Abstract

Coherent gradient sensing (CGS), a shear interferometry method, is developed to measure the full-field curvatures of a film/substrate system at high temperature. We obtain the relationship between an interferogram phase and specimen topography, accounting for temperature effect. The self-interference of CGS combined with designed setup can reduce the air effect. The full-field phases can be extracted by fast Fourier transform. Both nonuniform thin-film stresses and interfacial stresses are obtained by the extended Stoney’s formula. The evolution of thermo-stresses verifies the feasibility of the proposed interferometry method and implies the “nonlocal” effect featured by the experimental results.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. B. Freund and S. Suresh, Thin Film Materials; Stress, Defect Formation and Surface Evolution (Cambridge University Press, 2003).
    [PubMed]
  2. G. G. Stoney, “The tension of metallic films deposited by electrolysis,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character 82(553), 172–175 (1909).
    [CrossRef]
  3. L. B. Freund, J. A. Floro, and E. Chason, “Extensions of the Stoney formula for substrate curvature to configurations with thin substrates or large deformations,” Appl. Phys. Lett. 74(14), 1987–1989 (1999).
    [CrossRef]
  4. T. S. Park and S. Suresh, “Effects of line and passivation geometry on curvature evolution during processing and thermal cycling in copper interconnect lines,” Acta Mater. 48(12), 3169–3175 (2000).
    [CrossRef]
  5. L. B. Freund, “Substrate curvature due to thin film mismatch strain in the nonlinear deformation range,” J. Mech. Phys. Solids 48(6–7), 1159–1174 (2000).
    [CrossRef]
  6. Y. Huang and A. J. Rosakis, “Extension of Stoney's formula to non-uniform temperature distributions in thin film/substrate systems. The case of radial symmetry,” J. Mech. Phys. Solids 53(11), 2483–2500 (2005).
    [CrossRef]
  7. X. Feng, Y. G. Huang, H. Q. Jiang, D. Ngo, and A. J. Rosakis, “The effect of thin film/substrate radii on the stoney formula for thin film/substrate subjected to nonuniform axisymmetric misfit strain and temperature,” J. Mech. Mater. Struct. 1(6), 1041–1053 (2006).
    [CrossRef]
  8. D. Ngo, X. Feng, Y. Huang, A. J. Rosakis, and M. A. Brown, “Thin film/substrate systems featuring arbitrary film thickness and misfit strain distributions. Part I: analysis for obtaining film stress from non-local curvature information,” Int. J. Solids Struct. 44(6), 1745–1754 (2007).
    [CrossRef]
  9. M. A. Brown, A. J. Rosakis, X. Feng, Y. Huang, and E. Ustundag, “Thin film substrate systems featuring arbitrary film thickness and misfit strain distributions. Part II: experimental validation of the non-local stress curvature relations,” Int. J. Solids Struct. 44(6), 1755–1767 (2007).
    [CrossRef]
  10. X. Feng, Y. Huang, and A. J. Rosakis, “Stresses in a multilayer thin film/substrate system subjected to nonuniform temperature,” J. Appl. Mech. 75(2), 021022 (2008).
    [CrossRef]
  11. P. A. Flinn, D. S. Gardner, and W. D. Nix, “Measurement and interpretation of stress in aluminum-based metallization as a function of thermal history,” IEEE Trans. Electron. Dev. 34(3), 689–699 (1987).
    [CrossRef]
  12. E. Chason and B. W. Sheldon, “Monitoring stress in thin films during processing,” Surf. Eng. 19(5), 387–391 (2003).
    [CrossRef]
  13. H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental results,” Int. J. Fract. 48(3), 193–204 (1991).
    [CrossRef]
  14. H. V. Tippur, “Coherent gradient sensing: a Fourier optics analysis and applications to fracture,” Appl. Opt. 31(22), 4428–4439 (1992).
    [CrossRef] [PubMed]
  15. A. J. Rosakis, R. P. Singh, Y. Tsuji, E. Kolawa, and N. R. Moore, “Full field measurements of curvature using coherent gradient sensing: application to thin film characterization,” Thin Solid Films 325(1–2), 42–54 (1998).
    [CrossRef]
  16. M. A. Brown, T.-S. Park, A. Rosakis, E. Ustundag, Y. Huang, N. Tamura, and B. Valek, “A comparison of X-ray microdiffraction and coherent gradient sensing in measuring discontinuous curvatures in thin film: substrate systems,” J. Appl. Mech. 73(5), 723–729 (2006).
    [CrossRef]
  17. J. Tao, L. H. Lee, and J. C. Bilello, “Nondestructive evaluation of residual-stresses in thin-films via x-ray-diffraction topography methods,” J. Electron. Mater. 20(7), 819–825 (1991).
    [CrossRef]
  18. D. Post, B. Han, and P. Ifju, High Sensitivity Moire: Experimental Analysis for Mechanics and Materials (Springer-Verlag, 1994).
  19. Y. Y. Hung, “Shearography for non-destructive evaluation of composite structures,” Opt. Lasers Eng. 24(2–3), 161–182 (1996).
    [CrossRef]
  20. J. C. Murphy and L. C. Aamodt, “Photothermal spectroscopy using optical beam probing - mirage effect,” J. Appl. Phys. 51(9), 4580–4588 (1980).
    [CrossRef]
  21. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982).
    [CrossRef]
  22. T. S. Park, S. Suresh, A. J. Rosakis, and J. Ryu, “Measurement of full-field curvature and geometrical instability of thin film-substrate systems through CGS interferometry,” J. Mech. Phys. Solids 51(11–12), 2191–2211 (2003).
    [CrossRef]

