Islands in the Stream: Electromigration-Driven Shape Evolution with Crystal Anisotropy

  • Philipp Kuhn
  • Joachim Krug
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 149)


We consider the shape evolution of two-dimensional islands on a crystal surface in the regime where mass transport is exclusively along the island edge. A directed mass current due to surface electromigration causes the island to migrate in the direction of the force. Stationary shapes in the presence of an anisotropic edge mobility can be computed analytically when the capillary effects of the line tension of the island edge are neglected, and conditions for the existence of non-singular stationary shapes can be formulated. In particular, we analyse the dependence of the direction of island migration on the relative orientation of the electric field to the crystal anisotropy, and we show that no stationary shapes exist when the number of symmetry axes is odd. The full problem including line tension is solved by time-dependent numerical integration of the sharp-interface model. In addition to stationary shapes and shape instability leading to island breakup, we also find a regime where the shape displays periodic oscillations.


Two-dimensional shape evolution surface electromigration crystal steps crystal anisotropy 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    C.V. Thompson, J.R. Lloyd, Electromigration and IC interconnects. MRS Bulletin 18, No. 12 (1993), 19–25.Google Scholar
  2. [2]
    K.N. Tu, Recent advances on electromigration in very-large-scale-integration of interconnects. J. Appl. Phys. 94 (2003), 5451–5473.CrossRefGoogle Scholar
  3. [3]
    Z. Suo, Reliability of Interconnect Structures. In: Comprehensive Structural Integrity, I. Milne, R.O. Ritchie, B. Karihaloo, Editors-in-Chief, Vol. 8: Interfacial and Nanoscale Failure, W. Gerberich, W. Yang, Editors (Elsevier, Amsterdam 2003), 265–324.Google Scholar
  4. [4]
    E. Arzt, O. Kraft, W.D. Nix, J.E. Sanchez, Jr., Electromigration failure by shape change of voids in bamboo lines. J. Appl. Phys. 76 (1994), 1563–1571.CrossRefGoogle Scholar
  5. [5]
    O. Kraft, Untersuchung und Modellierung der Elektromigrationsschädigung in miniaturisierten Aluminiumleiterbahnen. PhD Dissertation (University of Stuttgart, 1995).Google Scholar
  6. [6]
    Y.-C. Joo, S.P. Baker, E. Arzt, Electromigration in single-crystal aluminum lines with fast diffusion paths made by nanoindentation. Acta Mater. 46 (1998) 1969–1979.CrossRefGoogle Scholar
  7. [7]
    M.R. Gungor, D. Maroudas, Theoretical analysis of electromigration-induced failure of metallic thin films due to transgranular void propagation. J. Appl. Phys. 85 (1999) 2233–2246.CrossRefGoogle Scholar
  8. [8]
    A.H. Verbruggen, Fundamental questions in the theory of electromigration. IBM J. Res. Develop. 32 (1988) 93–98.Google Scholar
  9. [9]
    R.S. Sorbello, Theory of electromigration. Solid State Phys. 51 (1998), 159–231.Google Scholar
  10. [10]
    O. Pierre-Louis, T.L. Einstein, Electromigration of single-layer clusters. Phys. Rev. B 62 (2000) 13697–13706.CrossRefGoogle Scholar
  11. [11]
    J.-J. Métois, J.C. Heyraud, A. Pimpinelli, Steady-state motion of silicon islands driven by a DC current. Surf. Sci. 420 (1999), 250–258.CrossRefGoogle Scholar
  12. [12]
    A. Saúl, J.-J. Métois, A. Ranguis, Experimental evidence for an Ehrlich-Schwoebel effect on Si(111). Phys. Rev. B 65 (2002) 075409.CrossRefGoogle Scholar
  13. [13]
    H. Mehl, O. Biham, O. Millo, M. Karimi, Electromigration-induced flow of islands and voids on the Cu(100) surface. Phys. Rev. B 61 (2000), 4975–4982.CrossRefGoogle Scholar
  14. [14]
    P.J. Rous, Theory of surface electromigration on heterogeneous metal surfaces. Appl. Surf. Sci. 175–176 (2001) 212–217.CrossRefGoogle Scholar
  15. [15]
