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

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...

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Autores principales:	Haijun Liu, Tao Yang, Jiang Han, Xiaoqing Tian, Shan Chen, Lei Lu
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	silicon wafer residual stress distribution continuity constraint multicollinearity problem Mechanical engineering and machinery TJ1-1570
Acceso en línea:	https://doaj.org/article/3b494ba16b144f569d58bf5c5ec6dde0
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id	oai:doaj.org-article:3b494ba16b144f569d58bf5c5ec6dde0
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spelling	oai:doaj.org-article:3b494ba16b144f569d58bf5c5ec6dde02021-11-25T18:12:17ZObtainment of Residual Stress Distribution from Surface Deformation under Continuity Constraints for Thinned Silicon Wafers10.3390/machines91102842075-1702https://doaj.org/article/3b494ba16b144f569d58bf5c5ec6dde02021-11-01T00:00:00Zhttps://www.mdpi.com/2075-1702/9/11/284https://doaj.org/toc/2075-1702Precision 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.Haijun LiuTao YangJiang HanXiaoqing TianShan ChenLei LuMDPI AGarticlesilicon waferresidual stress distributioncontinuity constraintmulticollinearity problemMechanical engineering and machineryTJ1-1570ENMachines, Vol 9, Iss 284, p 284 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	silicon wafer residual stress distribution continuity constraint multicollinearity problem Mechanical engineering and machinery TJ1-1570
spellingShingle	silicon wafer residual stress distribution continuity constraint multicollinearity problem Mechanical engineering and machinery TJ1-1570 Haijun Liu Tao Yang Jiang Han Xiaoqing Tian Shan Chen Lei Lu Obtainment of Residual Stress Distribution from Surface Deformation under Continuity Constraints for Thinned Silicon Wafers
description	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.
format	article
author	Haijun Liu Tao Yang Jiang Han Xiaoqing Tian Shan Chen Lei Lu
author_facet	Haijun Liu Tao Yang Jiang Han Xiaoqing Tian Shan Chen Lei Lu
author_sort	Haijun Liu
title	Obtainment of Residual Stress Distribution from Surface Deformation under Continuity Constraints for Thinned Silicon Wafers
title_short	Obtainment of Residual Stress Distribution from Surface Deformation under Continuity Constraints for Thinned Silicon Wafers
title_full	Obtainment of Residual Stress Distribution from Surface Deformation under Continuity Constraints for Thinned Silicon Wafers
title_fullStr	Obtainment of Residual Stress Distribution from Surface Deformation under Continuity Constraints for Thinned Silicon Wafers
title_full_unstemmed	Obtainment of Residual Stress Distribution from Surface Deformation under Continuity Constraints for Thinned Silicon Wafers
title_sort	obtainment of residual stress distribution from surface deformation under continuity constraints for thinned silicon wafers
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/3b494ba16b144f569d58bf5c5ec6dde0
work_keys_str_mv	AT haijunliu obtainmentofresidualstressdistributionfromsurfacedeformationundercontinuityconstraintsforthinnedsiliconwafers AT taoyang obtainmentofresidualstressdistributionfromsurfacedeformationundercontinuityconstraintsforthinnedsiliconwafers AT jianghan obtainmentofresidualstressdistributionfromsurfacedeformationundercontinuityconstraintsforthinnedsiliconwafers AT xiaoqingtian obtainmentofresidualstressdistributionfromsurfacedeformationundercontinuityconstraintsforthinnedsiliconwafers AT shanchen obtainmentofresidualstressdistributionfromsurfacedeformationundercontinuityconstraintsforthinnedsiliconwafers AT leilu obtainmentofresidualstressdistributionfromsurfacedeformationundercontinuityconstraintsforthinnedsiliconwafers
_version_	1718411492130291712

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

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