Interactions between directly and parametrically driven vibration modes in a micromechanical resonator
H. J. R. Westra, D. M. Karabacak, S. H. Brongersma, M. Crego-Calama, H. S. J. van der Zant, W. J. Venstra
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Interactions between directly and parametrically driven vibration modes in amicromechanical resonator
H. J. R. Westra, ∗ D. M. Karabacak, S. H. Brongersma, M.Crego-Calama, H. S. J. van der Zant, and W. J. Venstra Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628CJ Delft, The Netherlands Holst Centre / imec the Netherlands, High Tech Campus 31, 5656 AE, Eindhoven, The Netherlands (Dated: September 17, 2018)The interactions between parametrically and directly driven vibration modes of a clamped-clamped beam resonator are studied. An integrated piezoelectric transducer is used for direct andparametric excitation. First, the parametric amplification and oscillation of a single mode are ana-lyzed by the power and phase dependence below and above the threshold for parametric oscillation.Then, the motion of a parametrically driven mode is detected by the induced change in resonancefrequency in another mode of the same resonator. The resonance frequency shift is the result ofthe nonlinear coupling between the modes by the displacement-induced tension in the beam. Thesenonlinear modal interactions result in the quadratic relation between the resonance frequency ofone mode and the amplitude of another mode. The amplitude of a parametrically oscillating modedepends on the square root of the pump frequency. Combining these dependencies yields a lin-ear relation between the resonance frequency of the directly driven mode and the frequency of theparametrically oscillating mode.
PACS numbers: 85.85.+j, 05.45.-a, 46.32.+x
I. INTRODUCTION
Parametric amplification and oscillations occur whenin a resonant system, one of the system parameters(e.g. spring constant, effective mass) is modulated. Theprinciple is used in low-noise electronic amplifiers [1, 2]and to increase the broadband gain in fiber optics [3–5]. In mechanical resonators, parametric oscillationsare typically obtained by modulation of the springconstant [6–9]. Applications of parametric resonancesin nano- and micro electromechanics [10] (NEMS andMEMS) include quality (Q-)factor enhancement [11, 12]and bit storage and bit flips using the bistable phase ina parametric oscillator [13, 14]. Parametric amplifica-tion can also be used for noise-squeezing in a coupledqubit-resonator system [15] and was recently observedin carbon nanotube resonators [16].Another interesting phenomenon in NEMS is theinteraction between different vibration modes. Moti-vated by the trend towards large scale integration ofresonators, researchers study the interactions betweenseveral resonators [17]. Recently, nonlinear modalinteractions between two flexural modes in a clamped-clamped beam resonator [18–20] and a cantilever [21]have been reported: it has been shown that the reso-nance frequency of one mode depends quadratically onthe amplitude of another mode.Here, we explore the modal interactions betweena directly and a parametrically driven mode, yieldinga linear dependence of the resonance frequency of thedirectly driven mode on the pump frequency of the para- ∗ Electronic address: [email protected] metrically driven mode. In section II, the experimentalconditions are provided. The following section reportson a detailed analysis of the piezoelectrical parametricamplification of a single mode. Section IV discussesthe modal interactions between a directly driven anda parametrically pumped mode, and this is the centralresult of this work.
II. DEVICE DETAILS
The resonators are clamped-clamped beams fabricatedfrom 500 nm thick low-stress silicon nitride (SiN). Astack of platinum (Pt), aluminum nitride (AlN) and Pt(100-400-100 nm thick) is sputtered on top, to form anintegrated piezoelectric transducer. Fig. 1(a) shows ascanning electron micrograph of the device, the whitearrow indicates the transducer. The resonators arefreely suspended by a through-the-wafer etch. Twolengths are used: L = 500 and 750 µ m. The width ofboth resonators is 45 µ m. Details of the fabricationprocedure are described in Ref. [22]. An ac voltageon the piezo produces a force on the resonator and atthe same time modulates its spring constant. Boththe force and the spring constant depend linearlyon the voltage. The voltage on the piezo, V out , iscomposed of two frequencies, one to directly excitethe resonator and one to parametrically pump it, i.e. V out = V direct cos(Ω t ) + V pump cos(2Ω t + φ ), where Ω isthe drive frequency and φ the phase difference betweenthe two voltages.The motion of the resonator is measured usingan optical deflection setup, as depicted in Fig. 1(b). in(c) (d)
158 158.5 159 A f (kHz)
579 582 A f (kHz) f R,1 = 158.4 kHzQ = 6500 f R,2 = 580.1 kHzQ = 19600 (a) outdigital signal processor(b)AlNPtSiNSi
