Loss budget of a setup for measuring mechanical dissipations of siliconwafers between 300 and 4 K

J. P. Zendri,1 M. Bignotto,2 M. Bonaldi,3 M. Cerdonio,2 L. Conti,2 L. Ferrario,4 N. Liguori,2

A. Maraner,5 E. Serra,4 and L. Taffarello1

1INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padova, Italy2INFN, Sezione di Padova and Dip. di Fisica, Univ. Padova, Via Marzolo 8, I-35131 Padova, Italy3Istituto di Fotonica e Nanotecnologie CNR-FBK and INFN, gr. coll. Trento, I-38050 Povo, Trento, Italy4Fondazione Bruno Kessler, FBK-irst MicroTechnologies Laboratory, Via Sommarive 18, I-38050 Povo,Trento, Italy5INFN, Laboratori Nazionali di Legnaro, V.le dell’Università 2, I-35020 Legnaro, Padova, Italy

�Received 13 December 2007; accepted 26 January 2008; published online 7 March 2008�

A setup for measuring mechanical losses of silicon wafers has been fully characterized from roomtemperature to 4 K in the frequency range between 300 Hz and 4 kHz: it consists of silicon waferswith nodal suspension and capacitive and optical vibration sensors. Major contributions tomechanical losses are investigated and compared with experimental data scanning the fulltemperature range; in particular, losses due to the thermoelastic effect and to the wafer clamp aremodeled via finite element method analysis; surface losses and gas damping are also estimated. Thereproducibility of the measurements of total losses is also discussed and the setup capabilities formeasuring additive losses contributed by thin films deposited on the wafers or bonding layers. Forinstance, assuming that additive losses are due to an 80-nm-thick wafer bond layer with Youngmodulus about ten times smaller than that of silicon, we achieve a sensitivity to bond losses at thelevel of 5�10−3 at 4 K and at about 2 kHz. © 2008 American Institute of Physics.�DOI: 10.1063/1.2868810�

I. INTRODUCTION

A number of silicon wafer bonding methods1 have beendeveloped to obtain vacuum sealing and to realize complex,three-dimensional mechanical systems, overcoming the lim-its of photolithography. In basic research, bonding tech-niques are applied mainly to realize monolithic objects withhigh stability and mechanical strength at room temperature.For instance, in the field of gravitational wave experiments,stable platforms2 and quasimonolithic suspensions3 are con-structed by means of the hydroxy-catalyzed bonding, alsoknown as “silicate bonding.”4 Originally developed for join-ing fused-silica parts to realize the star-tracking telescope ofthe Gravity Probe-B space experiment, this technique is nowapplied with success to different materials such as sapphireand silicon. While the characterization of bonding techniquesis being pursued since several years, little has been done atlow temperatures.

In this paper we present an experimental apparatus de-signed for measuring the mechanical losses which are con-tributed at cryogenic temperatures and in the acoustic fre-quency range �100 Hz–10 kHz� by silicon wafer bondingtechniques. Our motivation comes from the need of develop-ing very large silicon masses �order of 10 ton� with ex-tremely low mechanical losses for the purpose of construct-ing a cryogenic DUAL detector of gravitational waves.5,6

Silicon is already known to possess very low mechanical lossangles ���10−8–10−9� from room7 to cryogenic8 tempera-tures but no measurements of losses contributed by bondingmethods are available at such low temperatures.

The loss of any elastic body is the sum of different con-tributions that can be generally divided into three main cat-

egories: fundamental losses �fund common to even perfectmaterials �i.e., materials with perfect crystal lattices�, intrin-sic losses �int due to material imperfections, and losses �ext

due to coupling to the external environment. The total lossescan thus be detailed as

�tot = �fund + �int + �ext. �1�

Low-loss measurements are often limited by �ext. In thework we report here we set up a standard procedure for me-chanical loss measurements: our main concern is to mini-mize �ext by isolating the sample under measurement fromthe lossy external environment and to understand the differ-ent contributions that sum up in the measured losses. Wecharacterize our setup in the whole temperature range be-tween 300 and 4 K stating the minimum losses contributedfurther by the bond layer and that we can measure on siliconwafers.

While our main concern regards mechanical losses dueto bond layers, this setup is also well suited for characteriz-ing additive losses contributed by other processes on silicon,such as surface coatings and other postproduction treatments.

II. THE EXPERIMENTAL APPARATUS

Our experimental apparatus consists of mechanical oscil-lators realized in the material under investigation, i.e., siliconfor the purpose of the work reported here: each oscillator issupported by a holder which is fixed to the final part of thesame mechanical suspension �see Fig. 1� and housed in theexperimental chamber �internal vacuum chamber �IVC�� of acryogenic facility; a system for exciting mechanically the

REVIEW OF SCIENTIFIC INSTRUMENTS 79, 033901 �2008�

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oscillators is available as well as nontouching sensors formeasuring the displacements of the oscillators.

The cryogenic facility was developed for the purpose oftesting the capacitive transducer and superconducting quan-tum interference device amplifier used in the detector ofgravitational waves AURIGA �Ref. 9� and thus it is namedTransducer Test Facility �TTF�; it is located at the LegnaroNational Laboratories of Istituto Nazionale di Fisica Nucle-are, near Padova, Italy. While the work reported in this papercovers the temperature range from 300 to 4 K, the TTF isalso equipped with a dilution refrigerator: extensions of themeasurements we report here down to 50 mK will be per-formed in the near future. The TTF has an experimentalchamber �IVC� of �420�700 mm2: this hosts mechanicalsuspensions optimized for frequencies around 900 Hz, wherethey provide −170 dB of attenuation from floor vibrationsalong the three orthogonal axes. The mechanical suspensionshung from the top: at their bottom they support the experi-mental apparatus. A view port on the center of the experi-mental chamber is available which is connected to the top ofthe TTF in air through a pipe line: it is used for the opticalsensor �see Sec. II C 1�. The TTF is described in more detailsin Ref. 10.

Figure 1 shows a view of the whole experimental appa-ratus: at the top of the TTF in air, a plate supports the opticalcomponents for the optical sensor described in Sec. II C 1.The lower part of the vacuum chamber is available for theexperiment: at the bottom of the suspension, threerectangular-section rods support an aluminum disk �height50 mm, diameter 300 mm� on top of which the oscillatorsand the capacitive sensor �see Sec. II C 2� are mounted. Werefer to this disk as the “base.” A second, lower base wasfixed to the first one by means of three copper rods: thislower level of the experimental setup was dedicated to mea-surements of mechanical dissipations on SiC samples but wedo not report them in this paper. In the following we describein more details the experimental apparatus which is specificto the research we report here.

One rectangular piezoelectric actuator is fixed to thebase �see Fig. 1�; it is mounted horizontally and a brass mass

of 330 g is glued to its top so to increase its efficiency asforce actuator. This actuator is used to excite the oscillators,as discussed in Sec. III.

