Electrostatically Actuated Cantilever

The elastic cantilever beam is an elementary structure in MEMS design. This example shows the bending of a beam due to electrostatic forces. The model uses the electromechanics multiphysics interface to solve the coupled equations for the structural deformation and the electric field. Such structures are frequently tested by means of a low frequency capacitance voltage sweep. The model predicts the results of such a test.

Model Definition

Figure 1 shows the model geometry. The beam has the following dimensions:

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Length: 300 μm

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Width: 20 μm

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Thickness 2 μm

Because the geometry is symmetric, only half of the beam needs to modeled. The beam is made of polysilicon with a Young’s modulus, E, of 153 GPa, and a Poisson’s ratio, ν, of 0.23. It is fixed at one end but is otherwise free to move. The polysilicon is assumed to be heavily doped, so that electric field penetration into the structure can be neglected. In this case, the Domain Terminal feature can be used to set up the Si domain. The beam resides in an air-filled chamber that is electrically insulated. The lower side of the chamber has a grounded electrode.

Figure 1: Model Geometry. The beam is 300 μm long and 2 μm thick, and it is fixed at x = 0. The model uses symmetry on the zx-plane at y = 0. The lower boundary of the surrounding air domain represents the grounded substrate. The model has 20 μm of free air above and to the sides of the beam, while the gap below the beam is 2 μm.

An electrostatic force caused by an applied potential difference between the two electrodes bends the beam toward the grounded plane beneath it. To compute the electrostatic force, this example calculates the electric field in the surrounding air. The model considers a layer of air 20 μm thick both above and to the sides of the beam, and the air gap between the bottom of the beam and the grounded layer is initially 2 μm. As the beam bends, the geometry of the air gap changes continuously, resulting in a change in the electric field between the electrodes. The coupled physics is handled automatically by the Electromechanics multiphysics interface.

The electrostatic field in the air and in the beam is governed by Poisson’s equation:

where derivatives are taken with respect to the spatial coordinates. The numerical model represents the electric potential and its derivatives on a mesh which is moving with respect to the spatial frame. The necessary transformations are taken care of by the Electromechanics multiphysics interface, which also contains smoothing equations governing the movement of the mesh in the air domain.

The cantilever connects to a voltage terminal with a specified bias potential, Vin. The bottom of the chamber is grounded, while all other boundaries are electrically insulated. The terminal boundary condition automatically computes the capacitance of the system.

The force density that acts on the electrode of the beam results from Maxwell’s stress tensor:

where E and D are the electric field and electric displacement vectors, respectively, and n is the outward normal vector of the boundary. This force is always oriented along the normal of the boundary.

Navier’s equations, which govern the deformation of a solid, are more conveniently written in a coordinate system that follows and deforms with the material. In this case, these reference or material coordinates are identical to the actual mesh coordinates.

Results and Discussion

There is positive feedback between the electrostatic forces and the deformation of the cantilever beam. The forces bend the beam and thereby reduce the gap to the grounded substrate. This action, in turn, increases the forces. At a certain voltage the electrostatic forces overcome the stress forces, the system becomes unstable, and the gap collapses. This critical voltage is called the pull-in voltage.

At applied voltages lower than the pull-in voltage, the beam stays in an equilibrium position where the stress forces balance the electrostatic forces. Figure 2 shows the beam displacement and the corresponding displacement of the mesh surrounding it. Figure 3 shows the electric potential and electric field that generates these displacements. In Figure 4 the shape of the cantilever’s deflection is illustrated for each applied voltage, by plotting the z-displacement of the underside of the beam at the symmetry boundary. The tip deflection as a function of applied voltage is shown in Figure 5. Note that for applied voltages higher than the pull-in voltage, the solution does not converge because no stable stationary solution exists. This situation occurs if an applied voltage of 6.2 V is tried. The pull-in voltage is therefore between 6.1 V and 6.2 V. For comparison, computations in Ref. 1 predict a pull-in voltage of

where c1 = 0.07, c2 = 1.00, and c3 = 0.42; g0 is the initial gap between the beam and the ground plane; and

If the beam has a narrow width (W) relative to its thickness (H) and length (L), Ê is Young’s modulus, E. Otherwise, E and Ê, the plate modulus, are related by

where ν is Poisson’s ratio. Because the calculation in Ref. 1 uses a parallel-plate approximation for calculating the electrostatic force and because it corrects for fringing fields, these results are not directly comparable with those from the simulation. However the agreement is still reasonable: setting W = 20 μm results in VPI = 6.07 V.

