Approaches to determining curvature of wafers by their topography
N.A. Djuzheva aNational Research University of Electronic Technology (MIET), sq. Shokina 1, Zelenograd, Moscow, 124498, Russian Federation bInstitute of Mathematical Problems of Biology, Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Prof. Vitkevich str. 1, Pushchino, Moscow Region, 142290, Russian Federation
We discuss peculiarities of the curvature analysis of’wafers considering the heterogeneity of their topography for a quantitative estimation and localization of irregularity, or for subsequent calculations of mechanical stresses. We analyze three approaches to calculating surface curvatures from digital elevation models. The first one is based on the analysis of wafer surface profiles using polynomial approximation; the calculation is based on the curvature radius determination of a curved line; mechanical stresses are calculated using the Stoney method. The second approach uses the second partial derivatives of an elevation function in the Cartesian or cylindrical coordinate systems to analyze irregular topography and then to calculate mechanical stresses. The third considers the entire wafer topography as a two-dimensional elevation matrix and uses the mathematical apparatus of differential geometry and the experience of geomorphometry to determine convex and concave areas of the surface as well as to perform a complete analysis of the surface curvature system. We demonstrate the implementation of these approaches on a wafer like a segment of a sphere and on a complex-shaped wafer.
Keywords: surface, topography, curvature, radius of curvature, deflection, mechanical stresses, deformation, Stoney equation, silicon wafer, optical profilometry, defectiveness, geomorphometry, digital elevation model, DEM PACS:68.35.Gy DOI:10.3367/UFNe.2021.10.039076 URL: https://ufn.ru/en/articles/2022/7/d/ Citation: Dedkova A A, Florinsky I V, Djuzhev N A "Approaches to determining curvature of wafers by their topography" Phys. Usp.65 706–722 (2022)