Tunable, Broadband Nonlinear Nanomechanical Resonator

Tunable, Broadband NonlinearNanomechanical ResonatorHanna Cho, Min-Feng Yu,* and Alexander F. Vakakis*

Department of Mechanical Science and Engineering, University of Illinois at UrbanasChampaign, 1206 West GreenStreet, Urbana, Illinois 61801

Lawrence A. Bergman, and D. Michael McFarland

Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, 104 South Wright Street,Urbana, Illinois 61801

ABSTRACT A nanomechanical resonator incorporating intrinsically geometric nonlinearity and operated in a highly nonlinear regimeis modeled and developed. The nanoresonator is capable of extreme broadband resonance, with tunable resonance bandwidth up tomany times its natural frequency. Its resonance bandwidth and drop frequency (the upper jump-down frequency) are found to bevery sensitive to added mass and energy dissipation due to damping. We demonstrate a prototype nonlinear mechanical nanoresonatorintegrating a doubly clamped carbon nanotube and show its broadband resonance over tens of MHz (over 3 times its natural resonancefrequency) and its sensitivity to femtogram added mass at room temperature.

KEYWORDS Nonlinear resonance, nanoresonator, broadband resonance, responsivity, carbon nanotube

Recent advances have seen the development of na-nomechanical resonators operating in the linearregime that are capable of detecting extremely small

physical quantities1-12 and even quantum interactions.13-15

However, the reduced device size reduces its dynamic range(down to the nanometer scale) for linear operation,16 whichmakes developing the required measurement system dif-ficult and accordingly limits its sensitivity, especially underambient and room-temperature environments.

The main element in most mechanical nanoresonatorsconsists of a nanoscale mechanical cantilever or a nanoscaledoubly clamped beam, which significantly reduces the ef-fective mass of the resonance system. A general feature ofsuch devices is that they operate predominantly in the linearregime and achieve high sensitivity to mass or chargethrough the realization of high quality-factor resonance athigh frequency. Most noticeably, their recent developmenthas allowed the sensing of mass down to the zeptogram (zg)level,7,12 for even a single molecule,9,11 and the transportof a single electron charge.17,18 Whereas the absolute mag-nitude of the involved resonance amplitude is small, therelative magnitude is actually quite significant when com-pared to the reduced device size. As a result, such nanoscaleresonance systems can easily transition from linear reso-nance operation to a nonlinear one through a slight increasein its dynamic operating amplitude.16-18 The importance ofnonlinearity in such nanomechanical resonance systems is

thus gaining more recognition. For example, electrostaticinteractions19 and coupled nanomechanical resonators20

were proposed for tuning the nonlinearity in nanoscaleresonance systems; noise-enabled transitions in a nonlinearresonator were analyzed to improve the precision in mea-suring the linear resonance frequency21 and a homodynemeasurement scheme for a nonlinear resonator was pro-posed for increasing the mass sensitivity and reducing theresponse time.22 In addition, the basins of attraction ofstable attractors in the dynamics of a nanowire-based me-chanical resonator were studied,23 and the nonlinear behav-ior of an embedded24 and a curved carbon nanotube25 wastheoretically investigated. Such studies increasingly offer anew conceptual understanding and thus strategies to dealwith and even exploit the increasingly prominent nonlinearbehavior in the development of nanomechanical resonators.

Herein, we design an intrinsically nonlinear nanome-chanical resonator defined by the inherent geometric non-linearity that can be readily incorporated into practicaldevice development, and we apply both theoretical model-ing and experimental validation to demonstrate its tunabil-ity, its capacity for broadband resonance, and its sensitivityto added mass and to energy dissipation due to damping.

The intrinsic nonlinearity is simply introduced into thenanoscale resonance system through a geometric design asdescribed in the following. Consider a fixed-fixed mechan-ical resonator employing a linearly elastic wire with negli-gible bending stiffness and no initial axial pretension. Whendriven transversely by a periodic excitation force appliedlocally to the middle of the wire, it will exhibit stronggeometric nonlinearity and become an intrinsically (purely)

