Models.mems.Electrostatically Actuated Cantilever

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    1 | E L E C T R O S T A T I C A L L Y A C T U A T E D C A N T I L E V E R

    E l e c t r o s t a t i c a l l y A c t u a t e d C an t i l e v e r

    Introduction

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

    Length: 300 m

    Width: 20 m

    Thickness 2 m

    Because the geometry is symmetric only half of the beam needs to modeled. The beam

    is made of polysilicon with a Youngs modulus,E, of 153GPa, and a Poissons ratio,, 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. The beam resides in an air-filled chamber that is electrically insulated.

    The lower side of the chamber has a grounded electrode.

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    Figure 1: Model Geometry. The beam is 300 m long and 2 m thick, and it is fixed atx = 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 m of freeair above and to the sides of the beam, while the gap below the beam is 2 m.

    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 20m 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 m. As the beam bends, the geometry of the air gap changes continuously, resultingin a change in the electric field between the electrodes. The coupled physics is handled

    automatically by the Electromechanics interface.

    The electrostatic field in the air and in the beam is governed by Poissons 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 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. Thebottom of the chamber is grounded, while all other boundaries are electrically

    V( ) 0=

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    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 Maxwells stress

    tensor:

    ,

    where Eand Dare the electric field and electric displacement vectors, respectively, and

    nis the outward normal vector of the boundary. This force is always oriented along

    the normal of the boundary.

    Naviers 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 thepull-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 2shows the

    beam displacement and the corresponding displacement of the mesh surrounding it.

    Figure 3shows the electric potential and electric field that generates these

    displacements. In Figure 4the shape of the cantilevers 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

    will 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 and6.2 V. For comparison, computations in Ref. 1predict a pull-in voltage of

    Fes1

    2--- E D( )n n E( )D+=

    VPI4c1B

    0L4c2

    21 c3

    g0W------+

    ----------------------------------------------=

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    where c1= 0.07, c2= 1.00, and c3= 0.42;g0is 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),is

    Youngs modulus,E. Otherwise,Eand, the plate modulus, are related by

    whereis Poissons ratio. Because the calculation in Ref. 1uses 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= 20m results in VPI= 6.07V.

    B EH3g03

    =

    E

    E---- 1

    2 W L( )1,37

    0,5 W L( )1,37+------------------------------------------

    0,98 L H( )

    0,056

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    Figure 2: z-displacement for the beam and the moving mesh as a function of position. Eachmesh 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 sectionsthrough the beam.

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

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

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    Figure 6: Device capacitance vs applied voltage V0.

    Figure 6shows 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 MechanicalProperty Measurements of Microelectromechanical Systems (MEMS), Ph.D. thesis,

    MIT, 1997.

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    Model Library path: MEMS_Module/Actuators/

    electrostatically_actuated_cantilever

    Modeling Instructions

    From the Filemenu, choose New.

    N E W

    1 In the Newwindow, click the Model Wizardbutton.

    M O D E L W I Z A R D

    1 In the Model Wizardwindow, click the 3Dbutton.

    2 In the Select physicstree, select Structural Mechanics>Electromechanics (emi).

    3 Click the Addbutton.

    4 Click the Studybutton.

    5 In the tree, select Preset Studies>Stationary.

    6 Click the Donebutton.

    G E O M E T R Y 1

    Use microns to define the geometry units.

    1 In the Model Builderwindow, under Component 1click Geometry 1.2 In the Geometrysettings window, locate the Unitssection.

    3 From the Length unitlist, choose m.

    Create the geometry so that a swept mesh can be used subsequently. 3 blocks are

    required as to sweep the mesh no change in the Y-Z cross section is allowed.

    Block 1

    1 On the Geometrytoolbar, click Block.2 In the Blocksettings window, locate the Sizesection.

    3 In the Widthedit field, type 320.

    4 In the Depthedit field, type 10.

    5 In the Heightedit field, type 2.

    6 Locate the Positionsection. In the zedit field, type 2.

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    7 Click the Build Selectedbutton.

    Block 2

    1 On the Geometrytoolbar, click Block.

    2 In the Blocksettings window, locate the Sizesection.

    3 In the Widthedit field, type 320.

    4 In the Depthedit field, type 40.

    5 In the Heightedit field, type 24.

    Block 31 On the Geometrytoolbar, click Block.

