Measurement of forces and spring constants of microinstruments …memsucsb/Research/... · 2005-01-16 · Thus, two pairs fracture two pillars. The cross-sectional areas of the pillars

$Page 1: Measurement of forces and spring constants of microinstruments …memsucsb/Research/... · 2005-01-16 · Thus, two pairs fracture two pillars. The cross-sectional areas of the pillars$
REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 69, NUMBER 3 MARCH 1998

Measurement of forces and spring constants of microinstrumentsM. T. A. Saifa) and N. C. MacDonaldSchool of Electrical Engineering and The Cornell Nanofabrication Facility, Cornell University,Phillips Hall, Ithaca, New York 14853

~Received 7 July 1997; accepted for publication 8 December 1997!

Micromachines, including micro scanning tunneling microscope and high force~'0.1 mN andhigher! actuators, are examples of a new class of silicon-based microinstruments for atomic scalesurface imaging and modification, as well as submicrometer scale material investigations. A largeclass of such microinstruments and sensors consist of actuators, such as interdigitated comb drives,that generate force,F, in the formF5bV2, whereb is a constant andV is the applied voltage. Suchactuators often move a large distance during actuation so that the restoring force,R, of the springsvaries nonlinearly withx, i.e., the springs behave as hard springs, andR5K1x1K3x31...~restoring force is similar for1ve and2ve x due to symmetry!. b andKi , i 51,3,..., of the micro-actuators usually differ from their design values due to processing nonuniformities. Hence,evaluation of these constants becomes necessary for each actuator, and is essential for theinstruments that are employed to study materials’ behavior on a small scale. In this article, amethodology is developed to calibrate microinstruments, i.e., to evaluate the values ofb andKi ,i 51,3,... . The method is based on buckling of a long slender beam of known dimensions, and madeof a material with known property~elastic modulus!. Buckling is achieved by an axial compressiveforce on the beam applied by the actuator of the microinstrument.b andKi are derived from therelation between the applied voltage on the actuator, and the post buckling deformation of the beam.The beam is designed and cofabricated with the actuator, and hence the calibration mechanism isintegrable with each microinstrument. An analysis is provided to estimate the possible errors incalibration due to errors in the measured dimensions of the calibrating beam. It is shown that thecalibration error increases linearly with the error in the measured linear dimensions. Theapplicability of the method is demonstrated by fabricating microinstruments, which, prior to theircalibration, are employed to apply torque on single crystal silicon bars~in the form of pillars!, untilthe bars fracture. The instruments are then calibrated, and the calibrated values ofb andKi are usedto evaluate the torques applied on the bars at different voltages. Stresses to fracture of the bars arealso estimated. The torsion experiment is an example of the application of integratedmicroinstruments for small scale material studies. ©1998 American Institute of Physics.@S0034-6748~98!00703-5#

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I. INTRODUCTION

Micromachines, including microscanning tunnelinmicroscope1–3 and high force actuators,4 are examples of anew class of silicon-based microinstruments for atomic scsurface imaging and modification, and submicron scaleterial investigations. A well-known problem for any instrument, including microinstruments, is their calibration. Fmany macroinstruments, the calibration scheme is wellveloped, such as, load cells, available commercially, tcalibrate the force generated by loading instruments forterial studies. The calibration of newly developed microstruments, based on microelectromechanical sys~MEMS! technology, is complicated due to their small sizMEMS microinstruments usually consist of a set of actuatthat generate force by electrical, thermal, or magnetic action. The actuators have a stationary and a movable comnent, the latter being held by a set of springs~usually struc-

a!Current address: Mechanical and Industrial Engineering, University olinois, Urbana, IL 61801.

1410034-6748/98/69(3)/1410/13/$15.00

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tural beams!. During actuation, the movable componemoves by deforming the springs. For small motion, tsprings behave linearly. For larger motions, frequentlycountered in MEMS, the nonlinear terms of the springscome dominant.5 Thus, calibration of a microinstrument involves evaluation of~1! the force generated due to a knowapplied input~voltage, temperature, magnetic field!, and~2!the spring constants including the nonlinear terms. Notethe microinstruments that are used as sensors may alsocalibration. Since calibration must be carried out for eainstrument independently, and since the small size scaleits the application of an external calibrating device, hencecalibration mechanism must be designed and integratedeach microinstrument.

In this article, a methodology is developed to calibrathe force generated by microinstruments and their sprconstants including the nonlinear terms. The method is baon buckling of a long slender beam of known dimensioand made of a material of a known elastic modulus. Acttors that generate force,F5bV2, by actuating a set of combcapacitors6 are considered. Hereb is a constant, to be deter

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1411Rev. Sci. Instrum., Vol. 69, No. 3, March 1998 M. T. A. Saif and N. C. MacDonald

mined, andV is the applied voltage between the movable athe fixed combs. In principle,b5e0Nh/d, wheree0 is thepermitivity constant,N is the number of comb capacitors,his the height of the comb capacitors, andd is the gap be-tween them.h and d for the all the combs are usually designed and fabricated identically, but deviation from thdesign values during fabrication rendersb unknown. Therestoring force,R, of the actuator spring is given byR5( i 51,3,...Kix

i , whereKi , i 51,3,... are the unknown sprinconstants, to be determined, andx is the actuator displacement. In principle,Ki depends on the material properties athe dimensions of the springs. Thus, in principle,Ki can bedesigned by a proper choice of materials and dimensioHowever, the deviation of the actual dimensions fromdesigned values renderKi unknown.