2008 (1)

X. Feng, Y. Huang, and A. J. Rosakis, “Stresses in a multilayer thin film/substrate system subjected to nonuniform temperature,” J. Appl. Mech. 75(2), 021022 (2008).
[CrossRef]

2007 (2)

D. Ngo, X. Feng, Y. Huang, A. J. Rosakis, and M. A. Brown, “Thin film/substrate systems featuring arbitrary film thickness and misfit strain distributions. Part I: analysis for obtaining film stress from non-local curvature information,” Int. J. Solids Struct. 44(6), 1745–1754 (2007).
[CrossRef]

M. A. Brown, A. J. Rosakis, X. Feng, Y. Huang, and E. Ustundag, “Thin film substrate systems featuring arbitrary film thickness and misfit strain distributions. Part II: experimental validation of the non-local stress curvature relations,” Int. J. Solids Struct. 44(6), 1755–1767 (2007).
[CrossRef]

2006 (2)

X. Feng, Y. G. Huang, H. Q. Jiang, D. Ngo, and A. J. Rosakis, “The effect of thin film/substrate radii on the stoney formula for thin film/substrate subjected to nonuniform axisymmetric misfit strain and temperature,” J. Mech. Mater. Struct. 1(6), 1041–1053 (2006).
[CrossRef]

M. A. Brown, T.-S. Park, A. Rosakis, E. Ustundag, Y. Huang, N. Tamura, and B. Valek, “A comparison of X-ray microdiffraction and coherent gradient sensing in measuring discontinuous curvatures in thin film: substrate systems,” J. Appl. Mech. 73(5), 723–729 (2006).
[CrossRef]

2005 (1)

Y. Huang and A. J. Rosakis, “Extension of Stoney's formula to non-uniform temperature distributions in thin film/substrate systems. The case of radial symmetry,” J. Mech. Phys. Solids 53(11), 2483–2500 (2005).
[CrossRef]

2003 (2)

E. Chason and B. W. Sheldon, “Monitoring stress in thin films during processing,” Surf. Eng. 19(5), 387–391 (2003).
[CrossRef]

T. S. Park, S. Suresh, A. J. Rosakis, and J. Ryu, “Measurement of full-field curvature and geometrical instability of thin film-substrate systems through CGS interferometry,” J. Mech. Phys. Solids 51(11–12), 2191–2211 (2003).
[CrossRef]

2000 (2)

T. S. Park and S. Suresh, “Effects of line and passivation geometry on curvature evolution during processing and thermal cycling in copper interconnect lines,” Acta Mater. 48(12), 3169–3175 (2000).
[CrossRef]

L. B. Freund, “Substrate curvature due to thin film mismatch strain in the nonlinear deformation range,” J. Mech. Phys. Solids 48(6–7), 1159–1174 (2000).
[CrossRef]

1999 (1)

L. B. Freund, J. A. Floro, and E. Chason, “Extensions of the Stoney formula for substrate curvature to configurations with thin substrates or large deformations,” Appl. Phys. Lett. 74(14), 1987–1989 (1999).
[CrossRef]

1998 (1)

A. J. Rosakis, R. P. Singh, Y. Tsuji, E. Kolawa, and N. R. Moore, “Full field measurements of curvature using coherent gradient sensing: application to thin film characterization,” Thin Solid Films 325(1–2), 42–54 (1998).
[CrossRef]

