ICSE2006 Proc. 2006, Kuala Lumpur, Malaysia

Investigation of the Torsion and Bending effects on StaticStability of Electrostatic Torsional Micromirrors

Ghader Rezazadeh, Faraz Khatami, Ahmadali TahmasebiMechanical Engineering Department.

Urmia University,Urmia, Iran

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Abstract- In this paper theelectromechanical behavior of a torsionalmicromirror was investigated using of a staticmodel with considering torsion and bendingcharacteristics of micro-beams. A set ofnonlinear equations based on the parallelplate capacitor model was derived torepresent the relationships between theapplied voltage, torsion angle, and verticaldisplacement of the torsional micromirror.Step by Step Linearization Method (Newton'sMethod) was used to calculate the rotationangle and vertical displacement of themicromirror due to the applied voltage. Thismethod is fast and gave acceptable andaccurate results which were in goodagreement with the experimental data.

I. INTRODUCTION

Today MEMS have evoked great interest in thescientific and engineering communities. Thingsbehave substantially differently in the microdomain. Due to many advantages of MEMs,microoptoelectromechanical systems (MOEMS)including optical switches, microscanners, digitalmicromirror device (DMD) etc, are investigatedfor emerging related fields like all-opticaltelecommunication networks. In the MOEMSsystems electrostatic micromirror is used widelyas an actuator. Various micromirror devices havebeen reported for various applications in openliteratures. Based on their motion types,micromirrors can be simply classified into fourcategories: deformable micromirror, movablemicromirror, piston micromirror, and torsionalmicromirror. The torsional micromirror has beenwidely used for applications because of its gooddynamic response and small possibility of ad-

hesion, for instance in digital projection displays,spatial light modulators and optical crossbarswitches. When the vertical displacement of thetorsional micromirror reaches a gap of 10% betweenthe micromirror and electrode plate, the phase ofreflected light beam is changed by half and even onewavelength of the lights. Thus, it seriously affectsthe design and usage of relevant optical applications,and may even result in the wrong design, i.e., inspatial light modulators. Therefore, accuratelyanalyzing of the torsional micromirror characteristicsis very important to optical devices and applications.Hence in this paper, a torsional micromirror ispresented. First, the parallel-plate model along withan effective torsional spring coefficient of thestructure to estimate the pull-in voltage isconsidered. When the vertical displacement andtorsion angle of the torsional micromirror are withinthe values of the same order, the bending and torsioneffect concurrently affect the static characteristics ofthe torsional micromirror, so as the secondconsideration, a parallel-plate model by assuming theeffect of torsional and bending characteristics isconsidered. In order to solve the nonlinear governingequations the step by step linearization method(Newton's method) is applied. This method is easy,fast and reliable to predict relationship betweenvoltage, angle, and displacement and is used tocalculate pull-in voltage of the torsional micromirror.The obtained results due to the effects of torsion andbending are compared well with experimental data.

II. MODEL DESCRIPTIONS AND ASSUMPTIONS

Figure 1 is a schematic 3D view of the micromirrorand Fig. 2 represent the cross-sectional view of themicromirror considering, a) torsion effect, b) torsionand bending effect. As it is shown in the foregoingfigures, assume a micromirror plate with length b,

0-7803-9731-2/06/$20.00 C2006 IEEE 72

ICSE2006 Proc. 2006, Kuala Lumpur, Malaysia

width a3, thickness t, beneath the micromirror,

there are two electrodes on the substrate, a1 isthe distance between the axes of rotation to theedge of the electrode and a2is width of theelectrode. The moving plate is suspended by twosimilar torsion beams with length 1, width w,thickness t, Young's modulus E, shear modulusG. In order to model the electrostatic torque, it isassumed that the plates are infinitely wide, sofringing fields (fields at the edges of the plates)are neglected, the deformation of the micromir-ror plate is very small, and the verticaldisplacement of the micromirror is mainlyattributed to the deflection of the microbeams, sothe micromirror can be regarded as a rigid body[1]. It is assumed that the torsion beams havenegligible residual stress and stress stiffeningeffects. Any non-uniformity in the electric fielddue to the curvature is neglected [2]. Themicromirror can be driven to rotate by addingpotential between the micromirror and one elec-trode, and can rotate in the reverse direction ifpotential is introduced between the micromirrorand another electrode instead. When the rotationangle is small it's assumed that sin 0 0 andcosO 1. This will give rise to less than I%error even at 10 .

