A Label-Free Measurement Method for Plane Stress States in Optical Isotropic Films with Spectroscopic Ellipsometry

Abstract

Background

Stress measurement for thin films is crucial in a variety of fields such as in semiconductor manufacturing, the optoelectronics industry, and biomedical science, among others. However, most measurement methods require surface treatment of the thin film.

Objective

A label-free measurement method for plane stress states in optical isotropic thin films based on spectroscopic ellipsometry analysis is proposed and verified in this paper.

Methods

The proposed method is based on the modulation of the stress-optic effect on reflected spectroscopic ellipsometry. A theoretical model is established to describe the relation between all components of the plane-stress state and the classic ellipsometric parameters (Ψ, Δ). An algorithm is developed to determine all components of a plane-stress state by fitting the model to the experiment data.

Results

In the verification experiment, we determined the plane stress state of a Cu film coated on a PI (polyimide) substrate. The results show a reasonable agreement between the experimental measurements from spectroscopic ellipsometry and the theoretical analysis based on the applied loading.

Conclusion

The results prove that our method can effectively measure the plane stress state of optical isotropic thin films.

Appendix

Theoretical Analysis

In this section, a theoretical analysis supporting the conclusions drawn in "Determination of Principal Stress Direction" section is provided. As depicted in Fig. 1(d), the rotation angle $\varphi$ represents the orientation of the optical principal axis relative to the incident plane. When $\varphi =0$, the optical principal axis is parallel to the incident plane, and in such an orientation, the refractive index tensor can be expressed as

$$n=\left(\begin{array}{ccc}{n}_{1}& & \\ & {n}_{2}& \\ & & {n}_{2}\end{array}\right),$$

(8)

where ${n}_{1}$ and ${n}_{2}$ are the refractive indices parallel and perpendicular to the incident plane, respectively. The corresponding dielectric tensor should be

$$\varepsilon =\left(\begin{array}{ccc}{\varepsilon }_{1}& & \\ & {\varepsilon }_{2}& \\ & & {\varepsilon }_{2}\end{array}\right),$$

(9)

where ${\varepsilon }_{1}={{n}_{1}}^{2}$, ${\varepsilon }_{2}={{n}_{2}}^{2}$.

As $\varphi$ changes, the dielectric tensor also changes. To determine the instantaneous dielectric tensor during rotation, the rotation tensor $B$ is introduced as

$$B=\left(\begin{array}{ccc}{\text{cos }}\varphi & -{\text{sin }}\varphi & 0\\ {\text{sin }}\varphi & {\text{cos }}\varphi & 0\\ 0& 0& 1\end{array}\right).$$

(10)

Thus, the instantaneous dielectric tensor during rotation is obtained by

$${\varepsilon }{\prime}=B\cdot \varepsilon \cdot {B}^{-1}=\left(\begin{array}{ccc}{\varepsilon }_{1}\;{{\text{cos}}}^{2}\;\varphi +{\varepsilon }_{2}{{\text{ sin}}}^{2}\;\varphi & \left({\varepsilon }_{1}-{\varepsilon }_{2}\right){\text{ cos }}\varphi {\text{ sin }}\varphi & 0\\ \left({\varepsilon }_{1}-{\varepsilon }_{2}\right){\text{ cos }}\varphi {\text{ sin }}\varphi & {\varepsilon }_{2}{{\text{ cos }}}^{2}\varphi +{\varepsilon }_{1}{{\text{ sin }}}^{2}\varphi & 0\\ 0& 0& {\varepsilon }_{2}\end{array}\right).$$

(11)

Since the difference of ${n}_{1}$ and ${n}_{2}$ is small, i.e., ${n}_{1}-{n}_{2}\approx 0$, we can get

$${\varepsilon }_{1}-{\varepsilon }_{2}={n}_{1}{}^{2}-{n}_{2}{}^{2}=({n}_{1}+{n}_{2})\cdot ({n}_{1}-{n}_{2})\approx 0.$$

(12)

So, in equation (11), the non-diagonal elements can be ignored compared with diagonal element, and it can be degenerated into

$${\varepsilon }{\prime}=\left(\begin{array}{ccc}{n}_{x}^{2}& & \\ & {n}_{y}^{2}& \\ & & {n}_{z}^{2}\end{array}\right)\approx \left(\begin{array}{ccc}{\varepsilon }_{1}{{\text{ cos}}}^{2}\;\varphi +{\varepsilon }_{2}{{\text{ sin}}}^{2}\;\varphi & 0& 0\\ 0& {\varepsilon }_{2}{{\text{ cos}}}^{2}\;\varphi +{\varepsilon }_{1}{{\text{ sin}}}^{2}\;\varphi & 0\\ 0& 0& {\varepsilon }_{2}\end{array}\right).$$

(13)

