Dynamic Interactions between Oscillating Cantilevers: Nanomechanical Modulation using Surface Forces
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Dynamic Interactions between Oscillating Cantilevers:Nanomechanical Modulation using Surface Forces
O. Basarir and K. L. Ekinci ∗ Mechanical Engineering Department and the Photonics Center,Boston University, Boston, Massachusetts 02215
Dynamic interactions between two oscillating micromechanical cantilevers are studied. In theexperiment, the tip of a high-frequency cantilever is positioned near the surface of a second low-frequency cantilever. Due to the highly nonlinear interaction forces between the two surfaces,thermal oscillations of the low-frequency cantilever modulate the driven oscillations of the high-frequency cantilever. The dissipations and the frequencies of the two cantilevers are shown to becoupled, and a simple model for the interactions is presented. The interactions studied here maybe useful for the design of future micro and nanoelectromechanical systems for mechanical signalprocessing; they may also help realize coupled mechanical modes for experiments in non-lineardynamics.
Miniaturized mechanical devices [1], called micro andnanoelectromechanical systems (MEMS and NEMS), aresteadily progressing toward attaining the speed and effi-ciency of their electronic counterparts. A mechanical sig-nal processor [2] may soon become a reality if the primaryoperations during the processing can be performed me-chanically — i.e. , using the mechanical motion of MEMSand NEMS. There is thus a focused research effort to re-alize micro- and nano-mechanical switching [3], mixing[9][4][8], amplification [5] and modulation[6, 7].Here, we study dynamic interactions between two os-cillating micromechanical cantilevers and harvest theseinteractions for mechanical signal modulation and de-tection. In the experiment, the carrier signal from ahigh-frequency microcantilever oscillator is modulated bylow-frequency thermal oscillations of a second microcan-tilever by simply bringing the two microcantilevers closetogether. The approach relies upon the strong inherentnonlinearity of the interaction force between two surfacesin close proximity and offers several advantages. Themodulation is purely mechanical, and mechanical signalsneed not be converted to electrical signals. The strengthof the coupling between the two mechanical signals, andhence the modulation index, can be adjusted by chang-ing the distance between the two microcantilevers. Con-servative and dissipative components of the interactionenable tuning of the frequencies and dissipation. Con-versely, monitoring the modulation on the carrier signalallows for sensitive mechanical displacement detection.Because the approach offers prospects for creating cou-pled mechanical modes [10] with tunable coupling, it maybe useful in fundamental investigations in nonlinear dy-namics.Our approach is derived from dynamic mode atomicforce microscopy (AFM). Related to our work here, var-ious AFM modalities have been used to detect the mo-tion of micro- and nano-mechanical resonators. In these ∗ Author to whom correspondence should be addressed; electronicmail: [email protected] experiments, a resonant mode of the small device un-der study is excited, e.g. , electrostatically or by usinga piezoelectric shaker. An AFM cantilever is broughtin close proximity of the resonator to probe its oscil-lations. Several groups have used contact interactions[11–13]; tapping mode AFM has also been employed forless intrusive probing [14–16]. In addition, other AFM-based approaches, including acoustic force microscopy[17] and electrostatic scanning probe microscopy [18],have also been applied to measuring small oscillations.In this work, we extend the above-mentioned efforts tonon-contact mode AFM and show that even non-contactAFM can perturb small resonators significantly.At the core of our experiment are two micro-cantilevers, which primarily interact through non-contactforces, as shown in Fig. 1(a). The cantilevers are of dif-ferent sizes and oscillate at their fundamental flexuralresonance frequencies. The smaller high-frequency one(hereafter labeled with the subscript “h”) has an unper-turbed fundamental flexural resonance at f h ≈ . f l ≈ . in vacuo parameters for both can-tilevers are listed in Table I. The cantilevers remain in-side an ultrahigh vacuum (UHV) chamber at a pressure p < × − Torr during the experiments so that gasdamping is not relevant [19]. The low-frequency (bot-tom) cantilever is fixed onto a sample holder and is ex-cited by thermal fluctuations at room temperature. Thehigh-frequency (top) cantilever sits on a nano-positionerand is driven by a piezo-shaker at its base. The responseof the high-frequency cantilever is measured using a stan-dard optical beam-deflection method [21]. The outputof the optical transducer is divided between a spectrumanalyzer, a self-oscillating loop and a detection-feedbackfeedback loop, as shown in Fig. 1(a). The self-oscillatingloop, shown by the dashed box in Fig. 1(a), maintainsthe high-frequency cantilever oscillating at resonance ata constant amplitude. The detection-feedback loop hasa large time constant (0.01 s . τ .
