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Pull-In of an RF MEMS Switch
Introduction
This model analyzes an RF MEMS switch consisting of a thin micromechanical bridge suspended over a dielectric layer. A DC voltage greater than the pull-in voltage is applied across the switch, causing the bridge to collapse onto the dielectric layer with a resulting increase in the capacitance of the device. A penalty-based contact force is implemented to model the contact forces as the bridge comes into contact with the dielectric. The capability of the Electromechanics multiphysics interface to handle both conducting and dielectric materials is also demonstrated.
Model Definition
Figure 1 shows the device geometry. The switch consists of a square polysilicon plate suspended 0.9 μm above a 0.1 μm thick thin film of silicon nitride (dielectric constant 7.5). Beneath the substrate is a silicon counter electrode that is grounded. The suspended plate is structurally anchored to the substrate by four rectangular flexures at its corners but is electrically isolated from the substrate.
Initially a small potential of 1 mV is applied to the polysilicon using the Domain Terminal feature. This voltage is sufficient to measure the DC capacitance of the device. After 25 μs the applied voltage is increased by 5 V with a step function that has a rise time of 10 μs. The applied voltage is greater than the pull-in voltage of the structure and the switch pulls down onto the nitride. This process results in an abrupt and significant change in the capacitance of the device.
Due to the symmetry of the device, it is possible to model only a single quadrant of the structure.
The gap between the polysilicon and the nitride presents a model setup challenge. Since it is not possible to collapse the mesh to zero thickness (when the gap is closed) for numerical reasons, the mesh is compressed into the nitride layer as the structure deforms. The nitride layer itself is consequently not represented explicitly within the geometry, but instead is represented by means of a spatially varying function for the dielectric constant within the domain that contains both the nitride and the gap. Specifically, the dielectric constant in the domain is represented by a smoothed step function. The midpoint of the smoothed step function is chosen to be slightly above the height of the dielectric, so that when the polysilicon is in contact with the nitride, the dielectric constant in the domain takes the value of the nitride dielectric constant throughout the domain.
It is also worth noting that the Electromechanics multiphysics interface is capable of treating both conductors (for example, the polysilicon) and dielectric materials (for example, the nitride), as demonstrated in this model.
Figure 1: Top: Device geometry showing anchor points and symmetry planes. Bottom: Model geometry. Due to symmetry only one quadrant of the device needs to be modeled.
The contact between the polysilicon and the nitride is handled by an approximate penalty or barrier method, as described in Ref. 1. Stiff, nonlinear springs are used to represent the surface of the nitride. When the polysilicon is away from the nitride surface these springs have low stiffness and consequently have a negligible influence on the deformation. As the gap is reduced and approaches a predefined distance the spring becomes much stiffer and resists further closure. The contact forces Fc are given by:
where tn is the input estimate of the contact force, en is the penalty stiffness, g is the gap, that is, the distance between the polysilicon and the nitride.
Note that when this method is employed, it is important to tune the elastic stiffness and the contact force. The formula for the contact force is an approximate one and the model does not correctly reproduce the details of the dynamics of contact. However the model is primarily concerned with estimating the time the switch takes to make contact and with computing the initial and final capacitance of the switch.
For this same reason, a small mechanical damping is added to the Si domain to suppress the ringing of the structure once contact is made, thus preventing the time dependent solver from taking very small steps to resolve the ringing, which is not the focus of this model. (Of course care must be taken to ensure that the magnitude of the extra damping should not change the switching time constant significantly.)
Results and Discussion
Figure 2 shows the spatial dependence of the total displacement when the device is pulled in. Most of the structure is in contact with the nitride and the bending occurs primarily in the flexures and in the vicinity of their attachment points. The form of the contact pressure, shown in Figure 3, is consistent with this observation and it is interesting to note that the largest forces occur in the vicinity of the flexures.
Figure 2: Displacement of the polysilicon when pulled in. Most of the polysilicon structure is in contact with the silicon nitride with a displacement of 0.9 μm.
Figure 3: Contact forces acting on the polysilicon when the structure is pulled in.
Figure 4 shows the displacement of the switch as a function of time. The switch takes significantly longer to close than the time scale on which the applied voltage changes, primarily due to its inertia. Figure 5 shows the capacitance of the device as a function of time. The capacitance increases by a factor of approximately 65. Note that the capacitance changes on a significantly shorter time scale than the displacement.