* Corresponding authors. E-mail: [email protected] and [email protected] for review: 02/09/2010Published on Web: 04/12/2010

pubs.acs.org/NanoLett

© 2010 American Chemical Society 1793 DOI: 10.1021/nl100480y | Nano Lett. 2010, 10, 1793–1798

nonlinear resonator (Figure 1A). In such a resonator, theforce-displacement dependence is described by the relationF ) kx[1 - L(L2 + x2)-1/2] ≈ (k/2L2)x3 + O(x5)26, where F isthe transverse point force applied to the middle of the wire,x is the transverse displacement in the middle of the wire,and L and k are the half-length and the effective axial springconstant of the wire, respectively. The total absence of alinear force-displacement dependence term (i.e., a term ofthe form kx) results in the realization of a geometricallynonlinear force-displacement dependence of pure cubicorder. This resonator has no preferential resonance fre-quency, and its resonance response is broadband,26 whichis conceptually different from typical linear mechanicalresonators. Moreover, the apparent resonance frequency iscompletely tunable through the instantaneous energy of thesystem. If the bending effects are non-negligible or if aninitial pretension exists in the wire, then a nonzero linearterm in the previous force-displacement relation is in-cluded, giving rise to a preferential resonance frequency.

However, as long as this preferential frequency is sufficientlysmall compared to the frequency range of the nonlinearresonance dynamics, the previous conclusions still apply.

Thus,weproceedtoanalyzeadoublyclampedEuler-Bernoullibeam having a foreign mass (mc) attached to its middle andexcited transversely by an alternating center-concentratedforce. Considering the geometric nonlinearity induced byaxial tension during oscillation, the vibration of the beam isdescribed by

where w(x, t) is the transverse displacement of the beamwith x and t denoting the spatially and temporally indepen-dent variables, E and F are the Young’s modulus and massdensity, A and L are the cross-sectional area and half lengthof the beam, I is the area moment of inertia of the beam, Qis the quality factor of the resonator in the linear dynamicregime, F is the excitation force applied to the middle of thebeam, ω() 2πf) is the driving frequency, and ωo () 2πfo) isthe linearized natural resonance frequency of the beam. Itis assumed that no initial axial tension exists when the beamis at rest, and shorthand notation for partial differentiationis used.

The transverse displacement of the beam can be ap-proximately expressed as w(x, t) ) ∑i ) 1

N Wi(x) φi(t), whereWi(x) is the ith mode shape of the linearized beam, φi(t) isthe corresponding ith modal amplitude, and N is the numberof beam modes considered in the approximation. Theleading modal amplitude, φ1(t), is then approximately gov-erned by a Duffing equation obtained by discretizing eq 1through a standard one-mode Galerkin approach27 (Support-ing Information):

Here, M ) [mc/(2FAL)]W12(L) ) (mc/m0)W1

2(L) is the ratio ofthe foreign mass to the overall mass of the beam multipliedby a factor due to the center-concentrated geometry of theforeign mass distribution (when the foreign mass is distrib-uted evenly on the beam, M ) mc/m0); the amplitude of thedriving force per unit mass in eq 2 is defined by q ) W1(L)F/mo, and the nonlinear coefficient is defined by

Following a harmonic balance approximation27 with asingle frequency ω, we find that the response spectrum of

FIGURE 1. Nanoresonator integrating intrinsic geometric nonlin-earity. (A) Schematic showing a simple doubly clamped mechanicalbeam (and its equivalent spring model) having an intrinsic geometricnonlinearity. (B) Tunability of the resonance bandwidth of a non-linear nanoresonator. The plot shows the dependence of the dropfrequency/natural frequency ratio on the applied drive force and thequality factor of the mechanical resonator. The plot in the insetshows the frequency response of a nonlinear resonator calculatedon the basis of the parameters listed for carbon nanotube B1 in theinset of Figure 2.

[FA + mcδ(x - L)]wtt + (mωo/Q)wt + EIwxxxx -

(EA/4L)wxx ∫0

2Lwx

2 dx ) F cos ωtδ(x - L) (1)

(1 + M)φ1

••+

ω0

Qφ1

•+ ω0

2φ1 + αφ1

3 ) q cos(ωt)

(2)

α ) - E

32FL4 ∫0

2LW1W1′′ dx∫0

2L(W1′)

2 dx

© 2010 American Chemical Society 1794 DOI: 10.1021/nl100480y | Nano Lett. 2010, 10, 1793-–1798

this Duffing oscillator forms a multivalued region when theoscillation amplitude is over a critical value as seen in theinset of Figure 1B. Specifically, there are two branches ofstable resonances that are connected by a branch of unstableresonances. As the frequency is swept up, the resonanceamplitude in the upper branch of stable resonances in-creases up to the maximum possible amplitude and thendrops abruptly to a lower value as the forced motion makesa transition to the lower stable branch. The drop frequency,fdrop, at which this jump phenomenon occurs is approxi-mately determined by the intersection of the Duffing re-sponse spectrum with the free oscillation or the “backbone”curve,27 and its ratio to the linearized natural frequency isgiven by

where Γ ) γ((FQ)/(E))2((2L)/(D))6((1)/(D4)) and γ ) 0.0303.From this equation, it is clear that the drop frequency of thisnonlinear resonator strongly depends on the attached centermass and damping, besides the geometry of the beam andthe applied excitation force. A similar computation can beperformed for the reverse jump-up frequency during adecreasing frequency sweep; in that case, the dynamicsfollows a transition from the lower stable resonance branchto the upper.