145 150 155 160 165 170-100-80-60-40 S p ec t r a l D e n s it y ( d B m / H z ) Frequency (kHz) x10 -22 x10 -22 x10 -22 x10 -22 S p ec t r a l D e n s it y ( m / H z ) Frequency (kHz) (a)(b)
SpectrumAnalyzerLaser Φ A A
AmplitudeControllerPD DemodulatorPID z h ( t ) z l ( t ) PZSNP z l Self-oscillating Circuit z h ( t ) z h z l ( t ) Set Point
Modulated SignalCarrierModulation Signal
FIG. 1: (a)
Schematic of the experimental setup. The op-tical transducer (PD: photodetector), amplifiers (A), phaseshifter (Φ) and amplitude controller are the components ofa positive feedback circuit (dashed box), which drives thehigh-frequency cantilever at resonance at a prescribed am-plitude via a piezo-shaker (PZS). The demodulator and PIDcontroller form a detection circuit, which keeps the frequencyshift (and hence the average gap between cantilevers) at a de-sired value. The inset shows the lumped mass models for thetwo cantilevers. (b)
Spectral density of the high-frequencycantilever oscillations. Two arrows at the upper and lowersidebands of the carrier at 153.8 kHz correspond to the ther-mal oscillations of the low-frequency cantilever. The insetshows the upper sideband in displacement units. average gap between the cantilevers at a prescribed valueand compensates for drifts.In the experiments, the high-frequency cantilever os-cillates coherently at a constant r.m.s oscillation ampli-tude of ∼
10 nm; the r.m.s. thermal oscillation am-plitude for the low-frequency cantilever remains around q k B Tk l ∼ . k B T is the thermal energy and k l is the (unperturbed) spring constant. The tip of thehigh-frequency cantilever is brought towards the free endof the low-frequency cantilever using the nano-positioner,and the spectrum of the oscillatory signal on the pho-todiode is measured. Fig. 1(b) shows a typical spec-tral density measurement. The dominant peak shown at f h ≈
153 kHz corresponds to the self-oscillations of the high-frequency cantilever. This can be regarded as thehigh-frequency carrier signal. At f h ± f l ≈ ± ∼ × − m/Hz / .Advancing the nano-positioner leads to changes inthe actual gap between the two cantilevers, resulting inchanges in the measured response. Returning to the insetof Fig. 1(a), we note that the time-dependent positionsof the two cantilevers are z h ( t ) and z l ( t ) with respect toa fixed reference point. The interaction force F , whichhas an attractive van der Waals component (see belowfor a details), results in changes in the average positions,¯ z l and ¯ z h . In our coordinate system, the average gapis ≈ ¯ z h − ¯ z l . Because the low-frequency cantilever istwo orders of magnitude softer than the high-frequencycantilever, we estimate that the average van der Waalsattraction mostly bends the low-frequency cantilever to-ward the high-frequency cantilever (upwards in Fig. 1(a))as the nano-positioner is advanced to decrease the gapbetween the cantilevers (i.e., to reduce ¯ z h ). The aver-age position of the high-frequency cantilever, ¯ z h , can betaken to be the same as that of the nano-positioner (upto an additive constant).When far away from each other, the cantilevers do notinteract and oscillate at their respective unperturbed res-onance frequencies, f h and f l . As they come closer,the coupling grows and the thermal oscillations of thelow-frequency cantilever become observable in the side-bands of the carrier. As the separation is reduced, thelinewidths and the frequencies of both cantilevers change.The motion of the high-frequency cantilever (carrier) re-mains mostly sinusoidal with relatively little perturba-tion to its resonance frequency and linewidth, since themodulating signal in the sideband is orders of magni-tude smaller than the carrier. The low-frequency can-tilever, on the other hand, suffers large changes in fre-quency and linewidth. Fig. 2(a) shows the sidebandpeaks corresponding to the low-frequency cantilever os-cillations. The zero in the frequency axis here corre-sponds to the resonance frequency f l , which decreasesas the nano-positioner advances to bring the two can-tilevers together. The modulation increases because the TABLE I: Unperturbed parameters for the two Silicon micro-cantilevers used in our experiments. The stiffness k valuesare provided by the manufacturer. The effective mass m iscalculated using k and f . Both k and m are approximate. l × w × t f = ω π Q k mµ m kHz N/m kg225 × . × ×