Figure 4: Displacement of the center of the device as a function of time.
Figure 5: Capacitance of the device as a function of time. The transients in this plot that occur after the point of contact are not physical. The capacitance of the structure changes from 0.1 pF to 6.5 pF as a result of the pull-in.
Reference
1. Crisfield M. A., Non-linear Finite Element Analysis of Solids and Structures, volume 2: Advanced Topics, John Wiley & Sons Ltd., England, 1997.
Application Library path: MEMS_Module/Actuators/rf_mems_switch
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  3D.
2
In the Select Physics tree, select Structural Mechanics>Electromagnetics-Structure Interaction>Electromechanics>Electromechanics.
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies>Stationary.
6
Click  Done.
Geometry 1
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, locate the Units section.
3
From the Length unit list, choose µm.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
V0
1[mV]
0.001 V
Initial voltage
Vstep
5[V]
5 V
Voltage step
insheight
100[nm]
1E-7 m
Insulator height
airheight
900[nm]
9E-7 m
Air height
en
1e15[Pa/m]
1E15 N/m³
Spring stiffness
tn
5e5[Pa]
5E5 Pa
Contact force
Definitions
Variables 1
1
In the Home toolbar, click  Variables and choose Local Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
gap
airheight+w
m
 
contactpressure
(gap<=0)*(tn-en*gap)+(gap>0)*tn*exp(-gap*en/tn)
N/m²
 
Va
V0+Vstep*step2(t/1[s])
 
 
Geometry 1
Work Plane 1 (wp1)
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, click  Show Work Plane.
Work Plane 1 (wp1)>Plane Geometry
In the Model Builder window, click Plane Geometry.
Work Plane 1 (wp1)>Rectangle 1 (r1)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 110.
4
In the Height text field, type 5.
5
Locate the Position section. In the xw text field, type -60.
Work Plane 1 (wp1)>Rectangle 2 (r2)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 50.
4
In the Height text field, type 60.
5
Locate the Position section. In the yw text field, type -10.
Work Plane 1 (wp1)>Rectangle 3 (r3)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 110.
4
In the Height text field, type 60.
5
Locate the Position section. In the xw text field, type -60.
6
In the yw text field, type -10.
Work Plane 1 (wp1)>Rectangle 4 (r4)
1
In the Work Plane toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 110.
4
In the Height text field, type 10.
5
Locate the Position section. In the xw text field, type -60.
6
In the yw text field, type -2.5.
Extrude 1 (ext1)
1
In the Model Builder window, right-click Geometry 1 and choose Extrude.
2
In the Settings window for Extrude, locate the Distances section.
3
In the table, enter the following settings:
Distances (µm)
1
2
4
4
Click  Build Selected.
Form Union (fin)
1
In the Geometry toolbar, click  Build All.
2
Click the  Wireframe Rendering button in the Graphics toolbar.
Definitions
Step 1 (step1)
1
In the Home toolbar, click  Functions and choose Local>Step.
2
In the Settings window for Step, locate the Parameters section.
3
In the Location text field, type 1.05*insheight.
4
Click to expand the Smoothing section. In the Size of transition zone text field, type 0.05*insheight.
Step 2 (step2)
1
In the Home toolbar, click  Functions and choose Local>Step.
2
In the Settings window for Step, locate the Parameters section.
3
In the Location text field, type 3e-5.
4
Locate the Smoothing section. In the Size of transition zone text field, type 1e-5.
Define selections.
Bridge
1
In the Definitions toolbar, click  Explicit.
2
Right-click Explicit 1 and choose Rename.
3
In the Rename Explicit dialog box, type Bridge in the New label text field.
4
Click OK.
5
Select Domains 8, 23, 26, and 29 only.
Gap
1
In the Definitions toolbar, click  Box.
2
In the Model Builder window, right-click Box 1 and choose Rename.
3
In the Rename Box dialog box, type Gap in the New label text field.
4
Click OK.
5
In the Settings window for Box, locate the Box Limits section.
6
In the z maximum text field, type 1.1.
7
Locate the Output Entities section. From the Include entity if list, choose Entity inside box.
Bridge surface
1
In the Definitions toolbar, click  Adjacent.
2
Right-click Adjacent 1 and choose Rename.
