## Measuring residual stress of anisotropic thin film by fast Fourier transform |

Optics Express, Vol. 18, Issue 16, pp. 16594-16600 (2010)

http://dx.doi.org/10.1364/OE.18.016594

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### Abstract

A new method for the measurement of anisotropic stress in thin films based on 2-D fast Fourier transform (FFT) is presented. A modified Twyman-Green interferometer was used for surface topography measurement. A fringe normalization technique was also used to improve the phase extraction technique efficiently. The measurement of anisotropic stress in obliquely deposited MgF_{2} thin film was demonstrated.

© 2010 OSA

## 1. Introduction

4. C. L. Tien, C. C. Lee, Y. L. Tsai, and W. S. Sun, “Determination of the mechanical properties of thin films by digital phase shifting interferometry,” Opt. Commun. **198**(4-6), 325–331 (2001). [CrossRef]

6. 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]

7. M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. **22**(24), 3977–3982 (1983). [CrossRef] [PubMed]

_{2}thin films produced by the electron-beam evaporation technique are investigated. The anisotropic stresses of MgF

_{2}thin films, deposited on a BK7 substrate, are determined using the proposed technique.

## 2. Principles

*a*(

*x*,

*y*) and

*b*(

*x*,

*y*) are the background and the contrast of the fringe coefficients, respectively;

*ϕ*(

*x*,

*y*) is the phase of the wave front; and

*f*

_{0}is the spatial carrier frequency. The intensity distribution can be rewritten in the following form:where the asterisk indicates the complex conjugate and

*u*and

*v*are the spatial frequencies in the

*x*and

*y*directions. The amplitude spectrum of Eq. (3) is a tri-modal function with A(

*u*,

*v*) forming a broad zero peak, and two peaks C(

*u*,

*v*) and C*(

*u*,

*v*) located symmetrically with respect to the origin. By means of bandpass filtering, the zero peaks A(

*u*,

*v*) and C*(

*u*,

*v*) are removed. The remaining spectrum is no longer symmetric and will yield a non-zero imaginary part after inverse transformation. Then, using inverse FFT for the phase distribution,

*ϕ*(

*x*,

*y*) can be calculated bywhere Re[

*c*(

*x*,

*y*)] and Im[

*c*(

*x*,

*y*)] are the real and imaginary parts of

*c*(

*x*,

*y*), respectively. With Eq. (4) based on FFT using only one image, a recovery of the phase

*ϕ*(

*x*,

*y*) can be obtained.

*σ*can then be determined by using the modified Stoney’s formula [9

9. 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]

*σ*is the residual stress in the thin film;

*R*and

_{1}*R*are the radius of curvature before and after thin-film is deposited on the substrate.

_{2}*R*is the radius of curvature of the film.

*E*

_{s}= 81GPa and

*ν*= 0.208 are the Young’s modulus and the Poisson’s ratio of the BK7 substrate, respectively;

_{s}*t*

_{s}is the thickness of the substrate and

*t*

_{f}(

*t*

_{f}<<

*t*

_{s}) is the film thickness. The film thickness is determined by the envelope method of optical transmission measurements. This treatment assumes an isotropic homogeneous stress distribution in the coating in which deformations of the substrate are small compared to the substrate thickness. By convention,

*σ*is negative for compressive stress and positive for tensile stress. The surface profile of the thin film can be used to judge a tensile stress or a compressive stress. The conventional approach to the stress calculation is to determine an average curvature radius of a whole sample and calculate the residual stress based on it.

11. E. van de Riet, “Deflection of a substrate induced by an anisotropic thin-film stress,” J. Appl. Phys. **76**(1), 584–586 (1994). [CrossRef]

*σ*

_{x}and

*σ*

_{y}are the biaxial stresses in the thin films;

*R*

_{x}and

*R*

_{y}are the radii of curvature of the film in the x-axis and y-axis directions, respectively. If an isotropic condition (

*R*

_{x}

*= R*

_{y}

*= R*) is satisfied, then Eqs. (7) and (8) can be simplified to Eq. (6).

