Acta Mechanica Solida Sinica

, Volume 23, Issue 2, pp 106–114 | Cite as

Local Bending of Thin Film on Viscous Layer

Article

Abstract

Effects of deposition layer position and number/density on local bending of a thin film are systematically investigated. Because the deposition layer interacts with the thin film at the interface and there is an offset between the thin film neutral surface and the interface, the deposition layer generates not only axial stress but also bending moment. The bending moment induces an instant out-of-plane deflection of the thin film, which may or may not cause the so-called local bending. The deposition layer is modeled as a local stressor, whose location and density are demonstrated to be vital to the occurrence of local bending. The thin film rests on a viscous layer, which is governed by the Navier-Stokes equation and behaves like an elastic foundation to exert transverse forces on the thin film. The unknown feature of the axial constraint force makes the governing equation highly nonlinear even for the small deflection case. The constraint force and film transverse deflection are solved iteratively through the governing equation and the displacement constraint equation of immovable edges. This research shows that in some special cases, the deposition density increase does not necessarily reduce the local bending. By comparing the thin film deflections of different deposition numbers and positions, we also present the guideline of strengthening or suppressing the local bending.

Key words

local bending deposition layer/dot thin film viscous layer constraint 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Medeiros-Ribeiro, G., Braykovsky, A.M., Kamins, T.I., Ohlberg, D.A.A. and Williams, R.S., Shape transition of Ge nanocrystals on Si(001): from pyramids to domes. Science, 1998, 279: 353–355.CrossRefGoogle Scholar
  2. [2]
    Floro, J.A., Chason, E., Sinclair, M.B., Freund, L.B. and Lucadamo, G.A., Dynamic self-organization of strained islands during SiGe epitaxial growth. Applied Physics Letters, 1998, 73(7): 951–953.CrossRefGoogle Scholar
  3. [3]
    Ross, F.M., Tersoff, J. and Tromp, R.M., Coarsening of self-assembled Ge quantum dots on Si(100). Physical Review Letters, 1998, 80(5): 984–987.CrossRefGoogle Scholar
  4. [4]
    Floro, J.A., Chason, E., Freund, L.B., Twesten, R.D., Hwang, R.Q. and Lucadamo, G.A., Evolution of coherent islands in Si1-xGex/Si(001). Physical Review B, 1999, 59: 1990–1998.CrossRefGoogle Scholar
  5. [5]
    Kim, H.J. and Xie, Y.H., Influence of the wetting-layer growth kinetics on the size and shape of Ge self-assembled quantum dots on Si(001). Applied Physics Letters, 2001, 79(2): 263–265.CrossRefGoogle Scholar
  6. [6]
    Jin, G., Liu, J.L. and Wang, K.L., Temperature effect on the formation of uniform self-assembled Ge dots. Applied Physics Letters, 2003, 83(14): 2847–2849.CrossRefGoogle Scholar
  7. [7]
    Lo, Y.H., New approach to grow pseudomorphic structure over the critical thickness. Applied Physics Letters, 1991, 59(18): 2311–2313.CrossRefGoogle Scholar
  8. [8]
    Yin, H., Huang, R., Hobart, K.D., Suo, Z., Kuan, T.S., Inoki, C.K., Shieh, S.R., Duffy, T.S., Kub, F.J. and Sturm, J.C., Strain relaxation of SiGe islands on compliant oxide. Applied Physics Letters, 2002, 91(12): 9716–9722.Google Scholar
  9. [9]
    Barvosa-Carter, W. and Aziz, M.J., Kinetically driven instability in stressed solids. Physical Review Letters, 1998, 81(7): 1445–1448.CrossRefGoogle Scholar
  10. [10]
    Gerling, M., Gustafsson, A., Rich, D.H., Ritter, D. and Gershoni, D., Roughening transition and solid-state diffusion in short-period InP/In0.53Ga0.47 as superlattice. Applied Physics Letters, 2001, 78(10): 1370–1372.CrossRefGoogle Scholar
  11. [11]
    Liu, F., Rugheimer, P., Mateeva, E., Savage, D.E. and Lagally, M.G., Response of a strained semiconductor structure. Nature, 2002, 416: 498.CrossRefGoogle Scholar
  12. [12]
