Extension of the one-dimensional Stoney algorithm to a two-dimensional case

TL;DR

This paper extends the one-dimensional Stoney algorithm to two dimensions by developing a new curvature estimation method that involves quadratic surface approximation, eigenproblem solving, and multiple algorithm versions for practical use.

Contribution

The paper introduces a novel two-dimensional Stoney algorithm extension with five variants, improving curvature estimation accuracy and practical applicability.

Findings

Five algorithm versions with different complexity and accuracy

Recommended practical version identified

Enhanced curvature estimation in two-dimensional cases

Abstract

This article presents the extension of the one-dimensional Stoney algorithm to a two-dimensional case. The proposed extension consists in modifying the method of curvature estimation. The surface profile of the wafer before deposition of the thin film and after its deposition was locally approximated by the quadric. From this quadric, a quadratic form and the first degree surface were separated. An eigenproblem was solved for the matrix of this quadratic form. From eigenvectors a new coordinate system was created in which a new formula of the quadric was found. In this new coordinate system, the two-dimensional problem of estimating the curvature tensor has been solved by solving two independent one-dimensional problems of curvature estimation. Returning to the primary coordinate system, in this primary coordinate system, a solution to the two-dimensional problem was obtained. The article proposed five versions of the two-dimensional Stoney algorithm, with diverse complexity and accuracy. The recommendation for the version of the algorithm that could be practically used was also presented.