Determination of mechanical properties of a MEMS directional … · 2020. 2. 10. · Determination of mechanical properties of a MEMS directional sound sensor using a nanoindenter

Determination of mechanical properties of a MEMS directional soundsensor using a nanoindenter

R.H. Downeya, L.N. Brewerb, G. Karunasiria

aDepartment of Physics, Naval Postgraduate School, Monterey, CA 93943bDepartment of Mechanical and Aerospace Engineering, Naval Postgraduate School, Monterey, CA 93943

Abstract

We use a nanoindenter to determine the mechanical properties of a microelectromechanical direc-tional sound sensor and to estimate the maximum sound pressure the sensor can tolerate. Becausethe sensor has bending and twisting components that act as springs in series, the overall stiffnessis the sum of several terms, each of which varies along the sensor’s surface. By fitting a curve to aplot of the measured overall stiffness at points along the sensor, we quantify the separate stiffnessterms and thereby estimate the resonant frequencies of the corresponding vibrational modes. Thefrequencies estimated by this method for the two modes are in reasonably good agreement withthe measured resonant frequencies. Finally, we establish a minimum failure strength of the sensor,from which we estimate that it can tolerate a sound pressure level greater than about 162 dBwithout damage.

1. Introduction

We demonstrate the use of a nanoindenterto determine the spring constants of a set ofcoupled springs incorporated into a microelec-tromechanical system (MEMS). Nanoindentersare commonly used on micro-scale cantileverbeams to investigate the elasticity and hard-ness of the beam material as originally de-scribed by Weihs et al. [1]. Nanoindenters havealso been used to mechanically actuate MEMScomponents, either to investigate behavior ofelectrical contacts [2, 3, 4], to investigate fail-ure modes of microstructures [5], or to assessmechanical stability [6]. A nanoindenter trans-ducer has even been incorporated into a hand-built apparatus to measure the stiffness of sin-gle cantilevers [7]. Here we use a nanoindenter

Email addresses: [email protected](R.H. Downey), [email protected] (L.N. Brewer),[email protected] (G. Karunasiri)

to analyze a more complex MEMS structurewhose stiffness varies across its surface due tosimultaneous bending and twisting motions ofseveral coupled components.

The MEMS structure in question is a direc-tional sound sensor inspired by the hearing or-gan of the parasitoid fly Ormia ochracea, whichuses hearing to find crickets as a food sourcefor its larvae [8]. Despite the handicap of be-ing much smaller than the wavelength of thecricket’s chirp, the fly is able to locate its preyby homing in on the sound. It accomplishesthis through mechanically coupled eardrums.

The eardrums can be modeled as two rigidbars connected by a flexible bridge. In this con-figuration the bars can vibrate in two normalmodes in response to incident sound: a “bend-ing” mode, in which the bars vibrate in phase,and a “rocking” mode, in which the bars vi-brate exactly out of phase. When the eardrumsare excited by the sound field, the amplitude

Preprint submitted to Sensors and Actuators A November 20, 2017

of the bending mode depends on the sum ofthe forces acting on the two eardrums, whilethe rocking mode depends on the difference be-tween the forces. As a result, the phase differ-ence between the two eardrums depends on thedirection of sound incidence [8].

The MEMS directional sound sensor mim-ics the eardrum of the fly. The sensor(see Figure 1) is constructed from the single-crystal (100) silicon device layer of a silcon-on-insulator (SOI) wafer, using a standardizedcommercial micromachining process [9]. Thelateral and longitudinal axes of the sensor areoriented along 〈110〉 crystal directions. The sil-icon layer is 9.5 µm thick as measured with aprofiler.

Legs

Bridge

Wing

Comb fingercapacitors

x

Figure 1: A photograph of the MEMS sensor. Thetwo wings are coupled by the bridge, which is con-nected to the substrate by the legs. The legs act astorsion springs and each end of the bridge acts asa flexible cantilever beam. Comb finger capacitorsenable electronic readout of the wingtip displace-ment from equilibrium. The sensor is 2.5 mm fromwingtip to wingtip.

