## Full-field measurement of nonuniform stresses of thin films at high temperature |

Optics Express, Vol. 19, Issue 14, pp. 13201-13208 (2011)

http://dx.doi.org/10.1364/OE.19.013201

Acrobat PDF (1668 KB)

### 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

## 1. Introduction

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]

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]

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]

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]

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]

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

## 2. The thermal effects on shear interferometry and the experimental setup

*G*and

_{1}*G*, with the same density (40 lines/mm) separated by a distance △. The diffracted beams from the two gratings are converged to interfere using a lens. Either of the ± 1 diffraction orders is filtered by the filtering aperture to obtain the interferogram recorded by a CCD camera.

_{2}*x*,

*y*) plane is set at the window of the temperature chamber and

*z = f*(

*x,y*) represents the shape function of the specimen in Cartesian coordinates as shown in Fig. 1(b). The refractive index of air is nonuniform, which is expressed as

*n*(

*x,y,z*). With the assumption

*S*(

*x,y*), can be calculated by considering the thermal effect [15

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]

*y*direction, partially differentiating

*S*(

*x*,

*y*) with

*y*leads to

*x*-direction shearing will give a similar result. When the temperature becomes stable, the refractive index of the air will distribute uniformly and can be expressed as [20

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]

*n*(

*t*) and

*n*

_{0}are the refractive indexes of the air at

*t*°C and 0°C, respectively, and

*a*is a constant that equals to 0.00367°C

^{−1}. Substituting Eq. (3) into Eq. (2) and considering both the

*x-*and

*y*-direction shearing give the phase of the interferogram [13

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]

*φ*(

^{(x)}*x,y*) and

*φ*(

^{(y)}*x,y*) are the phase distribution of the fringes obtained by shearing the reflected wavefront in the

*x*and

*y*directions, respectively, and

*p*is the pitch of the gratings

*G*and

_{1}*G*. Since

_{2}*n*

_{0}−1 is much smaller than 1, (

*n*

_{0}−1)/(1 +

*at*) is a higher-order term and can be neglected. It is important to notice that the higher the temperature is, the weaker the thermal effect is on the refractive index. Therefore, the CGS governing equation for high temperature can be given as

*x*direction,

*y*direction, and

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]

*φ*(

^{(x)}*x,y*) = arctan{Im[

*A*(

^{x}*x,y*)]/Re[

*A*(

^{x}*x,y*)]} and

*φ*(

^{(y)}*x,y*) = arctan{Im[

*A*(

^{y}*x,y*)]/Re[

*A*(

^{y}*x,y*)]} for

*x*and

*y*directions shear interferometry, respectively, where Im[

*A*(

*x,y*)] and Re[

*A*(

*x,y*)] denote the imaginary and real parts of complex amplitude

*A*(

*x,y*), and the superscripts

*x*and

*y*represent the shearing directions, respectively. The unwrapping algorithm is performed by MATLAB subroutine, and then the full-field curvatures are obtained from Eq. (5).

## 3. Experimental results and discussion

### 3.1 Substrate curvature measurement

_{2}film grown by thermal oxidation on Si substrate, which is the representative wafer structure widely used in semiconductor industry. The thicknesses of the SiO

_{2}film and Si substrate were 500nm and 500μm, respectively; their radius was 10mm. The geometry size agreed with the assumption

*h*≪

_{f}*h*≪

_{s}*R*. The specimen was placed vertically, as shown in Fig. 1(a). The back of the specimen was supported by a stiff frame through point contact. Moreover, the contact between the specimen and the bottom support was also point contact because the specimen was circled shaped. Therefore, the specimen could expand freely subjected to temperature, and there was no additional stresses induced by the boundary condition. As the temperature was elevated from room temperature to high temperature (e.g. ~300°C), the CGS interferograms were recorded by a CCD camera. Figure 3 shows the interferograms obtained at 300°C. The red fringes in Figs. 3(a) and 3(b) represent the contour curves of the specimen surface slope in lateral (

*x*direction) and vertical (

*y*direction) directions, respectively. The wrapped phase map is calculated by FFT method and shown in Figs. 3(c) and 3(d), respectively. As illustrated by the process flowchart in Fig. 2, unwrapping the phase map in Figs. 3(c) and 3(d) by using the standard MATLAB algorithm and then substituting the results into Eq. (5) would give the curvatures distribution of the substrate in Cartesian coordinates at 300°C. We used Zernike polynomials to fit the unwrapped phase maps and then differentiated them by using Eq. (5). Figures 4(a) and 4(b) show the corresponding system curvatures distribution in

*x*and

*y*directions, respectively, while Fig. 4(c) shows the twist curvature distribution. It is obvious that the curvature distribution is nonuniform and thus violates the Stoney’s formula assumption. The curvatures in the vicinity of the edge become much greater than those in the other area due to the edge effect.