In order to perform measurements at intermediate tem-peratures between room temperature and the liquid nitrogentemperature and between this and the liquid helium tempera-tures, we fixed three heaters to the curved surface of thebase, about 120° apart �see Fig. 1�: each heater consists of aManganin wire wounded around a copper core. The heatersare connected in parallel and the resulting resistance is 25 �.The heaters are the actuators of a feedback loop that vanishesthe difference between a set temperature and the reading of asilicon thermometer �T1 in Fig. 1� mounted on the circularsurface of the base: the loop is closed by a temperature con-troller �LakeShore 340� in air which regulates the currentthrough the heaters. We never exceed a power of 25 Wthrough the heaters to avoid damaging the heating device. Asecond silicon thermometer �T2 in Fig. 1� is placed on thelower base that supports the SiC samples.

After collecting the measurements of mechanical losseswe performed a cryogenic run to calibrate the sample tem-perature against the base temperature. For this purpose wefixed a thermometer �LakeShore, model CX-1050� close tothe outer diameter of the wafer with hole H2 �see Sec. II B�by means of silver paint; we also screwed a similar thermom-eter on the base close to the wafer holder. Then we per-formed a temperature scan and recorded the readings of thetwo thermometers, along with the reading of the temperaturefeedback thermometer T1. The temperatures relative to allmeasurements reported in the following are that of thesample, inferred from the reading of the base temperatureand this temperature calibration.

A. The oscillators

To perform an intensive experimental campaign we de-cided to adopt the geometry of thin circular disks �wafers�for the silicon samples: this simple geometry is the standardin the fabrication of semiconductor devices such as inte-grated circuits and it is thus largely available. In particular,we focused on single crystals of n-type silicon in the formof wafers, polished on both sides, with a radius of100.2�0.1 mm and a thickness of 500�15 �m. They weredoped with phosphor to an impurity concentration of1014–1015 cm−3 �resistivity �9–24 � cm�. The crystal ori-entation of the wafer, defined by the plane of its top surface,is �100�. According to production standards, each wafer wasshaped with a flat cut perpendicular to the �110� crystallo-graphic plane. In order to symmetrize the geometry andmake the center of the wafer circumference to coincide withthe oscillator center of mass, we refined the original cut andapplied a second, identical cut, parallel to the first one: weused a diamond saw blade in this process. We measured thedistance between the flats to be 94.8�0.1 mm. As discussedin Sec. II B, the symmetry of the oscillator is an importantdetail for realizing the central, nodal holder. The wafer ge-ometry is shown in Fig. 2. The wafers exhibit a number ofresonant modes in the acoustic frequency range of our inter-est �see Table I�.

FIG. 1. Left: Cross section of the cryogenic facility at LNL. At the top ofthe facility the horizontal plate �A� supporting the optical readout is visible.The liquid helium Dewar �B� and the IVC �C� are also shown; the four ringmasses in the top part of the IVC are part of the mechanical suspension.Right: Zoom of the bases supporting the silicon samples �top disk� and SiCsamples �bottom disk�. D Silicon oscillator with optical sensor. E Capacitivestrips holder. F Piezoelectric actuator. G Thermometer T2. H ThermometerT1. I One of the three heaters, about 120° apart.

033901-2 Zendri et al. Rev. Sci. Instrum. 79, 033901 �2008�

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Almost all thermomechanical properties of silicon, usedas the input values for the numerical analysis discussed inthe following, are taken from Ref. 11. The thermal conduc-tivity values of n-type silicon below 100 K was found inRef. 12. We also note that silicon is an anisotropic crystal,with cubic symmetry; so its Young’s modulus varies alongdifferent directions in the material relative to the crystal ori-entation. In the crystal axis coordinate system, its elasticproperties can be described by a 6�6 symmetric stiffnessmatrix Cij with only three independent constants:13 C11

=C22=C33=167.5 GPa, C12=C13=65.0 GPa, and C44=C55

=C66=80.1 GPa. These values determine the followingvalues for the Young modulus in the plane of our �100� wa-fer: parallel to flat, Y+=169 GPa and 45° diagonal to flat,Y�=130 GPa.

B. Oscillator holder

In order to minimize mechanical losses that might becontributed by the sample holder, we decided to minimize

the contact area between sample and holder and to realize anodal contact between the two. Nodal contact means that thesample is supported at a point which remains at rest when thesample oscillates; so the holder does not exert any reactionforce. In fact, the holder is an elastic body, which can applythe reaction force only by changing its dimensions, accord-ing to its stress-strain relations. If a lossy, not nodal holderparticipates in the motion, it dissipates part of the sampleenergy, thus increasing the total losses. The experimentalimplementation of a nodal contact between sample andholder is obviously approximated and hence we expect thelatter to contribute to the total mechanical losses: this issue isdiscussed in detail in Sec. V C.

To realize a central nodal holder we sandwiched the wa-fer between two aluminum plates, connected by two screws�see top of Fig. 3�; the wafer is kept in the horizontal posi-tion. Similar to Ref. 14, the contact between the plates andthe wafer is realized by two balls �1 mm in diameter� madeof sapphire, a very low-loss material:15 the bottom ball issimply placed on top of a small dip at the center of the lowerplate while a steel spring �elastic constant=0.75 N /mm�connects the top ball to the top plate �see top of Fig. 3�.Acting on the two screws the position of the top plate islowered by �2 mm: the spring is thus compressed and itexerts a vertical force of 1.5 N on the top ball, thus holdingthe wafer in place. We took great care to guarantee that theimaginary line passing through the center of the top andbottom balls is perpendicular to the wafer and passes throughits center: this is critical in order to avoid damaging the wa-fer when uncontrolled tilt is applied and to realize a centralsuspension. To this end we adopted two solutions �see Fig.3�: �H1� we drilled a through hole �0.5 mm in diameter� atthe center of the wafer using a dental tool �this is done afterthe flat cuts are applied as discussed in Sec. II A� and �H2�we excavated two cuts at the center of both sides of the wafervia chemical wet etching �this is done before the flat cuts areapplied as discussed in Sec. II A�. Since it follows the �111�crystal plane, the latter solution ends up in pyramidal cuts,

FIG. 2. Drawing of our �100� wafer �holes H1-2 not shown, see Fig. 3�: werefined the �110� primary flat and doubled it to make the center of masscoincide with the wafer center. The distance between the flat cuts is94.8�0.1 mm. The wafer dimensions are diameter 100.2�0.1 mm andthickness 500�15 �m. The families of crystal planes �100� and �110� areindicated for clarity.

TABLE I. Resonant frequency in hertz of the first acoustic modes of thewafer shown in Fig. 5: FEM estimation �second column, see Sec. 4� androom-temperature experimental measurement for the sample with H1 hole�third column� and H2 hole �fourth column�. The fifth column describes theacoustic modes as labeled in Fig. 5. The sixth column lists the loss factormeasured at about 7 K on the sample with hole H1.