Structurally, the sensor comprises two rela-tively stiff wings (rectangular plates) connectedby a flexible bridge which acts as a spring. Thisbridge is connected to the substrate by two thinlegs which, by twisting, enable the bridge as awhole to rock back and forth in a see-saw mo-tion. In this way the sensor is able to respondto sound pressure on the wings by oscillatingin rocking and bending modes analogous to thetwo vibrational modes of the fly’s ear, as illus-trated in Figure 2. Electronic readout of the

Table 1: Parameters of the MEMS sensor. Young’smodulus and shear modulus are treated as scalarquantities [12]. The comb finger capacitors aremodeled as a 100 µm extension of each wing tipwith one quarter the density of silicon.

thickness t 9.5 µmbridge length 500 µmbridge width wb 300 µmleg length l 100 µmleg width wl 40 µmwing length 1250 µmwing width 1500 µmYoung’s modulus E 169 GPashear modulus G 50.9 GPadensity 2330 kg/m3

(a) The “rocking” mode (b) The “bending” mode

Figure 2: The vibrational modes of the MEMSsensor, obtained from a COMSOL finite elementmodel. Displacement is greatly exaggerated forclarity; actual displacement of the wingtips is typ-ically much less than one percent of the length ofthe sensor. Capacitive comb fingers are not shownhere.

wings’ motion is enabled by interdigitated ca-pacitive comb fingers on the wingtips [10]. Thecomb finger capacitors are 100 µm long by 2 µmwide, separated by a gap of 2 µm from the op-posing interlaced fingers, and range along thewidth of the wing tip.

Since the vibrational response of the sensordepends on the stiffness of the bridge and legs,it is useful to measure the stiffness directly. Ac-curate measurements enable improved designof future prototypes as well as verification ofanalytical and computer-based models of thedevice. The nanoindenter technique enablesprecise, direct measurement of stiffness any-where on the sensor.

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2. Theory

The MEMS sensor can be modeled as a col-lection of springs acting in series. To simplifythe analysis, only points along the extendedcenterline of the bridge (defined as the x axis)are modeled. The overall stiffness k(x) is theindividual spring constants ki(x) added in se-ries:

1k(x) =

∑i

1ki(x) . (1)

The three springs of the sensor model repre-sent bending of the bridge, twisting of the legs,and vertical flexing of the legs and bridge. Thevertical flexing does not affect the sensor’s op-eration, but it is important to the nanoindenterstudy.

To estimate the bending stiffness, the bridgeis modeled as a cantilever beam of rectangularcross section, fixed at the legs and free at thepoint where the load is applied (Figure 3). Fora cantilever of width w and thickness t < w,acted on by a transverse load F a distance xfrom the fixed end, the deflection d is [11]

d = 4Fx3

Ewt3, (2)

where E is the Young’s modulus of the ma-terial. For a cubic material such as silicon,E may be taken as a scalar if all stresses arein the same crystal direction, as they are inthe simple beam bending model. Because bothof the MEMS sensor axes are oriented along〈110〉 directions, the scalar Young’s modulus isEx = Ey = 169 GPa [12]. The bending stiff-ness of the cantilever kb = F/d is then

kb = Ewbt3

4x3 . (3)

where wb is the width of the bridge.To model the rocking stiffness, the bridge is

treated as a rigid beam mounted on a pair oftorsion springs (the legs) with rectangular cross

F

dx

Figure 3: A simple model of the sensor’s bendingmode. The bridge is a flexible cantilever beam fixedon one end and free on the other end.

section (Figure 4). The angular deflection of apair of torsion springs acting in parallel is [11]

φ = 12τ l

GJ

where τ is the torque applied, l is the length ofeach spring, G is the shear modulus of the ma-terial, and J is the torsion constant, a functionof the cross-sectional dimensions of each springwith dimensions of length raised to the fourthpower [11].

ϕ

F d

x

Figure 4: A simple model of the sensor’s rockingmode. The legs twist while the bridge remains rigid.

If a transverse load F is applied to the bridgea distance x from the spring, the torque aboutthe rotation axis is τ = Fx. For a small twist-ing angle φ, the displacement of the cantileverat x is

d = φx = Flx2

2GJ .

Then the effective rocking stiffness kr = F/d is

kr = 2GJlx2 . (4)

For a rectangular torsion spring of cross-sectional width wl and thickness t with wl ≥ t,

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the torsion constant J is [11]

J = wlt3[

13 − 0.21 t

wl

(1− t4

12w4l

)].

As with the Young’s modulus, the shear mod-ulus G may be treated as a scalar in a siliconcrystal with the shear stresses about 〈110〉 di-rections. In this case G = 50.9 GPa [12].