1996 (1)

Y. Y. Hung, “Shearography for non-destructive evaluation of composite structures,” Opt. Lasers Eng. 24(2–3), 161–182 (1996).
[CrossRef]

1992 (1)

1991 (2)

J. Tao, L. H. Lee, and J. C. Bilello, “Nondestructive evaluation of residual-stresses in thin-films via x-ray-diffraction topography methods,” J. Electron. Mater. 20(7), 819–825 (1991).
[CrossRef]

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental results,” Int. J. Fract. 48(3), 193–204 (1991).
[CrossRef]

1987 (1)

P. A. Flinn, D. S. Gardner, and W. D. Nix, “Measurement and interpretation of stress in aluminum-based metallization as a function of thermal history,” IEEE Trans. Electron. Dev. 34(3), 689–699 (1987).
[CrossRef]

1982 (1)

1980 (1)

J. C. Murphy and L. C. Aamodt, “Photothermal spectroscopy using optical beam probing - mirage effect,” J. Appl. Phys. 51(9), 4580–4588 (1980).
[CrossRef]

1909 (1)

G. G. Stoney, “The tension of metallic films deposited by electrolysis,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character 82(553), 172–175 (1909).
[CrossRef]

Aamodt, L. C.

J. C. Murphy and L. C. Aamodt, “Photothermal spectroscopy using optical beam probing - mirage effect,” J. Appl. Phys. 51(9), 4580–4588 (1980).
[CrossRef]

Bilello, J. C.

J. Tao, L. H. Lee, and J. C. Bilello, “Nondestructive evaluation of residual-stresses in thin-films via x-ray-diffraction topography methods,” J. Electron. Mater. 20(7), 819–825 (1991).
[CrossRef]

Brown, M. A.

D. Ngo, X. Feng, Y. Huang, A. J. Rosakis, and M. A. Brown, “Thin film/substrate systems featuring arbitrary film thickness and misfit strain distributions. Part I: analysis for obtaining film stress from non-local curvature information,” Int. J. Solids Struct. 44(6), 1745–1754 (2007).
[CrossRef]

M. A. Brown, A. J. Rosakis, X. Feng, Y. Huang, and E. Ustundag, “Thin film substrate systems featuring arbitrary film thickness and misfit strain distributions. Part II: experimental validation of the non-local stress curvature relations,” Int. J. Solids Struct. 44(6), 1755–1767 (2007).
[CrossRef]

M. A. Brown, T.-S. Park, A. Rosakis, E. Ustundag, Y. Huang, N. Tamura, and B. Valek, “A comparison of X-ray microdiffraction and coherent gradient sensing in measuring discontinuous curvatures in thin film: substrate systems,” J. Appl. Mech. 73(5), 723–729 (2006).
[CrossRef]

Chason, E.

E. Chason and B. W. Sheldon, “Monitoring stress in thin films during processing,” Surf. Eng. 19(5), 387–391 (2003).
[CrossRef]

L. B. Freund, J. A. Floro, and E. Chason, “Extensions of the Stoney formula for substrate curvature to configurations with thin substrates or large deformations,” Appl. Phys. Lett. 74(14), 1987–1989 (1999).
[CrossRef]

Feng, X.

X. Feng, Y. Huang, and A. J. Rosakis, “Stresses in a multilayer thin film/substrate system subjected to nonuniform temperature,” J. Appl. Mech. 75(2), 021022 (2008).
[CrossRef]

M. A. Brown, A. J. Rosakis, X. Feng, Y. Huang, and E. Ustundag, “Thin film substrate systems featuring arbitrary film thickness and misfit strain distributions. Part II: experimental validation of the non-local stress curvature relations,” Int. J. Solids Struct. 44(6), 1755–1767 (2007).
[CrossRef]

D. Ngo, X. Feng, Y. Huang, A. J. Rosakis, and M. A. Brown, “Thin film/substrate systems featuring arbitrary film thickness and misfit strain distributions. Part I: analysis for obtaining film stress from non-local curvature information,” Int. J. Solids Struct. 44(6), 1745–1754 (2007).
[CrossRef]

X. Feng, Y. G. Huang, H. Q. Jiang, D. Ngo, and A. J. Rosakis, “The effect of thin film/substrate radii on the stoney formula for thin film/substrate subjected to nonuniform axisymmetric misfit strain and temperature,” J. Mech. Mater. Struct. 1(6), 1041–1053 (2006).
[CrossRef]

Flinn, P. A.