3
In the Rename Adjacent dialog box, type Bridge surface in the New label text field.
4
Click OK.
5
In the Settings window for Adjacent, locate the Input Entities section.
6
Under Input selections, click  Add.
7
In the Add dialog box, select Bridge in the Input selections list.
8
Click OK.
Base
1
In the Definitions toolbar, click  Box.
2
In the Model Builder window, right-click Box 2 and choose Rename.
3
In the Rename Box dialog box, type Base in the New label text field.
4
Click OK.
5
In the Settings window for Box, locate the Geometric Entity Level section.
6
From the Level list, choose Boundary.
7
Locate the Box Limits section. In the z maximum text field, type 0.1.
8
Locate the Output Entities section. From the Include entity if list, choose Entity inside box.
Box 3
1
In the Definitions toolbar, click  Box.
2
In the Settings window for Box, locate the Geometric Entity Level section.
3
From the Level list, choose Boundary.
4
Locate the Box Limits section. In the z minimum text field, type 0.9.
5
In the z maximum text field, type 1.1.
6
Locate the Output Entities section. From the Include entity if list, choose Entity inside box.
Bridge lower side
1
In the Definitions toolbar, click  Intersection.
2
Right-click Intersection 1 and choose Rename.
3
In the Rename Intersection dialog box, type Bridge lower side in the New label text field.
4
Click OK.
5
In the Settings window for Intersection, locate the Geometric Entity Level section.
6
From the Level list, choose Boundary.
7
Locate the Input Entities section. Under Selections to intersect, click  Add.
8
In the Add dialog box, in the Selections to intersect list, choose Bridge surface and Box 3.
9
Click OK.
Symmetry x
1
In the Definitions toolbar, click  Box.
2
In the Model Builder window, right-click Box 4 and choose Rename.
3
In the Rename Box dialog box, type Symmetry x in the New label text field.
4
Click OK.
5
In the Settings window for Box, locate the Geometric Entity Level section.
6
From the Level list, choose Boundary.
7
Locate the Box Limits section. In the x minimum text field, type 45.
8
Locate the Output Entities section. From the Include entity if list, choose Entity inside box.
Symmetry y
1
In the Definitions toolbar, click  Box.
2
Right-click Box 5 and choose Rename.
3
In the Rename Box dialog box, type Symmetry y in the New label text field.
4
Click OK.
5
In the Settings window for Box, locate the Geometric Entity Level section.
6
From the Level list, choose Boundary.
7
Locate the Box Limits section. In the y minimum text field, type 45.
8
Locate the Output Entities section. From the Include entity if list, choose Entity inside box.
Symmetry
1
In the Definitions toolbar, click  Union.
2
In the Settings window for Union, locate the Geometric Entity Level section.
3
From the Level list, choose Boundary.
4
Locate the Input Entities section. Under Selections to add, click  Add.
5
In the Add dialog box, in the Selections to add list, choose Symmetry x and Symmetry y.
6
Click OK.
7
In the Settings window for Union, type Symmetry in the Label text field.
All domains
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, locate the Input Entities section.
3
Select the All domains check box.
4
In the Label text field, type All domains.
Non-solid
1
In the Definitions toolbar, click  Difference.
2
In the Settings window for Difference, locate the Input Entities section.
3
Under Selections to add, click  Add.
4
In the Add dialog box, select All domains in the Selections to add list.
5
Click OK.
6
In the Settings window for Difference, locate the Input Entities section.
7
Under Selections to subtract, click  Add.
8
In the Add dialog box, select Bridge in the Selections to subtract list.
9
Click OK.
10
In the Settings window for Difference, type Non-solid in the Label text field.
Solid Mechanics (solid)
1
In the Model Builder window, under Component 1 (comp1) click Solid Mechanics (solid).
2
In the Settings window for Solid Mechanics, locate the Domain Selection section.
3
From the Selection list, choose Bridge.
Symmetry 1
1
In the Physics toolbar, click  Boundaries and choose Symmetry.
2
In the Settings window for Symmetry, locate the Boundary Selection section.
3
From the Selection list, choose Symmetry.
Boundary Load 1
1
In the Physics toolbar, click  Boundaries and choose Boundary Load.
2
In the Settings window for Boundary Load, locate the Boundary Selection section.
