Micro resonant force gauges - ris.utwente.nl

Sensors and Actuators A, 30 (1992) 35-53 35

Micro resonant force gauges

Harrle A C Tllmans, Mlko Elwenspoek and Jan H J Flmtman MESA Research Imtrtute, Unrversrty of Twente, P 0 Box 217, 75&I AE Enschede (Netherlamb)

(Remved September 5, 1991, accepted September 27, 1991)

Abstract

A review of micro resonant force gauges IS presented A theoretlcal descnptlon IS aven of gauges operahng m a flexural mode of vlbratlon, mcludmg a dlscusslon of non-hnear effects Gauge factor and quahty factor are defined and their relevance IS dIscussed Performance Issues such as sensltlvlty, stablhty and resolution are addressed Design aspects, mcludmg the means for excltahon and detection of the vlbratron, and examples of slhcon mlcrofabncatlon technolo@es are described

Introduction

A nucro resonant force gauge comprises a (small) vlbratmg mechamcal element, 1 e , the resonator, which converts an externally apphed (ax- lal) force mto a shift of the resonance frequency of the element It 1s used as a sensing device for measurmg quantities such as pressure, force (weight), mass flow, temperature or acceleration The term micro refers to the method of reahzatlon, I e , (sthcon) mlcrofabncatlon technologes Reso- nant force gauges can be used directly as sensors to convert the unknown applied force mto a frequency-stift output They can also form the heart of a sensor, for Instance, be apphed as strain gauges, where the measurand 1s first converted mto a stram, which 1s subsequently converted mto a frequency-shrft of the resonant gauge Generally spealung, rmcro resonant force gauges belong to the class of resonant sensors [ 1,2] Resonant sensors are very attractive in the precision measurement field, because of then high sensltlvlty and resolution, and further because of their semi-dig- tal output, 1 e , a frequency shft, which ehmmates problems such as intensity fluctuations associated with analog signals

In general, the resonance frequency of a mechamcal structure IS affected by a perturbation of tts potential or kmetlc energy [3] Potential energy perturbations can be caused by mechamcally strammg the resonator or by changmg the shape,

and thus the stiffness, of the resonator In a resonant force gauge, the frequency shift is a result of potential energy perturbations caused by mechamcal stram To avold error readmgs, kmetlc energy perturbations, e g , caused by a change of the resonator mass or by changes of the den&y of the surrounding medmm, should be ehmmated This can be done by hermetically seahng the resonator m an evacuated or inert cavity to provide a stable environment

Although a resonant force gauge can have a vanety of shapes and vibrate m various modes, the gauges considered m more detail m this paper are restncted to flexurally vlbratmg elements that are mounted at two end pomts Special attention 1s paid to pnsmatlc homogeneous beams that are clamped at both ends Analytlcal expressions are reviewed for the resonance frequencies of the beam, mcludmg a quantltatlve large-deflection analysis As a measure for the sensltlvlty of the gauge, gauge factors are defined and expressed m terms of dnnenslons and residual stress Further, the mechanisms of energy losses are discussed and ways of reducmg them are indicated A discussion of the vanous means of excltatlon and detection of the vibration 1s also included Fmally, slhcon fabncatlon technologes are bnefly reviewed and state of the art examples are gven on how to fabncate sealed resonators For more mformatlon regarding apphcatlons and other aspects of resonant sensors, reference 1s made to a number of review articles

0924-4247/92/$5 00 @ 1992 - Elsewer Sequora All nghts resewed

wrltten in recent years [l-6] Specifics of quartz resonating sensors are not a sub@ of this paper, but can be found m the hterature [7-91

Resonance frequencies

For an mltlally flat prismatic homogeneous beam, subJected to a (tensile) axial force N, the equation of motion governing the transverse deflection w(x, t) as a function of the posltlon x along the beam length and the time t 1s gven by the followmg linear partial differential equation [lo]

gz a4wcx, 0 _ N fwx, 0 ah+ t) ax4 ax2 +PA at2 = 4(x, 0

(1) where I? and p are the ‘effective’ Young’s modulus [ 1 l] and specific mass of the beam material, A and Z the cross-sectional area and second moment of inertia, respectively, and q(x, t) mQcates the dnvmg force per unit length To find the elgen- functions or natural modes together with the cor- responding elgenvalues or natural frequencies, the homogeneous part of eqn (1) 1s solved by employmg a separation of vanables approach m the form w(x, t) = 4(x) exp(Jot) Or, m other words, it 1s assumed that the beam IS osclllatmg at an angular frequency w with a mode shape C/I(X) For a clamped-clamped beam (~(0, t) = w(l, t) = 0 and ~‘(0, t) = w’(l, r) = 0, where 1 IS the length of the beam), non-trivial solutions are obtained if the followmg condition, 1 e , the charactenstlc equation, is satisfied

smh@l) sm(ll) = 1

where

AI = kl[(a2 + 1) “2 - a] ‘j2

@=kl[(a2+ l)l’2+,]“2

(2)

(3a)

(3b)

a = N/2i?Zk2 (44

k4 = pAw’/i?Z (4b)

This yields the followmg expression for the mode shape functions

cos(M) - coshbl) 4(x) = cos(lx) - cosh@x) + ,

i smh(pl) - sin(X)

x sm(lx) - i smh(px) >

The characterlstlc eqn (2) IS satisfied for a du- Crete set of solutions yielding a discrete number of values for k, le, k,, for n = 1,2, The correspondmg natural frequencies or elgenfre- quencles 0, are found from eqn (4) and the associated mode shapes & from eqn (5) Because of the transcendental nature of the charactenstlc equation, a closed-form expression for the natural frequencies as a function of the axial load N and the beam parameters cannot be obtained Ap- proximate solutions can be found, for example, by employmg Raylelgh’s energy method [ 121 The followmg expresslon for the natural frequencies can be obtained this way

where

(6)

