Solved with COMSOL Multiphysics 4.4Tunab l e MEMS Capa c i t o r

Introduction

In an electrostatically tunable parallel plate capacitor you can modify the distance between the two plates when the applied voltage changes. For tuning of the distance between the plates the capacitor includes a spring that attaches to one of the plates. If you know the characteristics of the spring and the voltage between the plates, you can compute the distance between the plates. This model includes an electrostatic simulation for a given distance. A postprocessing step then computes the capacitance.

The capacitor in this model is a typical component in various microelectromechanical systems (MEMS) for electromagnetic fields in the radio frequency range 300 MHz to 300 GHz.

Figure 1: The tunable MEMS capacitor consists of two metal plates. The distance between the plates is tuned via a spring connected to one of the plates. 1 | TU N A B L E M E M S C A P A C I T O R

Solved with COMSOL Multiphysics 4.4

2 | TU NModel Definition

To solve the problem, use the 3D Electrostatics physics in the AC/DC Module. The capacitance is available directly as a variable for postprocessing.

D O M A I N E Q U A T I O N S

The electric scalar potential, V, satisfies Poissons equation,

(1)

where 0 is the permittivity of free space, r is the relative permittivity, and is the space charge density. The electric field and the displacement are obtained from the gradient of V:

(2)

B O U N D A R Y C O N D I T I O N S

Potential boundary conditions are applied to the capacitor plates and bars. A terminal condition maintains the potential 1 V at the upper plate and the connecting bars, whereas the lower plate is kept at ground potential. For the surface of the surrounding box, apply conditions corresponding to zero surface charge at the boundary,

(3)

0r V( ) =

E V=D 0rE=

n D 0=A B L E M E M S C A P A C I T O R

Solved with COMSOL Multiphysics 4.4Results and Discussion

Figure 2 shows the computed electric potential distribution in the capacitor. The potential on each capacitor plate is constant, as dictated by the boundary condition.

Figure 2: The electric potential is shown as a surface plot.

The capacitance, C, obtained from the simulation is approximately 0.09 pF.

Model Library path: ACDC_Module/Capacitive_Devices/capacitor_tunable

Modeling Instructions

From the File menu, choose New.

N E W

1 In the New window, click the Model Wizard button.

M O D E L W I Z A R D

1 In the Model Wizard window, click the 3D button. 3 | TU N A B L E M E M S C A P A C I T O R

Solved with COMSOL Multiphysics 4.4

4 | TU N2 In the Select physics tree, select AC/DC>Electrostatics (es).

3 Click the Add button.

4 Click the Study button.

5 In the tree, select Preset Studies>Stationary.

6 Click the Done button.

G E O M E T R Y 1

1 In the Model Builder window, under Component 1 click Geometry 1.

2 In the Geometry settings window, locate the Units section.

3 From the Length unit list, choose m.

Create a number of Blocks by clicking on the Block button on the Geometry toolbar. Type the values in the following table:

Block Width Depth Height x y z

Block 1 22 60 8 0 240 46

Block 2 40 22 8 22 259 46

Block 3 176 262 8 62 19 46

Block 4 40 22 8 238 259 46

Block 5 22 60 8 278 240 46

Block 6 40 22 8 238 19 46

Block 7 22 60 8 278 0 46

Block 8 40 22 8 22 19 46

Block 9 22 229 8 0 41 0

Block 10 40 22 8 -40 139 0A B L E M E M S C A P A C I T O R

Solved with COMSOL Multiphysics 4.4At the end, click Build Selected, then click the Zoom Extents button on the Graphics toolbar.

Cylinder 11 On the Geometry toolbar, click Cylinder.

2 In the Cylinder settings window, locate the Size and Shape section.

3 In the Radius edit field, type 5.5.

4 In the Height edit field, type 38.

5 Locate the Position section. In the x edit field, type 11.

6 In the y edit field, type 250.

7 In the z edit field, type 8.

8 Click the Build Selected button.

Union 11 On the Geometry toolbar, click Union.

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

3 In the Union settings window, locate the Union section.

4 Clear the Keep interior boundaries check box.

5 Click the Build All button. 5 | TU N A B L E M E M S C A P A C I T O R

Solved with COMSOL Multiphysics 4.4

6 | TU NBlock 111 On the Geometry toolbar, click Block.

2 In the Block settings window, locate the Size section.

3 In the Width edit field, type 176.

4 In the Depth edit field, type 262.

5 In the Height edit field, type 8.

6 Locate the Position section. In the x edit field, type 62.

7 In the y edit field, type 19.

8 In the z edit field, type 8.

9 Click the Build Selected button.

Block 121 On the Geometry toolbar, click Block.

2 In the Block settings window, locate the Size section.

3 In the Width edit field, type 181.

4 In the Depth edit field, type 22.

5 In the Height edit field, type 8.

6 Locate the Position section. In the x edit field, type 139.

7 In the y edit field, type 139.

8 Click the Build Selected button.

Union 21 On the Geometry toolbar, click Union.

