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

Thin films deposited on various types of substrates are applied in many technologies, including electronic circuits, integrated optical devices, microelectromechanical systems (MEMS), systems-on-a-chip structures, as well as coatings used for thermal protection, oxidation, and corrosion resistance. The stresses in the films induced by fabrication or diverse processes are crucial to the performance and reliability of these devices. It is recognized that the mismatch in thermal expansion coefficients between the film and substrate subjected to a changing temperature environment is one of the dominant factors that cause the undesirable stresses. For instance, the interconnect wires or other function elements in integrated circuits (ICs) may fail because of the temperature cycling [1]. Consequently, the thin-film stresses measurements especially under high temperature conditions are important to improve the thin- film/substrate systems.

The most widely used method to determine the thin-film stresses at present is based on the measurement of the substrate curvature and Stoney’s formula [2]. However, the rigid assumptions of Stoney’s formula, such as uniform thin-film stress, uniform deformation over the entire system, and infinitesimal strains and rotations of the system cannot be satisfied in real situations. To infer thin-film stress by substrate curvature accurately, a number of extensions of Stoney’s formula have been derived to relax some assumptions [310]. Huang and Rosakis [6] studied the thin film/substrate system subjected to nonuniform but axisymmetric temperature distribution; they relaxed the uniform stress assumption. Recently, Feng and his associates [10] considered a circular multilayer thin-film/substrate system subjected to nonuniform and nonaxisymmetrical temperature distribution and derived an extension of Stoney’s formula that was more universal. There are a few techniques for curvatures measurement, such as the scanning laser method [11], a multibeam optical stress sensor (MOSS) [12], the coherent gradient sensing (CGS) method [1316], and x-ray diffraction [17]. Compared with other methods, CGS, one type of shear interferometry, has distinguished advantages, including full-field measurement and vibration insensitivity. Although Moire and shearography methods had been used for high-temperature displacement measurement, they were not specified for thin-film/substrate systems [18,19]. This paper presents an effective method based on extended CGS for full-field curvatures measurement in high-temperature environment, which can be insensitive to the disturbance of air flow resulting from the temperature. Moreover, the full-field curvatures are calculated by the fast Fourier transform (FFT) method, and nonuniform stresses of thin films at high temperature are obtained by the extension of Stoney’s formula. The “nonlocal” effect is also analyzed.

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

CGS method is a full-field curvature measurement technique that is sensitive to the surface slope of the specimen by laterally shearing the wavefront reflected from the sample. The CGS setup for high-temperature measurement is illustrated in Fig. 1(a) . A collimated laser beam passes through a beam splitter and is then directed to the reflecting specimen surface in the temperature chamber with a quartz window. The reflected beam from the specimen is further reflected by the beam splitter and then passes through two Ronchi gratings, G1 and G2, 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.

 

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.

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During the heating process, the air density varies owing to the thermal effect, which changes the refractive index of the air. Thus it is difficult to obtain the stable interferogram fringes, which is a critical challenge for optical measurement at high temperature. To analyze the thermal effect on CGS method at high temperature, we assume the (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 |2f|=f,x2+f,y21, the net change in optical path length, S(x,y), can be calculated by considering the thermal effect [15]

S(x,y)=20f(x,y)n(x,y,z)dz.
If the reflective wavefront is sheared in the y direction, partially differentiating S(x,y) with y leads to

S(x,y)y=20f(x,y)n(x,y,z)ydz+2n(x,f(x,y),z)f(x,y)y.

The 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]

n(t)=1+n011+at,
where 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]
{φx(x,y)=4πΔp(1+n011+at)f(x,y)xφy(x,y)=4πΔp(1+n011+at)f(x,y)y,
where 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 G1 and G2. Since 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
{κxx=2f(x,y)x2=p4πΔφ(x)(x,y)xκyy=2f(x,y)y2=p4πΔφ(y)(x,y)yκxy=κyx=2f(x,y)xy=p4πΔφ(y)(x,y)x,
where κxxis the curvature in x direction, κyyis the curvature in y direction, andκxyis the twist curvature.