    2 In the Blocksettings window, locate the Sizesection.

    3 In the Widthedit field, type 300.

    4 In the Depthedit field, type 40.

    5 In the Heightedit field, type 24.

    6 Click the Build All Objectsbutton.

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

    G L O B A L D E F I N I T I O N S

    Parameters

    1 On the Hometoolbar, click Parameters.

    2 In the Parameterssettings window, locate the Parameterssection.

    3 In the table, enter the following settings:

    The cantilever is assumed to be heavily doped so that it acts as a conductor, held at

    constant potential. The Linear Elastic Materialfeature is therefore used.

    E L E C T R O M E C H A N I C S

    Linear Elastic Material 1

    1 On the Physicstoolbar, click Domainsand choose Linear Elastic Material.

    2 Select Domain 2 only.

    Name Expression Value Description

    V0 5[V] 5.0000 V Bias on Cantilever

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    Fixed Constraint 1

    Fix one end of the cantilever.

    1 On the Physicstoolbar, click Boundariesand choose Fixed Constraint.

    2 Select Boundary 4 only.

    Symmetry 1

    Since only half of the cantilever is included in the model, the symmetry condition

    should be applied on the mid-plane of the solid. The electric field default condition

    (zero charge) is equivalent to a symmetry condition, so only the structural symmetry

    boundary condition needs to be applied.

    1 On the Physicstoolbar, click Boundariesand choose Symmetry.

    2 Select Boundary 5 only.

    Terminal 1

    Use the terminal feature to set the voltage on the exterior of the cantilever.

    1 On the Physicstoolbar, click Boundariesand choose Terminal.

    2 Select Boundaries 6, 8, 10, and 15 only.

    3 In the Terminalsettings window, locate the Terminalsection.

    4 From the Terminal typelist, choose Voltage.

    5 In the V0edit field, type V0.

    Ground 1

    Set up the ground plane underneath the cantilever.

    1 On the Physicstoolbar, click Boundariesand choose Ground.

    2 Select Boundaries 3 and 14 only.

    Prescribed Mesh Displacement 2

    Apply boundary conditions to constrain the mesh deformation.

    1 On the Physicstoolbar, click Boundariesand choose Prescribed Mesh Displacement.

    2 Select Boundaries 2, 7, 13, 16, 18, 23, and 24 only.

    3 In the Prescribed Mesh Displacementsettings window, locate the Prescribed Mesh

    Displacementsection.

    4 Clear the Prescribed z displacementcheck box.

    This way, you allow the mesh nodes to move in the z direction while they are fixed in

    the x and y directions.

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    Confirm that the default features have acquired the correct selections.

    5 Click the Wireframe Renderingbutton on the Graphics toolbar.

    6 Select Prescribed Mesh Displacement 1.

    The default mesh displacement feature constrains all the remaining boundaries to have

    zero displacement.

    7 Select Electromechanical Interface 1.

    The electromechanical interface feature applies forces to the exterior boundaries of the

    cantilever.

    8 Select Zero Charge 1.

    The zero charge feature applies symmetry conditions to the remaining boundaries

    where the electric potential is solved for.

    M A T E R I A L S

    Add Materials to the model.

    Mater ial 11 In the Model Builderwindow, under Component 1right-click Materialsand choose

    New Material.

    2 In the Materialsettings window, locate the Material Contentssection.

    3 In the table, enter the following settings:

    Mater ial 2

    1 In the Model Builderwindow, right-click Materialsand choose New Material.

    2 Select Domains 1, 3, and 4 only.

    Set the material to be non-solid, to ensure the interface solves the electrostatics

    equations in the spatial frame.

    3 In the Materialsettings window, click to expand the Material propertiessection.

    Property Name Value Unit Property group

    Relative permittivity epsilonr

    4.5 1 Basic

    Young's modulus E 153[GPa]

    Pa Basic

    Poisson's ratio nu 0.23 1 Basic

    Density rho 2330 kg/m Basic

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    4 Locate the Material Propertiessection. From the Material typelist, choose Non-solid.

    5 Locate the Material Contentssection. In the table, enter the following settings:

    M E S H 1

    Create a swept mesh.

    Mapped 1

    1 In the Model Builderwindow, under Component 1right-click Mesh 1and choose More

    Operations>Mapped.

    2 Select Boundary 4 only.

    Distribution 1

    1 Right-click Component 1>Mesh 1>Mapped 1and choose Distribution.

    2 Select Edge 5 only.

    Size

    1 In the Model Builderwindow, under Component 1>Mesh 1click Size.

    2 In the Sizesettings window, locate the Element Sizesection.

    3 From the Predefinedlist, choose Extremely fine.

    4 Click the Custombutton.

    5 Locate the Element Size Parameterssection. In the Maximum element sizeedit field,

    type 4.