Mode No. �FEM �H1 �H2 �a, n, s� �exp7 K

1 275.4 275.3 275.3 �0, 0, 0�2 386 380.1 383.0 �2,0,�� 1.1�10−7

3 483 489.3 484.6 �2, 0,��4 984 972.9 975.7 �3, 0,��5 997 985 989 �3, 0,��6 1518 1478 �0, 1, 0�7 1543 �1, 0,��8 1607 �1, 0,��9 1733 1721 1720 �4, 0,�� 1.3�10−8

10 1766 1784 1772 �4, 0,��11 2539 2500 2515 �2, 1,�� 1.1�10−7

12 2649 2633 2629 �5, 0,��13 2663 2636 2638 �5, 0,��14 2805 2846 2814 �2, 1,��15 3744 3701 3702 �6, 1,��16 3744 3703 3712 �6, 1,��17 3950 3924 3923 �3, 1,��18 3995 �3, 1,��

FIG. 3. �a� Drawing �not to scale� of the cross section of the sample with itssupport and of the displacement sensors: in the experiment the two sensorswere applied to two different samples. A, capacitive strip; B, spring; C, laserbeam of the optical readout; D, sapphire balls; and E, silicon sample. Bot-tom line: Scanning electron microscope photographs of the Si sample show-ing a zoom of �b� the through hole H1 and �c� the cuts H2.

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175 �m deep and with square section having a side of0.5 mm at the surface.

Hole H2 is obtained using the microelectromechanicalsystem technologies available at the Fondazione BrunoKessler-IRST in Trento, Italy: initially, a 500-nm-thick ther-mal silicon oxide layer and a 100-nm-thick low pressurechemical vapor silicon nitride layer are, respectively, grownand deposited on both wafer sides. Then a central hole isopened in the oxide/nitride layers via photolithography fol-lowed by a plasma dry etching; this etching stops selectivelyat the silicon surface. The silicon is then etched with wettetramethylammonium hydroxide �TMAH� 25% 90 °C solu-tion for several hours on each side: TMAH is highly selec-tive with respect to the oxide nitride mask. Once the centralholes �one per wafer side� have the desired size, the masklayers are removed.

C. Readouts

We employed two displacement sensors that do nottouch the wafers to minimize sensor contribution to me-chanical losses: one is based on the beam deflection tech-nique and one on capacitive strips. The sensors are vacuumcompatible and work in the full temperature range from300 to 4 K. The position of the sensors relative to the waferswas not optimized with respect to the acoustic modes undermeasurement. In the following the two sensors are described.

1. Beam deflection sensor

The beam deflection sensor16 consists in monitoring thedeflection of a laser beam by a displaced reflecting surfacewith a split photodetector. This technique works for any ma-terial but it is better suited for highly reflective ones so thatthe sample under measurement is not heated by absorptionof the incident laser beam or of scattered light. A 370�250 mm2 aluminum plate is placed horizontally on top ofthe TTF, in air �see Fig. 1�: it supports a 100 mW cwneodymium-doped yttrium aluminum garnet laser, focusingoptics and a split photodiode �partially shown in Fig. 1�. Thelight path is folded twice on the plate before entering a beamsplitter with 25% reflectivity: the transmitted beam is di-rected downward and, after passing through a 1 in. opticalwindow that realizes the vacuum sealing, it reaches the os-cillator with hole H2 about 2 m below. The upward beamreflected by the oscillator escapes from the same optical win-dow and the part transmitted by the beam splitter is directedto a quadrant photodiode �EG&G 30845�. The resulting fourphotocurrents are combined so to measure the change in theoptical density between the photodiode sections: thus a trans-verse deflection or vibration is converted into a difference ofelectrical signals of top/bottom or left/right parts of the pho-todiode. The laser spot size at the sample is about 0.2 mm.At 1 kHz the sensor is sensitive to about 1 nm of verticalmovement of the wafer reflecting surface. During the tem-perature calibration run we confirmed that the sample heat-ing due to the impinging laser beam is negligible at 4.2 K, asexpected from the high reflectivity of silicon.

The laser beam deflection method is widely employedbecause of the simple experimental setup and the easy opti-

cal beam alignment; in our case its usage is complicated bythe configuration of the experimental chamber and of theTTF �see above�. In particular, the beam must propagate onthe same 40-mm-diameter tube in both downward and up-ward directions and pass through the same 1-in.-diameterwindow; the minimum distance between the oscillator andthe nearest optics is about 2 m and for most part of it there isno access to the beam for monitoring its propagation. More-over it is not possible to perform measurements on differentoscillators during the same run since only one laser beam isavailable with fixed path. This is the main reason why wedecided to develop the capacitive strips.

2. Capacitive strips

In alternative to the optical sensor, we developed a ca-pacitive displacement sensor for measurements on dielectricmaterials. To this purpose we adopted the idea of the capaci-tive strips reported in Ref. 17, using them as sensor. Thisconsists in an array of 84 coplanar, rectangular strips, resem-bling a bar code �see Fig. 4�; it is obtained by machiningwith a circuit board plotter a 10-�m thick copper film ontop of a commercial fiberglass substrate of thicknessh=1.6 mm. The strips have width a=0.4 mm, lengthL=10 mm and are displaced by a period b=1.2 mm: the totalarea is 1�5.2 cm2. In the experiment only half of the stripswere faced to the silicon samples. The strip board is glued toan aluminum block which is fixed with screws to the sameplatform supporting the oscillators; we estimate the gap be-tween the strip plane and the wafer with hole H1 at its bot-tom to be 80�20 �m.

Figure 4 also shows the electrical scheme of the sensor:half of the strips are connected to ground while the other halfare connected to one side of a decoupling low-loss, high-voltage capacitor Cd=150 nF, which is placed on the lowestpart of the mechanical suspension of the TTF �see Fig. 1�:the voltage between this and ground constitutes the signal. Acharging line is also present �see Fig. 4�: at the beginning thesensor output is short-circuited and the charge line is con-nected �i.e., both switches in Fig. 4 are closed� to a high-voltage generator providing a bias voltage V0=80 V. Thenthe short across the sensor is removed and the high-voltagegenerator is disconnected, leaving a constant charge on thestrips.

A relative movement of the sample with respect to thesensor plane generates a change in the charge distribution on

FIG. 4. Top view �not to scale� of the strip sensor and its electrical scheme.Each strip is a L�a rectangle; the strips are patterned with periodicityb amounting to 84 strips �not all of them are shown here�. Cd is the decou-pling capacitor and V0 the bias voltage. The switches are closed only whencharging.

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the strip electrodes and in the capacity of the strip array,since the total charge is constant; the sensitivity is about2�104 V /m.

III. EXPERIMENTAL PROCEDURE

When measuring the elastic properties of solids throughmechanical oscillators, it is desirable to study more than asingle frequency. The first 18 modes of our silicon waferresonate at frequencies below 4 kHz, but only a few of themturn out to have small external losses �ext �see Sec. V C� andcan be actually used to evaluate the intrinsic properties of thematerial.