We estimate the mass responsivity (Rm), defined as theshift in drop frequency with respect to the change in theadded center mass, as

Compared with a mass sensor based on a linear resonatorfor which the responsivity is -fo/2mo, the nonlinear resona-tor utilizing the drop frequency as the measurant has a betterresponsivity by a factor of rdrop[1 -(rdrop

2 - 1)/(2rdrop2 - 1)]

when ignoring the term W12(L) and if rdropg 1.618. The mass

responsivities for three representative doubly clamped beamswith E ) 100 GPa and F ) 2600 kg/m3 and a single-walledCNT beam with E ) 1 TPa, for which the parameters arelisted in the inset table, are plotted in Figure 2A as a functionof the normalized frequency fdrop/fo. The value at fdrop/fo ) 1indicates the responsivity of a linear resonator. It is apparentthat the responsivity is enhanced not only by considering anonlinear resonator with a smaller intrinsic mass and ahigher resonance frequency but also by increasing the ratioof the drop frequency over the natural resonance frequency.This means that the performance of a mass sensor basedon a nonlinear nanoresonator can be considerably raised byincreasing its resonance bandwidth which, as we will showlater, is practically tunable.

For a nonlinear resonator to have such intrinsicallynonlinear behavior and a highly broadband resonanceresponse, several parameters, including the quality factor,the size of the mechanical beam, and the driving force, areto be optimized to provide a larger value of Γ according toeq 3. Here, it is noted that the resonance bandwidth can beextended by simply increasing the excitation force while

FIGURE 2. Sensing performance of a nonlinear nanoresonator to mass and energy dissipation due to damping. (A) Mass responsivities of fourdifferent doubly clamped beams as a function of the drop frequency/natural frequency ratio. (B) Shift in the drop frequency for a 1% changein the damping coefficient as a function of the drop frequency/natural frequency ratio. The inset table lists the parameters for the carbonnanotubes used in the calculation.

rdrop )fdrop

fo) (1 + √1 + (1 + M)Γ

(1 + M) )1/2

(3)

Rm ) lim∆mcf0

∆fdrop

∆mc) -

fo

2mordrop(1 -

rdrop2 - 1

2rdrop2 - 1)W1

2(L) (4)

© 2010 American Chemical Society 1795 DOI: 10.1021/nl100480y | Nano Lett. 2010, 10, 1793-–1798

keeping all other parameters of the resonator fixed. Figure1B shows the tunability of the bandwidth up to 2 orders ofmagnitude by simply changing the excitation force appliedto a nonlinear mechanical nanoresonator.

In addition, the drop frequency of the nonlinear nan-oresonator is very sensitive to the magnitude of dampingassociated with the resonance system under various ambi-ent conditions, according to eq 3. The damping responsivityof the drop frequency is estimated according to the changein the damping coefficient �, where � ) 1/(2Q):

The shift in drop frequency for a 1% change in the dampingcoefficient is plotted in Figure 2B and again shows the muchenhanced sensitivity offered by the intrinsically nonlinearnanoresonator compared to that offered by the linear one.

We fabricated a nonlinear nanoresonator using a doublyclamped carbon nanotube (CNT), of which a scanningelectron microscope (SEM) image is displayed in Figure 3A.The device was fabricated through micromachining andnanomanipulation. A silicon (100) wafer was coated with a500-nm-thick silicon nitride layer followed by 1.5-µm-thicksilicon dioxide. A thin Cr/Au layer was then sputter coatedonto the silicon wafer and subsequently patterned throughphotolithography to form a three-electrode layout. Thissilicon wafer was back etched in KOH to make a thinmembrane of silicon dioxide under the electrodes. Thewindow was then milled with a focused ion beam to createthree suspended electrodes. Three vertical platinum postswere fabricated onto these three electrodes through electron-beam-induced deposition. A high -quality multiwalled CNTproduced with an arc discharge was then selected andmanipulated inside an electron microscope and suspendedbetween two of the platinum posts with both ends fixed withelectron-beam-induced deposition of a small amount ofplatinum. The remaining platinum post was used as thedriving electrode for applying a localized oscillating electricfield to drive the oscillation of the CNT. The overall designof the device maximized the localization of the excitationforce applied to the CNT beam (Supporting Information).According to the previous discussion, the localization of theapplied force is necessary to create the strong geometricnonlinearity in the resonance system.