(7)

and the coefficients a, and yn can be found from

0 0

where C&(X) represents an approximate shape func- tlon for a particular mode n When &(x) corre- sponds to the exact shape function, eqn (6) yields the exact value for the natural frequency, otherwlse the result of eqn (6) w111 always be an upper bound of the natural frequency [ 121 A good approxlma- tton for the natural frequencies of a beam subjected to a non-zero axial load can be obtained If the shape functions of the same beam for zero apphed axial load are substituted for q,,(x) For a clamped-clamped beam, these functions are given by eqn (5) for ;1 = p = k, where k IS determmed from eqn (2), wluch for N = 0 reduces to cos(kl) cosh(kl) = 1 For the fundamental mode tlus yields ~1, = 4 730 and y, = 0 295, and for the first harmomc, a2 = 7 853 and y2 = 0 145 Tlus IS

31

In this case, addltlonal terms have to be included m the equation of motion [ 13,171 The effect of an elastic support on the resonance frequencies was discussed by Bouwstra and GelJselaers [ 131, and the analysis showed that the support forms a reasonable stiffness against bending, but not against axial elongations The latter implies that, m the case of an elastic support, the frequency shifts are smaller than predicted by the equations

2 0 2 4 6 above Torsional modes were discussed by Geyse- Rg 1 Normahzed angular resonance frequency of the fundamental mode and the first harmomc as a function of the normahzed axial

laers and Tijdeman [ 181 The effects of damping

force for an mrtlally flat pnsmatx clamped-clamped beam, see also terms and non-linear (large deflection) deforma-

eqn (6) tlon are not accounted for here, but will be bnefly discussed below

graphically illustrated m Fig 1 The error m the approxlmatlon for the natural frequency 1s smaller than 0 5% if yn Ni2/12& < 1 [ 131 An em- plncal correction m the form of an additional quadratic term m the axial force under the square root has been proposed by van Mullem et al [I41

In mlcromechamcal devices, very often beams with a rectangular cross section and a large width (b > 5h [ 151) are used and eqns (6) -( 7) can con- vemently be wntten as

x $ 1 f&(1-vZ) h (

1 2 112

0) (9)

where v and E are Poisson’s ratio and Young’s modulus of the beam matenal, respectively, and b and h are the wdth and thickness of the beam, respectively

In the model described above, effects due to shear deformation and rotary inertia are not taken mto account These effects are neghgble for slender beams, 1 e , beams with a large length/ thickness ratlo and for small values of the mode number n [ 12, 161 Besides the shift m the resonance frequency as a result of the induced axial force, the frequency also changes shghtly as a result of changes m the dlmenslons of the beam The latter effect, however, 1s small compared to the effect of the induced axial force (see also Section on gauge factors) Further, the model does not apply for beams wltb a non-zero lmtlal deflectlon, e g , beyond the Euler buckling point

Force, stress or strain?

The resonance frequencies of doubly supported beams respond to the apphed axial force N Ths IS easdy understood from an inspection of the equation of motion of the beam, eqn (1) The frequency dependence 1s expressed by eqn (6) If the beam IS used as a measurmg device it can therefore be designated as a ‘resonant force gauge’ An lllustratlve example IS the vlbratmg string of Wyman [ 191, where the external force, e g , a weight, comprises the axial force responsl- ble for the frequency shift of the strmg The force m this case IS the independent vanable, which Induces an axial elongation of the strmg If the cross-sectional area A does not change under the apphed load, the gauge could also be designated as a ‘resonant stress gauge’, since the axial stress IT m ths case 1s directly related to the axial force N o=N/A

In many apphcatlons, however, the axial force 1s a result of a displacement of the end points or elongation of the gauge, e g , see [20-291 Here, the Induced axial force IS directly related to the elongation AI or the stram Al/l of the gauge N = EA( Al/l), where 11s the length of the gauge The gauge truly senses a deformation of the supportmg structure and 1s therefore referred to as a ‘resonant elongation gauge’ or ‘resonant strain gauge’ to emphasize the measurmg prmaple and the ongm of the axial force It also hnks devices employmg resonant strain gauges to devices employmg the more conventional (piezo)resistive strain gauges In these apphcatlons, the gauges are embedded mto a supportmg structure, the so-called ‘load cell’, e g ,

38

a diaphragm [24], or a supporting frame [ 14,261, which converts the externally applied load, e g , a pressure, mto a strain m the gauge The contnbutlon of the stiffness of the gauge to the overall stiffness of the device 1s generally neghglble

Gauge factors

The gauge factor of a resonant gauge provides a measure of the gauge to the apphed axial load The gauge factor GNn, desmbmg the senatlvlty of the gauge, vlbratmg at o = CD,, to changes m the axial force N, and m the vlclmty of the operating point N = N,,, caused by the blaxlal residual stress o. or residual stram E,,, IS defined by

G = &?% Nn - [ 1 co,, aN (10) N=N~=(I-v)o+4=EAeO

or, for a (wide) beam with a rectangular cross sectlon (see eqn (9))

GN,, =i (11)

If the axial force 1s a result of an axial elongation of the beam, it is more useful to define a gauge factor G,, descnbmg the senslttvlty of mode II to changes m the strain E m the vlcmlty of the operating point e0 = (1 - v)oO/E

or using eqn (9) with N/EM = E,

(12)

G_=; (13)

If the effects of geometrical deformations are taken mto account I’ = I( 1 + E) and h’ = h( 1 - w), the gauge factor would be given by

G;, = -(2+v) +$(l +2Eo(l +“))

X r 1

0 2 m(l -v2)

h

1 +

Y”ECl(l - v2) 0 1 2 I (14) ;

The gauge factors G, and Gh are approximately the same For zero residual strain (Q, = 0), v = 0 3 and an aspect ratio l/h = 200, they differ by 0 04% Or m other words, the shift m the resonance frequency as a result of geometrical deformation of the gauges 1s small compared to the shift caused by the axial stlffemng This has an important consequence for the behavlour of resonant gauges that are SubJect to creep If creep occurs under a constant state of stress, for mstance as m the expenment described by Bethe et al [30], the resonance frequency will only be slightly affected On the other hand, d the input load of the gauge 1s an elongation, creep ~11 senously hrmt the performance of the device Also a POSSI- ble relaxation of the residual stress durmg operation will cause error readings Study of the relaxation of residual stress m thm films used for the fabrlcatlon of nucroresonators should therefore be an important topic of future research