96 100 A f (kHz) Q = 58
FIG. 1: Measurement setup. (a) False-colored scanning elec-tron micrograph of a SiN beam with the piezo actuator (whitearrow) on top (scale bar is 20 µ m). (b) An optical deflectionsetup is used to detect motion in air and vacuum. The piezo-active AlN layer is depicted in red. The piezo actuator andphoto diode are connected to a digital signal processor. (c,d)Typical frequency responses of the first (c) and second (d)mode (amplitude A ) in vacuum. The inset in (c) shows thefrequency response in air, of the resonator with length 750 µ m, with a resonance frequency of 98 kHz. The responseof a damped-driven harmonic oscillator is fitted through theresponses to obtain Q-factors and resonance frequencies. Frequency spectrum and network analyzer measure-ments are implemented in a digital signal processor.Measurements are conducted in vacuum at a pressureof 10 − mbar and at atmospheric pressure. For directdriving, the frequency responses at the first mode andsecond mode in vacuum are shown in Fig. 1(c) and (d),with Q = 6500 and Q = 19600 [23]. III. PARAMETRIC AMPLIFICATION OF ASINGLE MODE
The time-dependent part of the equation of motionof the piezoelectric resonator including parametric mod-ulation of the spring constant is described by: (a)(b) Q V pump (V)0.811.21.4 -1 -0.5 0 0.5 1 G φ ( π rad) FIG. 2: Characterization of the parametric amplification inair. (a) The Q-factor enhancement is proportional to theparametric pump voltage. (b) Measured gain-phase relation;the blue line represents Eq. 2, with fit parameter k p /k t =0 . m ¨ u + mω R Q ˙ u +[ mω + k p sin(2Ω t + φ )] u + αu = F cos(Ω t ) . (1)Here, u ( t ) is the amplitude of the mode, m is the ef-fective mass and F the direct drive force, and ω R is theresonance frequency. The dots denote taking the deriva-tive to time. The spring constant is modulated at twicethe drive frequency Ω with modulation strength k p . α accounts for the Duffing nonlinearity with α > G isdefined by the ratio between the amplitude of the motionwith and without parametric drive, and can be calculatedfrom Eq. 1 [25, 26]: G ( φ ) = s cos ( φ/ k p /k t ) + sin ( φ/ − k p /k t ) . (2)This equation holds for small amplitude vibrations,where the nonlinearity can be neglected. Dependingon φ , the motion is amplified ( G >
1) or attenuated(
G < k p > k t with k t = 2 mω /Q , the resonator is parametricallyoscillating.Parametric behavior is demonstrated for a resonatorwith length 750 µ m vibrating in air, with f R , = 98kHz and Q = 58 (frequency response in the inset ofFig 1c). To amplify the motion, the resonator is drivenparametrically at 2 f R , with φ = − . π . Figure 2(a)shows the Q-factor of the resonator as a function ofthe parametric pump voltage. The Q-factor increasesby a factor of 1.7 when the parametric pump is 10 V.Furthermore, the phase dependence of the gain at 10V parametric pump is plotted in Fig. 2(b). The gainvaries periodically with the phase difference with aperiod of 2 π . The minimum gain is smaller than one,indicating destructive interference by an out-of-phaseparametric signal. Eq. 2 fits the measured data wellwith k p = 0 . k t . In these experiments the parametricdriving is below the parametric threshold k t . A furtherincrease of the pump voltage is not possible as this woulddamage the piezo-stack. To study parametric oscillation,further experiments are conducted in vacuum. Herethe Q-factor improves by two orders of magnitude(Fig. 1(c)), enabling post-threshold driving. f (kHz) -0.500.51 φ ( π r a d ) (a) (b)V pump = 80 mV V pu m p ( m V )
158 158.5 159 159.5 160 (d) (c) f (kHz) V pump = 105 mV A V pump = 90 mV f (kHz) A f (kHz) FIG. 3: Parametric oscillations of the first flexural mode invacuum. (a) Frequency spectra at three pump voltages, theparametric oscillation becomes visible when V pump >
85 mV.(b) Parametric tongue, showing frequency responses when theresonator is driven directly ( V direct = 5 mV) and parametri-cally past the instability threshold. Color indicates the am-plitude of oscillation. (c) The hysteresis between the forward(red) and reverse sweep (green) when driving parametrically( V pump = 95 mV). The blue dashed line shows the square rootdependence of the amplitude ( A ) on the frequency f . (d) Thephase dependence of the parametric oscillations at V pump =95 mV. The color indicates the amplitude of oscillation. Figure 3 summarizes the measurements of theparametric oscillations performed in vacuum. A 500 µ mlong resonator is used, for which the frequency responseis plotted in Fig. 1(c). Frequency spectra are measuredfor three parametric pump voltages in Fig. 3(a). At 80mV no sign of oscillation is observed (lower panel), andthe onset of parametric oscillation is found around 85mV as shown in the middle panel. A further increaseof the pump voltage (upper panel) results in a largeroscillation amplitude. Here, the nonlinear term in Eq. 1 results in an amplitude-dependent resonance frequency.Fig 3(b) shows network analyzer measurements of theresonator amplitude (color scale) as a function of thepump voltage. The resonator is driven directly andparametrically with φ = − . π . A direct drive signal,weak enough to operate the resonator in the linearregime when V pump = 0, is applied to initiate themotion. The motion of the weakly driven resonatoris coherently amplified by the parametric excitationand the amplitude increases with V pump . The observedfrequency stiffening is expected for a cubic springconstant α >