It is clear from Equations 3 and 4 that bothbending and rocking stiffnesses diverge at thecenter of the bridge, where x becomes small.Clearly the actual stiffness of the sensor cannotbe infinite at x = 0. If measurements there areto be analyzed, a third stiffness term must beconsidered.

The flexing stiffness kf is independent of x,and is primarily due to vertical bending of thelegs, with a smaller component due to flexingof the bridge itself, as illustrated in Figure 5.Since the legs are cantilever beams, their bend-ing is described by an analysis similar to thatleading up to Equation 3. Here the two legs actin parallel, so vertical displacement in responseto a downward force F at the ends of the legsis half that given by Equation 2, or

dlegs = 2Fl3Ewlt3

,

where E is still 169 GPa, wl is the width ofeach leg, and the moment arm x is replaced bythe leg length l.

For a load applied at the center of the bridge,there is an additional displacement due to flex-ing of the bridge itself. For a thin rectangularplate simply supported on all sides, the dis-placement under a central point load is [13]

dbridge = αFw2b

Et3

where the bridge width wb is the shorter sidelength of the plate and the coefficient α de-pends on the length/width ratio of the plate.Since in our case the bridge ends are actuallyunsupported, we take the ratio to be infinite

leg bridge leg

dleg dbridge F

Figure 5: A simple model of the sensor’s flexingmode. The legs are cantilever beams and the bridgeis a rectangular plate supported by the legs. Adownward load at the center of the bridge causesthe bridge and legs to bend, and the total displace-ment at the center is the sum of the leg displace-ment and the bridge displacement.

so that α = 0.1851 [13]. The flexing stiffness isthen

kf = F

dlegs + dbridge= Ewlt

3

2l3 + αwlw2b

. (5)

With all three stiffness modes accounted for,Equation 1 becomes

1k(x) = 1

kb+ 1kr

+ 1kf

= 4x3

Ewbt3+ lx2

2GJ + 2l3 + αwlw2b

Ewlt3.

(6)

Once the numerical values for the various pa-rameters (listed in Table 1) are inserted, theanalytical model is complete. Figure 6 showsthe relative contributions of each mode to theoverall stiffness along the x axis. Note thatthe overall stiffness is dominated by the small-est individual stiffness term, and also that thecontribution from each mode becomes vanish-ingly small in the region where its model breaksdown, i.e. close to x = 0 for the rocking andbending modes, and at large x for the flexingmode.

From the individual modeled stiffness terms,it is possible to estimate the resonant frequen-cies of the corresponding modes. The resonantfrequency fr of the rocking mode can be esti-

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101

102

103

102

103

Distance from rotation axis (µm)

Sti

ffn

es

s (

N/m

)

Rocking

Bending

Flexing

Overall

Figure 6: A logarithmic plot of the modeled stiff-ness components and the overall stiffness in the an-alytical model of the MEMS sensor. At any givenposition on the sensor, the overall stiffness is gov-erned by the smallest of the individual terms.

mated using

fr = 12π

√κr

Ir(7)

where κr is the torsional stiffness and Ir =∫V ρr

2dV is the rotational moment of inertiaof both wings rotating together, which for thissensor is 7.84×10−14 kg ·m2 including the con-tribution from the comb fingers. It is easy toshow that the torsional stiffness κr associatedwith the rocking motion is related to the linearstiffness kr(x) by

κr = x2kr(x). (8)

In view of this relationship, it is clear fromEquation 4 that the rocking mode torsionalstiffness is independent of where the force isapplied, and is equal to 2GJ/l. Using the val-ues from Table 1, the rocking frequency is es-timated to be 1.79 kHz.

We can use a similar method to estimate theresonant bending frequency

fb = 12π

√κb

Ib, (9)

where both the torsional stiffness and the mo-ment of inertia have different values from the

rocking mode. Applying Equation 8 to thebending mode (Equation 3), the bending tor-sional stiffness becomes κb = Ewbt

3/4x, andto evaluate it we must choose a value for x.Since the acoustic pressure load is evenly dis-tributed over the sensor’s surface, and the wingcomprises the largest part of the surface, theacoustic load may be approximated by a pointload at the center of the wing. Hence for x weuse the distance from the rotation axis to thewing center, which is 875 µm.