Large gauge factors are achieved at low residual strain/stress levels and/or for slender beams Also, the gauge factor decreases with mcreasmg mode number This implies a larger contnbutlon of geometrical deformations to the shift m the resonance frequency for the lugher-order modes For the fundamental mode, the gauge factor gven by eqn (13) 1s graphically shown m Fig 2

Quality factor

The mechamcal quahty factor Q IS a measure of the energy losses of the resonator or, m other

Rg 2 Gauge factor G,, of the fundamental mode vs residual tenale stram sa for a clamped-clamped beam wth a rectangular cross section for three values of the slenderness ratlo I/h and v = 0 3, see also eqn (13)

39

words, a measure of the mechamcal dampmg The Q-factor 1s defined as

Q ~271 maxlmum energy stored m one penod

dissipated energy per penod

(15)

Low energy losses imply a high Q-factor The Q-factor cannot be determmed directly, but m- stead can be deduced from the response character- Istics of the resonator One common method of deterrmmng Q 1s from the steady-state frequency response plot of a resonator excited by a harmomc force with constant amplitude

Qg-F?!!L

AW-3 dB (16)

where w,, 1s the resonance frequency, defined as the frequency of maxlmum amphtude response, and Aw_~~~ 1s the half-power bandwidth of the frequency response Equation (16) mdlcates that Q 1s a measure of the sharpness or the frequency selectivity of the resonator A tigh Q-factor means a sharp resonance peak A high Q-factor for a resonant force gauge has several advantages The energy reqmred to mamtam osclllatlon 1s kept low, which reduces heat generation and opens up the posslblhty of powering the resonant gauges from alternative sources, e g , solar energy Further, a high Q means a good frequency stability and the effects on the osclllatlon frequency of the electronic circuitry used to sustain the osclllatlon are mmmuzed

Several mechanisms of energy loss can be lden- tlfied (a) losses mto the surrounding (fled) medium, caused by acoustic radiation and viscous drag, (b) losses mto the mount used to support the resonator due to motion of the mount, and (c) mtrmslc dampmg caused by energy losses mslde the material of the resonator The energy loss of each loss mechamsm separately can be described by a correspondmg quahty factor Q,, and the overall quahty factor Qtot can be found from

&=ci (17)

It 1s obvious from eqn (17) that Qtot cannot exceed the value of the smallest Ql The discussion below mainly deals with vlbratmg beams Quality factors of vlbratmg diaphragms are discussed else- where [31]

The losses mto the surrounding fled are due to viscous damping and/or ra&ation of sound waves propagating m a direction normal to the surface The latter effect will only be significant if the acoustic wavelength becomes equal to or less than a typical dunenslon of the resonator [32,33] Acoustic radiation can generally be ignored for mrcromechamcal resonators Viscous damping 1s characterized by two contrrbutions [ 34-361 One 1s due to the usual Stokes drag force for a body m umform motion through a viscous fluid The second (dynanuc) part, the so-called shear wave effect, IS characterized by a boundary layer around the vlbratmg structure and 1s dependent on the frequency of nbration w, and further on the den- slty p0 and viscosity q of the fluid medium The quality factor Qvlscous caused by viscous dampmg can be expressed as [37]

(18)

where c 1s the coefficient of the dlsslpatlve part of the drag force, R represents the relevant linear dlmenslon and 6 = ( 2q/p0w) ‘/* 1s the charactens- tic wdth of the boundary layer If S 1s much larger than the linear dunenslons of the resonator, the shear-wave effects can be ignored and c 1s a gven constant, dependent only on rl Qvmous= @o/c(q), stating that m ths case the quality factor IS linearly proportional to the frequency of vibration The VISCOUS quality factor of a wide beam with a rectangular cross sectlou and for zero applied axial force can then be expressed as (see also eqn (9))

(6 @RR) (19)

From the above expression it IS seen that the VISCOUS quahty factor 1s mversely proportional to the aspect ratio (l/h) of the beam Assummg c(q) w 24~ for a pnsmatlc beam with a rectangular cross section [38], a viscous quahty factor m air (q = 1 8 x lop5 N s/m”) of approximately 30 1s found for the fundamental mode of a clamped- clamped silicon beam 1 pm wide with an aspect ratlo of 100 If shear-wave effects cannot be ignored, the quality factor ~11 be lower and ~11 become du-ectly proportional to o ‘I2 This dependence has

been expenmentaliy observed by Lammermk et al [39] and Blom et al [ 371

In the dlscusslon above it was assumed that the resonator 1s isolated m space If the flexurally vibrating resonator 1s close to another stationary surface, damping forces increase due to a pressure built-up m the intervening space, I e , the (air) gap [ 38,40,41] This so-called squeeze-film damping effect becomes slgmficant f the gap spacing ap- proaches or becomes less than the width of the vibrating beam Whereas the viscous quality factor would be of the order of 100-2000 for an isolated beam, it rapidly drops below 1 if the gap spacing 1s too small TINS has been expenmentally observed by Howe and Muller [41] Squeeze-film damping can be slightly reduced by makmg ventl- latlon holes m the vibrating beam [41,42] or prac- tlcally ehmmated by placing the beam m an evacuated cavity [24,27,43]

