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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 16 — Aug. 2, 2010
  • pp: 16594–16600
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Measuring residual stress of anisotropic thin film by fast Fourier transform

Chuen-Lin Tien and Hung-Da Zeng  »View Author Affiliations


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 MgF2 thin film was demonstrated.

© 2010 OSA

1. Introduction

Films deposited using different coating techniques have a residual stress which may be tensile or compressive in nature. The stress behavior of thin films is very important in various optical coating applications. In many cases, the stresses in a deposited film are anisotropic due to the angle-of-incidence distribution of the depositing atom flux and/or the bombarding ion flux. In general, the methods for measuring residual stress always focus on an isotropic distribution in the thin film. There is very limited study on anisotropic stress measurement in the literature.

A number of methods have been developed to measure stress in thin films [1

1. M. Ohring, Materials Science of Thin Films: Deposition and Structure, 2nd ed. (Academic Press, 2004).

5

5. L. B. Freund, and S. Suresh, Thin Films:Materials: Stress, Defect Formation and Surface Evolution (Cambridge University Press, 2003).

]. Most of these methods determine the mean stresses by measuring the curvature of a substrate before and after coating. The substrate curvature techniques include the cantilever beam [1

1. M. Ohring, Materials Science of Thin Films: Deposition and Structure, 2nd ed. (Academic Press, 2004).

,2

2. 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).

], laser scanning method [3

3. W. D. Nix, “Mechanical properties of thin films,” Metall. Trans. A 20A, 2217–2245 (1989).

]and optical interferomtry [4

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]

] etc. On the other hand, X-ray diffraction has been used to measure changes in the lattice spacing [2

2. 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).

]. Although the X-ray diffraction method is suitable for measuring the stress behavior in thin films, it is an indirect method, where the stress is calculated from the elastic strain. All the above methods are somewhat elaborate and time consuming, and are not suitable for measuring the whole-field anisotropic stress distribution. To overcome the shortfall, a fast and accurate measurement method, based on fast Fourier transform (FFT) technique [6

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

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]

] combined with a modified Twyman-Green interferometer, was developed to measure the curvature changes and then to calculate anisotropic film stresses. To the best of our knowledge this is the first time that a method such as the proposed method has been discussed in the literature. The method not only measures isotropic residual stress but also anisotropic stress in a thin film. In addition, the obliquely deposited MgF2 thin films produced by the electron-beam evaporation technique are investigated. The anisotropic stresses of MgF2 thin films, deposited on a BK7 substrate, are determined using the proposed technique.

2. Principles

This paper presents a new approach for film stress measurement using optical interferometry, fast Fourier transformation and a digital filtering process. It has some advantages over traditional method where the surface geometry is not touched in that both multi-frame imaging and line-to-line scanning are not required.

The general interference equation for the intensity measured in a two-beam interferometer is
i(x,y)=a(x,y)+b(x,y)cos[2πf0x+ϕ(x,y)],
(1)
where 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:
i(x,y)=a(x,y)+c(x,y)exp(2πif0x)+c(x,y)exp(2πif0x),
(2)
where the asterisk indicates the complex conjugate and c(x,y)=12b(x,y)exp[iϕ(x,y)] is a complex fringe pattern. Now, Eq. (2) can be described by the 2-D fast Fourier transform (FFT) as follows:
I(u,v)=A(u,v)+C(u,v)+C*(u,v),
(3)
where the capital letters indicate the Fourier spectra; 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 by
ϕ(x,y)=tan1(Im[c(x,y)]Re[c(x,y)]),
(4)
where 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.

Generally, thin films deposited on a substrate will bend the substrate because of the stress. By measuring the radius of the curvature difference between the substrate before and after thin film deposition, and assuming that the stress is isotropic, the stress in a thin film deposited on a substrate results in substrate deformation proportional to the film stress. The film stress 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]

,10

10. 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).

]
σ=Ests26(1νs)tf(1R21R1)=Ests26(1νs)tfR,
(6)
where σ is the residual stress in the thin film; R1 and R2 are the radius of curvature before and after thin-film is deposited on the substrate. R is the radius of curvature of the film. E s = 81GPa and νs = 0.208 are the Young’s modulus and the Poisson’s ratio of the BK7 substrate, respectively; 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.

The above Stoney equation requires the basic assumption that the residual stress in a film is isotropic. Thus, the anisotropic stress [11

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]

] in thin films is calculated by
σx=16Es1νs2(1Rx+νsRy)ts2tf,
(7)
σy=16Es1νs2(1Ry+νsRx)ts2tf,
(8)
where 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

MgF2 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
Fig. 1 Schematic representation of the experimental setup.
. 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.

3.2 Measurement results

The stress distribution is considered anisotropic with respect to x and y, i.e., σ xσ 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
Fig. 3 Principal axes and their directions.
. 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.

We take MgF2 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
Fig. 4 Interferogram with carrier frequency
. The spectrum in the frequency domain can be obtained by a 2-D FFT process, as shown in Fig. 5(a)
Fig. 5 Proposed method used for: (a) Fourier transform spectrum; (b) wrapped phase obtained by IFFT.
. 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 xσ y). Figure 6 (a)
Fig. 6 3-D surface contour map: (a) before film deposition; (b) after film deposition.
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
Fig. 7 Curvature radius fitting: (a) in the x-axis direction for finding Rx; (b) in the y-axis direction for finding Ry.
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 MgF2 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 MgF2 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 MgF2 films were determined from SEM micrographs of the cross-sectional morphology.

3.3 Error analysis

To increase the measurement accuracy, two important corrections have to be taken into account. Firstly, the initial substrate may deviate from the ideal flatness by some amount. This can be eliminated by measuring the shape of the substrate before and after film deposition. Secondly, the sample may change its position between two sequential measurements. Any additional deformation must be carefully avoided. The sample placed on the substrate holder should coincide with the sample coordinates but rest freely on the three-axis translation stage without any external force. In this approach, the calibration step is crucial and requires careful attention. In this study, the relative uncertainty of film stress (Δσ /σ) depends on the relative uncertainties of the film thickness variation (∆tf/tf = 1.0 × 10−2), 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 by
Δσσ=(Δtftf)2+(Δϕϕ)2+(ΔTT)2+(Δnn)2,
(9)
where the uncertainty of the film stress is Δσ. 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

Table 1. Stress measuring results for three obliquely deposited MgF2 thin films

table-icon
View This Table
.)

4. Conclusions

A new method for film stress measurement based on the 2-D FFT technique is presented. Significant anisotropy values were observed in three obliquely-deposited MgF2 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, Materials Science of Thin Films: Deposition and Structure, 2nd ed. (Academic Press, 2004).

2.

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).

3.

W. D. Nix, “Mechanical properties of thin films,” Metall. Trans. A 20A, 2217–2245 (1989).

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]

5.

L. B. Freund, and S. Suresh, Thin Films:Materials: Stress, Defect Formation and Surface Evolution (Cambridge University Press, 2003).

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]

8.

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

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]

10.

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).

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]

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]

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

  1. M. Ohring, Materials Science of Thin Films: Deposition and Structure, 2nd ed. (Academic Press, 2004).
  2. 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).
  3. W. D. Nix, “Mechanical properties of thin films,” Metall. Trans. A 20A, 2217–2245 (1989).
  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]
  5. L. B. Freund, and S. Suresh, Thin Films:Materials: Stress, Defect Formation and Surface Evolution (Cambridge University Press, 2003).
  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]
  8. W. W. Macy., “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22(23), 3898–3901 (1983). [CrossRef] [PubMed]
  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]
  10. 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).
  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]
  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]

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