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ICSE2002 oc.2002,PenangNalaysiaAn Analytical Study on Diaphragm Behavior for Micro-machined Capacitive Pressu re Sensor

Norhayati Soin andBurhanuddin Yeop Majlis, Member, IEEEUKM-TMMicroelectronic centerFaculty of EngineeringUniversiti Kebangsaan Malaysia,43600,Bangi, Selangor, MALAYSIA.

... . MEMS Laboratory

Abstract Understanding the deflectionbehavior of micro-machined diaph ragm s isnecessary for designing mechanical sensorssuch as pressure sensors. An analylkalstudy on the diaphragm behavior withdifferent structures for micro-machinedcapacitive pressure sensor is presented inthis paper. In general, analytical solutionsfor diaphragm behavior are desirablebecause of their ease and the insight theyprovide to the designer. Specific geometriceffects can be ascertained form thesesolutions. However, these solutions aregenerally only applicable for smalldeflections. The behaviors of flat andcorrugated diaphragms with variousstructural parameters and properties areanalysed using the classical Timoshenkoplate theory respectively,1. INTRODUCTION

Diaphragms with a boss and corrugations arevery usefnl for micro-machined capacitivepressure sensors . Such diaphragms offerlonger linear travel and larger dynamic rangethan do planar and simple cormgateddiaphragms [1,2]. In many cases, the micro-machined silicon diaphragm s are in state ofinternal stress [3,4]. The internal stress cancause the diaphragms to be deformed and theload deflection changed.[5,6].It was found that the effects of the internalstress on the behavior of the diaphragms withdifferent structures, such as planar, simplecorrugated, and boss and corrugated, aredifferent. Understanding these issues anddeveloping appropriate processes to control,but generally takes the form of a square orcircle. These shapes behave similarly to anapplied the diaphragm deformation are veryimpottant for device design and fabrication.

In this paper, different types of diaphragmstructures, including flat and cormgateddiaphragm with various~cormgationdepths andinitial stresses. are- studied. T he analyticalobservations have been m ade by 'using ,theclassical Timoshenko plate theory[7]. Thisapproach have . been .used to explore .the.performance.of different diaphragm structures.The effects on load deflection, capacitance,non-linearity, and sensitivity performance dueto changes in diaphragm size, cormgationprofiles, and inte rnal shess a re also explored.It. THEORETICALAPPROACH

(i) Flat DiaphragmIn diaphragm based sensors, pressure isdetermine d by the deflection of the diaphragmdue to applied pressure. Fig.1 illustratesschematic section of a typical pressure se nsordiaphragm. The reference pressure can be asealed chamber or a pressure port so thatabsolute or gauge pressure are measured,respectively.

Ipplied Pressure

Fig. 1Schematic cross sectionof typical pressure sensordiaphragmThe shape of the diaphragm as viewed tkom the topis arhitrary, but generally takes the shape of a circleor square. These shapes behave similarly to anapplied stress. For the case of a clamped circularplate with small deflections (i.e., less than half ofthe diaphrag m thickness) the form of deflection is[71 :

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where w , r, a, and P are the deflection, radialdistance from the center of the diaphragm,diaphragm radius and applied pressurerespectively. D is the flexnral rigidity, givenbY

where E, h, and Y are the Young's modulus,thickness, and Poisson's ratio, respectively ofthe diaphragm. From he above equations, boththe shape and the amount of deflection can bedetermined. Moreover, it is readily apparentthat the amount of deflection is directlyproportional to the applied pressure. For thecase of a diaphra gm with larg e built in stress orlarge deflections this direct prop ortionality isno longer true. In general, it is desirable to usea deflection measurement scheme that is linearwith pressure, since such systems are simple tocalibrateand measure.

External pressure

and dielectricisoldion

Reference pressure inletFig. 2. Cross section schem atic of hulk micro-machined, capacitive pressure sensor.