The energy coupled mto the supporting structure can be mmlmlzed by mechamcally lsolatmg the resonator from the mount [ 1,7] An example of a geometry providing a means of decouphng a flexurally vlbratmg single beam from the support 1s an lsolahon system m between the beam and the support This consists of a large mass and an isolator beam, acting as a soft sprmg-mass system with a natural frequency much lower than the frequency of the beam and of the support [44,45] In this design, moment and shear reactlons are isolated from the support, resultmg m a reduction of energy losses The mass has to be large to give it enough rotary Inertia to form an efficient clamp- mg edge for the resonatmg beam A decouphng method for a torsional resonator was reported by Buser and de ROOIJ [46] In their structure, a frame was designed around the resonator which itself can vibrate m a torsional mode Again, the aim of the design was to have a high ratio of the resonance frequencies of the resonator and the mounting frame m order to mmlmlze the energy losses Quahty factors m vacuum as high as 600 000 were measured for a structure made out of angle-crystalline slhcon Another way of lowermg the energy losses mto the support 1s to use a specific resonator design A well-known example 1s the double-ended tuning fork (DETF), conastmg of two beams vlbratmg 180” out of phase, thereby cancelhng moment reactions at the beam roots and resulting m a reduction of energy losses, see Fig 3(a) [7,47] The tnple beam structure

(a)

(b) Fig 3 (a) Double-ended tunmg fork [471, (b) tnple beam structure [481

provides yet another way of cancelhng moments and shear forces at the clamped ends, see Fig 3(b) [48,49] It consists of three beams, the rmddle beam being twice the width of the outer beams The middle beam and the two outer beams vibrate 180” out of phase, resultmg m the desired cancella- tion of moments and shear forces Tlus structure IS very attractive from a fabncatlonal point of view It can easily be fabncated with planar nucroma- clumng technologes and the excltatlon/detectlon of the flexural transverse (perpendicular to the substrate) vibrations does not mean any more dlfficultles than for the angle-beam approach This 1s m contrast to the DETF, which reqmres m- plane exatatlon/detectlon schemes Another structure, conststmg of a balanced dual-haphragm, was reported by Stemme and Stemme [42] Here, a balanced torsional mode of vlbratlon 1s excited, where energy losses mto the mount are small because of the stationary centre of mass and because the vlbratmg element 1s suspended at the nodal lines, which numn-uzes the movement of the mount Balanced structures were revlewed by Stemme [ 51

Besides the external sources of energy losses described above, there still remains a large number of mechanisms through which vIbrationa energy can be dissipated wlthm the matenal that 1s cych- tally deformed They include magnetic effects

42

The dependence of the resonance frequency on the amplitude of vlbratton given by eqn (23) IS slrmlar m form to the frequency dependence of a simple spring-mass system with a restoring force havmg a cubic dependence on the vlbratlon amplitude An estimate for the coefficients /?, and b2 of the fundamental mode and first harmonic, respectively, can be found by substltutmg the mode shapes for zero axial force mto eqn (22) This yields /i’, = 0 528 and pZ = 1 228 for a prismatic clamped-clamped beam with a rectangular cross sectlon The coefficient Bn depends on the edge condltlons of the beam, e g , for simply supported edges &, will be higher In the case of movable axial end supports, the hardening effect will be smaller and can even revert to a softening effect for beams with a sufficiently small aspect ratio Z/h

[57] The dependence of the resonance frequency on the vibrational amplitude of the fundamental mode of a rlgldly clamped-clamped beam, together with a typlcal large-amphtude forced response plot, 1s shown m Fig 4 This Figure also mdlcates the effect of the magnitude of the dnvmg force on the response plot [ 6 l] If the magnl- tude of the driving force exceeds a crltlcal value F cr,, 9 the amphtude becomes three-valued within a range of frequencies This frequency interval defines the regon of hysteresis, with two stable points, one at a large amplitude and the other at a small amphtude To avoid hysteresis, the

16r

Frequency Ratlo o IO t(O,O)

Fig 4 Forced response plots, showmg the normahzed amphtude vs the normabzed frequency for a clamped-clamped beam with a rectangular cross s&Ion, lllustratmg the effect of a large vIbratIona amplitude If the magmtude Fof the excltatlon force exceeds a cntlcal value FCr,, . the amphtude becomes three-valued for a partrcular frequency Interval The dashed part of the curve m this Interval mdlcates unstable pomts [61]

magnitude of the dnvmg force should be smaller than the cntlcal load It can be shown that the critical load 1s inversely proportional to Q3’2, where Q IS the quality factor of the resonator [61] At the cntlcal load, the amplitude of vlbratron 1s defined as the crltlcal amplitude w,,,~~_.,~, being inversely proportional to Q ‘I2 Consldermg a clamped-clamped beam to behave as a simple sprmg-mass system with a restormg force having a cubic dependence on the amplitude, it can be shown [61] that the cntlcal amphtude can be ap- proximated by

wnaxcntlh = MQMl - v~))I”~ Hence, even though a high Q has several advantages as indicated m the previous Sectlon, it en- hances the chance of hysteresis occurrmg

Noise or other mstablhtles of the vlbratlonal amphtude will hmlt the ultimate frequency resolution or stability of the gauge Any frequency change caused by an amplitude change must be small compared to the mmlmum frequency change one wishes to resolve It can be denved from eqns (23) and ( 12) that a condltlon, which defines an upper limit for the vanatlons Aw,,, m the amphtude, 1s given by

(2) * (&)(&)‘(?),,. (w,,, < wmax .,,I (24)

where ( Aw /o),,, 1s the rmmmum resolution re- quu-ed For a clamped-clamped beam with an aspect ratlo l/h = 200, I/w,, = 105, zero residual strain (Ed = 0) and a rmmmum frequency resolution of 10P6, eqn (24) demands a relative vana- tlon of the maxlmum amplitude much smaller than 0 5 If the beam thckness h = 1 pm, this means that w,, = 2 nm and Aw,,, d 0 5 nm The critical amphtude m this case 1s approximately equal to 20 nm for Q = 10 000

Excitation and detection of the vibration

To measure the resonance frequency of a vlbratmg beam, it must first be excited mto vibration and subsequently the vibrational motion must be detected Mathematically, the dnvmg load q(x, t) used to excite the resonator can be accounted for m the mhomogeneous part of the dlfferenttal equation of motion, as Indicated by eqn (1) In