The methodology of calibration is demonstrated by faricating two pairs of microinstruments. Each pair has tidentical instruments. Each instrument consists of an actuwith 2000 comb capacitors and a buckling beam for calibtion. The pair is first employed to apply torque on a shsingle-crystal silicon~SCS! pillar until the pillar fractures.Thus, two pairs fracture two pillars. The cross-sectional arof the pillars are approximately 131 mm2 and 1.531.6mm2, measured by a calibrated scanning electroncroscopy~SEM!. The applied voltage on the actuators athe angle of twist of the pillars are recorded during theperiment. After the experiment, the actuators are calibraCalibration results are then used to evaluate the torqueplied on the samples as well as their failure stresses.torsion experiment is useful for studying the effect of larstrain gradients in small samples on material strength.7

II. METHOD OF CALIBRATION: MEASUREMENT OFFORCES AND SPRING CONSTANTS

A. Buckling of a long slender beam

A long slender beam subjected to an axial compressforce buckles when the force exceeds a critical value of8,9

Pcr5Hp2k/L2, ~1!

where the constantH depends on the boundary conditione.g.,H54 for beams with both ends clamped.k is the mini-mum flexural rigidity, andL is the length of the beam.k5EI when the beam is made of a single material with molus of elasticity,E, and with minimum area moment of inertia, I . I depends on the cross sectional geometry, sucI 5bh3/12 for a rectangular cross section with widthb andheighth. In this article, the fabricated beams have a trapezdal cross section, and the correspondingI is evaluated~seeSec. III!. If the beam has an initial imperfection, suchbeing slightly bent, then it does not buckle suddenly uncompression, but bends gradually along the transverse dtion with increasing load. The transverse deformationcreases rapidly with a small increase of load as the lapproachesPcr . Buckling of a beam with both ends clampehas been analyzed in Ref. 4. Here, the analysis is extenFor convenience, the notation adopted in Ref. 4 is used.

Consider the beam in Fig. 1. Its length, measured althe axis, isL, and its initial imperfection is modeled by it


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cosine shape with an amplitudej/2. L does not change duto buckling. Such assumption is reasonable since the efof axial shortening on buckling is small.9 The total deforma-tion of the buckled beam—initial deformation,j, plus thedeformation due to applied loadP—is approximated by acosine curve with an amplitudeD/2. The validity of the co-sine shape is verified experimentally later~see Sec. VI!. Letv52p/L, s be the coordinate along the beam axis from tpoint of inflexionC1 , y0(s) andy(s) be the initial and totaldeflections ats. Then

y05~j/2!sin~vs! and y5~D/2!sin~vs!. ~2!

At any section ofC1C2 , the internal momentM is related tocurvature8 by

M

k5

du

ds2

du0

ds, ~3!

whereu andu0 are defined in Fig. 1. Using

M52Py, sin u5dy

ds5y8,

du

ds5

y9

~12y82!1/2

in Eq. ~3!,

2Py

k5

y9

~12y82!1/22y09

~12y082!1/2 ~4!

'y9S 111

2y821

3

8y841

5

16y861...D

2y09S 111

2y08

213

8y08

415

16y08

6...D . ~5!

The second equality holds wheny82!1. Using Eq.~2! in Eq.~5!

P

kD sin~vs!5v2~D2j!sin~vs!1

1

2

v4

22 ~D32j3!

3sin~vs!cos2~vs!13

8

v6

24 ~D52j5!

3cos4~vs!sin~vs!15

16

v8

26 ~D72j7!

3sin~vs!cos6~vs!... . ~6!

FIG. 1. Schematic of an actuator and its calibration beam with initial iperfectionj. The total length of the beam along its axis isL. C1 andC2 arethe points of inflection.


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1412 Rev. Sci. Instrum., Vol. 69, No. 3, March 1998 M. T. A. Saif and N. C. MacDonald

Multiplying both sides of Eq.~6! by sin(vs) and integrating10

with respect tos from 0 toL/2, and usingPcr54p2k/L2, theload displacement relation is obtained:

P

Pcr5~12j/D !1

v2D2

32~12~j/D !3!1

3

64 S vD

2 D 4

3~12~j/D !5!125

1024 S vD

2 D 6

~12~j/D !7!1... .

~7!

The relative displacement,d, along the axial directionbetween the end points of the beam after buckling is giby4

d5v2D2L

16@12~j/D !2#. ~8!

The accuracy of Eq.~7! @with terms up to (j/D)3# and Eq.~8! have been verified by nonlinear 2D finite element~FE!analysis in Ref. 4. It is shown that, for a beam, 500mm long,1 mm wide, withj51.5mm, Eqs.~7! and~8! show excellentcorrespondence with FE results as long asD,50mm ~10%of L! and correspondingd,10mm.

B. Initial imperfection j

So far, imperfection of a long slender beam is modeby its initial bent shape that coincides with the first modebuckling with an amplitudej. A ‘‘perfect’’ beam must bestraight, uniform in cross section along the length, have znet stress on any cross section, and satisfy the requboundary conditions, such as clamped at both ends withslope. An imperfect beam buckles under a load less thanPcr .For example, an initially bent beam, deforms transversunder compressive load less thanPcr , but the rate of defor-mation with load increases rapidly as the load approacPcr . Hence the buckling is gradual. Whereas for an initiastraight beam, it is sudden, i.e.,dP/dD50 at P5Pcr .