    Free Quad 1

    1 In the Model Builderwindow, right-click Mesh 1and choose More Operations>Free

    Quad.

    2 Select Boundary 1 only.

    3 Click the Build Allbutton.

    Swept 1

    1 Right-click Mesh 1and choose Swept.

    2 In the Sweptsettings window, locate the Domain Selectionsection.

    3 From the Geometric entity levellist, choose Domain.

    4 Select Domains 1 and 2 only.

    Property Name Value Unit Property group

    Relative permittivity epsilonr 1 1 Basic

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    Distribution 1

    1 Right-click Component 1>Mesh 1>Swept 1and choose Distribution.

    2 In the Distributionsettings window, locate the Distributionsection.

    3 In the Number of elementsedit field, type 20.

    4 Click the Build Allbutton.

    Swept 2

    1 In the Model Builderwindow, right-click Mesh 1and choose Swept.

    2 In the Sweptsettings window, locate the Domain Selectionsection.

    3 From the Geometric entity levellist, choose Domain.

    4 Select Domains 3 and 4 only.

    Distribution 1

    1 Right-click Component 1>Mesh 1>Swept 2and choose Distribution.

    2 In the Model Builderwindow, under Component 1>Mesh 1>Swept 2right-click

    Distribution 1and choose Build Selected.

    Set up a parametric sweep over the applied voltage.

    S T U D Y 1

    Step 1: Stationary

    1 In the Model Builderwindow, under Study 1click Step 1: Stationary.

    2 In the Stationarysettings window, click to expand the Study extensionssection.

    3 Locate the Study Extensionssection. Select the Auxiliary sweepcheck box.4 Click Add.

    5 Click Range.

    6 Go to the Rangedialog box.

    7 In the Startedit field, type 1.

    8 In the Stepedit field, type 1.

    9 In the Stopedit field, type 6.10 Click the Addbutton.

    Add points at 6.05 and 6.1 V to the sweep by adding these points after the range

    statement. The table field should now contain: 'range(1,1,6) 6.05 6.1'

    11 On the Hometoolbar, click Compute.

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    R E S U L T S

    Displacement (emi)

    Create additional data sets for post processing. First create a mirrored data set.

    Data Sets

    1 On the Resultstoolbar, click More Data Setsand choose Mirror 3D.

    2 In the Mirror 3Dsettings window, locate the Plane Datasection.

    3 From the Planelist, choose zx-planes.

    Then create a data set with some surface selections.

    4 In the Model Builderwindow, under Results>Data Setsright-click Solution 1and

    choose Duplicate.

    5 Right-click Results>Data Sets>Solution 2and choose Add Selection.

    6 In the Selectionsettings window, locate the Geometric Entity Selectionsection.

    7 From the Geometric entity levellist, choose Boundary.

    8 Select Boundaries 36, 8, 10, 14, and 15 only.

    Finally, mirror this data set.

    9 On the Resultstoolbar, click More Data Setsand choose Mirror 3D.

    10 In the Mirror 3Dsettings window, locate the Datasection.

    11 From the Data setlist, choose Solution 2.

    12 Locate the Plane Datasection. From the Planelist, choose zx-planes.

    Displacement (emi)Edit the default displacement plot to show the z-displacement and the corresponding

    mesh deformation.

    1 In the Model Builderwindow, under Resultsclick Displacement (emi).

    2 In the 3D Plot Groupsettings window, locate the Datasection.

    3 From the Data setlist, choose Mirror 3D 1.

    4 In the Model Builderwindow, expand the Displacement (emi)node, then click Surface1.

    5 In the Surfacesettings window, locate the Expressionsection.

    6 Select Electromechanics (Solid Mechanics)>Displacement>Displacement field, Z

    component (w)in the Replace expression menu accessed in the upper-right corner of

    the section.

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    7 Locate the Coloring and Stylesection. Select the Reverse color tablecheck box.

    8 In the Model Builderwindow, right-click Displacement (emi)and choose Volume.

    9 In the Volumesettings window, locate the Datasection.

    10 From the Data setlist, choose Solution 1.

    11 Locate the Expressionsection. In the Expressionedit field, type z-Z.

    12 Click to expand the Shrink elementssection. Locate the Shrink Elementssection. In

    the Element scale factoredit field, type 0.8.

    13 Click to expand the Inherit stylesection. Locate the Inherit Stylesection. From the

    Plotlist, choose Surface 1.14 Right-click Displacement (emi)and choose Slice.