For each temperature point, we waited several hours toallow for thermalization of the samples, at a pressure of10−6 mbar. Then for each mode we excited the resonant vi-bration by feeding a monochromatic signal to the piezoelec-tric actuator fixed to the base �see Fig. 1�; amplitude of theexcitation wave ranges between 40 and 600 Vp.p. The outputof the employed displacement sensor is demodulated by anSR830 lock-in amplifier which mixes it with a reference sig-nal phase locked to the excitation signal; for the excitationand reference signals we use HP3325B synthesizer with highstability frequency reference. Before feeding it to the actua-tors, the excitation signal is amplified by a custom high-voltage amplifier. The demodulated signal is acquired viaGPIB �IEEE 488� with sampling rate ranging from 0.125 Hz�for lower-loss measurements� up to 4 Hz �for higher-lossmeasurements�. We actually have three such signal filteringsystems so that we can monitor up to three modes at thesame time. The fact that an acoustic mode can be excited byshaking the wafer holder indicates that the contact betweenwafer and holder is not perfectly nodal for that mode �seediscussion in Sec. V C 2�.

For each acoustic mode of the sample we obtain a firstestimation of the resonant frequency from the excitation fre-quency that minimizes the decay of the phase of the demodu-lated signal. Let �m be the resonant frequency of a mode; ifthis is excited by a monochromatic signal of frequency �s

and then at time t=0 suddenly the excitation signal is re-moved, the phase � of the demodulated signal evolves intime t as

��t� = �s + 2���s − �m�t , �2�

where �s is a constant. Correspondingly the mechanical vi-bration follows an exponentially damped decay whose enve-lope amplitude varies according to

u�t� = u0 exp�−t

. �3�

Once the first estimation of the resonant frequency is ob-tained for a given mode, we remove the excitation signal tothe piezoelectric actuator and acquire both the phase andamplitude of the demodulated signal. The final measurementof the mode resonant frequency is obtained from a fit of thetime evolution of the phase once the excitation signal isswitched off, according to Eq. �2�. The time constant of thelock-in is of the order of 1 s; since the linewidths that wemeasure are 0.05 Hz, the bandwidth of the lock-in ampli-fier is large enough to cover the whole mode linewidth with

large margins. Therefore our sensitivity to frequency drifts ofthe reference signals are much reduced in the amplitude sig-nal: still they would appear as a contribution to the timeevolution of the phase in addition to what expected fromEq. �2�. The experimental data are listed in Table I, alongwith their estimation via a finite element method �FEM�analysis �see Sec. IV�: the agreement of the numerical pre-dictions with experimental data is good. Table I also showsthe difference in the measured mode resonant frequenciesobtained between the two samples: one with hole H1 and theother with hole H2 �see Sec. II B�. By exchanging thesamples with respect to the sensors, we made sure that thereported differences are reproducible and do not depend onthe sensor technology. The relative difference varies with themode shape and never exceeds about 1%.

An exponential fit of the time evolution of the demodu-lated amplitude gives the decay time s �see Eq. �3�� andhence the mechanical loss angle at the frequency �s via thesimple formula

���s� =1

��ss. �4�

For all measurements the time span of the fitted data coversat least two decay times: for instance, this means that acqui-sition time for a single decay at a typical frequency of 1 kHzand with loss �=10−7 is �2 h. Therefore to speed up themeasurement campaign we measure the decay time of threedifferent modes of the same oscillator at the same time, hav-ing three lock-in amplifiers, as explained above: while amode is decaying we can excite a second mode, find itsresonant frequency, and follow its decay, without disturbingthe measurements of the first mode, provided the two modesdo not resonate at similar frequencies.

IV. ELASTIC MODEL OF THE OSCILLATOR

The equations describing oscillations of a thin circulardisk are usually written under the Kirchhoff hypotheses.18

�a� The straight lines, initially normal to the middle planebefore bending, remain straight and normal to themiddle surface during the deformation, and the lengthof such elements is not altered. This means that thevertical shear strains �xz and �yz are negligible. Thisassumption is referred to as the “hypothesis of straightnormals.”

�b� The stress normal to the middle plane, z, is smallcompared with the other stress components and may beneglected in the stress-strain relations.

Here z is the axis normal to the disk while x and y areorthogonal axes parallel to the disk, the origin of the coordi-nate system is chosen at the center of mass of the disk henceits middle plane is at z=0. We also note that because the diskis thin, the in-plane �tangential and extensional� vibrationsare decoupled from the out-of-plane �transverse� vibrations.

Let w�x ,y� represent the displacement of the middle

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plane along the z axis and h the disk thickness. The engineer-ing strain components in the disk due to this transverse dis-placement are given as18

�x

�y

�xy� = z kx

ky

kxy� , �5�

where −h /2zh /2 and the functions �k� represent themiddle surface curvature of the plate:

kx

ky

kxy� = − �2w/�x2

− �2w/�y2

− 2��2w/�x�y�� . �6�

The strain variations �Eq. �5�� determine the stress com-ponents through the relation

x

y

xy� = �Q11 Q12 Q16

Q12 Q22 Q26

Q16 Q26 Q66 �x

�y

�xy� , �7�

where Qij �i , j=1,2 ,6� are the reduced stiffnesses for a planestress state in the x-y plane, obtained by the full stiffnessmatrix Cij of the material under hypotheses �a� and �b�:

Qij = Cij −Ci3Cj3

C33. �8�

The constitutive equations �Eq. �7��, together with thedifferential equations of equilibrium and the proper boundaryconditions, represent the governing equations of a genericdisk. In the case of an isotropic material these equationsadmit an analytical solution19 which we review here toobtain a better insight into the normal modes of our disk.In this case the Qij can be simplified considerably and writ-ten in terms of the Young modulus Y and of the Poissonmodulus p:

�Q11 Q12 Q16

Q12 Q22 Q26

Q16 Q26 Q66 = Y� 1/�1 − p

2� p/�1 − p2� 0

p/�1 − p2� 1/�1 − p

2� 0

0 0 1/2�1 + p� . �9�

Then we search for solution where the displacement in the zdirection can be written as

w�r,�,t� = W�r,��ei�t, �10�

where r−� are the polar coordinates, t is the time coordinate,and � is the angular frequency. In this case the governingequation of the free transverse vibration writes as

�4W =�h�2

DW . �11�

Here � is the density and D is the flexural rigidity of the disk

D =Yh3

12�1 − p2�

. �12�

The solutions of Eq. �11� can be written as

W+�r,�� = cos�a���Bn�a�Ja��n

�a�r� + Cn�a�Ia��n

�a�r�� , �13�

W��r,�� = sin�a���Bn�a�Ja��n

�a�r� + Cn�a�Ia��n

�a�r�� ,

a = 0,1,2, . . . ,k , �14�

where Ja is the ath order Bessel function of the first kind andIa is the ath order modified Bessel function of the first kind.For each integer a, the constants Bn

�a�, Cn�a� and the eigenvalue

�n�a� are obtained when the boundary condition on the motion

is applied to the solution. In our experiment the disk center isat rest due to the reaction from the support, while the diskcircumference is free. The eigenfrequency of the mode isthen obtained by the relation �n

�a�=�4 �h��n�a��2 /D.

