An extended Stoney's formula including nonlinear deformation for large size wafer of multilayers with arbitrary thicknesses

Abstract

Applying the principle of minimum potential energy in an invariant plane reference frame, we obtain a concise curvature-strain equation for large size thick multilayered wafer, which incorporates strain due to nonlinear deformation. This equation manifests itself as Stoney's or other famous formulas for specific cases, and clarifies their relations and physical origins. We find nonlinear deformation greatly suppresses wafer bending and verify this effect experimentally in 2–4 in. GaN/sapphire systems. Such mechanical analysis provides a foundation of curvature control which is crucial in wafer fabrication for electronics and photonics.