Wafer Level Stress Distribution Impact on Dynamic Calibration Matrix Drift
Wafer-level residual stress gradients induce asymmetric die warpage that relaxes over time, causing dynamic calibration matrices to suffer severe cross-axis drift.
Stoney Formula
Mechanical Relationship
Mathematical relationships correlate the curvature of a thin substrate with the internal stress of a film deposited upon its surface. The Stoney formula provides a way to calculate the average biaxial stress in a thin-film layer by measuring the change in the radius of curvature of the underlying wafer.
Material Constant
Young’s modulus and the Poisson’s ratio of the substrate are the primary material constants used in the calculation. Application of the Stoney formula requires precise knowledge of the wafer thickness, as the stress value depends on the square of this dimension. Small errors in the measurement of the substrate thickness result in significant inaccuracies in the calculated stress.
Metrology Application
Optical profilers or laser scanning systems determine the shape of the wafer before and after the deposition process. The Stoney formula relates the change in curvature directly to the force exerted by the film on the lattice of the substrate. This method is used in semiconductor manufacturing to monitor the mechanical health of metal and dielectric layers.
Analytical Limit
Assumptions regarding the isotropy of the material and the uniformity of the film limit the applicability of the equation in complex geometries. While the Stoney formula is effective for flat wafers, it does not account for the localized stress gradients found near the edges of features or patterns.