Capacitive pressure sensors are basedupon parallel plate capacitors. A typicalhulk m icro-mach ined capacitive pressuresensor is shown in Fig. 2. Thecapacitance, C, of a parallel platecapacitor is given by

(3 )

,Penang,Malaysiawhere E. A , and d are the permittivity of the gap, the area ofthe plates, and the se paration of the plates, respectively. For amoving circular diaphragm sensor, the capacitance becomes

E rdrdb' (4)'= 5sd-u(r,b')where w is the deflection of the diaphragm.Using the deflection of a uniform thickness,circular diaphragm &om Equation ( I ) yields

solving the integral gives

which can be expand ed in a T aylor series to

(7)where CO is the undeflected capacitance givenby Equation 3 and WO s the center deflection ofthe diaphragm(i.e. w(r=O) from Equation (4).The capacitance with respect to appliedpressure, then is generally nonlinear due to thenonlinear deflected shape of the diaphragm.Thevalue of pressure sensitivity of the sensorfo r small values of measured pressure can becalculated from th e simplified formulaspresented in [8]. or pressure sensitivitywe canwrite

Pressure sensitivity depends on the membranethickness H, lectrode distance d at referencepressure and edge length 2R.The deflection of flat, clamped, circulardiaphragm is given approximatelyby (9) :

where P is the applied pressure, R is thediaphragm radius, h is the diaphragm thickness,Eis Young's Modulus, v is Poisson's ratio and WOis the center deflection of the diaphragm. Forcomparison pnrposes, the equivalent squarediaphragm deflection is given by [IO] :

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Pa' 4.20 WO 1.58 WO 'Eh' =Z ( T ) 1-,.G)I0 )-.where a is the half.sidelength .Thus a circulardiaphragm with a radius equal to the halfsidelength of a square diaphragm will be about

30% stiffer, for small deflections, and the non-linearity will be very similar, due to the nearlyequal ratios of linear to cubic coefficients. Not ethat in either case the non-linearity becomessignificant for deflections more than about 25% of the thichess of the diaphragm. This isgreat importance in the de sign of very sensitivemicro-machined diaphragms, particularly forcapacitive sensi ng applications.(ii) Corrugated Diaphrag mWith the introduction of corrugations into thediaphragm structure the situation can bechanged dramatically . For shallow, sinusoidalcorrugations the deflection is approximatelygiven by [IO] :

WO WO'E'h4 h h-_R4 -u,-+b,- ( 1 1 )where

b, = 1 6 5 k +0% +3 ) ) (13)q2(q+4xq +11X2q+1X3q+5 )

and

and for shallows, sinuso idal profiles :

with q the corrugation quality factor and H thecormgation depth. Thus q varies from 1, for aflat diaphragm, to a value that approaches 1.22times the ratio of corrugation depth todiaphragm thickn ess.

There is an ad ditional factor which needs to beconsidered depending on the method offabrication of the diaphragms. The ,m$itionalapproach has been to use,a heavily doped boronlayer as the etch stop used to form thediaphragm. It has-been shown [ I l l that'these . ':.etch sops introduce considerable residualtension in the resulting diaphragms. For largevalues of initial tension the deflection of a flatdiaphragm can he represented by :

PR' 4aRz WOEh' Eh' ( h ) (I6)-=- -

where (r is the initial s t ress . This resistance tobending due to initial stress can be added tothe terms given above using the principle ofsuperposition to g ive, for a flat diaphragm :PR4

(17)For a corrugated diao hram :i 2 o [ - b i ap h ']RZ h 2.83 4 RP = 4 - - u -+ -E - (18)- -The mechanical sensitivity of a circulardiauhramn is define d as

Therefore the mechanical sensitivity of thecormgated diaphragm with initial stress, forsmall deflections s given by" 2

(20)

