Abstract
A model of thermal stress in double-layer optical dielectric films on circular substrates was established based on the theory of double-layer composite beams. Here, considering the boundary conditions including force balance and bending moment balance, the distribution of stress and strain in the double-layer film-substrate system was analyzed following equivalence manipulation to determine a detailed formula for calculating the thermal stress in the equivalent film and substrate. The derived formula was not only effective in analyzing the stress and strain of the double-layer film-substrate system but was also applicable for predicting the distribution of thermal stress in the periodic elastic multilayer film-substrate system. According to the actual radius of curvature of the substrate measured via a profilometer before and after the deposition of the HfO\(_2\)/SiO\(_2\) double-layer films, the obtained residual stress of the film was − 79.33 MPa, whereas the thermal stress of the film was calculated to be −52.59 MPa using the theoretical formula. The calculations of the theoretical model were similar to the experimental results when the smaller intrinsic stresses were neglected and the double-layer film was only of nanometer thickness, thus verifying the effectiveness of the double-layer film-substrate model.
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The data that support the findings of this study are available from the corresponding author upon reasonable request.
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This work was funded by the Key Laboratory for Equipment Pre-research (No.6142207190407), International Science and Technology Cooperation and Exchange Plan Project of Shaanxi Province(Nos.2018KWZ-02).
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Shi, Y., Xu, J., Li, Y. et al. Theoretical Model of Thermal Stress in the Film-Substrate System of Optical Thin Film. J. Electron. Mater. 51, 5937–5945 (2022). https://doi.org/10.1007/s11664-022-09819-w
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DOI: https://doi.org/10.1007/s11664-022-09819-w