1. Residual Stress in a Thin Film Resonator

2. 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.

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

4. 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.

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

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

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