Due to optical lithography based fabrication, micromchanical beams are straight with high accuracy. Any appciable deviation~e.g., j.0.5mm of a 500mm long and 1mm wide beam! can be detected by SEM. The boundaconditions of micromechanical beams can be ensured bypropriate design of supports.

The major concern for micromechanical beams isinternal stress.11 These stresses arise from multiple layersthin films that constitute the beam. Internal stresses maybe imposed on the beam by the actuator through their inconnection, because the actuator itself might be subjecteresidual stresses. Hence, depending on the fabricationcess, one needs to take adequate measures~such asannealing12,13 and choosing appropriate location for the atachment of the beam with the actuator! to free the beamfrom internal stresses.

It has been shown in Ref. 4 that for a slightly bent beainitially free from any net axial force, the post buckling loadeformation relation is close to an initially straight beaThe post buckling deformation of the beam is not very ssitive to initial imperfections. The reason for such weak ssitivity to initial imperfection is described in Ref. 14. Thu


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in the rest of the paper, it is assumed thatj50, and that thebeam is initially free from any net axial force. Then, Eqs.~7!and ~8! reduce to

d5v2D2L/165p2D2/4L, ~9!

P5PcrF11d

2L13~d/2L !21

25

2~d/2L !31...G . ~10!

C. Force equilibrium

Consider again the actuator and the buckling beamFig. 1 withj50. Pcr of the beam is known from its geometrand material property@Eq. ~1!#. Here the constantsKi , i51,3,... andb of F5bV2 are unknowns. The displacemeof the actuator is denoted byd. Note that, for comb capacitors, the force does not depend on the displacement ofmoving comb, as long as the distance between the endthe movable and the fixed combs is large compared togap between them. With the increase of voltage, the actuapplies a compressive force on the buckling beam. Whenforce approachesPcr the beam buckles and the actuatmoves towards right by an amountd by stretching the springThe maximum transverse deformation,D, of the beam at themiddle is related to the axial deformationd by Eq.~9!. Forceequilibrium requires

F5bV25P1R, ~11!

whereP is the force shared by the buckled beam@Eq. ~10!#andR is the spring’s restoring force

R5 (i 51,3,...

Kidi , ~12!

i.e., the springs restoring force is similar for1ve and2ve d due to symmetry. From Eqs.~10!, ~11!, and~12!

V25Pcr

b S 11d

2L13~d/2L !21

25

2~d/2L !31••• D

1 (i 51,3,...

Ki

bd i ~13!

which implies that in a buckling experiment the square ofactuator voltage,V2, will functionally depend ond accordingto Eq. ~13!. From such an experiment, one measuresV andthe corresponding transverse deformation,D, of the buckledbeam. The axial deformation,d, can be calculated from Eq~9!. V2 can then be plotted as a function ofd. The data can bebest fitted to

V25a0S 11d

2L13~d/2L !21

25

2~d/2L !31••• D

1 (i 51,3,...

aidi , ~14!

where ~a0 , a1 , a3 ,...! are the least-squares parameteComparing Eqs.~13! and ~14!, the calibration parameters othe actuator are obtained:

b5Pcr /a0 and Ki5bai , i 51,3,... ~15!


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For most practical purposes, only the linear and the cuspring constants,K1 andK3 , are of interest in which case thfirst two terms in the parenthesis of Eqs.~13! and~14! wouldsuffice, sinced is expected to be several orders of magnituless than 2L.

D. Cantilevered buckling beam

In the discussion on the initial imperfection of the bucling beam, it was realized that internal stresses may beposed on the beam by the actuator through their intercontion. In order to ensure that the buckling beam is free frsuch stresses, the beam can be designed as a cantilevea small gap ofd1 between the actuator and the beam~Fig. 2!.

The actuator in Fig. 2, when actuated, first stretchesspring ~i.e., in reality, it bends the supporting structurbeams! and closes the gap,d1 , at a voltage ofV1 . Sinced1

is small, the spring restoring force is linear. Hence

bV125K1d1 .

With further increase of voltage, the additional force is cried by the buckling beam until the force on the beamproachesPcr . After buckling, the actuator moves to the righand the actuator force is equilibrated by the force,P, on thebuckling beam and the restoring force,R, of the spring,

bV25P1R,

where, from Eqs.~10! and ~12!

P5PcrF11d

2L13~d/2L !21

25

2~d/2L !31••• G ,

R5 (i 51,3,...

Kixi , x5d1d1 . ~16!

Thus

V25Pcr

b S 11~x2d1!

2L13S x2d1

2L D 2

125

2 S x2d1

2L D 3

1••• D1 (

i 51,3,...

Ki

bxi . ~17!

From a buckling experiment one obtains the data forV andd(D) @Eq. ~9!#, which, together with measuredd1 , gives abest-fit curve

FIG. 2. Schematic of an actuator and its calibration cantilever beam.total length of the beam along its axis isL.


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V25a0S 111~x2d1!

2L13S x2d1

2L D 2

125

2 S x2d1

2L D 3

1••• D1 (

i 51,3,...aix

i . ~18!

Comparing Eqs.~17! and ~18!

b5Pcr /a0 and Ki5bai , i 51,3,... ~19!

For most practical purposes, (x2d1)/2L!1 ~e.g., x2d1

510mm, 2L51000mm!, and onlyK1 and K3 will be ofinterest. Then the first two terms in the expansion ofP @Eq.~16!# will suffice. Equation~17! gives

V25Pcr

b S 12d1

2L D11

b S K11Pcr

2L D x1K3

bx3. ~20!

