution of the inclusion in the {110}-oriented single crystal of face-centered-cubic interconnects under electromigration. The bulk free energy density and the degenerate mobility applied in the present model are both constructed by the quartic double-well potential function. The validation of the program is verified by comparing the theoretical solution and the numerical solution. The effects of the misorientation, the anisotropic strength, and the conductivity ratio on the morphological evolution of inclusions are emphasized in detail. The results indicate that the morphological evolution is dependent on the misorientation, the conductivity ratio, and the anisotropic strength. And there are three evolution modes of inclusions: steady-state migration, oscillation, and unstable splitting. Small misorientation or conductivity ratio favors the steady-state migration, while a larger misorientation or conductivity ratio results in the process of merging and splitting. And the frequency of oscillation depends on the misorientation and the conductivity ratio. The migration velocity of the steady state is determined by both the conductivity ratio and the anisotropic strength.

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References

  1. J.R. Lloyd, Electromigration and mechanical stress. Microelectron. Eng. 49, 51–64 (1999)

    Article  Google Scholar 

  2. K.N. Tu, Recent advances on electromigration in very-large-scale-integration of interconnects. J. Appl. Phys. 94(9), 5451–5473 (2003)

    Article  ADS  Google Scholar 

  3. L. Xia, A.F. Bower, Z. Suo et al., A finite element analysis of the motion and evolution of voids due to strain and electromigration induced surface diffusion. J. Mech. Phys. Solids. 45(9), 1473–1493 (1997)

    Article  ADS  MATH  Google Scholar 

  4. E. Arzt, O. Kraft, W.D. Nix et al., Electromigration failure by shape change of voids in bamboo lines. J. Appl. Phys. 76(3), 1563–1571 (1994)

    Article  ADS  Google Scholar 

  5. C.M. Tan, A. Roy, Electromigration in ULSI interconnects. Mater. Sci. Eng. R. Rep. 58(1), 1–75 (2007)

    Article  Google Scholar 

  6. H. Ono, T. Nakano, T. Ohta, Diffusion barrier effects of transition metals for Cu/M/Si multilayers (M=Cr, Ti, Nb, Mo, Ta, W). Appl. Phys. Lett 64(12), 1511–1513 (1994)

    Article  ADS  Google Scholar 

  7. M. Hayashi, S. Nakano, T. Wada, Dependence of copper interconnect electromigration phenomenon on barrier metal materials. Microelectron. Reliab. 43(9), 1545–1550 (2003)

    Article  Google Scholar 

  8. C. Witt, C.A. Volkert, E. Arzt, Electromigration-induced Cu motion and precipitation in bamboo Al–Cu interconnects. Acta Mater. 51(1), 49–60 (2003)

    Article  ADS  Google Scholar 

  9. Q. Ma, Z. Suo, Precipitate drifting and coarsening caused by electromigration. J. Appl. Phys. 74(9), 5457–5462 (1993)

    Article  ADS  Google Scholar 

  10. H. Huntington, Electromigration in metals, ed. by A. S. Nowick, J. J. Burton (Elsevier, New York, 1975), pp. 303–352

  11. F. Cacho, X. Federspiel, Modeling of electromigration phenomena, ed. by C. Kim, (Elsevier, Cambridgeshire, 2011), pp. 1–43

  12. S. Strehle, J.W. Bartha, K. Wetzig, Electrical properties of electroplated Cu (Ag) thin films. Thin Solid Films 517(11), 3320–3325 (2009)

    Article  ADS  Google Scholar 

  13. M. Schimschak, J. Krug, Electromigration-induced breakup of two-dimensional voids. Phys. Rev. Lett. 80(8), 1674–1677 (1998)

    Article  ADS  Google Scholar 

  14. J. Cho, M.R. Gungor, D. Maroudas, Current-driven interactions between voids in metallic interconnect lines and their effects on line electrical resistance. Appl. Phys. Lett. 88(22), 221905 (2006)

    Article  ADS  Google Scholar 

  15. M.R. Gungor, D. Maroudas, Theoretical analysis of electromigration-induced failure of metallic thin films due to transgranular void propagation. J. Appl. Phys. 85(4), 2233–2246 (1999)