## 3. Experimental results

*3.1* Experimental setup

_{2}thin films were obliquely deposited using the electron-beam evaporation technique. The BK7 glass substrates were polished on one side to a flatness of one wavelength and ground on the other side. The stresses were measured for films deposited on the polished face of the BK7 glass substrates (25.4 mm in diameter, 1.5 mm in thickness). The stress measurement is using a modified Twyman–Green interferometer based on the FFT technique. The optical arrangement for film stress measurement is shown in Fig. 1 . A He-Ne laser is passed through a micro-objective and a pinhole which acts as a spatial filter to form a point source. This then propagates through a collimating lens to form a plane wavefront. The wavefront is divided in amplitude by a beam splitter. The reflected and transmitted beams travel to a reference mirror (flatness of λ/20) and the glass substrate. The glass substrate is mounted on a three-axis platform to generate the spatial-carrier frequency and acts as a test plate. After being reflected by both the reference mirror and the substrate, the beams are recombined by the beam splitter and travel toward a digital CCD camera. The camera has a resolution of 1280 × 960 pixels. The interference pattern can be seen on the LCD monitor attached to the CCD camera. The interferogram is recorded by a personal computer equipped with a self-developed stress analysis program.

8. W. W. Macy Jr., “Two-dimensional fringe-pattern analysis,” Appl. Opt. **22**(23), 3898–3901 (1983). [CrossRef] [PubMed]

12. C. L. Tien, S. S. Jyu, and H. M. Yang, “A method for fringe normalization by Zernike polynomial,” Opt. Rev. **16**(2), 173–175 (2009). [CrossRef]

**3.2** Measurement results

*x*and

*y*,

*i.e*.,

*σ*

_{x}≠

*σ*

_{y}. The biaxial stress state in an anisotropic film can be described by its two principal stresses,

*σ*

_{x}and

*σ*

_{y}. To evaluate the anisotropic stress of a thin film, deformation must be measured in different directions. The

*x*and

*y*axes serve as reference axes for the substrate as shown in Fig. 3 . The maximum gradient of deformation (corresponding to the principal axes) is initially set as

*x*' and

*y*' axes. The angle between

*x*'-axis and

*x*-axis is denoted by θ. For comparing with different samples, the

*x*'-

*y*' coordinates can be transformed to

*x-y*coordinates by a rotation through the angle. Thus

*x*'-

*y*' coordinates of the film coincide with

*x-y*coordinates of the substrate. Here

*σ*

_{x}and

*σ*

_{y}are the biaxial stresses in the thin films.

_{2}thin film with columnar tilt angle of 52° as an example. First, the interference fringe with carrier frequency is recorded by a high-speed CCD camera, as shown in Fig. 4 . The spectrum in the frequency domain can be obtained by a 2-D FFT process, as shown in Fig. 5(a) . The wrapped phase map is derived by shifting the + 1 order peak to the center of the frequency domain and inverting the 2-D FFT process, as illustrated in Fig. 5(b). Next, the unwrapped phase is obtained by using Macy’s phase unwrapping technique. To find the original surface contour, the unwrapped phase is transferred to the local height difference of the film or substrate. In particular, the Matlab algorithm can generate a 3D contour plot of film deformation and indicate isotropic (

*σ*

_{x}=

*σ*

_{y}) or anisotropic stress (

*σ*

_{x}≠

*σ*

_{y}). Figure 6 (a) shows the 3-D contour before film deposition (bare substrate). Figure 6 (b) indicates a 3-D surface contour after film deposition. The film deformation is obtained by subtracting the surface contour in Fig. 6(b) from Fig. 6(a). Finally, in order to determine isotropic film stress, a curvature fitting technique and a numerical method are used to calculate the radius of curvature of the bare substrate and coated substrate, respectively. A reference mirror is used as a standard surface (

*R*= infinity) to determine the radius of curvature of the test substrate. A curve fitting program is written to perform a least squares fit of the spherical surface (or circular) equation to the measured data. Figure 7 shows the measured and fitting curves in the

*x*-axis and

*y*-axis directions, respectively. The biaxial curvature radii of both

*R*

_{x}= 551.1 m and

*R*

_{y}= 470.2 m and the average curvature radius

*R*= 553.2 m are obtained by this approach. In this case, the results from the fitting routine were satisfactory. The anisotropic stresses of MgF

_{2}thin films with different tilt angles are measured by the proposed method. The measurement results of the anisotropic and average stresses for three obliquely deposited MgF

_{2}films are summarized in Tab. 1. The anisotropy of the thin film is defined as the ratio of |

*σ*

_{y}/

*σ*

_{x}|.The thickness and the columnar angles with respect to the substrate normal in columnar microstructures of the MgF

_{2}films were determined from SEM micrographs of the cross-sectional morphology.