43

Fig 5 SchematIc representation of excltatlon loads

general, for flexurally vibrating structures the load q(x, r) represents (harmomcally varying) bending moments or transverse forces

4(x, 0 = 4(x) exp( W = [. C q1 (x) + C <6(x - x, >

J

+ 2 MkS_,(x -xk) 1 exp(iwt) (25) k

where q,(x) IS a distributed force per unit length, F, is a point force applied at x = xJ, Mk is a point InOIYIent applied at x = xk, 0 1s the freqUenCy Of

excitation and 6_, 1s the derlvatlve of the Dlrac function (see also Fig 5)

An expression for the overall response can be obtained from a modal analysis [62-641 The dls- tnbutlon of the drlvmg load along the beam length determines the efficiency of excitation of a partlcu- lar mode of vlbratlon For instance, if the driving load 1s symmetric with respect to the centre of the beam, it 1s (theoretically) impossible to excite an anti-symmetric mode, e g , the first harmonic (n = 2) A measure for the efficiency of excitation of a particular mode 1s the generalized load P,, given by the inner product of the shape function &(x) of mode n and of the dnvmg load q(x)

[@I

A large value for P, implies a large contrlbutlon of mode II to the overall response of the resonator

Several excltatlon/detectlon schemes have been described m the literature

electrostatic excltatlon/capacltlve detection [21, 25,41,42,65-671 magnetic excitation/magnetic detection [ 19,24, 681 plezoelectrlc/plezoelectrlc [ 14, 26,69, 701 electrothermal/plezoreslsttve [ 49,71-741 optothermal/opfical [23,75-821 dlelectnc/capacltlve [ 831

For the first two methods the driving load can be modelled as a distributed transverse force q!(x) and the remaining four methods are modelled as a number of point moments M,d- I (x - xk) Sina a dlscusslon of the various means of excitation and detection has been given m the recent literature [5, 79,841, only a bnef summary w111 be given here

In the case of electrosfatzc excltatron, the dnvmg load 1s simply the attractive force between the two plates or electrodes of a capacitor One electrode 1s formed by (a part of) the beam and the other by a (stationary) surface located at a close distance from the beam Capacztroe detection IS based on the fact that an a c current will flow through a d c -biased capacitor if the distance between the capacitor plates, and therefore also the capacitance, fluctuates This scheme 1s very attractive from a technologcal point of new, fabrication of mlcromechamcal resonators and m- tegratlon with electronics 1s relatively easy [41,66,671 However, if the electrical feedthrough and/or parasltlc loads are too large, these effects will obscure the detection signal of the mechamcal resonance As a consequence, on-chip buffer- mg and/or amphficatlon 1s generally necessary Furthermore, for proper operation, a d c polar- ization voltage 1s required This voltage will reduce the dynamic stiffness of the structure and wdl therefore result m a lowering of the resonance frequency [65,85] Also, mstablhty will oc- cur if the polarlzatlon voltage exceeds the so-called pull-m voltage [65, 851 For preclslon measurements, the dependence of the resonance frequency on the polarlzatlon voltage and the m- stab&y problem require very stable voltages and amplitudes of vibration

The Lorentz force experienced by a current- carrying wire m the presence of a magnetic field forms the driving force for a magnetically exczted resonator Due to a change m area of the current loop, the magnetic flux passing through the loop will change if the beam vibrates This will induce a voltage m the loop, providing the detection signal Fabrication 1s relatively ample, but problems arise with packaging A magnet needs to be posltloned close to the resonator Another way to develop the static magnetic field 1s by means of an electrical current flowmg through a wire (or cod) that IS m close proximity to the resonator, e g , a design employing a double-ended tuning

44

fork [68] Problems anse with heat generation, resulting m a temperature difference between the beam and the support, which will induce compressive axial forces and thus results m a lowering of the resonance frequency [ 13,731

Plezoelectrlc excuatron and plezoelectrlc detectzon are based on the inverse plezoelectrlc effect and the direct plezoelectrlc effect, respectively [ 861 The inverse plezoelectnc effect manifests itself by a mechanical deformation m the material subjected to an electric field If the piezoelement 1s rigidly fixed on top of the beam, a bendmg moment will be developed which 1s used to excite the beam Conversely, through the direct plezoelectnc effect a mechanical stress or stram will generate a dlelectrlc displacement m the material, which can be detected as an electric current m an external circuit This method of excltatlon/detectlon has been widely used for quartz resonators, smce quartz itself 1s plezoelectrlc [7] Thm films of plezoelectnc materials such as zmc oxide (ZnO) or alummlum mtrlde (AlN) are very attractive for use m &con-based resonators ZnO has been investigated extensively for apphcatlons as an actuation material [87-911 Due to its strong piezoelectric coupling efficiency, it is a very attractive material for excltatlon and detection of the vibrational motion of a resonator A problem, however, 1s created because of the high d c con- ductivity, which prevents the build-up of an electric field Apphcatlons at low frequencies ( < 1 MHz) therefore require special configura- tions, such as the use of insulating layers or a depleted MOS-like structure [90], which make the fabrication process more complicated and the composite resonator more susceptible to temperature fluctuations as a result of differential thermal expansion effects Furthermore, ZnO 1s not an IC-compatible material and special precautions such as encapsulation of the ZnO layer m slhcon nitride layers [92] are necessary to avold zmc contammatlon and to protect the ZnO layer against the different etchants used Also, a lot of effort 1s necessary to study the reproduclblhty and the effects of heat treatment, humidity and deposl- bon condltrons on the mechamcal and electrical properties of the ZnO film

Electrothermal excltatton IS based on the thermal expansion of a reslstlve material due to heat generation by an electrical current If the resistor IS located m the upper fibres of the beam, a

temperature gradient m the thickness direction will develop This m turn creates a bending moment, which 1s used to excite the beam The motion 1s detected by plezoreslstors, made of doped poly- crystallme slhcon for example This method 1s very attractive from a technological point of view Standard IC processmg technologies can be used to fabncate the resonator An inherent problem IS the static heat generation causing compressive axial forces, which ~111 affect the resonance frequency [ 13,731 For precision measurements ths requires a good control of the heat flow to the surroundings