From a buckling experiment, the measured values ofV arerelated to the corresponding measured values ofx5d1

1d(D) according to

V25a01a1x1a3x3, ~21!

wherea0 , a1 , anda3 are the least-squares parameters ofbest-fit curve through the data. Comparing Eqs.~20! and~21!

b5Pcr

a0~12d1/2L !, K15a1b2Pcr/2L, K35a3b.

~22!

E. Displacement measurement using buckling beam

Displacement can be measured with nanometer restion using the buckling mechanism. A new novel methodproposed here as follows. The relative displacement ofends of the buckled beam,d, is related to the transversdeformation, D, by Eq. ~9!. Thus, if L5500mm, d50.1mm, D54.5mm, i.e., the displacement is amplified b45 times.15 A change ofD, dD, is related to a change ofd,dd, by

dd5~p2D/2L !dD. ~23!

The change ofdD can be measured using a micromechanivernier scale@see Figs. 5~a! and 6~a!# and an optical micro-scope with a resolution of less than 0.5mm. Thus atD54 mm, and forL5500mm, a change ofd can be measuredwithin 20 nm. Clearly, the sensitivity of the measurementdd can be increased significantly ifdD is measured in SEM,or measured electronically using a capacitive sensor.

III. ERROR ANALYSIS

The possible errors in the calibration of parameters,b,K1 , andK3 , obtained from the curve fitting algorithm, arestimated due to variations in measurements. There aremajor sources of errors. These correspond to measuremof: ~1! the cross-sectional dimensions of the calibratibeam, and~2! the initial gap,d1 , between the actuator anthe calibrating beam. The modulus of elasticity,E, of thecalibrating beam may also introduce errors and shouldaccounted for. In this article, however, SCS is used as

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beam material. Its modulus is well characterized in theerature which will be used, and its error will be considernegligible.

A. Error in moment of inertia, I

Figure 3 shows a typical cross section of a calibratbeam~see also an SEM micrograph in Fig. 9!. The minimummoment of inertia,I , about the centroidal verticle axis for thsection is given by

I 5hw3

121

hc3

181ch~w/21c/3!2, ~24!

wherew, h, and c are defined in Fig. 3. These values ameasured by a calibrated SEM~Stereoscan 440, Leica Cambridge Inc.! with 5 nm resolution. Error inI , dI, is due to theequal absolute errors inw, h, and w1 , namely, dw5dh5dw1 . Sincew15w12c, dc50. Thus

dI5S hw2

41

w3

121

c3

181cB21chBDdw, ~25!

whereB5w/21c/3.Error in the evaluation ofPcr is then given by

dPcr54p2EdI/L2. ~26!

B. Error in the coefficients of Eq. „21…

Errors in the coefficients,a0 , a1 , anda3 , are due to theerror in the linear dimensiond1 , dd15dw. The solid line in

FIG. 3. Sectional parameters of the calibrating beam are defined.

FIG. 4. Possible shift ofx vs V2 data due to error,dd1 , in the measuremenof d1 ~the gap between the actuator and the free end of the calibrabeam!.


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Fig. 4 represents Eq.~21! (V25a01a1x1a3x3), the best-fitcurve through the experimental data ofV2 and x5d1d1 .Due to error ind1 , dd1 , let the data be shifted to the right bdd1 . The dotted line is the best fit curve through the shiftdata, and is given by

Vshift2 5b01b1X1b3X3, X5x1dd1 . ~27!

The dotted line can also be represented in terms ofa0 , a1 ,anda3 by

V25a01a1~X2dd1!1a3~X2dd1!3. ~28!

Coefficients of Eq.~27! can be estimated by minimizing*d11dd1

D1dd1 (Vshift2V)2dX with respect tob0 , b1 , andb3 , i.e.,

]

]b0E

d11dd1

D1dd1~Vshift2V!2dX

5]

]b1E

d11dd1

D1dd1~Vshift2V!2dX

5]

]b3E

d11dd1

D1dd1~Vshift2V!2dX50,

where D is the maximum value ofd1d1 obtained experi-mentally.

Values ofb0 , b1 , andb3 can be obtained approximatelby noting that the initial slopes of the two curves of Fig.will be similar, i.e.,b0'a02a1dd1 , andb1'a1 . Thus theerrors ina0 anda1 due todd1 are

da05a1dd1 , da150 ~29!

and V2Vshift'a3(X2dd1)32b3X3. Henceb3 is obtainedfrom

]

]b3E

d11dd1

D1dd1@a3~X2dd1!32b3X3#2dX50 ~30!

giving the error ina3 ,

da35b32a3

5a3U Ed11dd1

D1dd1~~X2dd1!3X32X6!dXUY E

d11dd1

D1dd1X6dX

5a3@2X6dd1/213X5dd1

2/523X4dd13/4#ud11dd1

D1dd1

@X7/7#ud11dd1

D1dd1. ~31!

C. Errors in b, K 1 , and K 3

The worst case errors are obtained from Eq.~22!:

db5UdPcr

a0~12d1/2L !U1UPcr

a02 da0~12d1/2L !U

1UPcr

a0

dd1

2L U,~32!

dK15ua1dbu1udPcr/2Lu,

dK35ubda3u1ua3dbu.