The value of a in Eqs. �13� and �14� indicates the num-ber of nodal diameters, n the number of nodal circles, ands=+, � is the symmetry index of the displacement fields; asolution is fully identified by the set �a ,n ,s�. The orthogonaldisplacement fields W+�r ,�� and W��r ,�� represent modedoublet: they have the same radial distribution of the defor-mation, mutually rotated by � /2a, and share the same eigen-value �n

�a�. When a=0 the doublet merges in a single solutionwith rotational symmetry around the z axis.

In the case of silicon the solutions shown in Eqs. �13�and �14� are approximated, because the crystal is anisotropicand its Young’s modulus along a given direction depends onthe orientation with respect to the crystal axes. To take intoaccount the material’s anisotropy and the real geometry, wedeveloped a FEM model of the wafer using a commercialsoftware, using the stiffness matrix described in Sec. II A.Similar to the experimental apparatus, in the model the waferis constrained on both sides, on two small circular areas withdiameter 0.5 mm, along its z symmetry axis.

The stiffness anisotropy and the flat cuts induce a split-ting of the frequency of the mode doublets. This splitting isquite large for a=1, when the modes are mutually rotated by� /2 and the effect of the flat cuts is maximized, and fora=2, when the modes are mutually rotated by � /4 and theeffect of the stiffness anisotropy is maximized. We show inFig. 5 the model plots of the first 13 eigenmodes with solu-tion of the type described by Eq. �13� and �14�. The �a ,n ,s�indices of the corresponding mode in the isotropic case arealso shown. We note that the FEM analysis predicts the ex-

033901-6 Zendri et al. Rev. Sci. Instrum. 79, 033901 �2008�

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istence of “cantilever” modes of the disk pivoting about itssuspension point, which cannot be described by periodicfunctions in �; these modes resonate at frequencies that de-pend critically on the suspension details and they were notexperimentally investigated.

V. ESTIMATION OF LOSS CONTRIBUTIONS

A. Fundamental losses

In insulating solids such as silicon there are two funda-mental mechanisms that set the lower limit to the total me-chanical losses:20 the thermoelastic effect and the phonon-phonon interaction. Since the latter contributes a negligibleloss in our case, we do not investigate it further.

The thermoelastic effect was investigated first byZener:21 in the presence of a nonzero coefficient of thermalexpansion, when a solid undergoes a vibration other thanpure torsion, the strain field generates a thermal gradient andthus a heat flow which dissipates elastic energy. The ex-pected losses depend on the geometry of the elastic structure;it can be demonstrated that for pure flexure the thermoelasticloss is21

�th��� =Y�2T

�CV

�th

1 + �2th2 , �15�

where � is the thermal expansion coefficient, CV the specificheat per unit volume of the material, and T is the tempera-ture. The loss factor due to the thermoelastic effect dependson the oscillator dimensions through the relaxation time th:

th =h2�CV

�2�, �16�

where h is the oscillator thickness and � the thermal conduc-tivity. Given our geometry the formula of Eq. �15� cannot beapplied. To estimate the thermoelastic damping of all modesunder measurement, we performed a FEM analysis using acommercial software: thus we could take into account ofboth the material anisotropy and the mixture of flexural andtorsional strains. We performed a structural-thermal har-monic analysis at room temperature, forcing one mode at atime. The model is built using the coupled-field thermoelas-tic analysis options of a three-dimensional 20-node solid el-ement, and the convergence of the analysis was checkedagainst the mesh density.

The results are shown in Fig. 6, where the measuredlosses at room temperature are compared to that evaluated bythe FEM analysis. The agreement with experimental data iswithin 10% for the majority of the modes, confirming thatthermoelastic damping is the main dissipative phenomenonat room temperature in our oscillator.

FIG. 5. Model plots of the first 13 eigenmodes described by Eqs. �13� and�14�, evaluated by FEM analysis and ordered according to the �a ,n ,s� indi-ces. As customary in FEM analysis, the mode shapes are normalized to themass matrix.

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

23

4

5

11

13

1417

912

16

15

3

4 11

13

141617

2 91215

ϕ H2

×10

4 ϕ H2> ϕ FEM

ϕ H2< ϕ FEM

ϕ H1< ϕ FEM

ϕ H1> ϕ FEM

ϕ H1

×10

4

ϕFEM

× 104

FIG. 6. Comparison of the loss angle measured at room temperature on thesample with hole H1 ��H1� and hole H2 ��H2� and that evaluated by FEM��FEM�. The dashed lines represent the points at �H1,2=�FEM. The numbersclose to the experimental data refer to the mode number, as listed in Table I.

033901-7 Silicon wafer loss budget Rev. Sci. Instrum. 79, 033901 �2008�

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This FEM model allows us to compute the thermoelasticdamping at a given temperature simply by using the appro-priate thermomechanical properties of silicon: we get thesefrom Refs. 11 and 12 for the temperature range between 300and 4 K. In Fig. 7 we show as a gray line the losses expectedfor mode No. 9 as effect of the thermoelastic damping andcomputed via the FEM analysis: due to the large number ofcomputations involved here, a coarser mesh was employedwith respect to what was shown in Fig. 6. This accounts forthe discrepancy between the predictions for mode No. 9 atroom temperature between Figs. 6 and 7. The thickness ofthe gray line in Fig. 7 accounts for the uncertainty in ourestimates due to variation in thermal conductivity betweendifferent silicon samples. In Fig. 7 we also show the lossesmeasured for modes Nos. 2, 9, and 11 �as labeled in Table I�,obtained by scanning the sample temperature between 300and 4 K: overall the agreement of the experimental data formode No. 9 and our FEM prediction is very good above40 K, indicating that the thermoelastic damping originatesthe dominant loss contribution above 40 K. In this tempera-ture range the only deviation is observed at about 120 Kwhere the coefficient of linear expansion of silicon vanishesand so does the thermoelastic mechanism �see, for instance,Eq. �15��: the observed local minimum is set by a differentmechanism �see, for instance, Sec. V C 2�. Similarly below40 K the measured losses of mode No. 9 are larger thanthose expected from the thermoelastic damping, indicating adifferent dominant loss contribution. The losses of mode No.11 show a dependence with temperature similar to that ofmode No. 9, indicating a similar explanation for the domi-nant loss mechanism. Mode No. 2 has a smoother depen-dence with temperature, suggesting that it is dominated bydifferent damping mechanisms.

