# Stoney Equation

## Wafer Curvature

Customers interested in wafer stress are likely familiar with systems which measure stress in thin films deposited upon the surface of a wafer (often silicon). Naturally, the question is raised as to how these systems compare with stress maps from Polaritek Systems. This discussion shall attempt to highlight the key differences and advantages of Polaritek stress measurements.

## Background

Polaritek utilizes a non-contact optical infrared polariscopy technique to measure stress in silicon wafers. This technique is based on the transmission of infrared light through the thickness of the wafer to generate full-field stress maps. See Figure 1 for a sketch of the Polaritek approach.

Stress as a result of the deposition of thin films can be obtained from the curvature of the wafer. The amount of curvature provides a value for film stress by applying the Stoney Equation. This calculation is based upon a set of assumptions which often may limit the application of this technique. If the curvature is measured with a stylus, then contact must be made with the wafer. See Figure 2 for a sketch of the curvature-based approach.

## Discussion Points

- It must be acknowledged that a wafer may exhibit no curvature whatsoever, yet still contain significant stresses. Curvature is the result of unbalanced internal stresses and curvature-based stress measurement techniques are only able to estimate the
*net*imbalance of stresses imparted by deposition of a thin film. Polaritek is able to integrate the local stress field at each point in the wafer. - THIN FILM PRESENCE. It should be obvious that curvature-based stress measurements are only applicable
*in the presence of a thin film*. If the wafer does not possess a thin film, its stress cannot be measured by curvature-based methods. Polaritek technology has no such constraint. Many critical applications such as measuring stress imparted by the wafering process and monitoring surface damage removal cannot be solved by curvature-based techniques. - GLOBAL STRESS: Polaritek is able to quickly generate stress maps which plot
*local stress*variation across the full-field of the wafer. Curvature-based stress measurements can only compute*global stress*across the wafer. Local regions of high stress likely to be the site of failures can only be resolved by Polaritek technology. - STRESS STATE: Wafer curvature stress measurement methods require assumption of uniform curvature and an equibiaxial stress state. Research has shown this assumption is often violated in practice.
^{[1]}Polaritek does not require this assumption and can generate individual wafer maps of X and Y normal stress as well as the maximum shear stress. - FILM THICKNESS: Wafer curvature stress measurement methods require that the thickness of the thin film is much less than the thickness of the substrate. Polaritek requires so such assumption.
- STRESS LOCATION: Polaritek measures the sum of stresses through the thickness of the wafer (and thin film if present) at each X-Y point in the wafer. This presents a complete picture of stresses in both the wafer and the substrate. Wafer curvature stress measurement systems can only measure the stress in the thin film and cannot determine anything about stress variations within the substrate.

^{[1]} Feng, X., Y. Huang, and A. J. Rosakis. “On the Stoney formula for a thin film/substrate system with nonuniform substrate thickness.” Journal of Applied Mechanics 74.6 (2007): 1276-1281.

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