It is noted in Eqs.~25!–~29! that the errors inPcr anda0 arelinearly dependent ondw5dd1 . Equation~31! shows thatn


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FIG. 5. Microinstrument with torsion sample.~a! Schematic shows the instrument with the actuator and a calibrating cantilever buckling beam. The regR1

has a lower trench depth than the rest of the device. This reduces the height of the torsion pillars. Aluminum and silicon dioxide have been wet eregionR2 . The figure is not drawn to scale.~b! SEM micrograph.

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da3 , to first order, linearly varies withdd1 . Since dd1

!1 mm, the higher order terms ofdd1 in da3 are small.Thus the errors in the calibration parameters are also linedependent ondw5dd1 .

In the following, the applicability of the calibrationmethod is demonstrated by fabricating microinstrumeconsisting of actuators and their corresponding calibrabeams. The instruments are also employed to conduct aterial test, namely, twisting of SCS bars until fracture. Eabar is twisted by a pair of identical~nominally! microinstru-ments. After the completion of the torsion tests, the instments are calibrated, and their calibration coefficientsused to back calculate the stresses to failure of the silibars. The demonstration is an example of integrated miinstrumentation where the actuator, the material sample,the calibration mechanism are all designed, patterned,cofabricated.

IV. MICROINSTRUMENTS FOR TORSION TEST

Figure 5 shows the schematic of a pair of microinstments~L andR, not drawn to scale! for applying torque on aSCS pillar. The pillar is attached to the substrate, while attop it is attached to a lever armAB, released from the substrate. Each actuator is 1mm apart from the lever arm. Twoof such pillars, designated by samples 1 and 2, are twiand fractured by two pairs of microinstruments.

Each microinstrument consists of an actuator with 20comb capacitors and a calibrating buckling beam. The ac


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tor has a rigid backbone, released from the substrate, anheld by six beams~springs!. It ~backbone! supports cantile-ver beams which carry the movable combs. A rigid frame,F,is attached to the backbone of the actuatorL. The calibratingbuckling beam is nominally 500mm long. It is held by arigid support attached to the substrate. The free end ofbeam is nominally 1mm away from the frame,F, of theactuator. There is a vernier scale in the middle of the bewhich enables recording of the increments of the transvedisplacement,D, of the buckled beam with 0.8mm resolu-tion using an optical microscope. Thus, at a transverse demationD54 mm a change ofd can be measured with 20 nmresolution.

Two microinstruments,L andR of Fig. 5, are identicalto each other. The instruments as well as the samplesfabricated by the single-crystal reactive etching and metazation ~SCREAM! process.16,17 Torsion samples are formeby partial release of silicon pillars. RegionR1 in Fig. 5 has alower trench depth than the rest of the device. Thisachieved by maskingR1 and by avoiding the second anisotropic silicon etch of the SCREAM process. Thus the heigof the sample pillars are kept short in order to~1! reduce thetotal angle of twist required for fracture, and more impotantly, ~2! avoid metallization of the SCS torsion sample duing sputtering, since a short sample pillar is veiled by toverhang of the SCREAM sidewalls. SEM micrographsthe fractured samples, shown later~Fig. 7!, will reveal that itis indeed the case. After fabrication of the instruments, a


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minum and silicon dioxide are wet etched from the regR2 . Thus the buckling beams are made of only SCS. Becantilevered, they are free from any net axial force.

The distance between the loading arm of the actuaand the center of the torsion pillar is indicated byX in Fig. 5,whereX56 mm and 20mm for the sample pillars 1 and 2Figure 5~b! shows the SEM micrograph of a microinstrumewith the calibration beam~left end! and the torsion sample~right end!. Figure 6 shows the details of a calibrating beaand a close up of its left free end. Figure 7 shows the clups of a torsion sample and part of the actuators. Figure~a!shows that the sample floor is at higher elevation@regionR1

of Fig. 5~a!# than the floor of the device. Figure 7~b! showsthe torsion sample before and after wet etching of aluminand silicon dioxide from the top and sides for clarity. Testiis carried out with these films.

V. TORSION EXPERIMENT AND CALIBRATION

Torsion experiment and calibration are carried out unoptical microscope using a probe station. Two microinstments are employed to apply torque on a sample pillar~Fig.5!. With the increase of voltage on each instrument, itstuator approaches the calibrating beam. As the gap,d1 ,closes, the actuator also touches the lever arm of the torsample. The voltage is increased continuously on the actor until the calibrating beam just buckles. The voltageboth the actuators are then increased equally. Consequethe transverse deformation of the calibrating beam increa

FIG. 6. ~a! A calibrating beam and a part of the actuator that it calibratThe cantilever beam is 500mm long and 1mm wide. Aluminum and silicondioxide have been stripped off from the beam and the surrounding fraThus they are made of single-crystal silicon. The close up of the left freeof the beam is shown in~b!.


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rapidly, while the actuators advance along the axial directslowly, thus twisting the torsional sample. The applied voages on both the actuators, the transverse deformation,D, ofthe buckling beams, and the angle of twist of the sam@using the angular scale~Fig. 5!# are recorded. Voltages arincreased until the pillars fracture. Figure 8 shows the twing of a sample recorded under an optical microscope. Nthe actuators are calibrated. It is noted that for the torsexperiments the actuator displacement is small~x2d1

,10mm, 2L51000mm!, and hence onlyK1 andK3 of theactuator are of interest. Thus Eq.~22! will be used for cali-bration. Calibration of one of the actuators, denoted by 1L,is demonstrated in detail. The rest of the actuators~1L, 2L,and 2R! are calibrated identically.

