Modified Stoney's equation with anisotropic substrates undergoing large deformations

TL;DR

This paper extends Stoney's equation to account for large deformations in anisotropic substrates like silicon wafers, providing more accurate curvature predictions for thin film systems under non-linear deformation.

Contribution

It derives new relations between substrate curvature and film mismatch for anisotropic silicon wafers in the non-linear deformation regime, addressing limitations of the classical Stoney equation.

Findings

Derived relations for Si(001) and Si(111) substrates under large deformations.

Numerical results show significant deviations from classical predictions in non-linear regime.

Enhanced understanding of residual stress measurement in anisotropic substrates.

Abstract

Residual stresses in a thin film deposited on a substrate results in a curvature of the system, which can be measured using the well known Stoney equation. Isotropic elasticity of the substrate along with infinitesimal strains and rotations are two important assumptions used in the derivation of the Stoney equation. However, the transverse deflection in the substrate contributes significantly to the extensional strain in its plane, leading to non-linearity in its deformation. Moreover, Silicon wafers are predominantly used as substrate materials to measure the curvature of the system. In this paper, relations between normalized substrate curvature and normalized thin film mismatch are derived in the non-linear deformation regime, for substrates made of single crystal Si(001) and Si(111) wafers. Numerical results of curvature of thin film configurations with Si(001) and Si(111) wafer substrates, undergoing large deformations are presented and discussed.