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Abstract
In this study, an iterative method using polarized Raman spectroscopy to quantitatively determine all the in-plane components of the stress tensor in semiconductor structures is presented. Raman experiments were conducted on silicon at different stress states. The results obtained by the proposed method were in good agreement with the given stress state. In addition, the effect of random errors of Raman shifts on the stress component calculation is discussed. In contrast to the conventional analytical solution, our iterative method can significantly reduce the random errors.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Micro-Raman spectroscopy (μRS) is regarded as a powerful technique for mechanical measurement with advantages, such as high resolution (∼1μm), nondestructive, non-contact, and high stress sensitivity. A shift in the Raman wavenumber relative to the strain-free wavenumber arises for a strained sample. Based on the theory of lattice dynamics, the stress components are related to the Raman shift through complex tensorial algorithms. Regardless of whether the stress state is complex, one convenient way to quantitatively determine the stress is to neglect the shear stress and to adopt the uniaxial or equal biaxial stress assumption, that is,
where Δωobs is the experimentally observed Raman shift: Δωobs = ω - ω0 and ω and ω0 are the Raman wavenumbers in the presence and absence of strain, respectively; κ0 is the Raman shift-stress coefficient; σ is the uniaxial or the sum of equal biaxial stress. With this assumption, stress measurements based on μRS have been rapidly developed. De Wolf et al. studied the stress introduced by different processes in microelectronic structures using μRS [1–3]. Kang et al. applied μRS to systematically investigate the distribution of the residual stress of porous silicon specimens [4–6]. Naka et al. evaluated the local surface stress distributions on silicon microstructures using Raman spectroscopy under uniaxial tension [7]. μRS has been successfully applied in the field of 2D materials [8–10]. Raman spectroscopy also offers a novel way of measuring temperature during laser-assisted extreme manufacturing. Wang et al. discussed the mechanism of Raman-based temperature probing, its calibration, and sources of uncertainty/error, and how to control them in detail [11].However, the material surface is often under a complex stress state that contains three stress components (σx, σy, τxy) in practical applications. If the above assumption is adopted, the actual stress components cannot be obtained, and the measurement results are unreliable. Only several published papers mention the stress states, where the stress components are decoupled based on a theoretical hypothesis or numerical simulation. For example, Huang et al. analyzed the near-surface stress distribution in a through-silicon via (TSV) structure by combining Raman experiments and finite element analysis (FEA) [12]. Similarly, Chen et al. calculated the stress components induced by eutectic bonding and flip chip bumping by combining the Raman measurement results and FEA simulations [13]. Miyatake et al. measured the tensor elements of the residual stress in single-crystal silicon structures belonging to MEMS devices [14]. Narayanan et al. estimated the stress components in (111) silicon using the hypothesis of linear relationships between the observed Raman shift and stress components [15]. Loechelt et al. introduced an off-axis method to analyze the {100} in-plane stress tensor [16]. Ma et al. developed an analytical method for the quantitative measurement of plane stress components in (110) silicon [17]. However, the random error of the Raman shift during the experiment affects the accuracy of the stress using the analytical method. Uchida and Pezotti et al. presented a method for calculating the stress components with different Raman peaks [18,19]. For diamond-type materials, as only one Raman peak is observed, this method is not available. Therefore, there is still an absence of a general and accurate stress component measurement method using μRS.
In this study, an iterative method that combines angle-resolved Raman detection and an iteration algorithm was developed based on the least squares method. Raman experiments of silicon with two different stress states were conducted, and the experimental results using the proposed method were in good agreement with the given stress state, demonstrating the effectiveness of our iterative method. Compared to the other methods, the proposed iterative method can accurately obtain all the in-plane components of the stress tensor.
