Dynamic control and modal analysis of coupled nano-mechanical resonatorsDominik V. Scheible, Artur Erbe, and Robert H. Blick Citation: Applied Physics Letters 82, 3333 (2003); doi: 10.1063/1.1575491 View online: http://dx.doi.org/10.1063/1.1575491 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/82/19?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Intrinsic dissipation in a nano-mechanical resonator J. Appl. Phys. 116, 094304 (2014); 10.1063/1.4894282 Dynamic range of atomically thin vibrating nanomechanical resonators Appl. Phys. Lett. 104, 103109 (2014); 10.1063/1.4868129 Linear and nonlinear coupling between transverse modes of a nanomechanical resonator J. Appl. Phys. 114, 114307 (2013); 10.1063/1.4821273 Dynamic characteristics control of DLC nano-resonator fabricated by focused-ion-beam chemical vapordeposition J. Vac. Sci. Technol. B 29, 06FE03 (2011); 10.1116/1.3662493 The mechanical resonances of electrostatically coupled nanocantilevers Appl. Phys. Lett. 98, 063110 (2011); 10.1063/1.3553779

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Dynamic control and modal analysis of couplednano-mechanical resonators

Dominik V. Scheiblea)

Center for NanoScience and Fakulta¨t fur Physik, Ludwig-Maximilians-Universita¨t,Geschwister-Scholl-Platz 1, 80539 Mu¨nchen, Germany

Artur ErbeLucent Technologies, 600 Mountain Avenue, Murray Hill, New Jersey 07974

Robert H. BlickDepartment of Electrical & Computer Engineering, University of Wisconsin-Madison,1415 Engineering Drive, Madison, Wisconsin 53706

~Received 22 January 2003; accepted 19 March 2003!

We present measurements on nanomechanical resonators allowing fullin situ tuning of theirdynamic properties, including higher-order nonlinearities~up to fifth order! and the mechanicalquality factorQ. This is accomplished by gating electrodes and balancing resonators, similarily toa classical tuning fork. A detailed modal analysis is performed and reproducibility of the deviceresponse is verified. Eigenfrequencies are in the range of 40 to 70 MHz, and quality factor rises upto Q;63103. © 2003 American Institute of Physics.@DOI: 10.1063/1.1575491#

Ultrasmall mechanical systems have become an integralpart of mesosocopic physics over recent years. Nanoelectro-mechanical systems~NEMS! are studied and manufacturedto act as force detectors at the quantum limit,1,2 probe me-chanical quantum behavior,3 and for communicationapplications,4,5 to name only a few. However, limited tun-ability of these NEMS represents a certain challenge when itcomes to implementation as an experimental detector or tosensor/filter application. In recent work,6 we have shown thegradual transfer of a NEMS into a chaotic response. Here, weexpand the study of nonlinear excitation of a similar device,including full dynamic and modal control.

The integration of multiple nanomechanical beam reso-nators is the main concept the device is based on: one doublyclamped central beam (l 3w3h:2 mm3100 nm3200 nm)is appended by two singly clamped cantilevers (l 51mm) ateach end, incorporating, similar to a tuning-fork, about halfof the central beam’s mass@see Figs. 1~a! and 1~b!#. Wedenote the outer cantilevers as so-called Q-tips, due to theircrucial role in altering the system’s quality factorQ. Theentire system was manufactured from a Au-coated silicon-on-insulator chip, employing a sacrificial layer process,which we have demonstrated in detail elsewhere.7

In order to drive the resonator, the chip is mounted in acryostat equipped with a superconducting solenoid, provid-ing a magnetic field density up to 12 T and sample tempera-tures ofT5(1.52100) K. An ac signal, generated by a net-work analyzer ~HP-8751A!, causes a sinusoidal Lorentzforce in the wire due to the magnetic field. Displacement isdetected via the reflected powerPref as the system’s imped-ance changes when excited in mechanical resonance.8 To-gether with the resonator beams, we incorporated a set oftuning gates in the system. These gates, denoted by the let-

ters A through F, enabled us to place specific parts of theresonator in an electric ac/dc field@see Fig. 1~a!#.

We fabricated a series of resonators with equivalent lay-out, revealing similar spectra of mechanical excitation, asshown for a single structure in Fig. 1~c!. Each peak in thisspectrum corresponds to one eigenmode of the entire resona-tor M1, M2, and M3. In the case of a dc bias applied to a

a!Author to whom correspondence should be addressed; electronic mail:dominik.scheible@lmu.de

FIG. 1. ~a! Experimental setup to measure power absorption by mechanicaldisplacement; the ac-signal is generated and detected by an HP8751A net-work analyzer, which measures the ratio of incident and reflected power.~b!Electron-beam micrograph of the resonator system with the dotted line in-dicating ac-signal flow.~c! Spectrum of mechanical excitation for a mag-netic field densityB512 T and sample temperatureT54.2 K, featuring thethree main modes M1, M2, and M3. Inset shows the dispersion of threedifferent samples and the respective FEM simulation~!! for all observedmodes.