In the bending mode, each wing rotates inde-pendently and the moment of inertia Ib is thatof only one wing, or 3.92× 10−14 kg ·m2. Theestimated bending frequency is then 2.83 kHz.

To supplement the analytical model, wecreated a finite element model (FEM) ofthe MEMS sensor using COMSOL Multi-physics® 4.2 simulation software (2011, COM-SOL Inc.). The directional Young’s modulus,shear modulus, and Poisson’s ratio were en-tered using the 〈110〉 constants for silicon [12].To simplify the model geometry, each comb fin-ger capacitor bank was modeled as a solid pieceof reduced-density silicon. This simplificationgreatly reduced the complexity of the modelwithout altering the rotational moment of in-ertia.

Even though the nanoindenter tip is muchless than one micron in diameter, it may bemodeled as a much larger circle. It has beenshown [13] that the stresses in a thin platecaused by a load distributed over a small ra-dius r0 are similar to those that would obtainif the force were distributed over a larger radiusr′0 =

√1.6r2

0 + t2−0.675t, where t is the thick-ness of the plate. In this case, r0 � t and wemay use r′0 = 0.325t ≈ 3 µm. Using the largervalue for the tip size enabled coarser meshingwithout sacrificing accuracy.

We ran a series of simulations over differentnanoindenter tip locations along the lateral (x)axis of the sensor. For each location, a verti-cally downward boundary load was applied atthe location of the tip, and the composite stiff-

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ness k(x) was recorded as the load divided bythe displacement. Of note, the stiffness was in-variant at a given location over a wide rangeof applied loads, suggesting that Hooke’s lin-ear elasticity law is applicable to the MEMSsensor.

The FEM results were similar to those ofthe analytical model. At x = 0, the FEMpredicted a stiffness of 2.35 kN/m, comparedto 2.17 kN/m in the analytical model, a dif-ference of about 8%. The two models agreedwithin 2.5% at x = 100 µm, and continued toconverge with increasing x.

In addition to the stiffness study, we used theFEM to predict the eigenfrequencies of the sen-sor’s vibrational modes, obtaining 1.53 kHz forthe rocking mode and 2.91 kHz for the bend-ing mode. These values are within 15% and3%, respectively, of the analytical estimates.An isotropic FEM produced results that weresomewhat closer to the analytical model.

3. Experimental

Stiffness measurements were made using anAgilent G200 nanoindenter with the DynamicControl Module (DCM) head [14] and a dia-mond Berkovich tip with a radius specificationof ≤ 20 nm across. This tip was readily avail-able, but any other tip could be expected towork equally well for a study of MEMS surfacesin which the tip does not appreciably indentthe material. The G200 provides a motorizedstage that positions the test sample under thetip to an accuracy of 1 µm in each direction.

There were several instrumentation param-eters for this study. Because the compositestiffness of the MEMS sensor ranged from over2000 N/m at the center to just a few N/mnear the wingtips, one of the primary consider-ations was the sensitivity of the nanoindenterto surface contact. The G200’s “surface de-tection stiffness criteria” parameter allows theuser to determine the surface stiffness that willtrigger the instrument’s surface detection rou-tine. Repeated trials with various choices of

this parameter confirmed that the lowest sur-face stiffness that was reliably detected by theinstrument was about 20 N/m. Based on thislimitation, we restricted our study to the partsof the MEMS sensor where the modeled com-posite stiffness was higher than 25 N/m, corre-sponding to values of x less than 500 µm.

Another key testing parameter was the max-imum load applied during each test. Higherloads produced somewhat cleaner data, butalso greater deflection of the MEMS surface.Since excessive deflection would lead to theMEMS surface contacting the substrate be-low and could induce a nonlinear stiffness re-sponse, we limited the maximum load to thatresulting in a deflection less than the thick-ness of the silicon layer (9.5 µm) at the sen-sor’s wingtip. Based on the analytical and fi-nite element models, this maximum was deter-mined to be about 85 µN at x = 500 µm. Ul-timately, we settled on a conservative yet pro-ductive maximum load of 60 µN for each test.Measured vertical displacement of the sensorsurface ranged from 30 nm with the load ap-plied at the center of the bridge to over 600 nmwith the load applied beyond the end of thebridge. It should be noted that repeated testsusing higher or lower maximum loads (10 to100 µN) measured the same stiffness at a givenlocation. In other words, the response of theMEMS sensor was linear, even for loads corre-sponding to sound pressure levels much higherthan those normally experienced during oper-ation (see below for comparison of loads withsound pressure levels).