30 3 × − × × × . × − f l / f l Displacement (nm) (a) (b) (c) -0.04 -0.02 0.00 0.02 0.040.00.51.0 N o r m a li ze d S p ec t r u m Frequency (kHz) f l f h / f h Displacement (nm) -4 -3 -2 Q l - - Q l - Displacement (nm) -6 -6 -5 -5 Displacement (nm) Q h - - Q h - FIG. 2: (a)
Normalized sideband signals. The signals are normalized using the highest measured signal values. The data tracesare taken at the positions shown with the arrows in (b) . Because the resonance frequency f l shifts significantly, the frequencyaxis is displayed as measured from the resonance frequency f l . (b) The observed shift in the resonance frequencies of bothcantilevers. (c)
The change in the dimensionless dissipation of both cantilevers. The dissipation increases dramatically in theshaded region, suggesting that soft contact interactions start to become dominant. The error bars in all the data are smallerthan the symbol sizes unless shown explicitly. The snap to contact with accompanying instabilities in the high-frequency signaldetermines the position of zero in the x -axes of the plots in (b) and (c). mechanical coupling increases. In addition, the dissipa-tion (linewidth) increases.Figure 2(b) and (c) show results from systematic ex-periments as the nano-positioner brings the two can-tilevers together, i.e., the gap between the cantilevers ischanged. Returning to Fig. 1(a), we describe how theexperiment is performed. A frequency shift for the high-frequency cantilever is prescribed; the nano-positioner (inconjunction with the detection circuit) brings the twocantilevers together until this frequency set point is at-tained. The PID controller keeps this frequency shift (setpoint) fixed, thereby ensuring a constant average gap.At this set point, line-shape for the low-frequency can-tilever is recorded. At very small separations, the twocantilevers snap to hard contact, causing the carrier sig-nal to become unstable.In Fig. 2(b) (main), the frequency f l of the low-frequency cantilever is plotted as a function of the nano-positioner displacement. The inset of Fig. 2(b) similarlyshows f h of the high-frequency cantilever as a functionof the nano-positioner displacement. The zeros of the x -axes are taken to be the contact position, where thehigh-frequency cantilever can no longer oscillate stably.There is some degree of uncertainty in the position ofthis zero. The estimated region of soft contact betweenthe two cantilevers is shown by the shading around zeroin the main figure. This estimation is simply based onthe observation that the dissipation of both cantileversincreases more steeply for displacements . vs. tip-sampledistance curves taken in non-contact AFM work [20],where attractive forces are dominant. However, there is a significant difference. Because the low-frequency can-tilever is soft, the nominal displacement obtained fromthe nano-positioner cannot be related to the tip-samplegap in a straightforward manner. The interaction rangein Fig. 2(b) extends over 10 nm. Due to the attractiveforce between the two cantilevers, the soft cantilever fol-lows the stiffer high-frequency cantilever, as the two arebrought together. While both resonance frequencies f l and f h shift in a qualitatively similar fashion, the mag-nitudes of the changes in f l and f h are quite different.We confirm that the data possess the same features atlarger oscillation amplitudes .