APPLIED PHYSICS LETTERS VOLUME 82, NUMBER 19 12 MAY 2003

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gate, we observe a shift of the resonance frequency followingthe square of the applied voltageV for ideal coupling of theresonator to the respective gate.9 Applying separate gate volt-ages at gate pairs AB, CD, and EF, respectively, and observ-ing the resulting shift in the resonator’s natural frequency,allows one to determine the coupling strength of the specificpart of the resonator for each eigenmode. In concert withboth the simulation via finite element methods~FEM!10 andanalytic calculations, the experimental data allow one to de-termine the actual shape of each eigenmode M1–M3, asshown in Fig. 1~c!. The double clamping, the presence ofbalancing cantilevers, and a favorable aspect ratio (w,h)strongly supports in-plane modal stability.

In the inset of Fig. 1~c!, we show the dispersion of threeof these systems for all observed modes; due to the particular‘‘free-clamped-clamped-free’’ resonator configuration andmechanical attenuation by the clamping, a strong deviationfrom both straightforward beam (v;k2) and string (v;k)mechanics is obtained. Corresponding modes and frequen-cies have been obtained via FEM simulations@see Fig. 1~c!inset#.

Although energy is transferred from the driven centralresonator to theQ-tips, the latter possess different eigenfre-quencies. Their first-order resonance frequency is muchlarger than the excitation frequency of the central part. As aresult, we observe virtually no displacement~via capacitivedetection! of the outerQ-tips when the driven oscillator is inits lowest mode. From this and dc tuning experiments, wededuce the mode shape M1 to be actually as shown in Fig.1~c!, with an eigenfrequency off M1540.79 MHz. Once ex-citation frequencies reach the first eigenfrequency of theQ-tips, displacement is detected. The resulting mode M2 cor-responds to the second harmonic mode of the central systemwith f M2561.24 MHz.

A qualitative model of beam resonators can be obtainedby using the Duffing equation. This equation models the dis-placementy(t), where the restoring force of a driven anddamped oscillator is expanded in a Taylor series:11

] t2y~ t !12Gv0] ty~ t !1 (

odd n

N

knyn~ t !5K cos~vt !, ~1!

whereG is the attenuation,v0 the ~circular! eigenfrequency,v the frequency of the driving force, andK its amplitude. InEq. ~1!, we neglect asymmetric potential components.

In the linear regime of small displacements, a clearLorentzian line shape is found, corresponding to the solutionof the linear oscillator, only whenk15v0

2 is nonzero in Eq.~1!. Increasing the incident powerPin , the displacement be-comes so large that orders up tok3 have to be taken intoaccount. The respective solution features a hysteresis,12

which is reduced to the familiar sawtooth shape when exci-tation frequencies are swept monotonously. A positivek3

.0 renders a steep slope at the left side of the resonance,whereas a negativek3 yields a jump at the right.

A more detailed analysis of the second-order resonanceM2 revealed an even higher-order nonlinearity. In addition toa negativek3 , the Q-tips’ motion causes ak5y5(t) term toappear. This corresponds to a sixth order term in the poten-tial. Expansion in Eq.~1! up to N55 yields up to five pos-

sible solutions for a single frequency. As a result, a kink inthe resonance peak is observed~see Fig. 2!.

When tracing the peak at higher temperatures, the kinkvanishes gradually. The resonator gains thermal energy, andboth effective elasticity of the nanomechanical Si/Au con-figuration and the two clamping points substantially change,which, in turn, renders the influence of thek5 nonlinearityless pronounced. The thermal detuning of the kink can beseen in Fig. 2~a! for sample temperaturesT5(4212) K.Furthermore, the integrated tuning gates can be used to studythe nonlinearity; applying an electric potential across thecentral gate pair CD does not influence the kink, as shown inFig. 2~b!, since in mode M2 the central displacement at thegate-pair is negligible. On the contrary, the kink can be en-tirely detuned, when the outerQ-tips are placed in an addi-tional potential supplied by the gate pairs AB and EF@seeFig. 2~c!#. This gives further evidence for the strong influ-ence of the clamping; that is, the central beam’sk5 can bedetuned via the outerQ-tips.