III. MATHEMATICAL MODELING

For modeling by considering the torsion effect of themicrobeams, It is assumed that the effective area ofthe plate is A and for an element dA = bdx, thedistance between the rotational plate element andelectrode is considered as u = g - xO , when 0 isthe rotation about z axis and g is the initial gapbetween plates. Hence the total electrostatic torquecan be written as

TE ((D V) = V2Q1 ()1)where

Q1(O)) = E)2

a®)

- (1 - a(

+ ln(-

) (1-a®aaG

aaO-aaO)

I? be a1 a2 0,EA =2

, a= -,'a ,2O0 a2 g

000

(2)

Parameter 00 is the critical rotation angle that themicromirror plate takes to touch the electrodes plate

and is defined as 00= sin-'a3 a3

The

micromirror plate is suspended by beams andtherefore is subjected to a mechanical torqueopposing the electrostatic torque (considering nobending effect). The beam mechanical elastic torquecan be written as [3]:

TM =KTO, KT=

Fig. 1. Schematic 3D view of the torsional micromirrorx x

tial State i Initial State

d ~~~~~~~~~y

c2wt(tGI

(3)

Where w is the width, t is the thickness and G is theelastic shear modulus of the microbeam. Thecoefficient c2 depends on the ratio (w/t). In orderto study the equilibrium position the followingfunction as the difference of electrical andmechanical torque is introduced:

CD(O, V) = TE -TM (4)At the equilibrium position, the electrostatic torqueand the mechanical elastic torque are equal,meanwhile:

D(O, V) = 0

(a) (b)Fig.2. Cross sectional view of the torsional micromirror, a) Model

considering torsion effect, b) Model considering torsion andbending effects.

(5)The rotation angle of the micromirror can beobtained by solving the nonlinear Eq. (5) at aspecific applied voltage. For sufficiently lowvoltages, there are two physical exhibits of therotation angle, where only one of them is stable. For

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ICSE2006 Proc. 2006, Kuala Lumpur, Malaysia

a certain voltage, the two solutions of Eq. (5)coincide and pull-in phenomenon occurs. Forvoltages above the pull-in voltage, theelectrostatic torque is greater than the mechanicaltorque for any angle [4]. The pull-in state isfound at maximum of V(®). Using of theimplicit function theorem and Eq. (5), to reachthe maximum value of theV(®)), the equation8(D(O, V) 0 should be satisfied. By Solving

a®them, simultaneously, the pull-in parameterspull-in voltage (Vp,l ,) and angle of rotation(0 ) of micromirror can be calculated.For modeling by considering the effects oftorsion and bending of the microbeams, It can beseen from Fig. 2.b that u = g - (y + xO), wherey is the vertical displacement of the micromirror.Hence the total electrostatic torque as a functionof non-dimensional vertical displacement,normalized rotation angle and applied voltagecan be written as follow

TE (1, Y, V) = ,82V2Q2 (O) Y) (6)where

(1Q2(®IY)= 12 _ (1

ln(

a®Y - aO)

a a )Yy - aaO)1- Y - a )

1 - Y - aa® ,

be2 2002 g

The torsion and bending rigidity of themicrobeams have the same order of magnitude,therefore the electrostatic force causes themicrobeams to deflect and affects on themicromirror parameters and can not beneglected. Thus the total applied electrostaticforce on the micromirror plate can be noted asfollows:

FE(8, Y, V) = A3V Q3 (() Y) (8)whereas

Q 3(®,Y) ( -Y-aO) (9)

(I1 - Y - aaor;())b£

2g=

The mechanical elastic beam torque can bedefined as Eq. (3) and the mechanical elastic beamforce can be defined as follow

FM = KbY, Kb 24Eb 13

(10)

where Kb is the effective bending stiffness of themicrobeams. The parameter I is the moment ofinertia of the rectangular cross-section area of themicrobeam. To study the stability of themicromirror, the following functions are introduced

T1 ((0) Y¢V) = TE -TM (I11)T2(,Y,V) = FE -FM (12)

Hence at the equilibrium position the followingrelationships must be satisfied

T'P(®,Y,V) = 0 (13)