P. A. Flinn, D. S. Gardner, and W. D. Nix, “Measurement and interpretation of stress in aluminum-based metallization as a function of thermal history,” IEEE Trans. Electron. Dev. 34(3), 689–699 (1987).
[CrossRef]

Floro, J. A.

L. B. Freund, J. A. Floro, and E. Chason, “Extensions of the Stoney formula for substrate curvature to configurations with thin substrates or large deformations,” Appl. Phys. Lett. 74(14), 1987–1989 (1999).
[CrossRef]

Freund, L. B.

L. B. Freund, “Substrate curvature due to thin film mismatch strain in the nonlinear deformation range,” J. Mech. Phys. Solids 48(6–7), 1159–1174 (2000).
[CrossRef]

L. B. Freund, J. A. Floro, and E. Chason, “Extensions of the Stoney formula for substrate curvature to configurations with thin substrates or large deformations,” Appl. Phys. Lett. 74(14), 1987–1989 (1999).
[CrossRef]

Gardner, D. S.

P. A. Flinn, D. S. Gardner, and W. D. Nix, “Measurement and interpretation of stress in aluminum-based metallization as a function of thermal history,” IEEE Trans. Electron. Dev. 34(3), 689–699 (1987).
[CrossRef]

Huang, Y.

X. Feng, Y. Huang, and A. J. Rosakis, “Stresses in a multilayer thin film/substrate system subjected to nonuniform temperature,” J. Appl. Mech. 75(2), 021022 (2008).
[CrossRef]

M. A. Brown, A. J. Rosakis, X. Feng, Y. Huang, and E. Ustundag, “Thin film substrate systems featuring arbitrary film thickness and misfit strain distributions. Part II: experimental validation of the non-local stress curvature relations,” Int. J. Solids Struct. 44(6), 1755–1767 (2007).
[CrossRef]

D. Ngo, X. Feng, Y. Huang, A. J. Rosakis, and M. A. Brown, “Thin film/substrate systems featuring arbitrary film thickness and misfit strain distributions. Part I: analysis for obtaining film stress from non-local curvature information,” Int. J. Solids Struct. 44(6), 1745–1754 (2007).
[CrossRef]

M. A. Brown, T.-S. Park, A. Rosakis, E. Ustundag, Y. Huang, N. Tamura, and B. Valek, “A comparison of X-ray microdiffraction and coherent gradient sensing in measuring discontinuous curvatures in thin film: substrate systems,” J. Appl. Mech. 73(5), 723–729 (2006).
[CrossRef]

Y. Huang and A. J. Rosakis, “Extension of Stoney's formula to non-uniform temperature distributions in thin film/substrate systems. The case of radial symmetry,” J. Mech. Phys. Solids 53(11), 2483–2500 (2005).
[CrossRef]

Huang, Y. G.

X. Feng, Y. G. Huang, H. Q. Jiang, D. Ngo, and A. J. Rosakis, “The effect of thin film/substrate radii on the stoney formula for thin film/substrate subjected to nonuniform axisymmetric misfit strain and temperature,” J. Mech. Mater. Struct. 1(6), 1041–1053 (2006).
[CrossRef]

Hung, Y. Y.

Y. Y. Hung, “Shearography for non-destructive evaluation of composite structures,” Opt. Lasers Eng. 24(2–3), 161–182 (1996).
[CrossRef]

Ina, H.

Jiang, H. Q.

X. Feng, Y. G. Huang, H. Q. Jiang, D. Ngo, and A. J. Rosakis, “The effect of thin film/substrate radii on the stoney formula for thin film/substrate subjected to nonuniform axisymmetric misfit strain and temperature,” J. Mech. Mater. Struct. 1(6), 1041–1053 (2006).
[CrossRef]

Kobayashi, S.

Kolawa, E.

A. J. Rosakis, R. P. Singh, Y. Tsuji, E. Kolawa, and N. R. Moore, “Full field measurements of curvature using coherent gradient sensing: application to thin film characterization,” Thin Solid Films 325(1–2), 42–54 (1998).
[CrossRef]

Krishnaswamy, S.

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental results,” Int. J. Fract. 48(3), 193–204 (1991).
[CrossRef]

Lee, L. H.

J. Tao, L. H. Lee, and J. C. Bilello, “Nondestructive evaluation of residual-stresses in thin-films via x-ray-diffraction topography methods,” J. Electron. Mater. 20(7), 819–825 (1991).
[CrossRef]

Moore, N. R.