To acquire the response spectrum of the nanoresonator,the frequency of the applied ac driving voltage (Vac) wasswept up and then down while the oscillation amplitude inthe middle of the CNT was measured from the acquiredimages in a SEM at room temperature and at a vacuumpressure of ∼10-6 Torr. To evaluate the effect of added masson the dynamic behavior of the nanoresonator, a smallamount of platinum was deposited on the middle of the CNT

with electron-beam-induced deposition, and its mass wasestimated from the measured dimension.

Figure 3B shows the acquired response spectrum for anonlinear nanoresonator incorporating a CNT of 2L) ∼6.2µm and D ) ∼33 nm driven with an ac signal of 10 Vamplitude. The initiation of the oscillation started at around4 MHz, near the natural resonance frequency of this doublyclamped CNT. The amplitude of the resonance oscillationincreased continuously during the increasing frequencysweep up to 14.95 MHz, at which point the amplitudesuddenly dropped to zero. This response closely resembledwhat was modeled previously for an intrinsically nonlinearnanoresonator and corresponded to a resonance bandwidthof over 10 MHz. During the ensuing decreasing frequencysweep, the resonator stayed mostly in a nonresonance stateuntil reaching the neighborhood of the natural resonance

R� ) lim∆�f0

∆fdrop

∆�)

fo

�rdrop( rdrop

2 - 1

2rdrop2 - 1) (5)

FIGURE 3. Fabricated nonlinear carbon nanotube nanoresonator andits resonance response. (A) SEM images in the top view and tiltedview of a representative nanoresonator employing a CNT suspendedbetween and fixed at both ends on the fabricated platinum electrodeposts. The acquired response spectra of a CNT (2L ) ∼6.2 µm, D )∼33 nm) nonlinear nanoresonator driven with ac voltage signals of(B) 10 and (C) 5 V amplitude.

© 2010 American Chemical Society 1796 DOI: 10.1021/nl100480y | Nano Lett. 2010, 10, 1793-–1798

frequencies of the CNT, where transitions back to resonanceoscillations occurred. By fitting the obtained drop-jump andup-jump frequencies with the model prediction, the drivingforce was estimated to be ∼7 pN and the Q factor of thesystem was ∼260, which were in agreement with theestimate from an electrostatic analysis based on the experi-mental setup (Supporting Information) and the reportedQ-factorvaluesfortypicalCNT-basedresonators,28respectively.

The occurrence of multiple up-jump transitions during thedecreasing frequency sweep appears to be due to theexistence of multiple natural resonance frequencies in amultiwalled CNT and thus multiple modes of resonance. Intheory,29 there are the same number of fundamental fre-quencies and resonance modes as the number of cylindersin a multiwalled CNT. In a recent computational study,30 itwas shown that in the strongly nonlinear regime there canbe coupling between multiple radial and axial modes of adouble-walled CNT, with van der Waals forces provokingdynamic transitions between the modes of the inner andouter walls. Such strongly nonlinear modal interactions canbe studied using asymptotic techniques in the context ofcoupled nonlinear oscillators.31

The existence of multiple natural modes in this multi-walled CNT-based nonlinear resonator can also be revealedin an increasing frequency sweep when the driving force isreduced. Figure 3C shows the response spectrum acquiredfrom the same resonator when the applied ac amplitude wasreduced to 5 V. Two distinct resonance modes were excitedin this case. The first mode appeared at around 4 MHz, andits drop jump occurred at 7.05 MHz. The second mode wasthen initiated right after the drop jump of the first mode andjumped down at 14.15 MHz. As shown previously, when thedriving force was increased, it appeared that the first moderesonance became dominant and suppressed the initiationof the second mode in the increasing frequency sweepwhereas in the decreasing frequency sweep, because therewas no dominant mode, those modes were excited in theneighborhoods of their linearized resonance frequencies.Similar observations have been reported in coupled nonlin-ear resonators20 but not, until now, for a multiwalled CNTintentionally operated in a highly nonlinear regime.