T2(®,Y,V) = 0 (14)The angle of rotation and vertical deflection of

the micromirror can be calculated using thenonlinear Eqs. (13) and (14) at a given appliedvoltage. The pull-in state is found at maximumofV(®, Y). Using of the implicit function theoremand Eqs. (13) and (14), to reach the maximum valueof theV(®,Y), the following equation should besatisfied:

a'Y(O,Y,V) a'Y(O,Y,V)a® a 0 (15)2 (O, Y,IV) T2 (O,Y,IV)

ySolving Eqs. (13), (14) and (15) simultaneously, the

pull-in voltage (Vp,11j, ), angle of rotation (0 ) andvertical deflection (Y) of micromirror can becalculated.

IV. NUMERICAL SOLUTION

Because of nonlinearity of foregoing equations, thesolutions are complicated and elaborating. In order tosolve them, it is tried to linearize them. Because ofconsiderable value of 0 and Y respecting to initialgap and especially because of considerable value ofapplied voltage relative to its pull-in value, thelinearizing respect to initial state, may cause toappear some considerable errors [5]. Therefore, tominimize the value of errors, step by step byincreasing the applied voltage is proposed. It isproposed that the Y and 0,) are the non-dimensionalvertical displacement and normalized rotation angle

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ICSE2006 Proc. 2006, Kuala Lumpur, Malaysia

of micro mirror due to applied voltageVi . Tomodel of electrostatic micromirror consideringthe torsion effect using of Taylor's series and itsexpansion for function P(®E, Y) about3i,Vi®,we have

L (eV il I Vil2) =

La0 i +_SI

(Ei,vi) + Lee OcD]- Ea2v2 -

aV 2 (t) ad 20 Vdv ] 80 288a

L170V 817V2 1

(16)

dV j

It should be noted that both of (®i, §Vi) and

(®,+1, Vi+1 ) represent the equilibrium state of the

micromirror, so they satisfy 8D(®, V) = 0. Usinga

of the first order approximation of Eq. (16) we

have

01)i+l = Oi -aO oiVi aV oiVi )V (17)

Now using of Eq. (17), the angle of rotation ateach step of a given applied voltage can beobtained.To model of electrostatic micromirrorconsidering the effect of torsion and bendingusing Taylor's series expansion for functionT'(I( Y, V) and 2 (®, Y, V) about

(O3i,Y,V1), the following relationship can be

gained:

+ OTi1 OT, -al l! 1+La ay av 0,,V

a2T, a2T,oE2 a0ay

2 1 1 Ef0Ti3 a2Te 1 e 1

-ava® avaY(1i

(18)

a2ava0av

av2

andT2(9i+l Iy+l IVi+l) = T2 09i IYi I V)

aT2 aT2 aT2+ 2 2 211LaO ay av i,yiv

+I [soi syi sv]2a2T2a2w2a2T2alaOa2T2ava®

a2T2AOYa2T2ay2a2T2avaY

a2T2AOVa2T2ayava2T2av2

(19)

siHi +...

L(v

It must be noted that (®i,,V,1)and(O i+1 Y+1, vi+/ ) both represent the equilibrium stateof the micromirror so they satisfy the Eqs. (13) and(14). Using the first order approximation for Eqs.(18) and (19), we have:

0i+o 0

1 1

r®i[~P H ~j2 dV (20)[yi+ ]yi

IL IT2L 2O 2

OD(H DY DVHence using of the Eq. (20) at each step of the

applied voltage, the angle of rotation and verticaldisplacement of the micromirror can be obtained.

V. NUMERICAL RESULTS AND DISCUSSION

In this section, first it is tried to find the best stepsize for applying voltage for the Step by StepLinearization Method. Then related figures andtables for comparison between the torsion effect withtorsion and bending effect results are proposed. Thegeometrical and material properties of the model arelisted in Table 1 as:

Table 1: Parameters of the electrostatic micromirrorItems Parameters Values

Material propertiesE0 ,YV,v

Micromirror

Torsion beam

Shear modulus (Gpa)Young's modulus(Gpa)

Width, a3 (pm)

Length, b (,um)

Length, I (,um)Width, w (,um)Thickness, t (,um)

66

170.28

100

100

651.551.5

75

(t)

ICSE2006 Proc. 2006, Kuala Lumpur, Malaysia

0.1406 The calculated results and experimental data arelisted in Table 3 which shows that the calculated pull-

6 in voltage due to the consideration of torsion and84 bending effects is in good agreement with the

2.75 experimental data.