    Article  ADS  Google Scholar 

  16. J. Cho, M.R. Gungor, D. Maroudas, Theoretical analysis of current-driven interactions between voids in metallic thin films. J. Appl. Phys. 101(2), 023518 (2007)

    Article  ADS  Google Scholar 

  17. J.J. Métois, J.C. Heyraud, A. Pimpinelli, Steady-state motion of silicon islands driven by a DC current. Surf. Sci. 420(2–3), 250–258 (1999)

    Article  ADS  Google Scholar 

  18. C. Tao, W.G. Cullen, E.D. Williamst, Visualizing the electron scattering force in nanostructures. Science 328(5979), 736–740 (2010)

    Article  ADS  Google Scholar 

  19. A. Kumar, D. Dasgupta, C. Dimitrakopoulos et al., Current-driven nanowire formation on surfaces of crystalline conducting substrates. Appl. Phys. Lett. 108(19), 211–243 (2016)

    Google Scholar 

  20. P.S. Ho, Motion of inclusion induced by a direct current and a temperature gradient. J. Appl. Phys. 41(1), 64–68 (1970)

    Article  ADS  Google Scholar 

  21. W. Wang, Z. Suo, T.H. Hao, A simulation of electromigration-induced transgranular slits. J. Appl. Phys. 79(5), 2394–7403 (1996)

    Article  ADS  Google Scholar 

  22. M. Schimschak, J. Krug, Surface electromigration as a moving boundary value problem. Phys. Rev. Lett. 78(2), 278–281 (1997)

    Article  ADS  Google Scholar 

  23. T.H. Hao, Q.M. Li, Linear analysis of electromigration-induced void instability in Al-based interconnects. J. Appl. Phys. 83(2), 754–759 (1998)

    Article  ADS  Google Scholar 

  24. J. Cho, M. Gungor, D. Maroudas, Electromigration-driven motion of morphologically stable voids in metallic thin films: Universal scaling of migration speed with void size. Appl. Phys. Lett. 85(12), 2214–2216 (2004)

    Article  ADS  Google Scholar 

  25. Z. Li, N. Chen, Electromigration-driven motion of an elliptical inclusion. Appl. Phys. Lett. 93(5), 051908 (2008)

    Article  ADS  Google Scholar 

  26. S.K. Lin, Y.C. Liu, S.J. Chiu et al., The electromigration effect revisited: non-uniform local tensile stress-driven diffusion. Sci. Rep. 7(1), 3082 (2007)

    Article  ADS  Google Scholar 

  27. S.P. Riege, J.A. Prybyla, A.W. Hunt, Influence of microstructure on electromigration dynamics in submicron Al interconnects: real-time imaging. Appl. Phys. Lett. 69(16), 2367–2369 (1996)

    Article  ADS  Google Scholar 

  28. M. Genut, Z. Li, C.L. Bauer et al., Characterization of the early stages of electromigration at grain boundary triple junctions. Appl. Phys. Lett. 58(21), 2354–2356 (1991)

    Article  ADS  Google Scholar 

  29. T. Marieb, P. Flinn, J.C. Bravman et al., Observations of electromigration induced void nucleation and growth in polycrystalline and near-bamboo passivated Al lines. J. Appl. Phys. 78(2), 1026–1032 (1995)

    Article  ADS  Google Scholar 

  30. O. Kraft, E. Arzt, Electromigration mechanisms in conductor lines: void shape changes and slit-like failure. Acta Mater. 45(4), 1599–1611 (1997)

    Article  ADS  Google Scholar 

  31. A.V. Vairagar, S.G. Mhaisalkar, A. Krishnamoorthy et al., In situ observation of electromigration-induced void migration in dual-damascene Cu interconnect structures. Appl. Phys. Lett. 85(13), 2502–2504 (2004)

    Article  ADS  Google Scholar 

  32. C.S. Hau-Riege, An introduction to Cu electromigration. Microelectron. Reliab. 44(2), 195–205 (2004)

    Article  Google Scholar 

  33. A.W. Hunt, S.P. Riege, J.A. Prybyla, Healing processes in submicron Al interconnects after electromigration failure. Appl. Phys. Lett. 70(19), 2541–2543 (1997)

