Residual Stress in a Thin Film Resonator

• 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.

Introduction

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.

1

Er

T

Model Definition – Boundary Conditions

• The ends of the beam springs are attached to the substrate, and thus constrained to zero movement in all directions.

• As the resonator is very thin, the mesh is scaled in the z direction.

Model Definition – Mesh

Results

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