B. Intrinsic losses

Bulk silicon is demonstrated to possess extremely lowmechanical losses both at room temperature7 and 4 K.8 Thuswe do not expect that crystal defects contribute significantlyto the total losses of our samples. On the contrary surfacelosses might be a limiting factor: since the source of thisdamping is still not clear, in order to estimate it, we compareour oscillator to other silicon ones of different volume-to-surface ratios.

In Ref. 22 authors collect from the literature data ofroom-temperature losses for single-crystal silicon oscillatorsof different dimensions: in Fig. 5 of Ref. 22 they summarizesuch information and evidence a dependence of 1 /� versussize for oscillators such as ours which have not been an-nealed or for which reactive-ion etching has been involved.From this collection we infer that with a thickness of500 �m, surface losses at room temperature should be �s

10−5. For the modes that show the lowest losses at roomtemperature this figure is not far from the experimental val-ues �see Fig. 6�: thus for such modes, surface losses mightgive a not negligible contribution at room temperature,even though not the dominant one �see Fig. 6 and relateddiscussion�.

In Fig. 8 we plot the loss angle against the smallestoscillator dimension for a number of silicon oscillators ofdifferent shapes found in the literature:8,23–28 all measure-ments are taken at temperatures in the range 1–15 K. Foreach group of researchers �i.e., for each oscillator design� wequote the smallest reported loss angle. The dependence of themeasured losses on the size �or, equivalently, the volume-to-surface ratio� suggests a relevant damping mechanism fromthe surface. Our measurement of the losses of mode No. 9,also shown in Fig. 8, does not depart significantly from thistrend: thus we conclude that surface losses explain at leastpart of the total losses for mode No. 9 at liquid heliumtemperatures.

C. External losses

1. Gas damping

The residual gas present in the vacuum chamber thathosts the oscillators constitutes an additional source of me-chanical losses: in the free-molecule regime, where we per-formed all our measurements, they are due to the momentumwhich is transferred from the vibrating body to the gas mol-ecules during the collisions. According to the model byChristian29 which assumes a Maxwell–Boltzmann distribu-tion for the gas velocity, if this mechanism were the limitingone then we would measure loss angles given by

FIG. 7. Measurements as function of temperature of the losses of mode Nos.2 ���, 9 ���, and 11 ��� for the sample with hole H1. The experimentalerrors are not shown since they are smaller than the dimensions of thepoints. In gray the FEM predictions are shown for the losses of mode No. 9due to the thermoelastic damping: the line thickness accounts for the uncer-tainty in this prediction coming from uncertainty in the thermal conductivity.

10-1 100 101 102 103 104 105 10610-10

10-9

10-8

10-7

10-6

10-5 Mamin 01Gysin 04

Mihailovich 92

Zendri 08

Zimmer 07

Wago 96

Spiel 01

McGuigan 78

length (µm)

loss

angl

e

FIG. 8. Dependence of the loss angle vs the smallest oscillator dimensionfor a number of Si oscillators in the range 1–15 K: for each point the labelindicates the first author of the corresponding paper and the publication year�Refs. 8 and 23–28�. Our measurement for mode No. 9 is also shown �asZendri 08�. The dashed line is an eye guide.

033901-8 Zendri et al. Rev. Sci. Instrum. 79, 033901 �2008�

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�gas = � 2

�3/2 1

�h�m

�Mg

RTP �

P�mbar�

��Hz��T�K�, �17�

where R is the gas constant, Mg is the molar mass of theresidual gas, and P is the pressure. In the last equality weconsider helium to be the largest component of the residualgas since we use it to thermalize the cryogenic liquid bath ofthe TTF; P�mbar�, ��Hz�, and T�K� are, respectively, the valuesof pressure in mbars, of frequency in hertz, and of tempera-ture, in kelvins. Thus at T=4.2 K, P=10−6 mbar we get�gas=1�10−9 for mode No. 2 and �gas=3�10−10 for modeNo. 9.

The model of Ref. 29 assumes that the sample is placedin an infinitely large volume; if the distance d between thewalls and the sample becomes smaller than the typical di-mension L of the vibrating body, then a larger number ofcollisions should be expected and hence the damping shouldbe larger. In Ref. 30 for a rectangular plate the damping inthis so-called squeezed-film regime is expressed as

�sg = �gasL

16�d. �18�

This damping mechanism is present in the sample measuredby the strip sensor: using d=0.08 mm and L�25 mm, i.e.,respectively, the gap between the strips and the sample andthe length of the strip section which is faced to it, we calcu-late �sg�6�gas. At T=4.2 K, P=10−6 mbar we get �sg=8�10−9 for mode No. 2 and �sg=2�10−9 for mode No. 9.

Our estimates of the damping from the residual gas at4 K for mode Nos. 2 and 9 are about a factor of 10 below theexperimental measurements �see, for instance, Fig. 7�, thussuggesting a negligible contribution. On the other hand, itshould be noted that the above estimates should be refinedtaking into account the real geometry, the different modalshapes, and the surface outgassing; moreover, given theposition of our vacuum gauge which is on top of the TTF atroom temperature, we underestimate the residual gas pres-sure in Sec. IV C and thus our estimates of the gas dampingare affected by a systematic error.

Therefore to measure the size of the combined gasdamping effects and check the above estimates, we per-formed measurements of the loss angle at 77 K of the samplewith hole H1, i.e., that read by the capacitive strips, as func-tion of the pressure in Sec. IV C chamber for mode Nos. 2, 9,and 15 of Table I; the data are shown in Fig. 9. We infer thatat 77 K the gas damping is negligible below 10−4 mbar forfrequencies �1.7 kHz; resonating at a lower frequency,mode No. 2 shows a dependence of the loss angle on theresidual pressure even in the 10−6 mbar range where we per-formed most of the measurements reported in this paper. Forthis mode the data above 10−3 mbar are proportional to thepressure, as indicated by Eqs. �18� and �17�; at lower pres-sures the proportionality is lost, but a power-law trend �withexponent �1� is still present.

We also repeated the measurements at 77 K of the lossangle of mode No. 2 as function of pressure �in the range0.006–0.03 mbar and at 10−6 mbar� but on the sample readby the beam deflection sensor �sample with hole H2�: this isnot expected to suffer from the squeezed-film damping.

Above 0.006 mbar the losses measured with the optical read-out are about a factor of 0.2 smaller than the isobar measure-ments taken with the capacitive readout: this suggests in thelatter a significative contribution from the squeezed-filmmechanism. On the other hand at 77 K the loss angles ofmode No. 2 measured at 10−6 mbar with the two sensorsdiffer by less than 30% and amount to about 2.5�10−7. Thusat such pressure we can rule out the �squeezed-film� dampingfrom residual gas as dominant contribution even for modeNo. 2. To establish firmly the size of the gas damping effectat 4 K, a similar experimental campaign should be repeatedat that temperature; on the basis of the above estimates weexpect not a large effect.