**3.3** Error analysis

*σ*/

*σ*) depends on the relative uncertainties of the film thickness variation (∆

*t*/

_{f}*t*= 1.0 × 10

_{f}^{−2}), phase difference (∆

*ϕ*/

*ϕ*= 3.0 × 10

^{−4}), temperature variation (∆

*T*/

*T*= 1.2 × 10

^{−2}) and refractive index change (∆

*n*/

*n*= 1.0 × 10

^{−4}). If we are considering that the various uncertainties are independent, then the room mean square (rms) relative error of the film stress is given bywhere the uncertainty of the film stress is Δ

*σ*In many situations the measurement accuracy is limited by the environment. The effects of air flow and vibration can be reduced by making short exposure with a high-speed camera. Finally, with the values estimated for the different error sources an rms relative error of around 1.6% is claimed on the residual stress measurement with the proposed method. (See Table 1 .)

_{.}## 4. Conclusions

_{2}thin films. During the detailed evaluation of the shape of a thin film deformed by stress, the proposed method can be used to determine the anisotropic stress. It has the advantages of inherently fast measurement, high accuracy and easy operation. The proposed technique has non-contact and non-destructive features which can make it very useful in measuring stress in thin films.

## Acknowledgment

## References and links

1. | M. Ohring, |

2. | A. J. Perry, J. A. Sue, and P. J. Martin, “Practical measurement of the residual stress in coatings,” Surface Coatings Technol. |

3. | W. D. Nix, “Mechanical properties of thin films,” Metall. Trans. A |

4. | C. L. Tien, C. C. Lee, Y. L. Tsai, and W. S. Sun, “Determination of the mechanical properties of thin films by digital phase shifting interferometry,” Opt. Commun. |

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

6. | M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. |

7. | M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. |

8. | W. W. Macy Jr., “Two-dimensional fringe-pattern analysis,” Appl. Opt. |

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

10. | A. Brenner and S. Senderoff, “Calculation of stress in electrodeposits from the curvature of a plated strip,” J. Res. Nat’l. Bur. Stand. |

11. | E. van de Riet, “Deflection of a substrate induced by an anisotropic thin-film stress,” J. Appl. Phys. |

12. | C. L. Tien, S. S. Jyu, and H. M. Yang, “A method for fringe normalization by Zernike polynomial,” Opt. Rev. |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(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: June 15, 2010

Revised Manuscript: July 15, 2010

Manuscript Accepted: July 16, 2010

Published: July 22, 2010

**Citation**

Chuen-Lin Tien and Hung-Da Zeng, "Measuring residual stress of anisotropic thin film by fast Fourier transform," Opt. Express **18**, 16594-16600 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-16594

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### References

- M. Ohring, Materials Science of Thin Films: Deposition and Structure, 2nd ed. (Academic Press, 2004).
- A. J. Perry, J. A. Sue, and P. J. Martin, “Practical measurement of the residual stress in coatings,” Surface Coatings Technol. 8l, 17–28 (l996).
- W. D. Nix, “Mechanical properties of thin films,” Metall. Trans. A 20A, 2217–2245 (1989).
- C. L. Tien, C. C. Lee, Y. L. Tsai, and W. S. Sun, “Determination of the mechanical properties of thin films by digital phase shifting interferometry,” Opt. Commun. 198(4-6), 325–331 (2001). [CrossRef]
- L. B. Freund, and S. Suresh, Thin Films:Materials: Stress, Defect Formation and Surface Evolution (Cambridge University Press, 2003).
- 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]
- M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22(24), 3977–3982 (1983). [CrossRef] [PubMed]
- W. W. Macy., “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22(23), 3898–3901 (1983). [CrossRef] [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]
- A. Brenner and S. Senderoff, “Calculation of stress in electrodeposits from the curvature of a plated strip,” J. Res. Nat’l. Bur. Stand. 42, 105–123 (1949).
- E. van de Riet, “Deflection of a substrate induced by an anisotropic thin-film stress,” J. Appl. Phys. 76(1), 584–586 (1994). [CrossRef]
- C. L. Tien, S. S. Jyu, and H. M. Yang, “A method for fringe normalization by Zernike polynomial,” Opt. Rev. 16(2), 173–175 (2009). [CrossRef]

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