Thermal deflections in multilayer microstructures and athermalization

Exact and approximate analytical solutions are developed for calculating the thermally induced deformation of three-layer cantilever structures. The solution is derived from the closed-form solutions for multilayer films. Thermal deformation and athermalization conditions are derived using dimensionless parameters for film to substrate thickness ratios for three-layer structures. The analytical solution for a narrow beam is applied to a scan mirror plate suspended with two torsional flexures. The results agreed well with finite element method simulations and experiments. Tests are performed using a bulk-micromachined silicon microelectromechanical system scanner that has a thin gold (Au) coil layer on one side and an aluminum (Al) mirror layer on the other side. Useful figures using film-to-substrate thickness ratios and the material independent normalized parameters are introduced for easy thermal deformation computations and performance trades for three-layer structures.