3
From the Selection list, choose Bridge lower side.
4
Locate the Force section. Specify the FA vector as
contactpressure
z
Fixed Constraint 1
1
In the Physics toolbar, click  Boundaries and choose Fixed Constraint.
2
Select Boundary 24 only.
Linear Elastic Material 1
Add some small damping to reduce the transient effects at the final stage of the switch motion.
1
In the Model Builder window, click Linear Elastic Material 1.
Damping 1
1
In the Physics toolbar, click  Attributes and choose Damping.
2
In the Settings window for Damping, locate the Damping Settings section.
3
In the βdK text field, type 1e-6.
4
Locate the Domain Selection section. From the Selection list, choose Bridge.
Electrostatics (es)
The default Charge Conservationfeature was set to use solid material type. Add one more feature to represent the nonsolid (air) domains.
1
In the Model Builder window, under Component 1 (comp1) click Electrostatics (es).
Charge Conservation, Non-solid
1
In the Physics toolbar, click  Domains and choose Charge Conservation.
2
In the Settings window for Charge Conservation, type Charge Conservation, Non-solid in the Label text field.
3
Locate the Domain Selection section. From the Selection list, choose Non-solid.
Define the voltage applied to the bridge.
Terminal 1
1
In the Physics toolbar, click  Domains and choose Terminal.
2
In the Settings window for Terminal, locate the Domain Selection section.
3
From the Selection list, choose Bridge.
4
Locate the Terminal section. From the Terminal type list, choose Voltage.
5
In the V0 text field, type Va.
Define the voltage applied to the base.
Terminal 2
1
In the Physics toolbar, click  Boundaries and choose Terminal.
2
In the Settings window for Terminal, locate the Boundary Selection section.
3
From the Selection list, choose Base.
4
Locate the Terminal section. From the Terminal type list, choose Voltage.
5
In the V0 text field, type 0.
Terminal 1
Add another Domain Terminal to the bridge. This terminal is used for the first (Stationary) study step. It will be disabled in the transient step which will make Terminal 1 active.
Terminal 3
1
In the Model Builder window, under Component 1 (comp1)>Electrostatics (es) right-click Terminal 1 and choose Duplicate.
2
In the Settings window for Terminal, locate the Terminal section.
3
In the V0 text field, type V0.
Definitions
Deforming Domain 1
1
In the Model Builder window, under Component 1 (comp1)>Definitions>Moving Mesh click Deforming Domain 1.
2
In the Settings window for Deforming Domain, locate the Domain Selection section.
3
From the Selection list, choose Non-solid.
Symmetry/Roller 1
1
In the Model Builder window, click Symmetry/Roller 1.
2
In the Settings window for Symmetry/Roller, locate the Boundary Selection section.
3
From the Selection list, choose Symmetry.
Load the materials.
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in>Air.
4
Click Add to Component in the window toolbar.
5
In the tree, select MEMS>Semiconductors>Si - Polycrystalline silicon.
6
Click Add to Component in the window toolbar.
7
In the Home toolbar, click  Add Material to close the Add Material window.
Materials
Si - Polycrystalline silicon (mat2)
1
In the Settings window for Material, locate the Geometric Entity Selection section.
2
From the Selection list, choose Bridge.
Material 3 (mat3)
1
In the Model Builder window, right-click Materials and choose Blank Material.
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Selection list, choose Gap.
4
Click to expand the Material Properties section. From the Material type list, choose Nonsolid.
5
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Relative permittivity
epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0
1+6.5*(1-step1(z/1[m]))
1
Basic
Mesh 1
Mapped 1
1
In the Mesh toolbar, click  Boundary and choose Mapped.
2
Select Boundaries 10, 20, 30, 40, 50, 63, 73, 83, 93, and 103 only.
Swept 1
In the Mesh toolbar, click  Swept.
Distribution 1
1
Right-click Swept 1 and choose Distribution.
2
In the Settings window for Distribution, locate the Domain Selection section.
3
From the Selection list, choose Gap.
4
Locate the Distribution section. In the Number of elements text field, type 4.
Distribution 2
1
In the Model Builder window, right-click Swept 1 and choose Distribution.
2
In the Settings window for Distribution, locate the Distribution section.
3
In the Number of elements text field, type 2.