First, the buckling load,Pcr , for the calibrating beam of1L is evaluated. Cross-sectional dimensions of the buckbeam are measured using a calibrated SEM~Stereoscan 440

.

e.d

FIG. 7. ~a! A torsion sample and part of the actuators.~b! Close up of thetorsion pillar. Also, the sample and the actuators are shown after stripoff aluminum and silicon dioxide for clarity only. Testing is carried out wialuminum and SiO2.

FIG. 8. Twisting of a torsion sample recorded under an optical microsco


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Leica Cambridge Inc.!. Its resolution is less than 5 nm, verfied by measuring a standard grid. The cross section~Fig. 9,trapezoidal due to processing! can be obtained by cleavinthe calibrating beam. Instead, the dimensions are obtafrom a test cantilever beam, identical in design and plaadjacent to the calibrating beam. Figure 9 shows thecantilever beam. Note that the angle of tilt has been conered while measuring the section. From the cross-sectidimensions and the length of the calibrating beam,critical buckling load is calculated,Pcr54p2EI/L2532.55mN, where18 E5E$110%5168 900mN/mm2, I 51.19mm4,andL5494 mm, instead of 500mm ~there is a small grid athe middle of the beam spanning 6mm to hold the vernierscale!. Note that the cross sections of all the beams coincwith the ~110! crystal plane of SCS.E110 is the modulus onthe ~110! plane along the normal direction. The gap,d1 ,between the actuator and the free end of the calibrating bis measured using an SEM~Stereoscan 440, Leica Cambridge Inc.!. Note that the actuator approaches the be@Fig. 6~b!# at two locations. The gap for each locationmeasured. They are 0.81 and 0.83mm. d150.82mm is theiraverage value. Next, the buckling experiment is conduct

With actuation by voltageV, the actuator touches thcalibrating beam, and compresses it. The beam bucklethe compressive force exceedsPcr . The transverse deformation, D, of the buckled beam~using the vernier scale, Fig. 5!,and the corresponding voltages are recorded. Figure 1~a!shows such a buckled beam where the circles indicatedeformation predicted by Eq.~2! ~see also Sec. VI!. Figure10~b! shows a close up of a buckled beam near the verscales. The micrographs are recorded by an optical miscope. Figure 11 shows the values ofV as a function ofD.The end displacements,d, of the beam, and hence the actutor displacements,x5d1d1 , are calculated from Eq.~9!.Figure 12~a! showsV2 as a function ofx. Note that bucklingbegins abruptly atV2'560 V2. This implies that there wasvery little initial imperfection in the buckling beam. For,there were, buckling would be gradual. The solid line in F12~a! ~x5d150.82mm to x55 mm! is the best-fit curve,V25a01a1x1a3x3, where a05504 V2, a1578 V2/mm,anda350.94 V2/mm3. The vertical line,L1L2 , in Fig. 12~a!corresponds to the voltage required to generate the buckforce, Pcr , andL0L1 represents the force-displacement retion of the actuator prior to touching the calibrating bea

FIG. 9. The monitor beam for the calibration beam of actuator 1L.


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Using Eq.~22! and the values ofa0 , a1 , anda3 , the cali-bration parameters are evaluated:b50.065mN/V2, K1

55 mN/mm, K350.061mN/mm3. All the four actuators, 1L,1R, 2L, 2R, are calibrated similarly, and their parameteare shown in Figs. 12~a!–12~d!. The calibration parameterare then used to evaluate the force applied by the actuaon the torsion samples. The actuators 1L and 1R were em-ployed to fracture sample 1, while 2L and 2R to fracturesample 2.

A. Errors in the parameters of actuator 1 L

The cross-sectional dimensions of the calibrating bearew51.06,h57.15mm, c50.19mm ~Fig. 3!. Then, from

FIG. 10. ~a! Buckling of a calibrating beam. The circles denote the defmation predicted by Eqs.~36! and ~37!. ~b! Close up of a beam near thvernier scales. Both~a! and ~b! are recorded by an optical microscope.

FIG. 11. Maximum transverse deformation,D, of the calibration beam ofactuator 1L at different applied voltages. The deformation occurs at mlength of the beam.


eactuators’


FIG. 12. Actuator displacements obtained at different applied voltages~shown squared! for four actuators, 1L,..,2R. The solid lines are best-fit curves. Thverticle lines represent the voltages required to initiate buckling. The part of the curves on the left side of the verticle lines corresponds to thedisplacements prior to touching the calibrating beams.

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Eq. ~25!, dI52.98dw. The coefficients of Eq.~21! are a0

5504 V2, a1578 V2/mm, anda350.94 V2/mm3. The maxi-mum actuator displacement during calibration experimeD55.11d1 mm55.92mm. It is estimated that the SEMmeasurement errordw550 nm. Then from Eqs.~25!, ~26!,~29!, ~31!, and ~32!, and b50.065mN/V2, K155 mN/mm,K350.06mN/mm3, I 51.19mm4, Pcr532.55mN, and d1

50.82mm,

dI50.149 mm4, dPcr54.07 mN, da053.9 V2,

da150, da350.027 V2/mm3,

db50.0086 mN/V2, dK150.67 mN/mm,

dK350.0098 mN/mm3.