Accordingly, the temperature chamber is also designed deliberately in order to reduce the air effect. During measurement, the laser beam vertically passes through a quartz window on the side of the temperature chamber, where air convection is very weak at the stable temperature. The thermo-isolation materials are fixed around the window in order to reduce the temperature gradients near chamber window. CGS principle relies on the self-interference based on Eq. (4). Therefore, the thickness and the changes of refractive index of quartz window have only little effect on the interferometry.

The phase distribution can be calculated by FFT [21,22], such as φ(x)(x,y) = arctan{Im[Ax(x,y)]/Re[Ax(x,y)]} and φ(y)(x,y) = arctan{Im[Ay(x,y)]/Re[Ay(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).

Usually, the complicated process of wafer inevitably introduces the nonuniform deformation or misfit, which can result in serious thermo-stress due to temperature. However, classical Stoney’s formula considers only for the uniform situation, which cannot catch the real stresses status. Feng and his associates [10] had derived an extension of Stoney’s formula for a multilayer thin-film/substrate system subjected to nonuniform and nonaxisymmetrical temperature distribution. In the later part, we will use the cylindrical coordinates to present the stresses analysis. Then the nonuniform thin-film stresses from the nonuniform curvatures of the substrate can be expressed as [10]

σrr(f)+σθθ(f)=Eshs26(1νs)hf{κrr+κθθ¯+(1+νf)[(1+νs)αs2αf](1+νs)[(1+νs)αs(1+νf)αf](κrr+κθθκrr+κθθ¯)+[3+νs1+νs2(1+νf)[(1+νs)αs2αf](1+νs)[(1+νs)αs(1+νf)αf]]×m=1(m+1)(rR)m(Cmcosmθ+Smsinmθ)},
σrr(f)σθθ(f)=Eshs2αs(1νf)6(1νs)hf1(1+νs)αs(1+νf)αf                    ×{κrrκθθm=1(m+1)[m(rR)m(m1)(rR)m2]×(Cmcosmθ+Smsinmθ)},
σrθ(f)=Eshs2αs(1νf)6(1νs)hf1(1+νs)αs(1+νf)αf          ×{κrθ+12m=1(m+1)[m(rR)m(m1)(rR)m2](CmsinmθSmcosmθ)},
τr=Eshs26(1νs2){r(κrr+κθθ)1νs2Rm=1m(m+1)(rR)m1(Cmcosmθ+Smsinmθ)},
τθ=Eshs26(1νs2){1rθ(κrr+κθθ)+1νs2Rm=1m(m+1)(rR)m1(CmsinmθSmcosmθ)},
where hs and hf are the thickness of the substrate and thin film, respectively; R is the radius of the system; σrr(f) and σθθ(f) are the in-plane stresses of the thin film in the radial and circumferential directions, respectively; σrθ(f) is the film shear stress; and τr and τθ are the interfacial shear stresses between the substrate and thin film in the radial and circumferential directions, respectively. κrr+κθθ¯=1πR202π0R(κrr+κθθ)rdrdθ is the average curvature of the substrate as well as Cm=1πR202π0R(κrr+κθθ)(ηR)mcos(mφ)ηdηdφ and Sm=1πR202π0R(κrr+κθθ)(ηR)msin(mφ)ηdηdφ. Es is the Young’s modulus of the substrate. νs and νf are the Poisson’s ratio of the substrate and film, respectively. αs and αf represent the thermal expansion coefficients of the substrate and film, respectively. It should be noticed that the thin-film nonuniform stresses are not only dependent on the local curvatures of the substrate, but they are also related to the “nonlocal” curvatures (average curvature).

In summary, the measurement of nonuniform stresses of thin films at high temperature contains the following steps. First, use CGS method to obtain the interferograms of the specimen at high temperature. Second, calculate the phase distribution from the fringe pattern by FFT. Third, use Eq. (5) to obtain the curvatures of the substrate, and then transfer them into cylindrical coordinates. Finally, substitute the curvatures into Eqs. (6)(10) to obtain the nonuniform stresses of thin film. These steps are schematically illustrated in Fig. 2 .