    Article  ADS  Google Scholar 

  34. D.N. Bhate, A.F. Bower, A. Kumar, A phase field model for failure in interconnect lines due to coupled diffusion mechanisms. J. Mech. Phys. Solids. 50(10), 2057–2083 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. J.W. Barrett, N. Robert, V. Styles, Finite element approximation of a phase field model for void electromigration. SIAM J. Appl. Math. 42(2), 738–772 (2004)

    MathSciNet  MATH  Google Scholar 

  36. A.F. Bower, S. Shankar, A finite element model of electromigration induced void nucleation, growth and evolution in interconnects. Modell. Simul. Mater. Sci. Eng. 15(8), 923–940 (2007)

    Article  ADS  Google Scholar 

  37. N. Singh, A.F. Bower, S. Shankar, A three-dimensional model of electromigration and stress-induced void nucleation in interconnect structures. Modell. Simul. Mater. Sci. Eng. 18(6), 065006 (2010)

    Article  ADS  Google Scholar 

  38. T.O. Ogurtani, O. Akyildiz, 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(2), 59–78 (2008)

    Google Scholar 

  39. B. Sun, Z. Suo, A finite element method for simulating interface motion—II. Large shape change due to surface diffusion. Acta Mater. 45(12), 4953–4962 (1997)

    Article  ADS  Google Scholar 

  40. D. He, P. Huang, A finite-element analysis of in-grain microcracks caused by surface diffusion induced by electromigration. Int. J. Solids Struct. 62, 248–255 (2015)

    Article  Google Scholar 

  41. M. Mahadevan, R.M. Bradley, Simulations and theory of electromigration-induced slit formation in un-passivated single-crystal metal lines. Phys. Rev. B 59(16), 11037 (1999)

    Article  ADS  Google Scholar 

  42. P. Kuhn, J. Krug, F, Hausser, et al., Complex shape evolution of electromigration-driven single-layer islands. Phys. Rev. Lett. 94(16), 166105 (2005)

  43. D. Maroudas, Dynamics of transgranular voids in metallic thin films under electromigration conditions. Appl. Phys. Lett. 67(6), 798–800 (1995)

    Article  ADS  Google Scholar 

  44. M.R. Gungor, D. Maroudas, Current-induced non-linear dynamics of voids in metallic thin films: morphological transition and surface wave propagation. Surf. Sci. 461(1–3), 550–556 (2000)

    Article  ADS  Google Scholar 

  45. E.D. Koronaki, M.R. Gungor, C.I. Siettos et al., Current-induced wave propagation on surfaces of voids in metallic thin films with high symmetry of surface diffusional anisotropy. J. Appl. Phys. 107(7), 073506 (2007)

    Article  ADS  Google Scholar 

  46. D. Dasgupta, D. Maroudas, Surface nanopatterning from current-driven assembly of single-layer epitaxial islands. Appl. Phys. Lett. 103(18), 181602 (2013)

    Article  ADS  Google Scholar 

  47. D. Dasgupta, A. Kumar, D. Maroudas, Analysis of current-driven oscillatory dynamics of single-layer homoepitaxial islands on crystalline conducting substrates. Surf. Sci. 669, 25–33 (2018)

    Article  ADS  Google Scholar 

  48. A. Kumar, D. Dasgupta, D. Maroudas, Complex pattern formation from current-driven dynamics of single-layer homoepitaxial islands on crystalline conducting substrates. Phys. Rev. Appl. 8(1), 014035 (2017)

    Article  ADS  Google Scholar 

  49. J. Zhang, P. Huang, Phase field simulation of the void destabilization and splitting processes in interconnects under electromigration induced surface diffusion. Modell. Simul. Mater. Sci. Eng. 30(1), 015003 (2022)

    Article  ADS  MathSciNet  Google Scholar 

  50. Y. Li, X. Wang, Z. Li, The morphological evolution and migration of inclusions in thin-film interconnects under electric loading. Composites B 43(3), 1213–1217 (2012)