A. J. Rosakis, R. P. Singh, Y. Tsuji, E. Kolawa, and N. R. Moore, “Full field measurements of curvature using coherent gradient sensing: application to thin film characterization,” Thin Solid Films 325(1–2), 42–54 (1998).
[CrossRef]

Murphy, J. C.

J. C. Murphy and L. C. Aamodt, “Photothermal spectroscopy using optical beam probing - mirage effect,” J. Appl. Phys. 51(9), 4580–4588 (1980).
[CrossRef]

Ngo, D.

D. Ngo, X. Feng, Y. Huang, A. J. Rosakis, and M. A. Brown, “Thin film/substrate systems featuring arbitrary film thickness and misfit strain distributions. Part I: analysis for obtaining film stress from non-local curvature information,” Int. J. Solids Struct. 44(6), 1745–1754 (2007).
[CrossRef]

X. Feng, Y. G. Huang, H. Q. Jiang, D. Ngo, and A. J. Rosakis, “The effect of thin film/substrate radii on the stoney formula for thin film/substrate subjected to nonuniform axisymmetric misfit strain and temperature,” J. Mech. Mater. Struct. 1(6), 1041–1053 (2006).
[CrossRef]

Nix, W. D.

P. A. Flinn, D. S. Gardner, and W. D. Nix, “Measurement and interpretation of stress in aluminum-based metallization as a function of thermal history,” IEEE Trans. Electron. Dev. 34(3), 689–699 (1987).
[CrossRef]

Park, T. S.

T. S. Park, S. Suresh, A. J. Rosakis, and J. Ryu, “Measurement of full-field curvature and geometrical instability of thin film-substrate systems through CGS interferometry,” J. Mech. Phys. Solids 51(11–12), 2191–2211 (2003).
[CrossRef]

T. S. Park and S. Suresh, “Effects of line and passivation geometry on curvature evolution during processing and thermal cycling in copper interconnect lines,” Acta Mater. 48(12), 3169–3175 (2000).
[CrossRef]

Park, T.-S.

M. A. Brown, T.-S. Park, A. Rosakis, E. Ustundag, Y. Huang, N. Tamura, and B. Valek, “A comparison of X-ray microdiffraction and coherent gradient sensing in measuring discontinuous curvatures in thin film: substrate systems,” J. Appl. Mech. 73(5), 723–729 (2006).
[CrossRef]

Rosakis, A.

M. A. Brown, T.-S. Park, A. Rosakis, E. Ustundag, Y. Huang, N. Tamura, and B. Valek, “A comparison of X-ray microdiffraction and coherent gradient sensing in measuring discontinuous curvatures in thin film: substrate systems,” J. Appl. Mech. 73(5), 723–729 (2006).
[CrossRef]

Rosakis, A. J.

X. Feng, Y. Huang, and A. J. Rosakis, “Stresses in a multilayer thin film/substrate system subjected to nonuniform temperature,” J. Appl. Mech. 75(2), 021022 (2008).
[CrossRef]

D. Ngo, X. Feng, Y. Huang, A. J. Rosakis, and M. A. Brown, “Thin film/substrate systems featuring arbitrary film thickness and misfit strain distributions. Part I: analysis for obtaining film stress from non-local curvature information,” Int. J. Solids Struct. 44(6), 1745–1754 (2007).
[CrossRef]

M. A. Brown, A. J. Rosakis, X. Feng, Y. Huang, and E. Ustundag, “Thin film substrate systems featuring arbitrary film thickness and misfit strain distributions. Part II: experimental validation of the non-local stress curvature relations,” Int. J. Solids Struct. 44(6), 1755–1767 (2007).
[CrossRef]

X. Feng, Y. G. Huang, H. Q. Jiang, D. Ngo, and A. J. Rosakis, “The effect of thin film/substrate radii on the stoney formula for thin film/substrate subjected to nonuniform axisymmetric misfit strain and temperature,” J. Mech. Mater. Struct. 1(6), 1041–1053 (2006).
[CrossRef]

Y. Huang and A. J. Rosakis, “Extension of Stoney's formula to non-uniform temperature distributions in thin film/substrate systems. The case of radial symmetry,” J. Mech. Phys. Solids 53(11), 2483–2500 (2005).
[CrossRef]

T. S. Park, S. Suresh, A. J. Rosakis, and J. Ryu, “Measurement of full-field curvature and geometrical instability of thin film-substrate systems through CGS interferometry,” J. Mech. Phys. Solids 51(11–12), 2191–2211 (2003).
[CrossRef]

A. J. Rosakis, R. P. Singh, Y. Tsuji, E. Kolawa, and N. R. Moore, “Full field measurements of curvature using coherent gradient sensing: application to thin film characterization,” Thin Solid Films 325(1–2), 42–54 (1998).
[CrossRef]

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental results,” Int. J. Fract. 48(3), 193–204 (1991).
[CrossRef]

Ryu, J.