The mass-sensing capability of the nonlinear nanoreso-nator is evaluated by adding a small platinum deposit to themiddle of a suspended CNT, as shown in Figure 4. In thiscase, the CNT is ∼6.0 µm long and ∼26 nm in diameter.The added mass caused both a 2.0 MHz shift of the linearizednatural frequency, approximately defined as the frequencywhere the resonance oscillation was initiated, and a moresignificant 7.4 MHz shift of the drop frequency. The addedmass was estimated to be ∼7 fg on the basis of the dimen-sions of the deposit measured from the acquired SEMimages (Supporting Information). The corresponding massresponsivity calculated from the shift in the drop frequency(Rm,nonlinear ) 1.06 Hz/zg) was 3.7 times that calculated fromthe linearized natural frequency (Rm,linear ) 0.29 Hz/zg).

These mass responsivity values compare favorably with ourmodel prediction from which Rm,nonlinear ) 2.18 Hz/zg andRm,linear ) 0.60 Hz/zg. The magnitude of the shift in the dropfrequency increases with the increase in added mass whilein the meantime the bandwidth of the resonance decreases(Supporting Information).

This demonstration of a relatively simple nonlinear nan-oresonator incorporating intrinsic geometric nonlinearityoffers a model system for expanded studies of the nonlinearresonance behavior, which has been shown to be rich inphysics and in opportunities for practical applications on themacroscale, now down to the nanoscale. In this study, weshow a prototype as a mass sensor that can be applied toother types of high-sensitivity sensing applications. Com-pared with the sensing principles applied in nanoscale linearresonance systems, a nonlinear resonance system canexploit the instabilities intrinsically existing within the sys-tem that are very sensitive to external perturbation. Thelarge oscillation amplitude and the sharp transition at thesebifurcation points are all very favorable characteristics fromthe precision measurement point of view, which can poten-tially enhance the measurement sensitivity of a practicalsensing system; the large oscillation amplitude implies lesssusceptibility of the resonance system to thermal noise, anda sharp transition allows for a narrow measurement band-width. However, the use of such a nonlinear system for high-sensitivity sensing applications may ultimately rely on ourmore detailed understanding of the robustness of suchtransitions related to the instability and the effect of externalnoise (thermal noise and stochastic perturbations).21 Asshown in other studies,19-21,23 the complex dynamics of

FIGURE 4. Mass sensing with a nonlinear carbon nanotube nan-oresonator. (A) SEM image showing the Pt deposit in the middle ofa suspended CNT (2L ) ∼6.0 µm, D ) ∼26 nm). (B) The acquiredresponse spectrum of this CNT nonlinear nanoresonator (O) beforeand (•) after depositing a center mass with electron-beam-induceddeposition.

© 2010 American Chemical Society 1797 DOI: 10.1021/nl100480y | Nano Lett. 2010, 10, 1793-–1798

nonlinear systems and the instabilities associated with it aretheoretically predictable and are robust enough for practicaluse in sensing applications.

Moreover, nonlinear resonance systems have recentlybeen explored for more effective energy harvesting32 andmore efficient mechanical damping applications because oftheir broadband resonance nature and unique characteristicsfavoring directional energy transfer26 in coupled systems.As demonstrated in this study, such broadband resonancebehavior is preserved on the nanoscale and thus can bepotentially exploited for nanoscale energy-harvesting andenergy-transfer applications.

The design and demonstration of a simple nonlinearmechanical resonator, which operates on the nanoscale,expands the bandwidth of the resonance response, is tun-able over a broad frequency range, and provides the inher-ent instabilities that can be exploited for sensing applica-tions, offers new conceptual strategies for the developmentof nanoscale electromechanical devices. Such developmentis further facilitated by the inherent ease of realizing intrinsicgeometric nonlinearity in a nanoscale resonator and canthus be readily integrated into the ongoing development ofnanoscaleelectromechanicalsystemstoextendtheiroperation.

Supporting Information Available. Derivation of the dropfrequency. Estimation of the applied driving force. Young’smodulus and the natural frequency of carbon nanotubes.Added mass produced with electron-beam-induced Pt depo-sition. Shift in the drop frequency and decrease in thebandwidth with increasing center mass. This material isavailable free of charge via the Internet at http://pubs.acs.org.

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© 2010 American Chemical Society 1798 DOI: 10.1021/nl100480y | Nano Lett. 2010, 10, 1793-–1798