    J. Krug, Introduction to Step Dynamics and Step Instabilities (this volume).Google Scholar
  16. [16]
    M. Schimschak and J. Krug, Electromigration-driven shape evolution of two-dimensional voids. J. Appl. Phys. 87 (2000) 695–703.CrossRefGoogle Scholar
  17. [17]
    Z. Suo, W. Wang and M. Yang, Electromigration instabilities: transgranular slits in interconnects. Appl. Phys. Lett. 64 (1994) 1944–1946.CrossRefGoogle Scholar
  18. [18]
    P.S. Ho, Motion of an inclusion induced by a direct current and a temperature gradient. J. Appl. Phys. 41 (1970) 64–68.CrossRefGoogle Scholar
  19. [19]
    W. Wang, Z. Suo, T.-H. Hao, A simulation of electromigration-induced transgranular slits. J. Appl. Phys. 79 (1996) 2394–2403.CrossRefGoogle Scholar
  20. [20]
    M. Mahadevan, R.M. Bradley, Stability of a circular void in a passivated, current-carrying metal film. J. Appl. Phys. 79 (1996), 6840–6847.CrossRefGoogle Scholar
  21. [21]
    L. Xia, A.F. Bower, Z. Suo, C.F. Shih, A finite element analysis of the motion and evolution of voids due to strain and electromigration induced surface diffusion. J. Mech. Phys. Solids 45 (1997) 1473–1493.CrossRefGoogle Scholar
  22. [22]
    M. Schimschak, J. Krug, Electromigration-Induced Breakup of Two-Dimensional Voids. Phys. Rev. Lett. 80 (1998) 1674–1677.CrossRefGoogle Scholar
  23. [23]
    M. Mahadevan, R.M. Bradley, Simulations and theory of electromigration-induced slit formation in unpassivated single-crystal metal lines. Phys. Rev. B 59 (1999), 11037–11046.CrossRefGoogle Scholar
  24. [24]
    Z. Li, H. Zhao, H. Gao, A Numerical Study of Electro-migration Voiding by Evolving Level Set Functions on a Fixed Cartesian Grid. J. Comp. Phys. 152 (1999), 281–304.CrossRefGoogle Scholar
  25. [25]
    M. Ben Amar, L.J. Cummings, G. Richardson, A theoretical treatment of void electromigration in the strip geometry. Comp. Mater. Sci. 17 (2000), 279–289.CrossRefGoogle Scholar
  26. [26]
    D.N. Bhate, A. Kumar, A.F. Bower, Diffuse interface model for electromigration and stress voiding. J. Appl. Phys. 87 (2000) 1712–1721.CrossRefGoogle Scholar
  27. [27]
    L.J. Cummings, G. Richardson, M. Ben Amar, Models of void electromigration. Eur. J. Appl. Math. 12 (2001), 97–134.CrossRefGoogle Scholar
  28. [28]
    J.H. Kim, P.R. Cha, D.H. Yeon, J.K. Yoon, A phase field model for electromigration-induced surface evolution. Metals and Materials International 9 (2003), 279–286.Google Scholar
  29. [29]
    Z. Suo, Electromigration-induced dislocation climb and multiplication in conducting lines. Acta metall. mater. 42 (1994), 3581–3588.CrossRefGoogle Scholar
  30. [30]
    W. Yang, W. Wang, Z. Suo, Cavity and dislocation instability due to electric current. J. Mech. Phys. Solids 42 (1994) 897–911.CrossRefGoogle Scholar
  31. [31]
    M. Schimschak, Numerische Untersuchungen zur Elektromigration auf metallischen Oberflächen, PhD Dissertation (University of Essen, 1999).Google Scholar
  32. [32]
    L.F. Shampine, M.K. Gordon, Computer Solution of Ordinary Differential Equations — The Initial Value Problem (W.H. Freeman, San Francisco, 1975).Google Scholar
  33. [33]
    F. Hausser, A. Voigt (private communication).Google Scholar
  34. [34]
    P. Kuhn, J. Krug (unpublished).Google Scholar
  35. [35]
    J. Krug, H.T. Dobbs, Current-Induced Faceting of Crystal Surfaces. Phys. Rev. Lett. 73 (1994), 1947–1950.PubMedGoogle Scholar
  36. [36]
    M. Schimschak, J. Krug, Surface Electromigration as a Moving Boundary Value Problem. Phys. Rev. Lett. 78 (1997), 278–281.CrossRefGoogle Scholar
  37. [37]
    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 (2000), L550–L556.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • Philipp Kuhn
    • 1
  • Joachim Krug
    • 2
  1. 1.Fachbereich PhysikUniversität Duisburg-EssenEssenGermany
  2. 2.Institut für Theoretische PhysikUniversität zu KölnKölnGermany

Personalised recommendations