111. THEORETICAL ANALYSISThe capacitance is a non linear function of thepressure since it varies inversely with w,diaphragm deflection. If the non-linearity iscalculated for sensors of two different plate radiiand identical pressure ranges, it i s apparent thatthe smaller device is much more linear but haslow sensitivity, while the opposite is for thelarger device. Fig. 3 illustrates the difference incapacitance values, sensitivity and non-linearitybetween a large (R= 240 pm ) and small (&I30pm) device. This discrepancy can be exploitedin a linearity calibration technique, as developedin [13]. From the observation of Fig.3, it can beconcluded that as the sensor radius is reduced,the pressu re sensor sensitivity for a givenpressure range decreases, due to the relativeincrease n stiffne ss of the membrane for a given

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pressure load. Th e tradeoff in non-linearity vssensitivity can be exploited in developingschemes taking advantage of bothf.4 R=240umg 1 .31

0.2430.241n8= 0.239.-0.2370.235

IO 20 30 40 50 60Pressure (psi )

R- 130 um

10 20 30 40 50 60Pressure (psi )

Fig.3(a) Comparison of capacitanceperformance of tw o different sensor sizes.Both devices have h =IOW, d =2 pm10 1 R = 130 um

10 20 30 40 50 60Pressure (psi )

f0.0164 R, = 13 0 um0.0162.-2 0.0156 1I .0154

0.01520.015

10 20 30 40 50 60Pressure (psi )Fig. 3@) Comparison of capacitance performanceof hvo different sensor sizes. Both devices have h=IOpm,d =2 pm

10 20 30 40 50 60Pressure (psi )R = 13 0 um (a)0.2425

0.2405j3 0.2385n 0.2365' 0.2345:'Z L

10 20 30 40 -50 60Pressure (psi )

Fig. 3(c) Comparisonof non-linearity performanceof hvo different sensor sizes.Both devices have h =IOW, d =2 F : a) Linear Capaciatnce an d (b)Capacitance.aspects, by utilizing both large and small devicesto contribute bigb sensitivity and low non-linearity respectively. Specifically, for a givenfull-scale pressure a small diameter sensor can beused for linearity while the large diameter sensorprovid es high resolution and sensitivity[l3]. Bysuitably manipulating signals from the twosensors, it may be possible to continuouslylinearize the output.As we can see from Fig. 4, for the range ofpressures applied, the deflection of the center ofthe diaphragm has been found to vary linearlywith pressures. This shows that the pressuresapplied fall within the proportional limit ofsilicon. The large diaphragm is found to deflectgreater compared to the smaller one.

0 2 4 6 8 1 0Pressure (kPa)

Fig.4 Pressure-load deflection of a diaphragm withvarious size :(a) 6 5 0 p @) 600p, c) 550 p,nd(d) 500p

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1CSE2002Proc.Z002,Penang,MalaysiaFig. 5 shows the calculated central deflectionsversus the corrugation numbers (N) of arectangular cormgated diaphragm. As can beseen that sm aller cormg ation numbers normally

result in larger central deflections under definiteinitial stress, while for large corrugationnumber m=lOmeans the whole diaphragmarea is occupied by the corrugations), verysmall static deflection can be expected .Thus for the purpose of reducing the staticdeflection (as is preferred in capacit ivesensors), cor mg atio ns occupying the wholediaphragm area are necessary.Fig. 6 shows the calculated results ofmechanical sensitivities for differentcormgation depths of square and circularcormeated dianhraems. The initial stress and. IYoung 's Modulus are assumed to beMp a and 2OOGpa respectively.

3.5 1

0 10 20 30 40 50 60 70 80Pressu re (kPa))

F i g 3 C alculated pressure-cenml deflectionc w e s vs cormgation numbers (N) for arectangular cormgated diaphragms (Young'sModulus E=200 Gpa, initial stress IT =70 Mpa, H=1.2pm, H= O pm). : a) N=2@) N =5 and (c)N= IO..