40 nm (not shown) ofthe high-frequency cantilever. In all the measurementsreported here, non-contact or (intermittent) soft contactinteractions dominate, and the average force between thecantilevers remains attractive.Figure 2(c) shows the change in the dimensionless dis-sipation of each cantilever as a function of the nano-positioner displacement. Here, the change is obtainedby subtracting the intrinsic value of the dimensionlessdissipation, Q − , from the measured value Q − for eachcantilever. For the low-frequency cantilever, all the datapoints are obtained by fitting Lorentzians to resonanceline-shapes, such as those shown in Fig. 2(a). At largeseparations between cantilevers, the data appears nois-ier. This is because the signal size becomes smaller, andthe fits are not as accurate. For the high-frequency can-tilever, the data are extracted from the drive force (volt-age) applied to the piezo-shaker, given that the stiffnessof the high-frequency cantilever does not change appre-ciably and the amplitude controller keeps the oscillationamplitude constant[20]. The general trend is that dissi-pation increases as the separation decreases. However,the observed dissipation increase in the low-frequencycantilever is much more dominant.We now describe the coupled resonator dynamics. Thedynamic variables used in the equations below can beidentified in Fig. 1(a). Before analyzing the interact-ing cantilevers, we formulate the dynamics of individ-ual cantilevers far apart from each other. The one-dimensional lumped equation of motion for the high-frequency cantilever can be written as m h ¨ z h + m h ω h Q h ˙ z h + m h ω h ( z h − ¯ z h ) = F d ( t ), where the drive force is F d ( t ) = R ( z h ( t − t φ ) − ¯ z h ), with R being the gain, t φ being the loop delay of the (self-oscillating) loop, and m h being the effective mass of the cantilever. We usethe simplifying assumption that the cantilever always vi-brates sinusoidally at resonance at a constant amplitude,and the role of the external sustaining circuit is to sim-ply compensate for the energy losses. Thus, we can write z h ( t ) ≈ ¯ z h + A h sin ω h t , where A h remains constant and ω h does not change appreciably, consistent with experi-mental observations [Fig. 2(b) inset]. The low-frequencycantilever is driven by random thermal noise, but oscil-lates mostly sinusoidally because of its high quality factor(10 ≤ Q l . ). Modeling its displacement as narrow-band noise [22], we write z l ( t ) ≈ ¯ z l + A l ( t ) sin( ω l t + ψ l ( t )), where A l ( t ) and ψ l ( t ) are slowly-varying enve-lope and phase functions. Hence, both cantilevers canbe treated as simple one-dimensional oscillators when noperturbations are present: m i ¨ z i + m i ω i ( z i − ¯ z i ) ≈ i = l, h . Thus, both cantilevers oscillate sinu-soidally with ω h ≫ ω l , and each will tend to respondstrongly to the perturbation at its own resonance fre-quency. For our system, when the gap between the can-tilevers is large, the generalized non-contact interactionforce can be expressed in terms of the coordinates andtheir time derivatives: F = F ( z h , ˙ z h , z l , ˙ z l ) [20]. Thisforce can further be broken down into conservative anddissipative components as F = F diss + F cons [23].The dissipative forces F diss on the high-frequency andlow-frequency cantilevers can be approximated as [23] − γ ( ˙ z h − ˙ z l ) and − γ ( ˙ z l − ˙ z h ), respectively, based uponphenomenological arguments. Here, γ is a function ofgap: γ ( z h , z l ) = γ e − C ( z h − z l ) , where γ and C are em-pirical constants. The exponentially decaying form en-sures that F diss becomes weaker with increasing separa-tion. Interacting only via the dissipative force F diss , thetwo cantilevers can be described by the following coupledequations: m l ¨ z l + m l ω l ( z l − ¯ z l ) ≈ − γ ( ˙ z l − ˙ z h ) , (1a) m h ¨ z h + m h ω h ( z h − ¯ z h ) ≈ − γ ( ˙ z h − ˙ z l ) . (1b)To make further progress, we approximate the func-tion γ ( z h , z l ) with ¯ γ ≈ γ e − C (¯ z h − ¯ z l ) . Because of thediscrepancy in the two oscillatory time scales, the lowfrequency cantilever notices only the average position ofthe high-frequency cantilever, z h = ¯ z h . It may thus bejustifiable to set ˙ z h ≈ m l ¨ z l +¯ γ ˙ z l + m l ω l ( z l − ¯ z l ) ≈
0. Similarly, the dissipativeforce acting on the high-frequency cantilever is approxi-mately − ¯ γ ˙ z h because A h ω h ≫ A l ω l . Thus, we arrive at the approximation m h ¨ z h + ¯ γ ˙ z h + m h ω h ( z h − ¯ z h ) ≈ γ terms give rise to the energy dissipa-tion in both cantilevers. Thus, one should be able to re-late the measured dissipation changes in the coupled can-tilever system. In other words, m h ω h (cid:0) Q h − − Q h − (cid:1) ∼ m l ω l (cid:0) Q l − − Q l − (cid:1) at a given gap. At the largest gapvalues, where the perturbation is weak and the approxi-mations should hold better, we find the right hand sideand the left hand side to be of the same order of magni-tude (r . h . s ≈ × − kg/s and l . h . s ≈ × − kg/s)using the numbers from Table 1 and data from Fig. 2(c).Given that the values in Table 1 are approximate, this isquite satisfactory and suggests that our approximationsare reasonable.Returning now to the conservative component of theinteraction force, we take F cons = − HR ( z h − z l ) , as sug-gested by numerous AFM experiments [23]. Here, H is the Hamaker constant, and R is the tip radius (ofthe high-frequency cantilever). We emphasize that thissimple form is valid when the gap is large (non-contactregime), and the attractive van der Waals force dom-inates. Because the thermal oscillation amplitude (ofthe low-frequency cantilever) remains extremely small,we expand the force around ¯ z l , obtaining˜ F cons ≈ − HR ( z h − ¯ z l ) − HR ( z h − ¯ z l ) ( z l − ¯ z l ) . (2)Note that the sign of ˜ F cons must be adjusted such thatit remains attractive for both cantilevers. As above, weset z h = ¯ z h in ˜ F cons in the equation of motion of thelow-frequency cantilever: m l ¨ z l + m l ω l ( z l − ¯ z l ) ≈ ˜ F cons .This yields ω l ≈ ω l − HRm l (¯ z h − ¯ z l ) . (3)Finally, the source of the modulation can be identifiedas the HR ( z l − ¯ z l )( z h − ¯ z l ) term in the drive force in m h ¨ z h + m h ω h ( z h − ¯ z h ) ≈ − ˜ F cons . This term can be re-writtenas HR ( z l − ¯ z l )(¯ z h − ¯ z l ) (cid:16) zh − ¯ zh ¯ zh − ¯ zl (cid:17) and expanded, with the leading or-der term in ( z h − ¯ z h ) being − HR ( z l − ¯ z l )( z h − ¯ z h )(¯ z h − ¯ z l ) . Includ-ing this term in the equation of motion, we derive ω h ≈ ω h − HRm h (¯ z h − ¯ z l ) A l sin( ω l t + ψ l ) . (4)This is the source of the frequency modulation. The mod-ulation index can be found as [22] M = HRA l m h ω h ω l (¯ z h − ¯ z l ) ,with the ratio of the power in the carrier to that in a(single) sideband being M . For the measurement shownin Fig. 1(b), M ≈ . × − . Using H ∼ − J, R ∼
