Residual Stress in a Thin Film Resonator

Introduction Surface micromachined thin films are often subject to residual stress. In this model, the springs of a thin film resonator are fastened to a thick substrate. A temperature change causes thermal stress, due to the different thermal expansion coefficients in the film and the substrate. This alters the natural frequencies of the resonator.

Model Definition – Geometry The resonator can have either straight or folded springs.

Model Definition – Equations Stress-strain relation: The thermal strain equals the difference between deposition and operating temperature times the difference between the thermal expansion coefficients: The eigenfrequency equation is solved with the residual stress accounted for.

Model Definition – Boundary Conditions The ends of the beam springs are attached to the substrate, and thus constrained to zero movement in all directions.

Model Definition – Mesh As the resonator is very thin, the mesh is scaled in the z direction.

Results The eigenfrequencies of the folded resonator are less affected by the thermal expansion.