Figure 2: z-displacement for the beam and the moving mesh as a function of position. Each mesh element is depicted as a separate block in the back half of the geometry.

Figure 3: Electric Potential (color) and Electric Field (arrows) at various cross sections through the beam.

Figure 4: Displacement of the lower surface of the cantilever, plotted along the symmetry boundary, for different values of the applied voltage.

Figure 5: Cantilever tip displacements as a function of applied Voltage V0.

Figure 6: Device capacitance vs applied voltage V0.

Figure 6 shows the DC C-V curve predicted for the cantilever beam. To some extent, this is consistent with the behavior of an ideal parallel plate capacitor, whose capacitance increases with decreasing distance between the plates. But this effect does not account for all the change in capacitance observed. In fact, most of it is due to the gradual softening of the coupled electromechanical system. This effect leads to a larger structural response for a given voltage increment at higher bias, which in turn means that more charge must be added to retain the voltage difference between the electrodes.

Reference

1. R.K. Gupta, Electrostatic Pull-In Structure Design for In-Situ Mechanical Property Measurements of Microelectromechanical Systems (MEMS), Ph.D. thesis, MIT, 1997.

MEMS_Module/Actuators/electrostatically_actuated_cantilever

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

Geometry 1

Use microns to define the geometry units.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Create two blocks to represent the cantilever and air domains, respectively.

Block 1 (blk1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 300.

In the text field, type 10.

In the text field, type 2.

Locate the section. In the text field, type 2.

Click

Block 2 (blk2)

In the toolbar, click

In the window for , locate the section.

In the text field, type 320.

In the text field, type 40.

In the text field, type 24.

Add two more blocks to simplify meshing of the geometry.

Block 3 (blk3)

In the toolbar, click

In the window for , locate the section.

In the text field, type 20.

In the text field, type 40.

In the text field, type 24.

Locate the section. In the text field, type 300.

Block 4 (blk4)

In the toolbar, click

In the window for , locate the section.

In the text field, type 300.

In the text field, type 10.

In the text field, type 24.

Click

Add a parameter for the DC voltage applied to the cantilever.

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
V0	5[V]	5 V	Bias on cantilever

The cantilever is assumed to be heavily doped so that it acts as a conductor, held at constant potential. The feature is therefore used.

Solid Mechanics (solid)

In the window, under click .

Select Domain 2 only.

Electrostatics (es)

The default feature was set to use solid material type. Add one more feature to represent the nonsolid (air) domains.

In the window, under click .

Charge Conservation, Air

In the toolbar, click

and choose .

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

Select Domains 1 and 3–5 only.

Locate the section. Click

In the dialog box, type Air in the text field.

Click .

Definitions

Deforming Domain 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Fix one end of the cantilever.

Solid Mechanics (solid)

In the window, under click .

Fixed Constraint 1

In the toolbar, click

and choose .

Select Boundary 4 only.

Since only half of the cantilever is included in the model, the symmetry condition should be applied on the midplane of the solid. The electric field default condition () is equivalent to a symmetry condition, so only the structural symmetry boundary condition needs to be applied.

Symmetry 1

In the toolbar, click

and choose .

Select Boundary 5 only.

Definitions

Symmetry/Roller 1

In the window, under click .

Select Boundaries 2, 8, and 19 only.

Use the feature to set the voltage of the cantilever. Note: The Domain Terminal 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 Domain Terminal with the same effect. In addition, the computation load is reduced, because the electrostatic degrees of freedom within the Domain Terminal do not need to be solved for.

Electrostatics (es)

In the window, under click .

Terminal 1

In the toolbar, click