4
Select Domains 2, 5, 8, 11, 14, 23, 26, and 29 only.
5
Click  Build All.
Define the study steps. The first step (Stationary) is used to define the initial conditions of the transient problem (Step 2).
Study 1
Time Dependent
1
In the Study toolbar, click  Study Steps and choose Time Dependent>Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
In the Output times text field, type range(0,5e-7,5e-5).
4
Click to expand the Results While Solving section. Select the Plot check box.
5
From the Update at list, choose Time steps taken by solver.
6
Locate the Physics and Variables Selection section. Select the Modify model configuration for study step check box.
7
In the Physics and variables selection tree, select Component 1 (comp1)>Electrostatics (es)>Terminal 3.
8
Click  Disable.
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 1 (sol1) node, then click Time-Dependent Solver 1.
3
In the Settings window for Time-Dependent Solver, click to expand the Time Stepping section.
4
From the Steps taken by solver list, choose Intermediate.
5
Right-click Time-Dependent Solver 1 and choose Fully Coupled.
6
In the Model Builder window, under Study 1>Solver Configurations>Solution 1 (sol1)>Time-Dependent Solver 1 click Fully Coupled 1.
7
In the Settings window for Fully Coupled, click  Compute.
Results
Study 1/Solution 1 (sol1)
1
In the Model Builder window, expand the Results>Datasets node, then click Study 1/Solution 1 (sol1).
2
In the Settings window for Solution, locate the Solution section.
3
From the Frame list, choose Material  (X, Y, Z).
Mirror the solution to visualize the whole structure.
Mirror 3D 1
1
In the Results toolbar, click  More Datasets and choose Mirror 3D.
2
In the Settings window for Mirror 3D, locate the Plane Data section.
3
In the X-coordinate text field, type 50.
Mirror 3D 2
1
Right-click Mirror 3D 1 and choose Duplicate.
2
In the Settings window for Mirror 3D, locate the Data section.
3
From the Dataset list, choose Mirror 3D 1.
4
Locate the Plane Data section. From the Plane list, choose xz-planes.
5
In the y-coordinate text field, type 50.
Displacement (solid)
1
In the Model Builder window, click Displacement (solid).
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Dataset list, choose Mirror 3D 2.
4
In the Displacement (solid) toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Electric Potential (es)
1
In the Model Builder window, click Electric Potential (es).
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Dataset list, choose Mirror 3D 2.
Multislice 1
1
In the Model Builder window, expand the Electric Potential (es) node, then click Multislice 1.
2
In the Settings window for Multislice, locate the Multiplane Data section.
3
Find the x-planes subsection. In the Planes text field, type 5.
4
Find the y-planes subsection. In the Planes text field, type 0.
5
Find the z-planes subsection. In the Planes text field, type 0.
6
Click to expand the Quality section. From the Resolution list, choose No refinement.
7
In the Electric Potential (es) toolbar, click  Plot.
8
Click the  Zoom Extents button in the Graphics toolbar.
Contact force
1
In the Model Builder window, right-click Displacement (solid) and choose Duplicate.
2
Right-click Displacement (solid) 1 and choose Rename.
3
In the Rename 3D Plot Group dialog box, type Contact force in the New label text field.
4
Click OK.
Surface 1
1
In the Model Builder window, expand the Contact force node, then click Surface 1.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type contactpressure.
4
In the Contact force toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Displacement
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
Right-click 1D Plot Group 5 and choose Rename.
3
In the Rename 1D Plot Group dialog box, type Displacement in the New label text field.
4
Click OK.
Point Graph 1
1
Right-click Displacement and choose Point Graph.
2
Select Point 70 only.
3
In the Settings window for Point Graph, locate the y-Axis Data section.
4
In the Expression text field, type w.
5
In the Displacement toolbar, click  Plot.
6
Click the  Zoom Extents button in the Graphics toolbar.
Capacitance
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
Right-click 1D Plot Group 6 and choose Rename.
3
In the Rename 1D Plot Group dialog box, type Capacitance in the New label text field.
4
Click OK.
Global 1
1
Right-click Capacitance and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
4*es.Q0_1/es.V0_1
pF
Capacitance
4
Click to expand the Legends section. Clear the Show legends check box.
5
In the Capacitance toolbar, click  Plot.
6
Click the  Zoom Extents button in the Graphics toolbar.