T. S. Park, S. Suresh, A. J. Rosakis, and J. Ryu, “Measurement of full-field curvature and geometrical instability of thin film-substrate systems through CGS interferometry,” J. Mech. Phys. Solids 51(11–12), 2191–2211 (2003).
[CrossRef]

Sheldon, B. W.

E. Chason and B. W. Sheldon, “Monitoring stress in thin films during processing,” Surf. Eng. 19(5), 387–391 (2003).
[CrossRef]

Singh, R. P.

A. J. Rosakis, R. P. Singh, Y. Tsuji, E. Kolawa, and N. R. Moore, “Full field measurements of curvature using coherent gradient sensing: application to thin film characterization,” Thin Solid Films 325(1–2), 42–54 (1998).
[CrossRef]

Stoney, G. G.

G. G. Stoney, “The tension of metallic films deposited by electrolysis,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character 82(553), 172–175 (1909).
[CrossRef]

Suresh, S.

T. S. Park, S. Suresh, A. J. Rosakis, and J. Ryu, “Measurement of full-field curvature and geometrical instability of thin film-substrate systems through CGS interferometry,” J. Mech. Phys. Solids 51(11–12), 2191–2211 (2003).
[CrossRef]

T. S. Park and S. Suresh, “Effects of line and passivation geometry on curvature evolution during processing and thermal cycling in copper interconnect lines,” Acta Mater. 48(12), 3169–3175 (2000).
[CrossRef]

Takeda, M.

Tamura, N.

M. A. Brown, T.-S. Park, A. Rosakis, E. Ustundag, Y. Huang, N. Tamura, and B. Valek, “A comparison of X-ray microdiffraction and coherent gradient sensing in measuring discontinuous curvatures in thin film: substrate systems,” J. Appl. Mech. 73(5), 723–729 (2006).
[CrossRef]

Tao, J.

J. Tao, L. H. Lee, and J. C. Bilello, “Nondestructive evaluation of residual-stresses in thin-films via x-ray-diffraction topography methods,” J. Electron. Mater. 20(7), 819–825 (1991).
[CrossRef]

Tippur, H. V.

H. V. Tippur, “Coherent gradient sensing: a Fourier optics analysis and applications to fracture,” Appl. Opt. 31(22), 4428–4439 (1992).
[CrossRef] [PubMed]

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental results,” Int. J. Fract. 48(3), 193–204 (1991).
[CrossRef]

Tsuji, Y.

A. J. Rosakis, R. P. Singh, Y. Tsuji, E. Kolawa, and N. R. Moore, “Full field measurements of curvature using coherent gradient sensing: application to thin film characterization,” Thin Solid Films 325(1–2), 42–54 (1998).
[CrossRef]

Ustundag, E.

M. A. Brown, A. J. Rosakis, X. Feng, Y. Huang, and E. Ustundag, “Thin film substrate systems featuring arbitrary film thickness and misfit strain distributions. Part II: experimental validation of the non-local stress curvature relations,” Int. J. Solids Struct. 44(6), 1755–1767 (2007).
[CrossRef]

M. A. Brown, T.-S. Park, A. Rosakis, E. Ustundag, Y. Huang, N. Tamura, and B. Valek, “A comparison of X-ray microdiffraction and coherent gradient sensing in measuring discontinuous curvatures in thin film: substrate systems,” J. Appl. Mech. 73(5), 723–729 (2006).
[CrossRef]

Valek, B.