Because in normal operation the deforma-tion of the MEMS sensor is purely elastic, wewere interested in measuring the elastic stiff-ness. In a nanoindenter test, the elastic stiff-ness is simply the slope of the load vs. displace-ment curve just after unloading begins. Whenthis curve does not have a well-defined slope,a power law fit can produce more precise mea-surements of elastic stiffness [15], but for thisMEMS study the curve is linear enough to use

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a linear fit. Figure 7 shows this curve duringthe loading and unloading steps of a test takenat the stiffest point on the MEMS sensor, i.e.x = 0. At large x, the load-displacement curvewas quite linear.

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

Displacement into sample (nm)

Lo

ad

on

sa

mp

le (

µN

)

Nanoindentation dataSlope of unloading portion of curve

Loading

Unloading

Figure 7: A test conducted at the stiffest point onthe MEMS sensor. The elastic stiffness is the slopeof a line fit to the data just after unloading begins.At points on the sensor where the stiffness was less,this curve was straighter.

To establish the stiffness as a function of po-sition on the sensor, we conducted tests at in-tervals of 5 µm along the x-axis and computedthe elastic stiffness for each test. Figure 8shows the results plotted as a function of dis-tance from the rotation axis, along with the re-sults from the analytical model and FEM. Thediscrepancy between measurement and modelranges from 12% at x = 0 to only 1.4% atx = 500 µm.

To evaluate the individual stiffness compo-nents, we fit a curve to the measured data us-ing Matlab with the Ezyfit open-source fittingtool. The curve fit was based on the analyticalexpression in Equation 6 and given by

1k(x) = (x− x0)3

a3+ (x− x0)2

a2+ 1a1, (10)

where a1, a2, and a3 are parameters and an xoffset is included to account for any systematicerror in the coordinates of the stiffness tests.

0 100 200 300 400 500

102

103

Distance from rotation axis (µm)

Sti

ffn

ess (

N/m

)

Analytical

FEM

Measured

Figure 8: Agreement between the analytical model,the finite element model, and the nanoindenter re-sults.

The curve in Equation 10 was fit to two sep-arate data sets, one from the left side of theMEMS sensor and the other from the rightside, as shown in Figure 9. Table 2 lists thevalues for the parameters and the correlationcoefficients R for both fits. From the curve fitparameters, it is possible to estimate the res-onant frequencies of the rocking and bendingmodes. By matching the fit parameters to thecorresponding values in the analytical model,it is straightforward to estimate the torsionalstiffness for each mode and thence the resonantfrequencies.

Table 2: Parameters for the curves fit to the datafrom the left and right sides of the MEMS sensor.Values are in SI units except x0.

Fit parameter Left side fit Right side fita1 (= kf ) 2.09 × 103 2.10 × 103

a2 (= 2GJ/l) 9.05 × 10−6 8.95 × 10−6

a3 (= Ewbt3/4) 1.72 × 10−8 1.94 × 10−8

x0 3.64 µm -2.25 µmR 0.99994 0.99997

The resonant frequencies estimated from themeasured stiffness are listed in Table 3 alongwith the estimates obtained from the analyti-

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100 200 300 400 500

102

103

Distance from rotation axis (µm)

Sti

ffn

ess (

N/m

)

Measured stiffness

Curve fit

(a) Left side of sensor

100 200 300 400 500

102

103

Distance from rotation axis (µm)

Sti

ffn

ess (

N/m

)

Measured stiffness

Curve fit

(b) Right side of sensor

Figure 9: Equation 10 fit to the nanoindenter mea-surements. In both cases the correlation coefficientR is greater than 0.9999 using the fit parameterslisted in Table 2.

Table 3: Resonant frequencies from the analyticalmodel, from the FEM, estimated from the mea-sured stiffness, and directly measured using a laservibrometer.

Rockingfrequency(kHz)

Bendingfrequency(kHz)

Analytical model 1.79 2.83FEM 1.53 2.91Estimated fromstiffness

1.71 3.67

Vibrometer 1.41 2.80

cal and finite element models. Also listed forcomparison are the resonant frequencies of anidentical sound sensor as measured by a laservibrometer with the sensor excited by an ex-ternal sound source. The sensor used in thenanoindenter study could not be removed in-tact from the mount for laser vibrometer test-ing. Because there is variation in resonant fre-quencies even between ostensibly identical sen-sors, the vibrometer results should be viewedas another estimate rather than an exact mea-surement.