2. Clamping losses

A rough estimation of the interaction between eachacoustic mode and the holder can be quantified by consider-ing the elastic energy dissipated by the holder when thismoves. A displacement in the holder can be induced by a notperfectly nodal suspension or by a change in the thickness hof the wafer where it is in contact with the sapphire balls. Areliable FE model of the complete system cannot be devel-oped, due to the presence of pressed contacts and springs.Thus for each mode we estimate the goodness of the nodalsuspension by evaluating the maximum thickness changeduring one period induced when only one face of the disk isconstrained in the central area. The surface of the wafer isconstrained on one side over a circular area of diameter0.3 mm. The displacement of the points along the symmetryaxis of the wafer is constrained along the z axis, to avoidrocking movements. The model, with the constraints and themeasured area, is shown in Fig. 11. To catch reliably secondorder effects we took care in meshing the central area of thedisk where we used about 2000 elements. The model consistsof 16 500 elements in all, and we checked the convergenceof the results against the mesh density.

With this model we performed a number of harmonicanalysis, forcing one mode at a time, measuring the rmsvalue Hc of the displacement of a circular area of diameter0.3 mm on the free side of the wafer �where we placed thepressed sapphire ball during the experiment�. In Fig. 10 weplot for each mode the losses measured at 121�1 K versusthe square of the computed thickness change normalized tothe mode energy Emode: this quantity is an estimation of the

10-6 10-5 10-4 10-3 10-2 10-110-7

10-6

10-5

10-4 10-3 10-2 10-1 100 101

mode 2mode 9mode 15

loss

angl

e

Pressure (mbar)

Pressure (Pa)

FIG. 9. Dependence of the loss angle on the residual gas pressure for modeNos. 2 ���, 9 ���, and 15 ���: the measurements are taken at 77 K with thecapacitive strips.

033901-9 Silicon wafer loss budget Rev. Sci. Instrum. 79, 033901 �2008�

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interaction between the wafer and the holder. The experi-mental data were taken for both H1 and H2 samples and thechosen temperature is such that the coefficient of linear ex-pansion of silicon vanishes: thus the thermoelastic dampingis negligible and we estimate clamping losses to arise thedominant loss contribution. As expected lower losses aremeasured for the modes that induce little thickness variationin correspondence of the holder. The trend evident from thegraph suggests that holder losses dominate at this tempera-ture except for the lowest-loss mode �mode No. 9�. For a fewmodes we show two different measurements taken on thesame oscillator with both sensors in two different runs, afterthe oscillator is removed and replaced back in the apparatus.This is an estimation of the reproducibility of our measure-ments at a frequency where the thermoelastic damping isnegligible. Figure 10 also shows that no major differencesare observed in the losses between the samples with differentholes.

In Fig. 11 we plot the modal shapes of the doubletformed by modes �4,0,�� �mode No. 10� and �4,0,�� �modeNo. 9�. The zoom of the deformation in the central areashows that the free gray area, opposite to the constraint, ismuch more displaced in mode No. 10 than in mode No. 9:the former is more subject to clamp losses. This explains thelarge difference in their loss angles observed at 121 K �re-spectively, 2.0�10−5 and 8.2�10−8�.

We also note that clamping losses are temperature de-pendent as shown by the variation with temperature of theloss factor of mode No. 10 which is dominated by clampinglosses: for this mode we measure �=1.5�10−4 at 300 K and�=2.0�10−5 at 121�1 K.

VI. EXPERIMENTAL RESULTS AND LOSS BUDGET

As explained in Sec. I, the goal of experimental appara-tus characterized in the above is to measure the loss contrib-uted by post-treatments of the silicon samples, in particular,by layers of silicon-onto-silicon bonds. The minimum bondlosses that add to those discussed in Sec. V and that the setupis sensitive to depend on the ratio between the energy Ebond

stored in the bond layer and the total energy Etot in the body.If �0 indicates the losses of the untreated sample and con-sidering only intrinsic mechanisms, the total losses �1 of thebonded sample, with overall dimensions identical to the un-treated sample, can be written as

�1 = �0 +Ebond

Etot�bond. �19�

In the multilayer case, Eq. �5� maintains its validity andthe strain is proportional to the distance from the neutralplane through the disk curvature. On the contrary the stressvariation through the thickness is not necessarily linear, andstresses must be evaluated in each layer by Eq. �7�, using thereduced stiffness matrix relative to the layer material. Theelastic energy stored in an element dV is

dE =1

2 l�ldV =

1

2z2Q̂lmklkmdV , �20�

where we imply summation over repeated indices �l ,m=1,2 ,3� and adopt the contracted notation for stress, strain,and curvature:

�1

�2

�3� � �x

�y

�xy�, k1

k2

k3� � kx

ky

kxy�, 1

2

3� � x

y

xy� .

�21�

The matrix Q̂ is given in terms of the reduced stiffnesses Qij

�see Eq. �8�� as

�Q̂11 Q̂12 Q̂13

Q̂12 Q̂22 Q̂23

Q̂13 Q̂23 Q̂33

� �Q11 Q12 Q16

Q12 Q22 Q26

Q16 Q26 Q66 . �22�

10-7

10-6

10-5

10-4

10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-1010-8

10-7

10-6

10-5

23

4

10

12

14

15

17

2

1

9

1

4

9

3 4

9

10

1112

14

15

16

1

2

5

17

ϕ H2

ϕ H1

Hc

2/Emode

( m2/J )

FIG. 10. Loss angle at the temperature of 121�1 K as a function of thenormalized squared thickness change at the center of the wafer, Hc

2 /Emode.The experimental data refer to the sample with hole H2 �top plot� and H1�bottom plot�. As thermoelastic damping is negligible at this temperature,the effect of the holder dissipation is seen as a monotone trend with respectto Hc

2 /Emode. The numbers close to the points refer to the mode labels ofTable I. For a few modes two different measurements are shown, taken onthe same oscillator with both sensors in two different runs and after theoscillator is removed and replaced back in the apparatus.

FIG. 11. Cross section �top row� and zoom �bottom row� of the sampleundergoing vibration corresponding to the modes �4,0,�� �left; mode No.10� and �4,0,�� �right; mode No. 9�. The cross section at rest is shown bydotted lines as an eye guide. For each mode the vibration is plotted at thephase which maximizes the displacement in the suspension area �shown ingray�. The surface of the sample is constrained on the bottom side over acircular area of diameter 0.3 mm.