M. A. Brown, T.-S. Park, A. Rosakis, E. Ustundag, Y. Huang, N. Tamura, and B. Valek, “A comparison of X-ray microdiffraction and coherent gradient sensing in measuring discontinuous curvatures in thin film: substrate systems,” J. Appl. Mech. 73(5), 723–729 (2006).
[CrossRef]

Acta Mater. (1)

T. S. Park and S. Suresh, “Effects of line and passivation geometry on curvature evolution during processing and thermal cycling in copper interconnect lines,” Acta Mater. 48(12), 3169–3175 (2000).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

L. B. Freund, J. A. Floro, and E. Chason, “Extensions of the Stoney formula for substrate curvature to configurations with thin substrates or large deformations,” Appl. Phys. Lett. 74(14), 1987–1989 (1999).
[CrossRef]

IEEE Trans. Electron. Dev. (1)

P. A. Flinn, D. S. Gardner, and W. D. Nix, “Measurement and interpretation of stress in aluminum-based metallization as a function of thermal history,” IEEE Trans. Electron. Dev. 34(3), 689–699 (1987).
[CrossRef]

Int. J. Fract. (1)

H. V. Tippur, S. Krishnaswamy, and A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental results,” Int. J. Fract. 48(3), 193–204 (1991).
[CrossRef]

Int. J. Solids Struct. (2)

D. Ngo, X. Feng, Y. Huang, A. J. Rosakis, and M. A. Brown, “Thin film/substrate systems featuring arbitrary film thickness and misfit strain distributions. Part I: analysis for obtaining film stress from non-local curvature information,” Int. J. Solids Struct. 44(6), 1745–1754 (2007).
[CrossRef]

M. A. Brown, A. J. Rosakis, X. Feng, Y. Huang, and E. Ustundag, “Thin film substrate systems featuring arbitrary film thickness and misfit strain distributions. Part II: experimental validation of the non-local stress curvature relations,” Int. J. Solids Struct. 44(6), 1755–1767 (2007).
[CrossRef]

J. Appl. Mech. (2)

X. Feng, Y. Huang, and A. J. Rosakis, “Stresses in a multilayer thin film/substrate system subjected to nonuniform temperature,” J. Appl. Mech. 75(2), 021022 (2008).
[CrossRef]

M. A. Brown, T.-S. Park, A. Rosakis, E. Ustundag, Y. Huang, N. Tamura, and B. Valek, “A comparison of X-ray microdiffraction and coherent gradient sensing in measuring discontinuous curvatures in thin film: substrate systems,” J. Appl. Mech. 73(5), 723–729 (2006).
[CrossRef]

J. Appl. Phys. (1)

J. C. Murphy and L. C. Aamodt, “Photothermal spectroscopy using optical beam probing - mirage effect,” J. Appl. Phys. 51(9), 4580–4588 (1980).
[CrossRef]

J. Electron. Mater. (1)

J. Tao, L. H. Lee, and J. C. Bilello, “Nondestructive evaluation of residual-stresses in thin-films via x-ray-diffraction topography methods,” J. Electron. Mater. 20(7), 819–825 (1991).
[CrossRef]

J. Mech. Mater. Struct. (1)

X. Feng, Y. G. Huang, H. Q. Jiang, D. Ngo, and A. J. Rosakis, “The effect of thin film/substrate radii on the stoney formula for thin film/substrate subjected to nonuniform axisymmetric misfit strain and temperature,” J. Mech. Mater. Struct. 1(6), 1041–1053 (2006).
[CrossRef]

J. Mech. Phys. Solids (3)

L. B. Freund, “Substrate curvature due to thin film mismatch strain in the nonlinear deformation range,” J. Mech. Phys. Solids 48(6–7), 1159–1174 (2000).
[CrossRef]

Y. Huang and A. J. Rosakis, “Extension of Stoney's formula to non-uniform temperature distributions in thin film/substrate systems. The case of radial symmetry,” J. Mech. Phys. Solids 53(11), 2483–2500 (2005).
[CrossRef]

T. S. Park, S. Suresh, A. J. Rosakis, and J. Ryu, “Measurement of full-field curvature and geometrical instability of thin film-substrate systems through CGS interferometry,” J. Mech. Phys. Solids 51(11–12), 2191–2211 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lasers Eng. (1)

Y. Y. Hung, “Shearography for non-destructive evaluation of composite structures,” Opt. Lasers Eng. 24(2–3), 161–182 (1996).
[CrossRef]

Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character (1)

G. G. Stoney, “The tension of metallic films deposited by electrolysis,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character 82(553), 172–175 (1909).
[CrossRef]

Surf. Eng. (1)

E. Chason and B. W. Sheldon, “Monitoring stress in thin films during processing,” Surf. Eng. 19(5), 387–391 (2003).
[CrossRef]

Thin Solid Films (1)

A. J. Rosakis, R. P. Singh, Y. Tsuji, E. Kolawa, and N. R. Moore, “Full field measurements of curvature using coherent gradient sensing: application to thin film characterization,” Thin Solid Films 325(1–2), 42–54 (1998).
[CrossRef]

Other (2)

D. Post, B. Han, and P. Ifju, High Sensitivity Moire: Experimental Analysis for Mechanics and Materials (Springer-Verlag, 1994).