0.25$ 0.2 1 7

It can be seen that both diaphragms exhibit apeak at the corrugation depth,H =4 - p m atwhich the mechanical sensitivity reaches themaximum.. There are' no obvious peaks f or -Hlarger that 5-7 ' - k m. For 'the cnmigateddiaphragms with initial stress the mechanicalsensitivity is higher for small cormgation depthscompared to relatively large cormgation depth. Itwas found from the.calculated results that f0r.asquare cormgated diaphragm, the curve remainrelatively flat i.e ., the diaphragm mechanicalsensitivity is not sensitive to the largecormgation depths of at least IO p m.Fig.7 shows that the mechanical sensitivity of acorrugated diaphragm with two values of initialstress. It can he seen that the mechanicalsensitivity is increased when the initial stress isincreased for small corrugation depth. In thiscase it may be corrugated diaphragm with acormgation depth of at least 8p.

ta l.30.2-$ 0.1

3go'a.-m-0.1

-0.2m

-0.3 corrugation Depth (urn)Fig.7 The calculated mechanical sensitivity of a circularcormgated diaphragm versus corrugation depths (H)under different values of initial s t r ess : (a) Initial stress =10 Pa an d @) IO' Pa

0.3

3._e.-..*0.05 *....

3 5 7 10 12 15(a)2 4 6 8 1 0 1 2 q w l u e srn

Corrugation Depth (urn) Fig. 8Calculated relationshipof sensitivity versus q valuesCorrugated diaphragms with different q (2.65, 5,7.42, 9.89, 12.2, and 14.7) values have beeninvestigated. The results are shown in figure 8.Aswe can see, the larger value of q would producethe lower m echanical sensitivity.

Fig. 6 . Calculated mechanical sensitivities versuscormgation depth of different diaphragm : (a)'Omgated diaphragm and @) flatcormgated diaphragm.

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iV . CONCLUSION

The development of high-performancediaphragm structures i s of critical importancein the successful realization of micro-machined pressure sensors. Pressure sensorsfor different pressure ranges can be designedhy adjusting the parameters such as diaphragmthickness and cormgation depth. Thesensitivity of the diaphragm can be optimizedby choosing appropriate structure parameters.Th e q value (cormgation quality factor) is animportant parameter that determines themechanical sensitivity and the rigidity of thehack-plate. Optimal structures can be obtainedhy choosing a n appropriate q value.V. ACKNOWLEDGEMENTS

The authors would like to thank the ElectricalDepartment of University Malaya for theirsupport and contributions.VI.REFERENCES

[ I ] M. Giovanni,Flat and CorrugatedDiaphragm Design Handbook, MarcelDekker, Inc., New Y ork, 1982.[2] I. Jerman,The Fabrication and Use ofMicro-machined Cormgated SiliconDiaphragms, Sensors and Actuators, A23(1990) 988.[3] K. E. Peterson,Silicon as a mechanicalmaterial, Proceedings o the IEEE pp.420-457 (M q 1982)[4 ] X . Ding, W. KO nd J. Mansour,Residualstress and mechanical properties of BoronDoped P - Silic on Films, Sensor andActuators, A23 (1990) 866.[ 5 ] X. Ding, W. Ko,Buckling Behavior ofBoron Doped PI - Silicon Diaphragms,Transducers 91, Digest of Technical Papers,pp. 201.[6] F. Maseeh and S . Senturia,PlasticDeformation of highly Doped Silicon,Sensors and Actuators, A23 (1990) 861.[7] S. P. Timoshenko, Theory of Plates AndShells New York: M c Cra w Hill, 1959.[8] Husak,M, On-Chip Integrated ResonanceCircuit with the Capacitive PressureSensorJonma1 of Micromechanics and Micro-engineering 7(1997),pp. 173.178.[9] Bert, C.W. and Martindale, J.L., AnAccurate, Simplified Method for AnalyzingThin Plates Undergoing Large Deflections,J.AIAA, vol. 26, n. 2,Feb. 1998, p.235.

[IO] Haringx, J.A., Design of CorrugatedDiaphragms, ASME Transactions, vol. 79,1957 ,~ . 5.[ l ] Hin-Leung Chau and Wise, loc. Cit.[I21 J. A. Llyod, Phd theses& IntegratedCircuit Pressure Sensing System with AdaptiveLinearity Calibration., Massachussets instituteofTechnology, Jun 1997

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