2. Theory
To obtain the strain/stress information from the Raman experiment, it is necessary to develop a measurement theory between the Raman wavenumber shift and the strain/stress tensor. In the condition of approximately free vibration, the secular equation of different materials whose solutions (λ) yield the Raman shift of the optical phonons in the presence of strain can be obtained. In the case of the diamond-type structure, the Raman spectrum is represented by triply degenerate optical phonons, which include two transverse optical modes (TO) and one longitudinal optical mode (LO). The secular equation is as follows [20,21]:
By combining Eqs. (2) and (3), three Raman shifts of the optical phonons in the presence of stress were obtained. The Raman effect is represented by the Raman scattering tensor (Ri) of optical phonons. Loudon pioneered the investigation of the Raman effect in crystals and derived the Raman tensors for each of the 32 crystal classes [22]. The Raman tensors and corresponding polarization vectors of diamond-type or zinc-blende-type semiconductors, and thus, c-Si in the absence of stress, are given as
By solving the secular equation, the eigenvalues (λi) and the corresponding eigenvector (ni) of this matrix can be obtained. The Raman tensor (Rnj) in the presence of strain is given by [23]
The observed Raman modes and corresponding Raman scattering intensities were determined by the Raman selection rule. For the given polarization vectors of the incident and scattered light: ei = (cosθ, sinθ, 0) and es = (cosφ, sinφ, 0), as shown in Fig. 1, in which θ and φ are defined as the polarized angle of the incident light (θ) and the Raman scattered light (φ) in sample coordinate system (X1´X2´X3´), respectively, the Raman intensity, Ij, which is a function of θ and φ, can be expressed in the following form,
where Ij is the Raman scattering intensity of the phonon j, and Q is a constant.Because the strain is too small to cause a clear splitting of doubly or triply degenerate Raman modes, only one Raman peak of these degenerate modes can be observed during the experiment. When more than two of the degenerate Raman modes are observed in a single Raman peak, the observed Raman shift (Δωobs) induced by strain can then be obtained according to Eq. (7) [21]:
Based on the theoretical framework, the relationship between the observed Raman shift and stress for an arbitrary crystal orientation under a plane stress state can be obtained. For a given polarized angle of θ and φ, the Raman shift Δωobs can be expressed as Eq. (8), which is a function of θ and φ at the same location, as shown in Fig. 1. However, Eq. (8) is complicated when shear stress is applied:
3. Method
It can be seen from Eq. (8) that the Raman shift is related to the polarized angle under the plane stress state. To obtain multiple Raman shifts at the same location, a polarized device can be used. With at least three measured Raman shifts in different polarized directions, the three in-plane stress components (σx, σy, τxy) of the measured surface can be calculated using the theoretical relationship of the Raman shift stress. In our previous work, the analytical and linear Raman wavenumber shift-stress relationship for (110) silicon with a plane stress state takes the form [16]:
However, Eq. (8) is complicated when shear stress is applied [24]. As a result, an analytical solution was not available for all crystal orientations. A practical technique was introduced with an iterative method that is applicable for any arbitrary crystal orientation. In this solution, the theoretical Raman shift Δωth(θi, φi) under different polarization configurations (θi, φi) can be obtained based on the derivation process when the initial stress value (σx, σy, τxy)|0 is given. Subsequently, the least squares method is used to calculate the variance χ2 of the theoretical Raman shift Δωth(θi, φi) and the measured Raman shift Δωobs(θi, φi), as shown in
Fig. 2. Flow chart of iterative method for stress component decoupling under complex in-plane stress state.
4. Experiment and discussion
A series of experiments were performed to verify the iterative method. Due to the influence of laser stability, ambient temperature and other factors, random errors will inevitably be introduced during the experiment, resulting in the deviation between the measured value and the real value. The influence of the random error on the results obtained by the analytical solution and iterative method will be discussed. The samples were cut from (110) c-Si wafers with geometric sizes of 5 mm × 3 mm × 0.40 mm. The sample coordinate system of the (110) surface is given by X1´ = [−110], X2´ = [001], and X3´ = [110]. The loading angle (ψ) is defined between the width direction and the X1´ axis in the sample coordinate system, as shown in Fig. 3. In our experiments, the loading angles of samples A and B were Ψ1=7.5° and Ψ2=35.0°, respectively.