    15 In the Slicesettings window, locate the Expressionsection.

    16 In the Expressionedit field, type z-Z.

    17 Click to expand the Inherit stylesection. Locate the Inherit Stylesection. From the

    Plotlist, choose Surface 1.

    18 On the 3D Plot Grouptoolbar, click Plot.

    Potential (emi)

    Edit the default potential plot.

    1 In the Model Builderwindow, under Resultsclick Potential (emi).

    2 In the 3D Plot Groupsettings window, locate the Datasection.

    3 From the Data setlist, choose Mirror 3D 1.

    4 In the Model Builderwindow, under Results>Potential (emi)click Slice 1.

    5 In the Slicesettings window, locate the Plane Datasection.

    6 In the Planesedit field, type 7.

    7 In the Model Builderwindow, right-click Potential (emi)and choose Surface.

    8 In the Surfacesettings window, locate the Expressionsection.

    9 Select Electromechanics (Electical Quasistatics)>Electric>Electric potential (V)in the

    Replace expressionmenu, which is accessed in the upper-right corner of the section.

    10 Locate the Datasection. From the Data setlist, choose Mirror 3D 2.

    11 Click to expand the Inherit stylesection. Locate the Inherit Stylesection. From the

    Plotlist, choose Slice 1.

    12 Right-click Potential (emi)and choose Arrow Volume.

    13 In the Arrow Volumesettings window, locate the Expressionsection.

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    14 Click Electromechanics (Electical Quasistatics)>Electric>Electric field (Spatial)

    (emi.Ex,...,emi.Ez)in the Replace expressionmenu, which is accessed in the

    upper-right corner of the section.15 Locate the Arrow Positioningsection. Find the y grid pointssubsection. In the Points

    edit field, type 15.

    16 Locate the Coloring and Stylesection. From the Arrow lengthlist, choose Normalized.

    17 On the 3D Plot Grouptoolbar, click Plot.

    1D Plot Group 3

    Add a plot to show the deformed shape of the underside of the cantilever.

    1 On the Hometoolbar, click Add Plot Groupand choose 1D Plot Group.

    2 On the 1D Plot Grouptoolbar, click Line Graph.

    3 Select Edge 6 only.

    4 In the Line Graphsettings window, locate the y-Axis Datasection.

    5 Select Electromechanics (Solid Mechanics)>Displacement>Displacement field, Z

    component (w)in the Replace expression menu accessed in the upper-right corner ofthe section.

    6 Click to expand the Legendssection. Select the Show legendscheck box.

    7 In the Model Builderwindow, click 1D Plot Group 3.

    8 In the 1D Plot Groupsettings window, click to expand the Legendsection.

    9 From the Positionlist, choose Lower left.

    10 Click to expand the Titlesection. From the Title typelist, choose Manual.11 In the Titletext area, type Shape of cantilever displacement for different

    applied voltages.

    12 Right-click 1D Plot Group 3and choose Rename.

    13 Go to the Rename 1D Plot Groupdialog box and type Displacement vs Applied

    Voltagein the New nameedit field.

    14 Click OK.

    15 On the 1D Plot Grouptoolbar, click Plot.

    1D Plot Group 4

    Add a plot of tip displacement vs. applied DC voltage.

    1 On the Hometoolbar, click Add Plot Groupand choose 1D Plot Group.

    2 On the 1D Plot Grouptoolbar, click Point Graph.

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    3 Select Point 10 only.

    4 In the Point Graphsettings window, locate the y-Axis Datasection.

    5 Select Electromechanics (Solid Mechanics)>Displacement field>Displacement field, Z

    component (w)in the Replace expression menu accessed in the upper-right corner of

    the section.

    6 In the Model Builderwindow, right-click 1D Plot Group 4and choose Rename.

    7 Go to the Rename 1D Plot Groupdialog box and type Tip Displacement vs

    Applied Voltagein the New nameedit field.

    8 Click OK.9 On the 1D Plot Grouptoolbar, click Plot.

    1D Plot Group 5

    Finally, plot the DC capacitance of the device vs voltage.

    1 On the Hometoolbar, click Add Plot Groupand choose 1D Plot Group.

    2 On the 1D Plot grouptoolbar, click Global.

    3 Locate the y-Axis Datasection. In the table, enter the following settings:

    4 In the Model Builderwindow, right-click 1D Plot Group 5and choose Rename.

    5 Go to the Rename 1D Plot Groupdialog box and type DC C-V Curvein the New name

    edit field.

    6 Click OK.

    7 On the 1D Plot Grouptoolbar, click Plot.

    Expression Unit Description

    2*emi.C11 fF Capacitance

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