The discrepancy between modeled, esti-mated, and measured values is greater for thebending mode frequency, which is to be ex-pected since the measurements were taken inregions where the bending mode makes only asmall contribution to the overall stiffness (asmentioned earlier, we restricted our measure-ments to locations where the composite stiff-ness was above 25 N/m and was dominated bythe rocking mode). Nevertheless, the estimatesare close enough to suggest that the nanoinden-ter method is a valid means of estimating theresonant frequencies of a MEMS device.

The final test conducted with the nanoinden-ter was to estimate the maximum sound pres-sure level the sensor can tolerate without frac-turing. For this measurement, we replaced thenanoindenter’s DCM head with the XP headto allow for loads up to 500 mN [14]. The loadwas applied at the center of one of the sen-sor’s wings, and was set to an arbitrary high

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value in the expectation that the sensor wouldfracture during the loading process. This frac-ture would be indicated by the load suddenlydropping to zero as the displacement contin-ued to increase. However, the expected frac-ture did not occur. Instead, as the load wasincreased, the displacement eventually stoppedincreasing, as shown in Figure 10. The maxi-mum load attained was 4.5 mN, and the ulti-mate displacement was about 32 µm. The haltin displacement suggests that the sensor hadmade contact with the side walls of the trenchbeneath.

0 5 10 15 20 25 30 350

1

2

3

4

5

6

Displacement into sample (µm)

Lo

ad

on

sam

ple

(m

N) Wingtip

contactssubstrate

Figure 10: Failure testing of the MEMS sensor.Loads are two orders of magnitude higher than inthe stiffness testing, and the response here is clearlynonlinear (compare to Figure 7). At about 32 µm,the load spikes, suggesting that the sensor has madecontact with the substrate below.

The sensor remained intact, and subsequentinspection with an optical microscope showedno visible damage. Moreover, repeated simi-lar measurements showed identical results, sug-gesting that the sensor was not permanentlyaffected by the test loads. Unfortunately, itwas not possible to remove the sensor intactfrom the nanoindenter mount in order to di-rectly test its acoustic response after the failuretesting.

Even though the exact fracture load was notestablished, these results show that the sensor

can withstand a sound pressure level (SPL) atleast equivalent to the nanoindenter test load.The equivalent SPL is

SPL = 20 log(Pequiv

Pref

)(11)

where Pequiv is the equivalent acoustic pres-sure amplitude and Pref is the standard refer-ence pressure of 20 µPa. The acoustic pressureequivalent to the test load can be estimated asthe load divided by the wing area, in this caseabout 2400 Pa. The resulting SPL of about162 dB suggests that the MEMS sensor cantolerate extremely high sound levels. Also, itis important to note that the actual failure loadfor the sensor must be higher than the test loadused in the calculation. Thus the failure SPLfor the sensor is higher than 162 dB.

Because the nanoindenter load is static,it does not perfectly simulate the dynamicstresses from actual acoustic pressure. Never-theless, this technique can be used to estimatethe sensor’s maximum tolerated sound pressurelevel.

4. Conclusions

Nanoindentation is a useful tool for investi-gating basic properties of simple MEMS struc-tures. Here we have demonstrated the use of ananoindenter to analyze several characteristicsof a MEMS device with multiple coupled flex-ible components. First, direct measurement ofthe stiffness can confirm an overall linear re-sponse (i.e. Hooke’s law) over the range ofloads likely to be experienced by the sensorduring operation. Second, by measuring the lo-cal stiffness at various points across the devicesurface, it is possible to extract the spring con-stants of individual components of the deviceand thereby estimate the resonant frequenciesfor each vibration mode. Third, the nanoin-denter can be used to find the failure load ofthe MEMS device and thence the maximumtolerable sound pressure level. Similar studies

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can be useful in analyzing other MEMS withcoupled mechanical components.

Acknowlegments

Special thanks to Y. Kwon for helpful discus-sions on plate bending theory, and to J. Rothand D. Grevenitis for their previous work onthe MEMS directional sound sensor. This workis supported by NCMR/NSF.

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