The basic pnnclple of optothermal excttatlon IS the same as for electrothermal excltatlon The difference 1s the heat source, winch 1s now formed by the absorption of light by bare slhcon or an absorbing layer, e g , alummmm [93] Several techniques are used for optical detection, such as mten- slty modulation by means of a shutter controlled by the resonator [81], use of an optical proximity or displacement sensor [94], or mterferometrlc techniques [23, 58,78, 801 Optical sensors are very attractive for apphcatlons where there 1s no reh- able electrical solution, such as under extreme environmental or hazardous condltlons Fabnca- tlon of the resonator 1s relatively simple, but the integration with the optical (fibre) system makes the reahzatlon more complicated Further, there 1s a frequency dependence on the optical dnve power as a result of thermally induced axial forces [73,951

Dlelectrrc excltatlon IS based on the lateral deformation of a dlelectnc thm film, sandwiched between a top and bottom electrode, due to an electrostatic force ansmg if a voltage 1s applied across the electrodes The lateral deformation will cause bending moments that are used to dnve the structure The detectzon IS capacitive, based on the change of the capacitance of a dielectric capacitor If the dlelectnc 1s deformed So far, the method seems promlsmg, but the signals appear to be extremely small Only if materials with a very high permlttlvlty, such as PZT, having a reported per- mltfivlty as large as 2500 for a thm film [96], are used 1s this method interesting for future apphcations

It 1s clear that none of the exatatlon/detection methods described above 1s perfect They all have certain advantages and disadvantages The choice of a particular method 1s determmed by several

45

crltena first of all, the kmd and number of mater+ als that are required for fabtlcatlon In this respect, conslderatlons concermng aging of material properties, fatigue, yield strength, stress relaxation, etc , and differential thermal expansion effects are important A homogeneous structure, preferably consisting of a single-crystalline matenal, seems to be Ideal Another relevant aspect IS the slgnal- to-noise ratio A big problem 1s caused by noise and other interference signals (e g , electrlcal feed-through), which can obscure the mechamcal resonance This also requires special attention to the overall design of the structure, mcludmg shielding, mmlmlzatlon of parasltlc loads, on-chip electromcs for buffering, etc An efficient transduction mechanism, e g , using ZnO, 1s preferred to obtam a large signal level Another constraint 1s formed by the complexity of the fabrlcatlon process with respect to cost, yield and through-put, and further by the technology that 1s available, or whether It can be used m an IC environment or not

Fabrication technology

The technology described m this Sectlon refers to the nucromachmmg of mhcon, &con-based matenals, metals and special transducer materials such as ZnO A global overview will be given and for processmg detads, reference IS made to the literature

Several technologies have been developed over the past three decades to fabricate mlcromechamcal resonators In general, two mam technologies can be dlstmgmshed bulk mlcromachmmg [97- 99] and surface mlcromachmmg [ lOO- 1021 Com- binations of both have also been used

In the case of bulk machmmg, a piece of &con 1s sculptured mto a three-dlmenslonal structure usmg sophisticated etching techniques such as an- lsotroplc etching [ 103-1071, high boron etch stops [ 105, 1081, electrochemical etching [ 109- 1141 or HF anodlc etching [ 115, 1161 Examples of resonating structures built this way are resonating pressure sensors [21,23,28,29,72,81,94], a resonating force sensor [ 14,261, resonatmg accelerometers [49,80, 1171, a resonating mass-flow sensor [ 118, 1191 and a vibration sensor [ 1201

Surface mlcromachmmg 1s generally associated with the deposition of thm films and sacrlficlal layer

etching techniques [ 102, 12 I] The first prototype of a resonator fabricated this way was demonstrated by Nathanson et al m 1967 [65] They used metal films for both the construction material and the sacrlficlal layer More recently, ‘construction matenal/sacnfiaal layer’ combmatlons of polyslhcon/ oxide [41,43, 66, 122, 123, 1241, polyahcon/mtnde [124], mtnde/(poly)sdlcon [118, 119, 12551271, oxlde/sdlcon [ 1281 and polylmlde/alummmm [ 1291 have been used Prototypes of resonating structures include a resonant vapour sensor [41], a resonatmg mass-flow sensor [ 1181, a resonant force sensor [25,60], a resonating accelerometer [ 1171 and resonating polyslhcon mlcrobrldges [ 41,43,66, 1231

As mentioned before, resonant sensors are very attractive m the precision measurement field However, m order to exploit the advantageous properties, unwanted disturbances have to be ehmmated and a high Q-factor must be obtained This requires vacuum encapsulation An elegant way of accomphshmg this was first demonstrated by Ikeda et al m 1988 [24, 1301 Figure 6 shows the basic structure of a differential pressure sensor using a sealed resonator as the strain-sensing element This new concept requires an extended defi- nition of the gauge The term resonant force/strain gauge not only includes the vibrating element itself, but also the evacuated cavity and the shell surrounding the cavity The shell forms an integral part of a resonant force gauge

The fabrication process of sealed resonators as described by Ikeda et al 1s based on selective epltaxlal growth of boron-doped smgle-crystalline slhcon [ 1311, high boron etch stops and selective electrochemical amsotroplc etching Very heavily boron-doped (p”) sthcon layers are used as the construction material and heavily boron-doped (p’) &con layers as the sacrificial layer The p++ layers are not attacked by the amsotroplc etchant

Fig 6 Example of a mechamcal sensor conslstmg of a diaphragm wth an encapsulated budt-rn resonator that can be used as a dtfferentlal pressure sensor