    Asaro, R.J. and Tiller, W.T., Interface morphology development during stress corrosion cracking — Part I. via surface diffusion. Metallurgical and Materials Transactions B, 1972, 3(7): 1789–1796.CrossRefGoogle Scholar
  13. [13]
    Srolovitz, D.J., On the stability of surfaces of stressed solids. Acta Metallurgica, 1989, 37(2): 621–625.CrossRefGoogle Scholar
  14. [14]
    Liau, Z.L., Strained interface of lattice-mismatched wafer fusion. Physical Review B, 1997, 55(19): 12899–12901.CrossRefGoogle Scholar
  15. [15]
    Zhang, Y., Analysis of dislocation-induced strain field in an idealized wafer-bonded microstructure. Journal of Physics D: Applied Physics, 2007, 40: 1118–1127.CrossRefGoogle Scholar
  16. [16]
    Sridhar, N., Srolovitz, D.J. and Suo, Z., Kinetics of buckling of a compressed film on a viscous substrate. Applied Physics Letters, 2001, 78(17): 2482–2484.CrossRefGoogle Scholar
  17. [17]
    Yang, W.H. and Srolovitz, D.J., Cracklike surface instabilities in stressed solids. Physical Review Letters, 1993, 71(10): 1593–1596.CrossRefGoogle Scholar
  18. [18]
    Huang, R. and Suo, Z., Wrinkling of a compressed elastic film on a viscous layer. Journal of Applied Physics, 2002, 91(3): 1135–1142.CrossRefGoogle Scholar
  19. [19]
    Huang, R. and Suo, Z., Instability of a compressed elastic film on a viscous layer. International Journal of Solids and Structures, 2002, 39: 1791–1802.CrossRefGoogle Scholar
  20. [20]
    Huck, W.T.S., Bowden, N., Onck, P., Pardoen, T., Hutchinson, J.W. and Whitesides, G.M., Ordering of spontaneously formed buckles on planar surfaces. Langmuir, 2000, 16(7): 3497–3501.CrossRefGoogle Scholar
  21. [21]
    Bowden, N., Brittain, S., Evans, A.G., Hutchinson, J.W. and Whitesides, G.M., Spontaneous formation of ordered structures in thin films of metals supported on an elastomeric polymer. Nature, 1998, 393: 146–149.CrossRefGoogle Scholar
  22. [22]
    Zhang, Y. and Zhao, Y., Applicability range of Stoney’s formula and modified formulas for a film/substrate bilayer. Journal of Applied Physics, 2006, 99: 053513.CrossRefGoogle Scholar
  23. [23]
    Timoshenko, S. and Woinosky-Krieger, S., Theory of Plates and Shells, 2nd edn. New York: McGraw-Hill Book Company, 1959.Google Scholar
  24. [24]
    Zhang, Y. and Zhao, Y., Modelling analysis of surface stress on a rectangular cantilever beam. Journal of Physics D: Applied Physics, 2004, 37: 2140–2145.CrossRefGoogle Scholar
  25. [25]
    Zhang, Y. and Murphy, K., Crack propagation in structures subjected to periodic excitation. Acta Mechanica Solida Sinica, 2007, 20(3): 236–246.CrossRefGoogle Scholar
  26. [26]
    Evans, D.R. and Craig, V.S.J., The origin of surface stress induced by adsorption of iodine on gold. Journal of Physical Chemistry B, 2006, 110: 19507–19514.CrossRefGoogle Scholar
  27. [27]
    Zhang Y. and Zhao, Y., An effective method of determining the residual stress gradients. Microsystem Technologies, 2004, 12: 357–364.CrossRefGoogle Scholar
  28. [28]
    Zhang, Y. and Zhao, Y., Static study of cantilever beam stiction under electrostatic force influence. Acta Mechanica Solida Sinica, 2004, 17(2): 104–111.Google Scholar
  29. [29]
    Timoshenko,S., Theory of Elastic Stability, 1st edn. Engineering Societies Monographs, 1936.Google Scholar
  30. [30]
    Colwell, D.J. and Gillett, J.R., A property of the Dirac delta function. International Journal of Mathematical Education in Science and Technology, 1986, 18: 637–638.Google Scholar
  31. [31]
    Johnson, H.T. and Freund, L.B., Mechanics of coherent and dislocated island morphologies in strained epitaxial material system. Journal of Applied Physics, 1997, 81(9): 6081–6090.CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2010

Authors and Affiliations

  1. 1.State Key Laboratory of Nonlinear Mechanics (LNM)Institute of Mechanics, Chinese Academy of SciencesBeijingChina
  2. 2.Faculty of Information and AutomationKunming University of Science and TechnologyKunmingChina

Personalised recommendations