Stationary Analysis of a Biased Resonator

Stationary Analysis of a Biased Resonator — 3D

Silicon micromechanical resonators have long been used for designing sensors and are now becoming increasingly important as oscillators in the consumer electronics market. This sequence of models analyzes in detail a surface micromachined MEMS resonator, designed as part of a micromechanical filter. The resonator is based on that developed in Ref. 1.

This model performs a stationary analysis of the resonator, with an applied DC bias. It serves as a basis for all the subsequent analyses.

Model Definition

The model consists of a poly-silicon resonator, which is manufactured through a surface micromachining process. Initially, a silicon wafer is coated with 0.75 μm of oxide and 0.15 μm of silicon nitride to isolate the micromachined parts from the wafer ground plane. Polysilicon electrodes with a thickness of 0.3 μm are deposited next. A sacrificial layer of oxide is then deposited to a thickness of 198.5 nm. Note that in Ref. 1 the sacrificial oxide is actually 1.3 μm, but the gap thickness was adjusted to this value for the purposes of simulation to account for the depletion layer in the silicon. This model uses the same adjustment to enable the simulations to be directly compared with those presented in the paper. Holes are etched in the sacrificial layer (to provide anchor points for the resonator) and the structural polysilicon is deposited with a thickness of 1.9 μm.

The structure has a plane of symmetry, so it is possible to model only half of the geometry explicitly, although care must be taken to mirror the geometry before performing a modal analysis. Figure 1 shows the geometry.

Figure 1: Top: Device geometry. The wafer itself is not shown explicitly, but is represented in the model by a ground plane on the underside of the geometry. Bottom: The model geometry as it appears in COMSOL.

The layers of deposited material from the ground plane up are: silicon oxide, silicon nitride (too thin to see clearly), polysilicon electrodes/air gap (etched sacrificial oxide), polysilicon resonator, and air.

In operation, both the silicon resonator and the underlying wafer are grounded and an electric voltage is applied to the driving electrode, which is bisected by the symmetry plane.

Typically a DC bias of 35 V is applied in normal operation of the device. The assumption is made that the polysilicon is a perfect conductor, so the bias voltage can be applied on the resonator using the Domain Terminal feature, which is much easier than selecting the many boundaries for setting the electric potential on them.

In this model, the deformation of the structure is computed with the applied DC bias. Note that the silicon oxide and nitride are assumed to be rigid for the solid mechanics simulations, so the structure is anchored at the base of its electrode, and these domains are not included in the solid mechanics equations.

Electromechanical Forces

Within a vacuum or other medium, forces between charged bodies can be computed on the assumption that a fictitious state of stress exists within the field. The Electromagnetic or Maxwell stress tensor can be used to compute the induced stresses in a material as a result of an electric field as well as surface forces acting on bodies in air or vacuum. In this model, it is assumed that the polysilicon is doped sufficiently heavily that it can be treated as a perfect conductor. The electric field is assumed to be zero inside the resonator, which means that the Maxwell stress tensor is zero inside the material and there are no volumetric electrical forces. The Maxwell stress tensor in the medium surrounding the resonator, where the electric field is nonzero is (Ref. 2)

A net force on the surface typically results from the discontinuity of the stress tensor at the interface. However, because it is undesirable to apply a stress term throughout the vacuum, the force is only computed on the surface of the resonator, and is applied by the Electromechanical Interface node. The surface force is given by

where n1 is the surface normal, pointing out from the mechanical body.

Results and Discussion

Figure 2 shows the z displacement of the structure with an applied DC bias. As expected the structural displacement is maximal on the symmetry plane at the center of the device. The maximum displacement is 13 nm. Electric potential isosurfaces are also shown in Figure 2. As expected, the isobars are uniformly distributed and closest together between the resonator and the electrode. This corresponds to a region of uniform electric field. Around the electrode the fringing fields can also be seen. Note that the surface of the resonator is assumed to be perfectly grounded. This is a result of the potential boundary condition used and is equivalent to the assumption that the polysilicon is a perfect conductor.