Single-Side Warpage Estimation and Residual Stress Identification in Silicon Wafers

Background Evaluation of warpage and residual stress in silicon wafers is essential for semiconductor manufacturing. Conventional warpage measurement requires double-side surface profiling, increasing system complexity and cost. Objective This study proposes methods to estimate warpage from single-side surface measurements of a horizontally supported wafer, where the measured profile includes both gravitational deflection and intrinsic warpage. Methods Two approaches are developed to separate deflection from the measured profile. The first estimates deflection by averaging accumulated measurements, assuming wafer-to-wafer warpage randomness. The second exploits the approximate 60-degree symmetry of the deflection distribution to estimate deflection from a single measurement. Results Compared with the reliable three-point-support inverting method, warpage is identified within approximately 10% using accumulated data and within about 20% using a single measurement. Residual stress is subsequently identified from the estimated warpage using a superposition-based inverse analysis with stress-gradient regularization. The regularization parameter is determined objectively using the Akaike and Bayesian information criteria. Numerical validation and application to measured data demonstrate that plausible residual stress distributions can be obtained from single-side measurements. Conclusions The proposed method enables warpage and residual stress estimation without double-side inspection, offering simplified instrumentation and reduced cost for silicon wafer evaluation.