Table 2 shows the obtained results for torsioneffect, and torsion and bending effectconsiderations, containing calculated pull-in!voltage for different step sizes of applied voltage.It can be seen that decreasing of the step size ofthe applied voltage causes pull-in voltagechange, up to a point where favorite accuracy issatisfied and after that, by changing the step sizeof applied voltage, pull-in voltage remains=constant.

Table 2: The obtained pull-in voltages for differentsteps of the applied voltage

Step size of the appliedvoltage (V) 1 0.5 0.1 0.05

Torsion

effect 20 20 19.8 19.8Pull-in voltage Torsion(V) and

bendingeffect 18 17.5 17.4

Based on the results of Table 2, the optimumstep size for applied voltage may be takenaccount as 0.05(V). Figure 3. shows the rotationangle versus the applied voltage. As it is seen, thecalculated results due to the considering of thetorsion and bending effects are in good agreementwith the experimental results of Ref.[6], wherethe difference between results of consideration ofthe torsion effect respect to the experimental ones

may not be neglected.

0.8

0.6

0.4

0.2

0 2 4 6 8 10 12 14 16 18 20Applied voltage [V]

Fig. 3. Comparison between calculated andexperimental results [6] for the rotation angle.

Table 3: Comparison between calculated results withexperimental data at the pull-in pointPull-in S E1 V E2 Y E3iracteristics (%) (V) (%) (%)chE

Ex0.4198

Torsion

effectTorsionand

bendingeffect

17.4

0.5187 23.5 19.8

0.4224 0.6 17.5

0.0778

13.8

0.6 0.0763 1.2

Notes: El =[( 0 -Oexp)/( 0exp)] *100%, E2 =[(VVexp)/ Vexp] * 00%, and E3 = [(Y-Yexp)/ Yexp] * 00%-

V. CONCLUSION

The primary of this work was to derive therelationships between the applied voltage, torsionangle, and vertical displacement, using the parallel-plate capacitor model for the torsional micromirror,then the normalized governing nonlinear equationswere solved by Step-by-Step Linearization Methodwhich was fast, reliable and easy to calculate thepull-in voltage with approximately small differencerespecting to the experimental data for the foregoingmodel. The results showed that assuming the effectsof the torsion and bending gives better results thanthe only torsion consideration.

REFERENCES

[1]. J. P. Zhao, H.L. Cheu, J.M. Huang, A.Q. Liu, A study ofdynamic characteristics and simulation of MEMS torsionalmicromirrors, Sensors and Actuators A, Vol. 120, pp. 199-210, 2005.

[2]. J.M. Huang, A.Q. Liu, Z.L. Deng, Q.X. Zhang, J. Ahn, A.Asundi, An approach to the coupling effect between torsionand bending for electrostatic torsional micromirrors ,Sensorsand Actuators A, vol. 115, pp. 159-167, 2004.

[3]. S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, 3d ed,McGraw-hill New York, sec 109, 1970.

[4]. 0. Degani, E. Socher, A. Lipson, T. Leitner, D. J.Setter, S.Kaldor, Y. Nemirovskey, Pull-in study of an Electrostatictorsion Micromirror, Journal Of MicroelectromechanicalSystems, VOL. 7, NO. 4, DECEMBER, 1998.

[5]. Gh. Rezazadeh, A. Tahmasebi and Mikhail Zubtsov,Application of Piezoelectric Layers in Electrostatic MEMActuators: Controlling of Pull-in Voltage, Journal ofMicrosystem Technologies, Vol.12, No. 12. , pp 1163-1170,2006.

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Electrode

Coefficient, C2

Width, a1 (pm)

Width, a2 (tm)

Gap, h (Utm)-1

ICSE2006 Proc. 2006, Kuala Lumpur, Malaysia

[6]. J.M. Huang, A.Q. Liu, Z.L. Deng, Q.X. Zhang, J. Ahn,A. Asundi, An approach to the coupling effect betweentorsion and bending for electrostatic torsionalmicromirrors ,Sensors and Actuators A, vol. 115, pp.159-167, 2004.

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