COMSOL 6.4 - Stationary Analysis of a Biased Resonator

Stationary Analysis of a Biased Resonator — 2D

Silicon micromechanical resonators have long been used for designing sensors and are now becoming increasingly important as oscillators in the consumer electronics market. In this sequence of models, a surface micromachined MEMS resonator, designed as part of a micromechanical filter, is analyzed in detail. 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 is used as a basis for all the subsequent analyses.

Model Definition

Ref. 1 describes a polysilicon resonator, which is manufactured through a surface micromachining process. The details of this process are outlined in the Stationary Analysis of a Biased Resonator — 3D documentation. For this 2D study, a simplified version of the 3D geometry is considered. For simplicity, the resonator is modeled as a 2 μm thick rectangular beam with a length of 45 μm. A boundary is applied to each end of the resonator to act as the anchor points at which the resonator is attached to the substrate wafer. The wafer substrate is not explicitly modeled, instead only a 0.1985 μm thick air gap between the resonator and the substrate is included. The effects of the driving electrode are included using boundary conditions applied directly to the underside of the air gap, as shown in Figure 1.

Note that although the structure has a plane of symmetry, which vertically bisects the device, we do not use a symmetry boundary condition. A subsequent model considers the normal modes of the structure, and a symmetry condition eliminates the anti-symmetric modes from this analysis.

Figure 1: The model geometry. The rectangular resonator and the air gap are highlighted in red and green outline, respectively. The boundaries to which the driving voltage is applied are highlighted in blue. Note that the vertical dividing lines are not part of the physical geometry of the resonator, but are included to allow a suitable swept mesh to be easily created.

In operation the silicon resonator is grounded (using the feature) and a driving electrode applies an electric potential to the central portion of the air gap, as shown in Figure 1. Typically a DC bias of 35 V is applied in normal operation of the device. In this model the deformation of the structure is computed with the applied DC bias.

ElectromechanicS Coupling

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. Within a material, COMSOL Multiphysics uses the following form of the stress tensor TEM,S, which is appropriate for isotropic materials (Ref. 2):

where E is the electric field, D is the electric displacement field, I is the identity tensor, and ε0 is the permittivity of free space and a1 and a2 are material parameters that specify the electrostrictive properties of the material (for this device, assume a1 = a2 = 0 because the field is in any case very low within the material). This additional stress is applied to the material by the coupling. Note that mechanical stresses are usually induced in the material as a result of the net forces acting on the surfaces, in addition to the stress induced by the electric field.

The forces on the surfaces of a solid body can be computed by applying a similar stress term within the vacuum of the form:

A net force on the surface typically results from the discontinuity of the stress tensor at the interface. However, since it is undesirable to apply a stress term throughout the vacuum, the force is only available on the surface of solid bodies, via the coupling. 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 y displacement of the structure with an applied DC bias. As expected the structural displacement is maximal at the center of the geometry. The maximum displacement is 11 nm.

The electric potential contours are shown in Figure 3. The fringing fields extend approximately 1 μm into the gap either side of the driving electrode. Note that the fringing fields are not well resolved due to the structure of the swept mesh. In order to investigate these fields the mesh must be refined on either side of the electrode. Also note that the silicon is assumed to be a perfect conductor. Although the author’s of Ref. 1 do not explicitly give the doping in the polysilicon, it is likely that this assumption is relatively poor given the estimated depletion region width of approximately 0.7 μm that is quoted. This model could be extended to include the effects of semiconductor transport and an improvement on this assumption could be made by adding the electric currents which are induced inside the resonator.

Figure 2: The y-displacement of the resonator as a function of position. The maximum displacement occurs in the center of the resonator, immediately over the biasing electrode.

Figure 3: Electric potential contours in the gap between the grounded resonator and the biased driving electrode.

References

1. F.D. Bannon III, J.R. Clark, and C.T.-C. Nguyen, “High-Q HF Microelectromechanical Filters,” IEEE Journal of Solid State Circuits, vol. 35, no. 4, pp. 512–526, 2000.

2. J.A. Stratton, Electromagnetic Theory, McGraw Hill, New York, 1941.

MEMS_Module/Actuators/biased_resonator_2d_basic

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > > .

Click .

Click

In the tree, select > .

Click

Define and enter the values for the following parameters.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
Vdc	35[V]	35 V	DC bias voltage
l_b	22.5[um]	2.25E-5 m	Length of beam
t_b	2[um]	2E-6 m	Thickness of beam
l_e	10[um]	1E-5 m	Length of electrode
gap	0.1985[um]	1.985E-7 m	Air gap

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type l_b.

In the text field, type t_b.

Locate the section. In the text field, type -l_b.

In the text field, type gap.

Rectangle 2 (r2)

In the toolbar, click

In the window for , locate the section.

In the text field, type l_b.

In the text field, type gap.

Locate the section. In the text field, type -l_b.

Rectangle 3 (r3)

In the toolbar, click

In the window for , locate the section.

In the text field, type l_e.

In the text field, type gap+t_b.

Locate the section. In the text field, type -l_e.

Mirror 1 (mir1)

In the toolbar, click

and choose .

Click in the window and then press Ctrl+A to select all objects.

In the window for , locate the section.

Select the checkbox.

Click

Click the

button in the toolbar.

Add materials to the model.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Materials

Si - Polycrystalline silicon (mat1)

Select Domains 2, 4, 6, and 8 only.

Definitions

Air

In the toolbar, click

In the window for , type Air in the text field.

Select Domains 1, 3, 5, and 7 only.

Solid Mechanics (solid)

In the window, under click .

Select Domains 2, 4, 6, and 8 only.

Moving Mesh

Deforming Domain 1

In the window, under > click .

In the window for , locate the section.

From the list, choose .

Set up symmetry/roller boundary conditions needed for convergence when gap becomes very narrow.

Symmetry/Roller 1

In the window, click .

Select Boundaries 7 and 12 only.

Set up the solid mechanics and electrostatics boundary conditions.

Solid Mechanics (solid)

Fixed Constraint 1

In the toolbar, click

and choose .

Apply a to both ends of the resonator.

Select Boundaries 3 and 22 only.

Electrostatics (es)

With the assumption that the silicon membrane is a good conductor, use the feature to ground the Si domains. Note: The feature will be very handy for a conducting domain with a complex shape and many exterior boundaries - instead of selecting all the boundaries to set up the ground, terminal, or electric potential boundary condition, we only need to select the domain to specify the feature with the same effect. In addition, the computation load is reduced, because the electrostatic degrees of freedom within the feature do not need to be solved for.

In the window, under click .

Domain Terminal 1

In the toolbar, click