 

Fig. 2 The flow chart of the measurement of nonuniform film stresses.

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3. Experimental results and discussion

3.1 Substrate curvature measurement

The specimen consists of SiO2 film grown by thermal oxidation on Si substrate, which is the representative wafer structure widely used in semiconductor industry. The thicknesses of the SiO2 film and Si substrate were 500nm and 500μm, respectively; their radius was 10mm. The geometry size agreed with the assumption hfhsR. 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.

 

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).

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Fig. 4 The substrate curvatures measured at 300°C: (a) curvature κxx in lateral direction, (b) curvature κyy in vertical direction, (c) twist curvature κxy.

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3.2 Nonuniform stresses of the thin film

To calculate the film stresses at high temperature, we select the room temperature as a reference state and use κrr(h)κrr(r), κθθ(h)κθθ(r), and κrθ(h)κrθ(r) to replace κrr, κθθ, and κrθ in Eqs. (6)(10), where the superscripts h and r represent the curvatures obtained at high temperature and room temperature, respectively. The physical parameters of the system are Es = 170GPa, νs = 0.22, αs = 0.25 × 10−6°C−1, Ef = 71GPa, νf = 0.16, and αs = 0.5 × 10−6°C−1 [1]. The thin film stresses for 300°C are shown in Fig. 5 . Figures 5(a), 5(b), and 5(c) show the film stresses σrr(f)(radial direction), σθθ(f) (circumferential direction), and σrθ(f) (shear stress), respectively. Figures 5(d) and 5(e) show the interfacial shear stresses τr(radial direction) and τθ(circumferential direction) between the film and the substrate, respectively. It is found that the magnitude order of thin-film stresses is at GPa. For most areas, σrr(f) is not equal to σθθ(f), and the shear stress σrθ(f) is large. The nonuniformity of the film stresses becomes more severe owing to the nonlocal effect shown in Eqs. (6)(10). In addition, the interfacial stresses τrand τθwith the magnitude of a few MPa are rather smaller compared with the film stresses. Actually, the shear stresses on interface are more dangerous in most cases.

 

Fig. 5 The nonuniform stresses of the thin film measured at 300°C: (a) stress σrr(f) in radial direction, (b) stress σθθ(f) in circumferential direction, (c) shear stress σrθ(f), (d) interfacial shear stress τr in radial direction, (e) interfacial shear stress τθ in circumferential direction.

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To investigate the thermo-stresses of thin film subjected to varied temperature, we conducted the experiment from room temperature to 300°C with the step of 50°C. Then the full-field stresses can be obtained at the different temperatures following the same process as above. The film stresses of the central point in the specimen are selected to illustrate the thermo-stress evolution, as shown in Fig. 6 . σrr(f)is −130MPa (in compression) at the beginning room temperature then increases to 45MPa (in tension) at 200°C; however, σrr(f) drops down to −190MPa at 250°C again and then reaches to 130MPa at 300°C. The fluctuation of σrr(f)may result from the nonuniformity and the nonlocal effect. Meanwhile σθθ(f) monotonically decreases from tension to compression with the increase of temperature.

 

Fig. 6 The film stresses in radial and circumferential directions at the central point of the specimen vs. temperature.

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4. Conclusions

The results presented here demonstrate to use a coherent gradient sensing method to measure the thin-film/substrate system curvature at high temperature and calculate the nonuniform stresses of the film by the extension of Stoney’s formula. This optical technique is featured as full-field nonuniform curvatures measurement and vibration insensitivity. A SiO2 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

We gratefully acknowledge the support from the National Natural Science Foundation of China (Grant Nos. 90816007, 10902059, 10820101048, 10832005) and the Foundation for the Author of National Excellent Doctoral Dissertation of China (FANEDD) (No. 2007B30).

References and links

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]  

References

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  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

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

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

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

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

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

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

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

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

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

1992

1991

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

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

1980

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

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.

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.

Appl. Phys. Lett.

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.

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.

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.

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.

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.

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.

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.

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

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.

Opt. Lasers Eng.

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

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.

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

Thin Solid Films

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

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]

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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 ) } ,

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