Thus the percentage errors inb, K1 , andK3 are 13%, 13.5%,and 16%, respectively.


t,

B. Estimation of the linear spring constant, K 1

The linear spring constant,K1 , of the actuator can beestimated from the stiffness of the individual beamssprings, and compared with the calibrated value. The sness,k1 , of a beam, clamped at both ends, is given by8 k1

512EI/L3, whereL is the length of the beam, andE is themodulus of elasticity of the material of the beam,I is themoment of inertia of the cross section about an axis norto the direction of displacement. For a composite beam,EIshould be replaced by( iEi I i , whereEi andI i are the modu-lus of elasticity, and the moment of inertia of thei th materialsection. Here, one of the beams of the actuator 1L is cleavedto evaluate its spring constant and assume that the rest ofive beams have similar constants. Figure 13 shows thecrograph of the cleaved section. The inner core is Scoated on top and sides by SiO2 and sputtered aluminumThe modulus of Al and SiO2 are close to 70 GPa4 and hence


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they are taken as one material. The cross-sectional dimsions of the beam are estimated from the micrograph whtogether withL5500mm, ESCS5168.9 GPa~normal to 110plane!, and the values of the elastic modulii of Al and SiO2,give k1'0.65mN/mm. Thus the spring constant of the atuator, K156k153.9mN/mm, in contrast to the calibratevalueK155 mN/mm. Note that the purpose of this compason is simply to have an independent estimation of calibtion, not validation.

C. Applied torque

Force,p, applied by the actuator at voltage,V, on thelever arm of the torsion sample is

p5bV22P2R

5bV22Pcr2Pcr

d

2L2K1~d1d1!2K3~d1d1!3. ~33!

In the torsion experiment, the axial motion of the actuawas small prior to fracture, and hence the contribution ofcubic term in Eq.~33! is small. Torque,T, applied by thetwo actuators~e.g., 1L and 1R! on the torsion sample is

T5~pL1pR!X, ~34!

where pL and pR are the forces due to the left and rigactuators. HereX56 mm for sample 1, and520mm forsample 2. Figure 14 shows the torque applied on thesamples as a function of the angle of twist. The anglemeasured directly from the angular scale~Fig. 5! as well asfrom the value ofd. The two angles match each other.

Voltages applied on the actuators 1L and 1R prior tofracture of the sample 1 are 46 and 43.6 V, respectively.corresponding displacements,d, of the actuators are 0.5 an0.425 mm. Equations~33! and ~34!, together with the cali-brated actuator parameters~Fig. 12!, give the torque appliedon sample 1 prior to failure:T151174mN mm. Similarly,prior to fracture of sample 2, the applied voltages on acttors 2L and 2R are 36 V ~same for both actuators!. The

FIG. 13. Cleaved section of one of the six supporting beams~springs! ofactuator 1L. Its cross-sectional dimensions are estimated to evaluatelinear spring constant of the actuator, which is compared with the calibravalue.


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corresponding actuator displacements are 0.8mm and 0.75mm, respectively. Thus the applied torque isT2

511848mN mm.An approximate shear stress to failure of the sample

evaluated as follows. It is noted that the cross section~par-allel to the wafer surface! of the torsional sample pillars coincide with the~100! direction of SCS. Also at mid-height othe pillars, their cross-sectional area is minimum and isproximately a square. The minimum cross-sectional dimsions are estimated prior to fracture by a calibrated S~Stereoscan 440, Leica Cambridge Inc.!. They are (131 mm2) and (1.531.6mm2) for the samples 1 and 2, respectively.

The maximum shear stress on the cross section~~100!plane!, tmax, along the edge of the rectangle is given by19

tmax5TG

b3 , ~35!

whereG54.803 for a square section, andb is the width ofthe section. Thustmax55653 MPa for sample 1 andtmax

52630 MPa for sample 2~usingb51.5mm!. Figures 15 and16 show the fractured surfaces of samples 1 and 2 fromdifferent angles. Note that there is no aluminum on tsamples except near the bottom. Aluminum did not reachsamples during sputtering due to the overhang of the swalls ~Fig. 7!. From Figs. 15 and 16 it is clear that the failuoccurred along an inclined plane, most likely the~111! planefor which the surface energy is 1.23 J/m2 in contrast to~100!plane, where it is20 2.13 J/m2. Due to the inclined fractureplane, two cusps are formed in case of the shorter sampThese cusps are part of the lever arms of the torsion samwhich were released from the substrate prior to the torsexperiment~Fig. 17!. When the sample is tall and/or narrowas in case of sample 1, such cusp are small~Fig. 15!, or theymay not form at all. The entire fracture surface is containwithin the length of the pillar.

Figure 14 shows that the torque-twist curve for the Storsion samples are linear until failure, implying very little ono nonlinear elastic or plastic deformation. This is attributto apparent low stress to failure@less than 10% of the shea

en

FIG. 14. Angle of twist of two torsion samples at different applied torquuntil fracture. The torques are computed from the applied voltages onactuators and their calibration parameters.


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modulus~80 GPa! for ~100! plane of SCS21,22#. The failuremight have initiated from a surface defect causing stress ccentration.

Finally, note that the forces applied by the two actuat

FIG. 15. Fracture surface of sample 1, seen~a! from 30° angle with verticleand ~b! from 45° with verticle.

FIG. 16. Fracture surface of sample 2, seen~a! from top, and~b! from 45°angle with verticle.


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on the lever arm of a given torsion sample are unequal,example,pL598.2mN andpR597.9mN for sample 1 priorto fracture. The force differential,Dp5upL2pRu, induces asmall additional uniform shear stress on the cross sectioalso induces a small moment,M5DpDz, on the pillar~Fig.18!. However, the moment arm,Dz, is small (,0.5mm),because the SCREAM beams are slightly tapered andpoint of contact between the lever arm of the torsion samand the loading arm of the actuator is towards the lowerof the beam, as shown in Fig. 18~also see SEM micrographof Fig. 7!. Thus the effect of moment is also small comparto the applied torque (pL1pR)X, X56 or 20 mm, and theabove experiment generates almost pure torsion onsample.