    Article  Google Scholar 

  51. J. Santoki, A. Mukherjee, D. Schneider et al., Role of conductivity on the electromigration-induced morphological evolution of inclusions in {110}-oriented single crystal metallic thin films. J. Appl. Phys. 126(16), 165305 (2019)

    Article  ADS  Google Scholar 

  52. J. Santoki, A. Mukherjee, D. Schneider et al., Effect of conductivity on the electromigration-induced morphological evolution of islands with high symmetries of surface diffusional anisotropy. J. Appl. Phys. 129(2), 025110 (2021)

    Article  ADS  Google Scholar 

  53. M.E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Phys. D 92(3–4), 178–192 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  54. J.W. Cahn, C.M. Elliott, A. Novick-Cohen, The Cahn–Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature[J]. Eur. J. Appl. Math. 7(3), 287–301 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  55. C.L. Liu, J.M. Cohen, J.B. Adams et al., EAM study of surface self-diffusion of single adatoms of fcc metals Ni, Cu, Al, Ag, Au, Pd, and Pt. Surf. Sci. 253(1–3), 334–344 (1991)

    Article  ADS  Google Scholar 

  56. C.L. Liu, Diffusion mechanisms at fcc metal surface-embedded atom method calculations. Int. J. Mod. Phys. B 9(1), 1–44 (1995)

    Article  ADS  Google Scholar 

  57. D. Gaston, C. Newman, G. Hansen et al., MOOSE: a parallel computational framework for coupled systems of nonlinear equations. Nucl. Eng. Des. 239, 1768–1778 (2009)

    Article  Google Scholar 

  58. T. Rabczuk, H. Ren, X. Zhuang, A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem. CMC. 59(1), 31–55 (2019)

    Google Scholar 

  59. H. Ren, X. Zhuang, T. Rabczuk., A higher order nonlocal operator method for solving partial differential equations. Comput. Methods Appl. Mech. Eng. 367, 113132 (2020)

  60. H. Ren, X. Zhuang, N.T. Trung et al., Nonlocal operator method for the Cahn–Hilliard phase field model. Commun. Nonlinear. Sci. Numer. Simulat. 96, 105687 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  61. N. Valizadeh, T. Rabczuk., Isogeometric analysis for phase-field models of geometric PDEs and high-order PDEs on stationary and evolving surfaces. Comput. Methods Appl. Mech. Eng. 351, 599–642 (2019)

  62. T. Yu, J. Zhao., Semi-coupled resolved CFD-DEM simulation of powder-based selective laser melting for additive manufacturing. Comput. Methods Appl. Mech. Eng. 377, 113669 (2021)

  63. N. Valizadeh, T. Rabczuk., Isogeometric analysis of hydrodynamics of vesicles using a monolithic phase-field approach. Comput. Methods Appl. Mech. Eng. 388, 114191 (2022)

  64. P. Yue, C. Zhou, J. J. Feng., Spontaneous shrinkage of drops and mass conservation in phase-field simulations. J. Comput. Phys. 223(1), 1–9 (2007)

  65. R. Almgren, Second-order phase field asymptotics for unequal conductivities. SIAM J. Appl. Math. 59(6), 2086–2107 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  66. N. Moelans, B. Blanpain, P. Wollants, An introduction to phase-field modeling of microstructure evolution. Calphad 32(2), 268–294 (2008)

    Article  Google Scholar 

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Acknowledgements

The work is supported by the Natural Science Foundation of Jiangsu Province of China under Grant No. BK20141407 and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

Author contributions

CD: methodology, software, validation, formal analysis, investigation, data curation, writing—original draft, and visualization. PH: conceptualization, resources, writing—review and editing, and supervision. JZ: methodology and software.

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  1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, No. 29 Yudao Street, Nanjing, 210016, China

    Congcong Dong, Peizhen Huang & Jiaming Zhang

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Dong, C., Huang, P. & Zhang, J. Finite-element simulation of a phase-field model for inclusion electromigration in {110}-oriented single crystal metal interconnects due to interface diffusion anisotropy. Appl. Phys. A 128, 617 (2022). https://doi.org/10.1007/s00339-022-05754-z

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