CAP RESONATOR EVACUATED

\ \

Cc) Fig 7 Fabncatlon process of sealed resonant force gauges usmg selectwe epltaxy [130] (a) After oxldatlon, pattermng and HCl etching of sthcon, (b) after subsequent selective epltaxml growth of p+, p++, p+ and p++ slhcon, (c) after removmg the oxide, selectwe etchmg of p+ slhcon, reactive sealing with n-epi and anneabng m N,

(a mixture of hydrazme and water) due to the high boron concentration [ 1081 The n-type slhcon substrate 1s protected from the etchant through electrochemical passlvatlon The cavity 1s sealed by closing off the etch channels with n-epl Sub- sequent high-temperature annealing m a nitrogen atmosphere produces a vacuum cavity ( < 1 mTorr) by out-diffusion of the residual hydrogen through the slhcon cavtty walls The basic steps of the process are illustrated m Ftg 7(a) -(c)

An alternative way of fabrrcatmg sealed resonators has been described by Guckel et al [43] Their process 1s based on sacnficlal layer etchmg and reactive sealing techniques [ 1321 In this way a hermetically sealed cavity can be formed with a low residual pressure, as indicated by the obtamed quality factors of around 35 000 Fine-gramed ten- slle polyslhcon, grown using low-pressure chemical vapour deposition (LPCVD) [ 133,134], 1s used as a construction material and sillcon oxide and/or mtnde can be used as a sacnficlal layer The

Slllcon Subsirate

(W

CAP RESONATOR NACUATED

(Cl

Fig 8 Fabncatlon process of sealed resonant force gauges usmg LPCVD polys~hcon [43] (a) After deposmon and patternmg of the first sacrdiaal layer (CVD or thermal oxide) and the first polyslhcon layer, (b) after deposmon and patternmg of the second sacnficlal layer (CVD oxide or rutride), thm thermal oxide for the etch channels and the second polysd~con layer, (c) after the sacrdklal layer etch III HF and reactive seahng with polyslhcon or slhcon mtrlde

sacnficlal etchant used IS hydrofluonc acid ‘Stlck- mg’ of the resonator and the seahng cap to the substrate 1s avolded by freeze-subhmatlon proce- dures [ 1241 The basic steps for the formation of sealed polyslhcon resonators are schematically shown m Ag 8

A thu-d way of fabncatmg sealed resonators [ 1351 is based on HF ano&c etchmg [ 115,116] and slhcon fusion bonding [ 1361 The results are prehmmary but very encouragmg

In deciding which technology to use for the fabncatlon of sealed resonators, several aspects are worth consldermg In the case of selective epltaxy and HF anodlc etchmg, single-crystallme ahcon 1s used as the resonator material It 1s expected that a smgle-crystalhne matenal 1s more

suitable than polycrystalhne materials due to tJle superior material properties with respect to aging, dnft, hysteresis, fatigue, creep, yielding, etc , of the former A definite statement, however, cannot be made at this point since the study of the matenal properties of polysilicon is still in a prehmmary stage Selective epitaxy seems to be more comph- cated and less flexible For instance, the sealing cap always closely surrounds the resonator This precludes the sealing of more than one resonator m a smgle cavity, e g , as proposed for a dlfferen- teal design for common-mode reJection purposes [ 1371 In addition, as a result of the high deposltlon rate (300 nm/min) of epitaxially grown sihcon, it will be difficult to realize very thm ( < 1 pm) resonators m an accurately controlled and repro- duclble way For very thm resonators, LPCVD polyslhcon (deposition rate of the order of 7 nm/ mm) seems to be the best candidate On the other hand, the low deposltlon rate for polyslhcon means long deposltlon times, which makes this technology less attractive If the resonator thickness exceeds 2-3 pm, LPCVD polyslhcon 1s no longer practical

It 1s pointed out here that m the process descnp- tlons given above, no special attention 1s paid to the excltatlon/detectlon mechanism used Ikeda’s resonator 1s driven magnetically and no addltlonal processmg on the resonator 1s reqmred, since the P ++ resonator 1s already conductive enough Guckel’s resonator 1s dnven electrothermally or electrostatlcally and detection IS done pelzoresls- tlvely This requires additional processmg on the resonator to form the polyreslstors and the electrodes This can be done m a convenient way by means of local Implantation of impuntIes, e g , boron The fabncatlon of a plezoelectncally driven resonator, on the other hand, creates new chal- lenges to the process engineer A question that arises 1s how subsequent heat treatments affect the properties of the plezoelectrlc layer This might exclude epltaxlal growth of slhcon, smce this 1s done at relatively high temperatures (lOOO- 1150 “C) Also, the plezoelectrlc layer should either be resistant to the sacrdiclal layer etchant, or It should be protected against the etchant by encapsulating the plezomaterlal m a passlvatlon layer

Another relevant aspect 1s the final stress state of the (composite) resonator caused by the residual stresses m the applied matenals In general, it 1s

preferred to have the resonator under small ten- sion This mmlmlzes the chance of buckling and allows long beams to be fabncated A large ten- sion 1s not attractive, since this will lower the gauge factor and thus the sensltlvlty of the device, see Ftg 2

Design aspects and performance issues

The sensltlvlty of a resonant force/strain gauge to an applied (mechanical) load depends on the gauge factor of the resonator Large gauge factors require slender beams with a low residual strain

In a sensor system, the resonant force gauge will be the frequency-determmmg bulldmg block of an electronic oscillator As Indicated by eqn (24), automatic gam control (AGC) 1s necessary to mmlmlze error readings due to a variable amphtude of vibration Improved behavlour of a feedback oscillator as a result of AGC has been experimentally demonstrated by Ikeda et al [59]