2 Select the objects blk11 and blk12 only.

3 In the Union settings window, locate the Union section.

4 Clear the Keep interior boundaries check box.

5 Click the Build Selected button.A B L E M E M S C A P A C I T O R

Solved with COMSOL Multiphysics 4.46 Click the Zoom Extents button on the Graphics toolbar.

Block 131 On the Geometry toolbar, click Block.

2 In the Block settings window, locate the Size section.

3 In the Width edit field, type 360.

4 In the Depth edit field, type 340.

5 In the Height edit field, type 94.

6 Locate the Position section. In the x edit field, type -40.

7 In the y edit field, type -20.

8 In the z edit field, type -20.

9 Click the Build Selected button. 7 | TU N A B L E M E M S C A P A C I T O R

Solved with COMSOL Multiphysics 4.4

8 | TU N10 Click the Zoom Extents button on the Graphics toolbar.

D E F I N I T I O N S

Explicit 11 On the Definitions toolbar, click Explicit.

2 In the Model Builder window, under Component 1>Definitions right-click Explicit 1 and choose Rename.

3 Go to the Rename Explicit dialog box and type Electrode in the New name edit field.

4 Click OK.

5 In the Explicit settings window, locate the Output Entities section.

6 From the Output entities list, choose Adjacent boundaries.

7 Select Domain 2 only.

Explicit 21 On the Definitions toolbar, click Explicit.

2 In the Model Builder window, under Component 1>Definitions right-click Explicit 2 and choose Rename.

3 Go to the Rename Explicit dialog box and type Ground Plane in the New name edit field.A B L E M E M S C A P A C I T O R

Solved with COMSOL Multiphysics 4.44 Click OK.

5 In the Explicit settings window, locate the Output Entities section.

6 From the Output entities list, choose Adjacent boundaries.

7 Select Domain 3 only.

Explicit 31 On the Definitions toolbar, click Explicit.

2 In the Model Builder window, under Component 1>Definitions right-click Explicit 3 and choose Rename.

3 Go to the Rename Explicit dialog box and type Dielectric in the New name edit field.

4 Click OK.

5 Select Domain 1 only.

M A T E R I A L S

Material 11 On the Home toolbar, click New Material.

2 Right-click Material 1 and choose Rename.

3 Go to the Rename Material dialog box and type Dielectric in the New name edit field.

4 Click OK.

5 In the Material settings window, locate the Geometric Entity Selection section.

6 From the Selection list, choose Dielectric.

7 Locate the Material Contents section. In the table, enter the following settings:

E L E C T R O S T A T I C S

1 In the Electrostatics settings window, locate the Domain Selection section.

2 From the Selection list, choose Dielectric.

Terminal 11 On the Physics toolbar, click Boundaries and choose Terminal.

2 In the Terminal settings window, locate the Boundary Selection section.

Property Name Value

Relative permittivity epsilonr 4.2 9 | TU N A B L E M E M S C A P A C I T O R

Solved with COMSOL Multiphysics 4.4

10 | TU3 From the Selection list, choose Electrode.

4 Locate the Terminal section. From the Terminal type list, choose Voltage.

Ground 11 On the Physics toolbar, click Boundaries and choose Ground.

2 In the Ground settings window, locate the Boundary Selection section.

3 From the Selection list, choose Ground Plane.

M E S H 1

The default mesh gives a sufficiently accurate solution for the purposes of this example. To examine mesh convergence you can optionally go back later and re-solve the model with a finer mesh.

S T U D Y 1

On the Home toolbar, click Compute.

R E S U L T S

Electric Potential (es)The default plot shows slices of the electric potential. You can add a surface plot to visualize the electric potential on the electrode surfaces and the enclosing boundaries.

3D Plot Group 21 On the Results toolbar, click 3D Plot Group.

2 On the 3D Plot Group 2 toolbar, click Surface.

3 On the 3D Plot Group 2 toolbar, click Plot.

To see the capacitor, suppress the display of the enclosing boundaries.

Data Sets1 In the Model Builder window, under Results>Data Sets right-click Solution 1 and

choose Add Selection.

2 In the Selection settings window, locate the Geometric Entity Selection section.

3 From the Geometric entity level list, choose Boundary.

4 From the Selection list, choose All boundaries.

5 Select Boundaries 3 and 578 only.

This is easiest done by removing boundaries 1, 2, and 4 from the list once you have selected all.N A B L E M E M S C A P A C I T O R

Solved with COMSOL Multiphysics 4.43D Plot Group 2

Derived ValuesHaving solved the model, you can now extract the capacitance.

1 On the Results toolbar, click Global Evaluation.

2 In the Global Evaluation settings window, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Electrostatics>Terminals>Capacitance (es.C11).

3 Click the Evaluate button.

The capacitance evaluates to 0.09 pF. 11 | TU N A B L E M E M S C A P A C I T O R

Solved with COMSOL Multiphysics 4.4

12 | TU N A B L E M E M S C A P A C I T O R

Tunable MEMS CapacitorIntroductionModel DefinitionResults and DiscussionModeling Instructions