50 nm and experimental values for the remainingparameters, we find ¯ z h − ¯ z l ∼ A h staysconstant and the cantilevers oscillate sinusoidally. Tofully account for the nonlinear interaction, a better modelmust allow for the amplitudes to be affected, with somedegree of amplitude modulation as well as nonlinearityin the oscillations. Furthermore, the presented modelis expected to become inaccurate as the perturbationgrows ( i.e. , the gap becomes smaller), and the dynam-ics becomes complicated due to stronger non-linearities, hysteresis and larger fluctuations. One can incorporatecontact effects by using Derjaguin-M¨uller-Toporov in-teraction. Regardless, the data and the simple modelpresented here may be useful for designing MEMS andNEMS devices for future applications. Given that theinteraction between the two cantilevers can be tuned ef-ficiently by reducing the gap between them, one can alsostudy non-linear dynamics of coupled oscillators.We acknowledge support from the US NSF (throughGrant Nos. ECCS-0643178 and CMMI-0970071). [1] K. L. Ekinci and M. L. Roukes, Rev. Sci. Instrum. ,061101 (2005).[2] R. H. Blick, H. Qin, H.-S. Kim, R.Marsland, New J.Phys. , 241 (2007).[3] J. Zou, S. Buvaev, M. Dykman, and H. B. Chan, Phys.Rev. B , 155420 (2012).[4] R. Lifshitz and M. C. Cross, Nonlinear dynamics ofnanomechanical and micromechanical resonators, Re-views of nonlinear dynamics and complexity 1 (2008):1-48.[5] I. Kozinsky, H. W. Ch. Postma, O. Kogan, A. Husain,and M. L. Roukes, Phys. Rev. Lett. , 207201 (2007).[6] U. Kemiktarak, T. Ndukum, K. C. Schwab, K. L. Ekinci,Nature , 85 (2007).[7] M. R. Kan, D. C. Fortin, E. Finley, K-M. Cheng, M. R.Freeman, and W. K. Hiebert, Applied Physics Letters 97,no. 25 (2010): 253108-253108.[8] A. T. Alastalo, M. Koskenvuori, H. Seppa, J. Dekker, InMicrowave Conference, 2004. 34th European (Vol. 3, pp.1297-1300) IEEE (2004, October).[9] I. Bargatin, E. B. Myers, J. Arlett, B. Gudlewski, andM. L. Roukes, Appl. Phys. Lett. , 133109 (2005).[10] C. D. Honig, M. Radiom, B. A. Robbins, J. Y. Walz,M. R. Paul, and W. A. Ducker, Appl. Phys. Lett. ,053121-053121 (2012).[11] H. Safar, R. N. Kleiman, B. P. Barber, P. L. Gammel,J. Pastalan, H. Huggins, L. Fetter, and R. Miller, Appl. Phys. Lett. , 136 (2000).[12] S. Ryder, K. B. Lee, X. Meng, and L. Lin, Sens. Actua-tors, A , 135 (2004).[13] A. S. Paulo, J. Bokor, R. T. Howe, R. He, P. Yang, D.Gao, C. Carraro, and R. Maboudian, Appl. Phys. Lett. , 053111 (2005).[14] A. S. Paulo, J. P. Black, R. M. White, and J. Bokor,Appl. Phys. Lett. , 053116 (2007).[15] D. Garcia-Sanchez, A. S. Paulo, M. J. Esplandiu, F.Perez-Murano, L. Forro, A. Aguasca, and A. Bachtold,Phys. Rev. Lett. , 085501 (2007).[16] B. Ilic, S. Krylov, L. M. Bellan, and H. G. Craighead, J.Appl. Phys. , 044308 (2007).[17] F. Sthal and R. Bourquin, Appl. Phys. Lett. , 1792(2000).[18] K. M. Cheng, Z. Weng, D. R. Oliver, D. J. Thomson, G.E. Bridges, J. MEMS , 1054 (2007).[19] K. L. Ekinci, V. Yakhot, S. Rajauria, C. Colosqui and D.M. Karabacak, Lab Chip , 3013 (2010).[20] S. A. Morita, R. A. Wiesendanger, E. A. Meyer, (2002).Noncontact atomic force microscopy (Vol. 1). Springer.[21] G. Meyer and N. M. Amer, Appl. Phys. Lett.53