The two samples were then loaded using a uniaxial compression force along the width direction of the samples. The uniaxial compression stresses were 267 MPa and 171 MPa for samples A and B, respectively. The stress tensors in the sample coordinate system are given as
Holding on the compression force, the sample was detected using a Renishaw InVia micro-Raman spectrometer with a 50X (n.a. 0.8) lens and a 532 nm laser. For an area of 12 × 12 µm2 with a horizontal and vertical spacing of 2 µm, 49 measurement points were collected for each (θ, φ) with a preset φ= θ + 90°. The black dots in Fig. 4 show the Raman shifts of the two samples under a polarization angle of θ with an interval of 10° from 0° to 180° by averaging the 49 measurement points for each (θ, φ). Based on the measured Δωobs at different θ and φ, three stress components for samples A and B can be determined using the iterative method. The red lines in Fig. 4 show the regression curve obtained using the iterative method for the two samples. Overall, the regression curve was in good agreement with the measured results. For sample A, the prediction results of the three stress components are σx=-262.63 MPa, σy=-7.82 MPa, and τxy=-34.00 MPa. For sample B, the prediction results of the three stress components are σx=-118.26 MPa, σy=-62.98 MPa, and τxy=-70.83 MPa.
Table 1 shows a comparison of the actual given stress and the present prediction stress. It can be seen that all three calculated (measured) stress components of the two samples are close to the actual given stress components, proving the effectiveness of the iterative method.
Table 1. Comparison between the actual loadings and results by iterative method
In our experiment, a Renishaw InVia micro-Raman spectrometer was used. During the experiment, the Raman wavenumber is a direct measurement result, and the stress components are indirect calculation results. The errors in the indirect calculated value y resulting from the direct measurement values xi are formulated as follows:
With different polarizations of θ and φ, the analytical expressions of the stress components are in different forms. By analyzing the experimental data shown in Fig. 4 with the analytical solution and our iterative method, the maximum absolute errors of the different methods are listed in Table 2. Using the analytical method, the maximum absolute errors of the σ'11, σ'22, and τ'12 are 1643.35, 2041.19, and 84.38 MPa for the two samples, respectively. The errors in the experimental results far exceeded the stress sensitivity of the Raman spectrum. From Table 2, it can be seen that all three calculated (measured) stress components using the iterative method are within a 10 MPa error range, in comparison with the actual given stress state, which is less than the stress sensitivity of the Raman measurement.
Table 2. The maximum absolute errors of the results obtained by different methods
Considering a random error in the Raman shift, the accuracy of the analytical results depends on the confidence of the single-point measurement, and the random error of the Raman shift has a significant influence on the stress component decoupled by the analytical method. Although the calculation results of the iterative method are affected by the random error, compared with the actual stress state, the error is minimal (< 10 MPa). These calculated results were within the allowable error range. Therefore, the iterative method proposed in this study can effectively alleviate the impact of random errors on the calculation of the stress components.
5. Conclusions
In this study, a practical technique that can effectively and accurately calculate all the stress components through polarized μRS was developed. Multiple Raman shifts at the same location can be obtained by adjusting the polarized angles of the incident and scattered light. With the multiple Raman shift data under different polarized angles, an iterative algorithm can be applied to determine each stress component in actual applications based on the least squares method. The analytical solution for the Raman-stress relationship is sensitive to the random error of the Raman shift, and a small bias in the Raman shift may lead to a large error. The proposed iterative method, however, can effectively reduce the impact of random errors on all stress components. The proposed method can not only be applied to silicon structures with arbitrary crystal orientations but can also be extended to other material systems.
Funding
National Natural Science Foundation of China (11827802, 11890682, 12021002).
Acknowledgments
Lulu Ma and Wei Qiu would like to acknowledge the support of the National Natural Science Foundation of China.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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