033901-10 Zendri et al. Rev. Sci. Instrum. 79, 033901 �2008�

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If h is the total thickness and hb and zb are, respectively,the thickness of the bond layer and its mean distance fromthe middle plane �see Fig. 12�, the total energy is obtained asthe sum of the energy stored in each layer:

Etot =1

2�

−h/2

zb−hb/2

z2dz� klkmQ̂lm�bulk�dxdy

+1

2�

zb−hb/2

zb+hb/2

z2dz� klkmQ̂lm�bond�dxdy

+1

2�

zb+hb/2

h/2

z2dz� klkmQ̂lm�bulk�dxdy ,

where the second term is the energy �Ebond� stored in thebonding layer. If the bonding layer is thin, we approximatethe energy ratio as

Ebond

Etot��

zb−hb/2

zb+hb/2

z2dz� klkmQ̂lm�bond�dxdy

�−h/2

h/2

z2dz� klkmQ̂lm�bulk�dxdy

. �23�

If the bulk material and the bond are both isotropic and havethe same Poisson ratio, from Eqs. �9� and �22� we see that thetwo surface integrals differ only by the Young moduli �weuse the apexes bulk and bond to distinguish between thetwo�, and we can write

Ebond

Etot�

12

h3

Ybond

Ybulk �zb2hb +

hb3

12 � 12

Ybond

Ybulk

zb2hb

h3 . �24�

Then the energy coupling is minimum when the bondedsample is symmetric with respect to the bond layer �i.e.,zb=0�. This result is a good approximation for isotropic ma-terials in the Kirchhoff hypothesis.

Even if FEM analysis should be done to obtain preciseresults for �100� silicon, the estimate �Eq. �24�� can be usedto evaluate the minimum �bond that our apparatus is sensitiveto, computed as the minimum difference in the total lossesthat we can observe between the unbonded and bondedsamples, both having identical overall dimensions. If T

represents the standard deviation of loss measurements at atemperature T and for a given frequency, with a confidencelevel of 99.73% we can write the minimum detectable lossesas

�bond,T �Etot

Ebond3 T. �25�

If the bonded sample is composed by three identical slices instack �thus zb� �h /6� and hb�h �see Fig. 12�, we get

�bond,T � 4.5Ybulk

Ybond

h

hb T. �26�

Estimation of the standard deviation of the loss measure-ments is a delicate issue. For mode No. 9, at room tempera-ture and at 77 K, where losses are dominated by the ther-moelastic mechanism, measurements are well reproducible�within 10%�: we checked this to be the case even after theoscillator is removed and then replaced back in the apparatusand even by interleaving the measurements by a full thermalcycle to 4 K. So we estimate the standard deviation of theloss angle of mode No. 9 to be 300 K=1�10−6 at 300 K and 77 K=5�10−8 at 77 K. For the same mode, measurementsat 4 K are less reproducible: the standard deviation of mea-surements on mode No. 9 is 4 K=1�10−8, obtained afterremoving and replacing back the sample in the apparatus andby interleaving a full thermal cycle. In the literature wefound that loose reproducibility is associated with clampinglosses:27 this is also one of the dominant loss mechanisms formode No. 9 at 4 K. We note that at all temperatures theabove quoted scatters are much larger than the errors fromthe exponential decay fit of the data, in the literature usuallyreported as the error in the loss factor; moreover at all tem-peratures the measured losses are better reproducible withina sequence of consecutive measurements with respect to theabove reported scatters. Finally we note that the above re-ported scatters refer to a thermalized oscillator: in fact, thescatter between the measurements is larger when not enoughtime is allowed for the system to thermalize.

To give a number of our sensitivity we consider the caseof a sample of total thickness h=500 �m obtained by stack-ing three identical wafers �see Fig. 12� via hydroxy-catalysisbonding, a technique for which data are already available: inthis case hb=80 nm and Ybond=7.9 GPa and �bond is of theorder of 10−1 at room temperature.31 Thus for the mode�4,0,�� at about 1.7 kHz �mode No. 9� we are sensitive to�bond,300 K�5�10−1 at room temperature, �bond,77 K�3�10−2 at 77 K, and �bond,4 K�5�10−3 at 4 K.

The maximum bond losses that would allow the con-struction of a sensitive DUAL detector in silicon dependcritically on the detector configuration, presently still understudy, and construction procedure. However, we can give arough approximation of the maximum tolerable losses byconsidering the following case: it is pessimistic since it en-hances the effect of bond losses, assuming a uniform distri-bution of the elastic energy. Let us consider to build a siliconbody of volume Vtot=1 m3 by bonding together 103 cubeswith edge length of 0.1 m; let us also consider the facingfaces of adjacent cubes bonded over their whole surfaces andlet hb=100 nm be the bond thickness. The total volume of

FIG. 12. �Color online� Schematic cross section of the multilayer waferwhich we refer to in order to compute the sensitivity of our apparatus: threeidentical wafers are stacked for a total thickness h. hb is the thickness ofeach bond layer between the wafers �shown as dark lines�. The middle planeof the multilayer wafer is shown as dashed line: the distance between thisand the bond layers is zb.

033901-11 Silicon wafer loss budget Rev. Sci. Instrum. 79, 033901 �2008�

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the bond layers is thus Vbond�3�10−7 m3. Then, if such amass must show losses at the level of �M �10−8 at 4 K, thebond losses �bond should be

�bond �Vtot

Vbond�M = 3 � 10−3. �27�

Our sensitivity is thus enough for the purpose of constructinga DUAL gravitational wave detector, if the hydroxy-catalyzed bond is a good approximation. By the way, we notethat the sensitivity could be optimized by choosing a clevergeometry for the bonded sample. For instance, at room tem-perature a thicker sample would suffer less from the ther-moelastic damping and thus lead to a better sensitivity, pro-vided the relative error in the loss measurement is constant.

VII. CONCLUSIONS

We have performed a through characterization of a setupcapable of measuring mechanical losses contributed by post-processing treatments on silicon wafers, in the temperaturerange between room temperature and 4 K and in the fre-quency range between 300 Hz and 4 kHz. To get a full un-derstanding of the measured total losses we compared theexperimental data to estimations of the losses expected frommajor contributing mechanisms: the thermoelastic damping,computed for our anisotropic wafers by a detailed FEMmodel, was shown to dominate our lowest-loss mode in al-most the entire temperature range from ambient down to40 K. Below such temperature surface losses and residualclamping losses are estimated to give the largest dampingcomponent. Overall, we were able to measure losses as lowas 10−8 at 4 K for our 100-mm-diameter, 500-�m-thick wa-fers. We also investigated the reproducibility of our lossmeasurements at 300, 77, and 4 K: thus we were able toestimate our sensitivity to further losses contributed by post-production treatments on silicon wafers. With a confidencelevel of 99.7%, in the case of a 100-nm-thick lossy layerwith Young modulus a factor of 10 smaller than that of sili-con, we are able to measure additive loss contributions at thelevel of a few 10−3 at 4 K: this figure is sufficient for char-acterizing losses at cryogenic temperatures due to silicon wa-fer bonding methods to be employed in the construction of aDUAL detector of gravitational waves. This same apparatuscan be used to investigate the losses due to other types ofthin films on silicon, such as optical coatings.

ACKNOWLEDGMENTS

This work was supported partially by EuropeanCommunity �project ILIAS, Contract No. RII3-CT-2004-

506222� and by Provincia Autonoma di Trento �project QL-Readout�.

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