## 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 [3–10]. 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 [13–16], 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, *G _{1}* and

*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}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 $\left|{\nabla}^{2}f\right|={f}_{,x}^{2}+{f}_{,y}^{2}\ll 1$, the net change in optical path length, *S*(*x,y*), can be calculated by considering the thermal effect [15]

*y*direction, partially differentiating

*S*(

*x*,

*y*) with

*y*leads to

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*) 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]

*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, ${\kappa}_{yy}$is the curvature in

*y*direction, and${\kappa}_{xy}$is 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[

*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).

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]

*h*and

_{s}*h*are the thickness of the substrate and thin film, respectively;

_{f}*R*is the radius of the system; ${\sigma}_{rr}^{(f)}$ and ${\sigma}_{\theta \theta}^{(f)}$ are the in-plane stresses of the thin film in the radial and circumferential directions, respectively; ${\sigma}_{r\theta}^{(f)}$ is the film shear stress; and

*τ*and

_{r}*τ*are the interfacial shear stresses between the substrate and thin film in the radial and circumferential directions, respectively. $\overline{{\kappa}_{rr}+{\kappa}_{\theta \theta}}=\frac{1}{\pi {R}^{2}}{\displaystyle {\int}_{0}^{2\pi}{\displaystyle {\int}_{0}^{R}({\kappa}_{rr}+{\kappa}_{\theta \theta})r\text{d}r\text{d}\theta}}$ is the average curvature of the substrate as well as ${C}_{m}=\frac{1}{\pi {R}^{2}}{\displaystyle {\int}_{0}^{2\pi}{\displaystyle {\int}_{0}^{R}({\kappa}_{rr}+{\kappa}_{\theta \theta}){\left(\frac{\eta}{R}\right)}^{m}\mathrm{cos}(m\phi )\eta \text{d}\eta \text{d}\phi}}$ and ${S}_{m}=\frac{1}{\pi {R}^{2}}{\displaystyle {\int}_{0}^{2\pi}{\displaystyle {\int}_{0}^{R}({\kappa}_{rr}+{\kappa}_{\theta \theta}){\left(\frac{\eta}{R}\right)}^{m}\mathrm{sin}(m\phi )\eta \text{d}\eta \text{d}\phi}}$.

_{θ}*E*is the Young’s modulus of the substrate.

_{s}*ν*and

_{s}*ν*are the Poisson’s ratio of the substrate and film, respectively.

_{f}*α*and

_{s}*α*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).

_{f}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 .

## 3. Experimental results and discussion

#### 3.1 Substrate curvature measurement

The specimen consists of SiO_{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

To calculate the film stresses at high temperature, we select the room temperature as a reference state and use ${\kappa}_{rr}^{(h)}-{\kappa}_{rr}^{(r)}$, ${\kappa}_{\theta \theta}^{(h)}-{\kappa}_{\theta \theta}^{(r)}$, and ${\kappa}_{r\theta}^{(h)}-{\kappa}_{r\theta}^{(r)}$ to replace ${\kappa}_{rr}$, ${\kappa}_{\theta \theta}$, and ${\kappa}_{r\theta}$ 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 *E _{s}* = 170GPa,

*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]. The thin film stresses for 300°C are shown in Fig. 5 . Figures 5(a), 5(b), and 5(c) show the film stresses ${\sigma}_{rr}^{(f)}$(radial direction), ${\sigma}_{\theta \theta}^{(f)}$ (circumferential direction), and ${\sigma}_{r\theta}^{(f)}$ (shear stress), respectively. Figures 5(d) and 5(e) show the interfacial shear stresses ${\tau}_{r}$(radial direction) and ${\tau}_{\theta}$(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, ${\sigma}_{rr}^{(f)}$ is not equal to ${\sigma}_{\theta \theta}^{(f)}$, and the shear stress ${\sigma}_{r\theta}^{(f)}$ 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 ${\tau}_{r}$and ${\tau}_{\theta}$with 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.

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 . ${\sigma}_{rr}^{(f)}$is −130MPa (in compression) at the beginning room temperature then increases to 45MPa (in tension) at 200°C; however, ${\sigma}_{rr}^{(f)}$ drops down to −190MPa at 250°C again and then reaches to 130MPa at 300°C. The fluctuation of ${\sigma}_{rr}^{(f)}$may result from the nonuniformity and the nonlocal effect. Meanwhile ${\sigma}_{\theta \theta}^{(f)}$ monotonically decreases from tension to compression with the increase of temperature.

## 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 SiO_{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

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).

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