### 3.2 Nonuniform stresses of the thin film

*h*and

*r*represent the curvatures obtained at high temperature and room temperature, respectively. The physical parameters of the system are

*E*= 170GPa,

_{s}*ν*= 0.22,

_{s}*α*= 0.25 × 10

_{s}^{−6}°C

^{−1},

*E*= 71GPa,

_{f}*ν*= 0.16, and

_{f}*α*= 0.5 × 10

_{s}^{−6}°C

^{−1}[1

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

## 4. Conclusions

_{2}film grown on a Si wafer is used to verify the proposed method, which can be potentially extended to higher temperature. These results provide a fundamental approach to understand the thin-film stresses and the feasible measurement method for high temperature.

## Acknowledgments

## References and links

1. | L. B. Freund and S. Suresh, |

2. | G. G. Stoney, “The tension of metallic films deposited by electrolysis,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character |

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. |

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. |

5. | L. B. Freund, “Substrate curvature due to thin film mismatch strain in the nonlinear deformation range,” J. Mech. Phys. Solids |

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 |

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. |

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. |

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. |

10. | X. Feng, Y. Huang, and A. J. Rosakis, “Stresses in a multilayer thin film/substrate system subjected to nonuniform temperature,” J. Appl. Mech. |

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. |

12. | E. Chason and B. W. Sheldon, “Monitoring stress in thin films during processing,” Surf. Eng. |

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. |

14. | H. V. Tippur, “Coherent gradient sensing: a Fourier optics analysis and applications to fracture,” Appl. Opt. |

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 |

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. |

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. |

18. | D. Post, B. Han, and P. Ifju, |

19. | Y. Y. Hung, “Shearography for non-destructive evaluation of composite structures,” Opt. Lasers Eng. |

20. | J. C. Murphy and L. C. Aamodt, “Photothermal spectroscopy using optical beam probing - mirage effect,” J. Appl. Phys. |

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. |

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 |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.6780) Instrumentation, measurement, and metrology : Temperature

(240.0310) Optics at surfaces : Thin films

(310.4925) Thin films : Other properties (stress, chemical, etc.)

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: March 21, 2011

Revised Manuscript: May 20, 2011

Manuscript Accepted: June 2, 2011

Published: June 23, 2011

**Citation**

Xuelin Dong, Xue Feng, Keh-Chih Hwang, Shaopeng Ma, and Qinwei Ma, "Full-field measurement of nonuniform stresses of thin films at high temperature," Opt. Express **19**, 13201-13208 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13201

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### References

- L. B. Freund and S. Suresh, Thin Film Materials; Stress, Defect Formation and Surface Evolution (Cambridge University Press, 2003). [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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. 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]
- 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]
- E. Chason and B. W. Sheldon, “Monitoring stress in thin films during processing,” Surf. Eng. 19(5), 387–391 (2003). [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]
- H. V. Tippur, “Coherent gradient sensing: a Fourier optics analysis and applications to fracture,” Appl. Opt. 31(22), 4428–4439 (1992). [CrossRef] [PubMed]
- 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]
- 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. 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]
- D. Post, B. Han, and P. Ifju, High Sensitivity Moire: Experimental Analysis for Mechanics and Materials (Springer-Verlag, 1994).
- Y. Y. Hung, “Shearography for non-destructive evaluation of composite structures,” Opt. Lasers Eng. 24(2–3), 161–182 (1996). [CrossRef]
- J. C. Murphy and L. C. Aamodt, “Photothermal spectroscopy using optical beam probing - mirage effect,” J. Appl. Phys. 51(9), 4580–4588 (1980). [CrossRef]
- 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]
- 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]

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