L. B. Freund and S. Suresh, Thin Film Materials; Stress, Defect Formation and Surface Evolution (Cambridge University Press, 2003).
[PubMed]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

The experimental setup and the thermal effect: (a) schematic of CGS setup for high temperature measurement, (b) thermal effect on the optical path length.

Fig. 2
Fig. 2

The flow chart of the measurement of nonuniform film stresses.

Fig. 3
Fig. 3

Interferograms at 300°C and their wrapped phase maps: (a) interferogram obtained by shearing laterally, (b) interferogram obtained by shearing vertically, (c) wrapped phase map for Fig. 3(a), (d) wrapped phase map for Fig. 3(b).

Fig. 4
Fig. 4

The substrate curvatures measured at 300°C: (a) curvature κ x x in lateral direction, (b) curvature κ y y in vertical direction, (c) twist curvature κ x y .

Fig. 5
Fig. 5

The nonuniform stresses of the thin film measured at 300°C: (a) stress σ r r ( f ) in radial direction, (b) stress σ θ θ ( f ) in circumferential direction, (c) shear stress σ r θ ( f ) , (d) interfacial shear stress τ r in radial direction, (e) interfacial shear stress τ θ in circumferential direction.

Fig. 6
Fig. 6

The film stresses in radial and circumferential directions at the central point of the specimen vs. temperature.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

S ( x , y ) = 2 0 f ( x , y ) n ( x , y , z ) d z .
S ( x , y ) y = 2 0 f ( x , y ) n ( x , y , z ) y d z + 2 n ( x , f ( x , y ) , z ) f ( x , y ) y .
n ( t ) = 1 + n 0 1 1 + a t ,
{ φ x ( x , y ) = 4 π Δ p ( 1 + n 0 1 1 + a t ) f ( x , y ) x φ y ( x , y ) = 4 π Δ p ( 1 + n 0 1 1 + a t ) f ( x , y ) y ,
{ κ x x = 2 f ( x , y ) x 2 = p 4 π Δ φ ( x ) ( x , y ) x κ y y = 2 f ( x , y ) y 2 = p 4 π Δ φ ( y ) ( x , y ) y κ x y = κ y x = 2 f ( x , y ) x y = p 4 π Δ φ ( y ) ( x , y ) x ,
σ r r ( f ) + σ θ θ ( f ) = E s h s 2 6 ( 1 ν s ) h f { κ r r + κ θ θ ¯ + ( 1 + ν f ) [ ( 1 + ν s ) α s 2 α f ] ( 1 + ν s ) [ ( 1 + ν s ) α s ( 1 + ν f ) α f ] ( κ r r + κ θ θ κ r r + κ θ θ ¯ ) + [ 3 + ν s 1 + ν s 2 ( 1 + ν f ) [ ( 1 + ν s ) α s 2 α f ] ( 1 + ν s ) [ ( 1 + ν s ) α s ( 1 + ν f ) α f ] ] × m = 1 ( m + 1 ) ( r R ) m ( C m cos m θ + S m sin m θ ) } ,
σ r r ( f ) σ θ θ ( f ) = E s h s 2 α s ( 1 ν f ) 6 ( 1 ν s ) h f 1 ( 1 + ν s ) α s ( 1 + ν f ) α f                      × { κ r r κ θ θ m = 1 ( m + 1 ) [ m ( r R ) m ( m 1 ) ( r R ) m 2 ] × ( C m cos m θ + S m sin m θ ) } ,
σ r θ ( f ) = E s h s 2 α s ( 1 ν f ) 6 ( 1 ν s ) h f 1 ( 1 + ν s ) α s ( 1 + ν f ) α f            × { κ r θ + 1 2 m = 1 ( m + 1 ) [ m ( r R ) m ( m 1 ) ( r R ) m 2 ] ( C m sin m θ S m cos m θ ) } ,
τ r = E s h s 2 6 ( 1 ν s 2 ) { r ( κ r r + κ θ θ ) 1 ν s 2 R m = 1 m ( m + 1 ) ( r R ) m 1 ( C m cos m θ + S m sin m θ ) } ,
τ θ = E s h s 2 6 ( 1 ν s 2 ) { 1 r θ ( κ r r + κ θ θ ) + 1 ν s 2 R m = 1 m ( m + 1 ) ( r R ) m 1 ( C m sin m θ S m cos m θ ) } ,

Metrics