Inversion of Residual Stresses in Silicon Wafer from Surface Deflection Measurements

Obata, Yuri; Yoneyama, Satoru

doi:10.1007/978-3-030-30098-2_11

Cited by 3 publications

(5 citation statements)

References 3 publications

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“…There exists a strict mapping relationship between silicon wafer deformation and polishing-induced residual stress. Therefore, it is possible in theory that the residual stress could be calculated from the deformation measurements [28]. For silicon wafers, the Stoney formula is widely used for relating the curvature of the wafer to the residual stress [29,30].…”

Section: Residual Stress Distribution Calculation Of Polished Silicon...mentioning

confidence: 99%

Evaluation of polishing-induced subsurface damage based on residual stress distribution via measured global surface deformation for thinned silicon wafers

Liu¹,

Zhou²,

Han³

et al. 2021

Surf. Topogr.: Metrol. Prop.

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Polishing is widely used to obtain smooth thinned wafer surfaces and the process-induced subsurface damage is often evaluated by scanning electron microscope or transmission electron microscope images. However, the difference of microscope images is very slight and subtle between wafers and affected by the location selection of the measured points. In this study, we explored the method of evaluating the polishing-induced subsurface damage based on residual stress distribution via measured global surface deformation. The surface deformation was first measured and then the polishing-induced residual stress distribution was calculated from the surface topography data by solving the system of linear equations. The polishing-induced subsurface damage could be evaluated independent of the dimension of the silicon wafer by the proposed method. How the thinned silicon wafer was supported during measurement process greatly affected polishing-induced wafer deformation and gravity-induced deformation should be separated from the measured data. Regularization method should be introduced in the calculation of residual stress to reduce the errors caused by the multi-collinearity of the system of linear equations. The calculated residual stress distribution was found to be an obvious centrosymmetric figure and exhibited a quasi-axisymmetric pattern with the axis of symmetry correlated with certain crystalline directions, indicating the effect of anisotropy of monocrystalline silicon in the polishing process. The proposed method could also be used in the evaluation of polishing-induced subsurface damage of other large and thin substrate plates.

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Section: Residual Stress Distribution Calculation Of Polished Silicon...mentioning

confidence: 99%

Evaluation of polishing-induced subsurface damage based on residual stress distribution via measured global surface deformation for thinned silicon wafers

Liu¹,

Zhou²,

Han³

et al. 2021

Surf. Topogr.: Metrol. Prop.

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show abstract

“…There were n linear equations corresponding to the n points and there were m variables to be determined in the system of linear equations. The system of linear equations could be transformed into a matrix equation [13], which is shown in Equation (3).…”

Section: Principle Of Residual Stress Obtainmentmentioning

confidence: 99%

“…In the polishing process, the surface of the polishing pad was smooth and the density of abrasive grain trajectories was uniformly distributed along the wafer surface. Therefore, the residual stress obtained from wafer deformation by solving Equation ( 5) which was adopted in reference [13] was not true and new methods needed to be explored to obtain reasonable solutions. The total wafer deformation was simulated using the calculated x and compared with the measurement result to verify the method.…”

Section: Principle Of Residual Stress Obtainmentmentioning

confidence: 99%

“…However, there exists a multi-collinearity problem and small measurement errors would lead to large changes in residual stress calculation results. Besides, the residual stress distribution is not continuous across the wafer surface, which does not agree with actual situations [13]. The conventional regularization method using an Identify matrix as the Tikhonov matrix was used in our previous research.…”

Section: Introductionmentioning

confidence: 96%

“…The residual stress leads to the wafer deformation and there is a definite correspondence between them. The residual stress could be obtained inversely from the wafer deformation data [13]. Stoney equation is often used to relate the residual stress to the geometric parameters of the wafer deformation [14,15].…”

Section: Introductionmentioning

confidence: 99%

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Obtainment of Residual Stress Distribution from Surface Deformation under Continuity Constraints for Thinned Silicon Wafers

et al. 2021

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Precision machining (e.g., fine grinding, polishing) induced residual stress is very small and often not constant across the wafer and it is difficult to be directly obtained by stress testing equipment or Stoney equation. The residual stress could be obtained theoretically based on the principle of superposition in which the entire wafer deformation is taken as the sum of all deformations induced by the residual stresses of different positions on the wafer surface. However, the solved residual stress is affected greatly by deformation measurement errors and fluctuates greatly across the wafer surface. To solve the problem, a regularization method with continuity constraints was proposed in this study. The mechanisms for the discontinuity of the residual stress distribution and the sensitivity of calculation results to the measurement errors were studied. The influences of the number of subareas of the silicon wafer were investigated and the continuity constraint term was constructed based on the positional relationship of different subareas. Stable and continuous residual stress distribution was successfully obtained after using the proposed regularization method. The method may also be applied to estimate the residual stress from surface deformation for thin substrate plates of other materials.

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Estimating residual stresses of silicon wafer from measured full-field deflection distribution

Obata

Sekino

Takeo

et al. 2022

Optics and Lasers in Engineering

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Inversion of Residual Stresses in Silicon Wafer from Surface Deflection Measurements

Cited by 3 publications

References 3 publications

Evaluation of polishing-induced subsurface damage based on residual stress distribution via measured global surface deformation for thinned silicon wafers

Evaluation of polishing-induced subsurface damage based on residual stress distribution via measured global surface deformation for thinned silicon wafers

Obtainment of Residual Stress Distribution from Surface Deformation under Continuity Constraints for Thinned Silicon Wafers

Estimating residual stresses of silicon wafer from measured full-field deflection distribution

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