The fundamental property characterizing a mechanicalresonator is the quality factorQ, defined as the ratio of theoscillation energyeosc and the amount of energy which isdissipatededis during one period:

Q5eosc

edis5

f 0

2G. ~2!

FIG. 2. Nonlinear excitation of the system in eigenmode M2 at a magneticfield density ofB512 T reveals a sixth-order term in the restoring potential.~a! Suppression of the kink by increasing bath temperature from 4.2 to 12.5K. ~b! Application of an additional electric potential across the gate pair CDup toVCD53 V leads to no substantial change of the mode M2 in nonlinearresponse.~c! Complete detuning of the kink by the electric potential acrossthe outer gate pairs AB and EF occurs already atVAB/EF>2 V. Differentresponse for central and outer gate-pairs supports modal analysis of M2.

3334 Appl. Phys. Lett., Vol. 82, No. 19, 12 May 2003 Scheible, Erbe, and Blick

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For Q@1, the approximationQ5 f 0 /D f 0 is valid, wheref 0

is the eigenfrequency, andD f 0 the full width at half-maximum of the resonance peak.

The present three-beam system is a design that mini-mizes energy loss;13 this is achieved by adopting the conceptof a tuning fork and adding electrodes gating the mechanicalresonator, as shown. In addition to our previous work,12 weemploy the two outerQ-tips as ‘‘counterweights.’’ The outerbeams compensate for the central resonator’s motion, render-ing the center of mass fixed at all times. Consequently, weexpect a decreased attenuation and an enhanced quality fac-tor Q. In the classical tuning fork, two parallel fins ensurelargeQ.14

Apart from enhancing the quality factorQ, the presentsetup also allows us to tune this fundamental property of thesystem. As mentioned earlier, the clamping prevents theQ-tips from oscillating at base frequency. In this mode, com-pensation of the central resonator’s motion is therefore sup-pressed. Hence, the relatively poorQ;600.15 By ac-drivingthe gates, however, it is possible to prepare the desired modeshape.7 The Q-tips are excited capacitively by applying alarge ac signal at the outer pairs AB and EF. At the sametime, magnetomotive excitation of the central beam is per-

formed. When both frequencies match in a way that thewhole resonator moves in phase,12 the resonator’s center ofmass is fixed in space. Dissipation is reduced and the qualityfactor Q is strongly increased~see Fig. 3!.

We have to point out that the resonance shape thenclearly deviates from the usual Lorentzian. We explain thiswith an increased nonlinear behavior due to multiple excita-tion by two sources. Furthermore, the driving term in Eq.~1!must be replaced byKA cos(vAt)1KB cos(vBt1w). Conse-quently, we estimate15 a maximum value ofQ;60006600with an increased error.

In conclusion, we are able to fully control the dynamicresponse of a coupled nanomechanical resonator by featuringa design similar to a classical tuning fork. By additionallybiasing the resonator with gating electrodes the specific me-chanical modes can be identified, and nonlinearities can bedetuned at constant driving power. Finally,Q-tuning withinone order of magnitude is achieved by a combination ofcapacitive and magnetomotive excitation.

We like to thank J. P. Kotthaus for his continuous sup-port and A. Kriele and K. Werhahn for excellent technicalhelp. This work was funded in part by project Bl-487/1 of theDeutsche Forschungsgemeinschaft~DFG!.

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10SOLVIA, v. 95.2~a finite element system!.11F. K. Kneubuhl, Oscillations and Waves~Springer, Berlin, 1997!.12A. Kraus, G. Corso, A. Erbe, K. Richter, and R. H. Blick, Appl. Phys. Lett.

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Conference on Solid State Sensors & Actuators, 2001, p. 1110.14H. R. Schwarz,Methode der finiten Elemente~B. G. Teubner, Stuttgart,

1991!.15Quality factorsQ have been obtained via a Lorentzian fit of experimental

data and applying approximation of Eq.~2!; that is,Q5 f 0 /D f 0 . Result-ing errors are less than 2Q/100 for Lorentzian behavior.

FIG. 3. Tuning of the quality factorQ by preparation of a high-Q modeM1* (B512 T, T54.2 K). TheQ-tips are capacitively driven at frequen-cies f AB/EF . The best match occurs when maximumQ;6000 is attained.Reflected powerPref is given in arbitrary units relative to reference~dotted!lines. The inset showsQ vs frequency mismatch of both systems~Ref. 15!,normalized to the frequency of maximumQ. The increased error of the peakvalueQ560006600 is a result of the non-Lorentzian resonance shape.

3335Appl. Phys. Lett., Vol. 82, No. 19, 12 May 2003 Scheible, Erbe, and Blick

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