Unwanted frequency shifts are not only caused by spreads m the vibrational amplitude, but can also be caused by a variety of physical and chemical loads, e g , mass loading due to surrounding fluid, temperature, humidity, vapour adsorption, dust, etc Most of these disturbing loads can be eliminated by lsolatmg the gauge from its surroundings, e g , through vacuum encapsulation Temperature very often remams as a major source of errors Material properties such as Young’s modulus and the resonator dlmenslons are all temperature dependent and, as a result, a temperature change will induce a shift of the resonance frequency Moreover, differential thermal expansion effects within the composite resonator or between the sensor chip and the mount can strongly degrade the performance of the device Solutions to this problem are found m electromc temperature compensation (requiring an on-chip temperature sensor), accurate temperature regula- tion of the sensor environment and/or by employmg a differential resonator design [ 1371 Other common-mode errors that are suppressed m a differential design are variations m the reference clock frequency, aging of material propertles, even-order non-hneanty effects and, last but not least, mounting or package-induced stresses A further reduction of mounting strains 1s achieved

48

if stress-free assembly techniques [ 1381 or mechanical decouphng zones [ 1391 are used

Factors that nught hmlt the resolution are the exatatlon/detectlon scheme used and, of course, the noise m the system [140] If the resonator 1s regarded as a simple harmomc oscillator with an angular frequency of resonance oo, and if the feedback oscillator 1s locked at the frequency where the phase shift 1s x/2 radians, I e , at w = wo, the mmlmum relative frequency resolu- tlon IS given by

(A4~)rnm = ( WQ) Wmn

where Q 1s the mechanical quality factor of the resonator and A&,,, the mmlmum phase shift that can be detected by the oscillator Hence, the resolution IS inversely proportional to the quality factor Q The nummum phase shift that can be resolved is determined by the resolution of the detection mechanism and by the noise m the entire system

As already indicated above, the ultimate resolution of the resonant force gauge 1s determmed by several factors temperature sensltlvlty and van- altons, stab&y of electrical components of the feedback oscillator, stability of the environment and the quality factor of the resonator These factors actually determine the short-term resolution or stability (Aw/o)s, m,n, which can be expressed mathematlcally as

where max 1s the ‘maximum’ function, the first argument between the brackets can be obtained from eqn (24), GT = (l/o)@w/8T), the relative temperature sensltivlty of the gauge and Gt = (l/o)@w/iFJL), the relative sensltlvlty of the gauge to any other unwanted load L, e g , hmmdlty, AT,,, and AL,,, are the mmlmum vanatlons m temperature and other unwanted loads seen by the resonator, respectively The resolution Ap,,,,” , defined as the smallest change m load, e g , a pressure change, that can be dlstmgmshed by the sensor structure (e g , the pressure sensor as shown m Fig 6) 1s Bven by

A~m,n = (1&W&& mm

where GP = (l/w)(&~/dp), the relative frequency sensltlvlty of the sensor to the applied mechanical load p

In speaking of long-term resolution or stab&y, agmg phenomena, creep and stress relaxation become relevant Long-term dnft requires frequent recahbratlon of the system This generally con- cerns dnft of matenal properties and It IS expected that single-crystalline matenals have more stable properties than polycrystalhne or amorphous materials Therefore, single-crystalhne materials are preferred Also, the mtrmslc quality factor of these materials 1s higher

According to Karrer and Ward [ 1411, the response time 1s determmed by (1) the time the resonator 1s maintained at the old load, (2) the magnitude of the load change, and (3) the desired degree of stablhty It 1s expected that the response time 1s small compared to the time reqmred for counting the frequency shift, especially m hlgh- preciaon sensors

Finally, mode interference, or m other words interference of unwanted modes with the desired mode, 1s a relevant Issue For complex resonator shapes, the frequencies of the different modes of vibration can be very close to each other and they all have different sensltlvltles to the axial load, e g , [49,142] The dynanuc range or measurement window of the gauge will be hrmted to the frequency range where modes do not mterfere, since only here will the frequency be a smgle- valued function of the axial load The best way to increase the dynamic range 1s to keep the resonator structure simple and, moreover, to design and place the excltatlon/detection elements m such a way that the unwanted modes are not excited (see also eqn (26)) and/or cannot be detected

Conclusions

Resonant force gauges are very attractive for preaslon measurements because of then high senn- tlvlty, high stablhty and frequency output Mlcro- machmmg of the gauges allows batch fabrication to take place, wth possible on-&up electronics, manu- facturmg costs to be reduced and reproduclbhty to be improved

To be effectively employed m high-performance sensors, resonant force gauges must be isolated from the envn-onment to ehmmate frequency shifts caused by unwanted loads A very prormsmg way of domg thus, which nught define the &rectlon of future research, IS by means of local vacuum en- capsulatlon of the gauge m a batch fabncatlon process From tis point of wew the cavity and the enclosing shell form an integral part of the gauge Temperature wdl always remam as a dls- turbmg load and special attention must be gven not only to the force gauge, but also to the design of the entire sensor chtp, mcludmg packagmg aspects to reduce dlfferentlal thermal expansion effects

Single-crystalline matenals such as smgle-crystalhne slhcon and quartz have highly stable properties with low hysteresis and creep, which makes them very suitable for the fabncatlon of resonant force/strain gauges The potential apphcation of other materials such as thm films of polycrystalhne slhcon and sllrcon mtnde should be further mvesti- gated with respect to stability and repeatablhty of mechanical properties, because they are very attractive from a nucromachmmg point of view Complex mechamcal structures, mcludmg sealed resonators, can relatively easily (as compared to quartz and single-crystallme &con) be fabricated Furthermore, the thm-film/sacnficlal layer technology 1s very flexible and versatile Also, more effort must be put mto the development of new technologes for mlcromachmmg single-crystalline slhcon, such as dopant-selective electrochemical etchmg and HF anodlc etching A challenge 1s formed by the proper selection and lmplementa- tlon of a smtable exatatlon/detectlon scheme that has a mmlmal degrading effect on the resonator performance

The feaslblllty of &con resonant force gauges has already been demonstrated by a large number of laboratory prototypes, but as indicated m this paper, a long way still has to be gone to demon- strate their apphcablhty m a real environment

Acknowledgements

The authors would like to thank Dr Slebe Bouwstra for many fruitful dlscusslons and his valuable comments on the manuscnpt

49

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