VI. REMARKS

Here the major assumptions of the calibration methits limitations, and the design considerations of the calibring beam are highlighted.

FIG. 17. The formation of two cusps on the fracture surface~Fig. 16! ofsample 2 is explained.~a! Micrograph of a thick and short torsion sampl~b! schematic of the sample and the fracture surface. The pointS on thecross section has the highest shear stress.

FIG. 18. The cross section of the torsion sample and the loading arm ofof the two actuators are shown. Due to a small tapering of the SCREbeams, the point of contact between the loading arm and the lever arm otorsion sample is close to the neck region of the sample. Hence the flemoment arm,Dz, is small.


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~1! It is assumed that the first mode of buckling of the cabrating beam is a cosine curve, and an initially straigbeam (j50) deforms after buckling according to@Eq.~2!#

y5D

2@cos~vs!21#, ~36!

wheres is measured along the axis of the beam, withs50 at the left support. The horizontal projection,x(s),of the beam can be evaluated from

x~s!5E0

s

dx5E0

sA12~dy/ds!2ds. ~37!

Figure 10~a! shows the coordinates of a buckled beamcircles, evaluated from Eqs.~36! and ~37!, where L5500mm and D533.5mm are used.D is obtainedfrom the deformation (AB) of the experimentally buck-led beam shown in the micrograph. The actual buckbeam is also shown in the figure. Clearly the cosbuckled shape is a reasonable assumption even whetransverse deformation,D, is large. It is, however, foundthat x(s5500mm)5494.41mm, whereas the distancbetween the ends of the buckled beam measured fthe micrograph isX5494.15mm. The overestimation othe predicted value is expected, since the axial compsion of the beam due to the buckling load is neglectClose correspondence between the predicted and theperimental deformations of the buckled beam implthat the beam deforms primarily at its first mode. Ahigher modes, if present due to imperfections, have nligible components in the deformation.

~2! In the model presented in this article, it is assumed tthe length of the calibrating beam does not change ding buckling due to compression, i.e., the axial deformtion of the beam is negligible compared to the axial dplacement,d. For long slender beams this is a reasonaassumption.9 However, if axial deformation becomeimportant, then one needs to conduct a nonlinearnite element analysis of buckling to obtain the relatiP5P(d) and d5d(D) numerically instead of Eqs.~7!and ~8!. Thus bV25P1R5P(d)1( i 51,3,..Kid

i . Theexperimentally obtained data ofV andd(D) can be bestfitted toV25a0P(d)1( i 51,3,...aid

i giving b51/a0 , andKi5aib.

~3! The accuracy of the calibration depends on the precisof the calibrating beam. Thus, its prescribed boundconditions, and its uniformity along the length mustensured. The beam must be free from any axial compsive force other than the force applied by the actuaduring actuation. The elastic modulus of the matermust also be well defined.

~4! It is assumed that the compliance of the backbone ofactuator of Fig. 5 is negligible, i.e., the springs do ndeform when the actuator applies compressive forceP,Pcr on the calibrating beam. Such deformation isdeed negligible compared to the post buckling deformtion, d, of the springs, provided the backbone andattached frameF ~Fig. 5! are rigid andPcr is small. Onecan, however, reasonably account for such smallplacement by usingd5de1p2D2/4L, wherede is the


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displacement of the actuator prior to buckling of thbeam. Note that if there is an initial gap,d1 , between theactuator and the buckling beam, then the actuatorplacement isd1d1 .

~5! While calibrating the actuator 1L for b, K1 , and K3 ,only the first two terms in the parenthesis of Eqs.~17!and ~18! are retained, and the termG53@(x2d1)/2L#2

125/2@(x2d1)/2L#3 as well as the higher order termare ignored. Such minor simplification is justified sinK3x3@GPcr within the range of interest ofx(0 – 5mm).

~6! The methodology developed here is readily applicableactuators that generate force,F5bI , whereI is a func-tion of the input, e.g.,I 5V2 for comb actuators, andtemperature for thermal actuators.

~7! The calibrating beam should be designed such thatbuckling load,Pcr , is well within the force range of theactuator.

~8! The fracture experiments reported in this paper aretended to demonstrate the application of microinstments for material characterization. The data of the faure stresses should not be considered as conclusive

~9! An optical microscope and a probe station were usedcarry out the calibration and torsion experiments to deonstrate the simplicity and flexibility of the experimentsetup. An SEM could be used to carry out thmeasurements,23 or capacitine sensors can be designedmeasure the displacements.

~10! The method of calibration presented here needs vecation by an independent check using a direct fosensor. Work is currently underway, and will be rported shortly.

ACKNOWLEDGMENTS

The work was supported by ARPA and the National Sence Foundation. The computing resources of the MaterScience Center, Cornell University, were used to carrythis research.

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Vac. Sci. Technol. B7, 775 ~1989!.13T. Randazzo, H. Guckel, and D. W. Burns, J. Appl. Phys.57, 1671~1985!.


oa–

al


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fl

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Measurement of forces and spring constants of microinstruments …memsucsb/Research/... · 2005-01-16 · Thus, two pairs fracture two pillars. The cross-sectional areas of the pillars