It can be deduced that if the optical anisotropy of the specimen is induced by stress, the dielectric tensor can always be approximated as a diagonal tensor. As outlined in Ref. [18], when the dielectric tensor is diagonal, the ellipsometric parameters $\Psi$ and $\Delta$ can be calculated directly by

$$\begin{array}{c}{r}_{\text{ss}}=\frac{{n}_{i}{\text{ cos }}\theta -{\left({n}_{y}^{2}-{n}_{i}^{2}{{\text{ sin}}}^{2}\;\theta \right)}^{1/2}}{{n}_{i}{\text{ cos }}\theta +{\left({n}_{y}^{2}-{n}_{i}^{2}{{\text{ sin}}}^{2}\;\theta \right)}^{1/2}}\\ {r}_{\text{pp}}=\frac{{n}_{x}{n}_{z}{\text{ cos }}\theta -{n}_{i}{\left({n}_{z}^{2}-{n}_{i}^{2}{{\text{ sin}}}^{2}\;\theta \right)}^{1/2}}{{n}_{x}{n}_{z}{\text{ cos }}\theta +{n}_{i}{\left({n}_{z}^{2}-{n}_{i}^{2}{{\text{ sin}}}^{2}\;\theta \right)}^{1/2}}\\ \rho =\tan\;\Psi {e}^{i\Delta }={r}_{\text{pp}}/{r}_{\text{ss}}\end{array},$$

(14)

where ${n}_{i}$ is the refractive index of air, which is equal to 1, and $\theta$ is the incident angle. Substituting equation (13) into equation (14), we can obtain

$$\rho =\frac{\left(\sqrt{{\varepsilon }_{2}}{\text{ cos }}\theta \sqrt{\left({\varepsilon }_{1}{{\text{ cos}}}^{2}\;\varphi +{\varepsilon }_{2}{{\text{ sin}}}^{2}\;\varphi \right)}-\sqrt{{\varepsilon }_{2}{}^{2}-{{\text{ sin}}}^{2}\;\theta }\right)\left(\sqrt{{\text{ cos }}\theta }+\sqrt{{\left({\varepsilon }_{2}{{\text{ cos }}}^{2}\;\varphi +{\varepsilon }_{1}{{\text{ sin}}}^{2}\;\varphi \right)}^{2}-{{\text{ sin}}}^{2}\;\theta }\right)}{\left(\sqrt{{\varepsilon }_{2}}{\text{ cos }}\theta \sqrt{\left({\varepsilon }_{1}{{\text{ cos}}}^{2}\;\varphi +{\varepsilon }_{2}{{\text{ sin}}}^{2}\;\varphi \right)}+\sqrt{{\varepsilon }_{2}{}^{2}-{{\text{ sin}}}^{2}\;\theta }\right)\left(\sqrt{{\text{ cos }}\theta }-\sqrt{{\left({\varepsilon }_{2}{{\text{ cos}}}^{2}\;\varphi +{\varepsilon }_{1}{{\text{ sin}}}^{2}\;\varphi \right)}^{2}-{{\text{ sin}}}^{2}\;\theta }\right)}.$$

(15)

Subsequently, the partial derivative of equation (15) with respect to rotation angle $\varphi$ is

$$\begin{array}{c}\frac{\partial \rho }{\partial \varphi }=\sin2\varphi \cdot \cos\theta \cdot f({\varepsilon }_{1},{\varepsilon }_{2},\theta ,\varphi )\\ \frac{{\partial }^{2}\rho }{\partial {\varphi }^{2}}=\cos\theta \left[{f}^{\prime}({\varepsilon }_{1},{\varepsilon }_{2},\theta ,\varphi ){\text{ sin }}2\varphi +2f({\varepsilon }_{1},{\varepsilon }_{2},\theta ,\varphi ){\text{ cos }}2\varphi \right]\end{array},$$

(16)

where $f({\varepsilon }_{1},{\varepsilon }_{2},\theta ,\varphi )$ is a complicated function, which is too long to be presented here. It can be found from equation (16) that when $\varphi$ is an integral multiple of ${}^{\pi}/_{2}$, $\frac{\partial \rho }{\partial \varphi }=0$ and $\frac{{\partial }^{2}\rho }{\partial {\varphi }^{2}}=0$, which indicates that $\rho$ attain the extreme values. Since the ellipsometric parameters Ψ and Δ are monotone functions of $\rho$ according to equation (14), they will attain extreme values, too. Therefore, we have demonstrated that when the ellipsometric parameters attain their extreme values, the incident plane aligns with the optical principal axis, which corresponds to the stress principal axis.

DIC Measurement

The experiment employs a high-speed camera, as depicted in Fig. 13(a), provided by Beijing Daheng Image Vision Co., Ltd. The dimensions of each pixel are 1.4 $\mu m \times 1.4 \mu m$, and the frame rate is 13 frames per second ($fps$). Speckles are applied onto the masking tape using a black marking pen. As the ellipsometer requires a smooth, reflective surface for measurement, for the uniaxial stress state, speckles are applied to both edges of the Cu film, as illustrated in Fig. 13(b), but not on the measurement point itself. Under uniaxial stress, the area covered by the Cu film experiences uniform stress. This allows for the stress values at the measurement point to be determined by calculating the relative displacement between speckles using 1D-DIC. In the case of biaxial stress condition, this method is inapplicable due to the non-uniform strain field. To measure the strain at the measurement point (the center of the specimen), speckles were sprayed as shown in Fig. 13(c). Utilizing the Finite Element Method, the stress values at the measurement point can be ascertained based on the strain in the areas with speckles using 1D-DIC. 

Fig. 13

(a) The high-speed camera employed in DIC, (b) the specimen employed in uniaxial stress condition, (c) the specimen employed in biaxial stress condition