0. The oscillation sustains over a few kHzwhen the frequency is swept forward. The amplitudeshows a hysteretic response when the frequency is sweptback, see Fig. 3(c). The amplitude of the oscillationdepends on the square root of the frequency (dashedblue line) [25]. To study the relation between theparametric oscillation amplitude and the phase φ , theresonator is parametrically excited above the threshold.Fig. 3(d) shows the amplitude of the oscillation whenthe direct drive and frequency is swept while varyingthe phase difference. Depending on the phase betweenthe direct initiator drive and the parametric excitation,constructive or destructive interference occurs whichresults in amplification or attenuation of the motion in-duced by the initiator signal. The maximum parametricamplification is found at a phase difference of − π and π .The experiments described above clearly demonstratethe parametric behavior. IV. COUPLING BETWEEN PARAMETRICAND DIRECT DRIVEN MODES
We now investigate the interactions between thedifferent vibrational modes of the same mechanicalresonator, when one of the modes is parametricallyoscillating. This requires to monitor the response ofone mode while another mode is parametrically excited.In particular, the modal interactions between the firstand second mode are considered. First, we study theeffect of the parametric oscillations of the first mode,characterized in the previous section, on the resonancefrequency of the second mode. Fig. 4(a) shows frequencyresponses of the second mode, when the first mode isparametrically pumped around its resonance frequency.The first mode is only parametrically pumped and nodirect drive at the resonance frequency is applied. Belowthe resonance frequency of the first mode, no changein resonance frequency of the second mode is observed.Pumping at twice the resonance frequency, the firstmode starts to oscillate parametrically. This oscillationinduces a significant shift in resonance frequency of thesecond mode. By parametrically exciting the first mode,the resonance frequency of the second mode is tunedover more than 200 times the resonator linewidth. Thereis a linear relation of f R , on f pump , with sensitivity f R , /f pump , = 1 . f R ,i ∼ | A j | for modes i and j . Theamplitude of the parametric oscillation depends on thesquare root of the pump frequency | A j | ∼ p f pump , j [25],as experimentally verified in Fig. 3(c). Combining thesetwo dependencies, one expects f R ,i ∼ f pump ,j . This lin-ear dependence is clearly observed in the measurements,see Fig. 4(a).
308 310 312f pump , (kHz)580582584 f ( k H z ) pump , (MHz)154154.4154.8155.2 f ( k H z ) (a)(b)
1 22 1
FIG. 4: Interactions between a directly and parametricallydriven mode. (a) Frequency responses of the second modewhile varying pump frequency of the first mode. Color scaleindicates the amplitude of the second mode. The linear de-pendence of f R , on f pump , is observed as explained in thetext. (b) Reversed experiment; frequency responses of thefirst mode for varying the pump frequency of the second mode. We have also studied the influence of the paramet-rically excited second mode on the resonance frequencyof the first mode, i.e., the first mode is now probing thesecond mode, which is parametrically oscillating. Again,a linear dependence of the resonance frequency on theparametric pump frequency is found, as is shown inFig. 4(b). In this case, the sensitivity f R , /f pump , = 79mHz/Hz. As the pump frequency f pump , is increasedabove 1.165 MHz the parametric oscillation disappears,and the resonance frequency of the first mode jumpsback to its original value. At this point, the nonlinearitycauses the oscillation of the second mode to jump tothe low amplitude state, which is reflected by the sharptransition of the resonance frequency of the first mode.The large difference in sensitivity with the reversedexperiment in Fig. 4(a) indicates that parametric pump-ing of the second mode is less effective to change theresonance frequency of the first mode than vice versa.This can be understood since the first mode has thelargest oscillation amplitude and can provide the largesttension in the beam. V. CONCLUSION
The interactions between a directly and a parametri-cally oscillating mode of the same mechanical resonatorare studied. The parametric amplification and oscilla-tions of a clamped-clamped resonator with an integratedpiezoelectric transducer are investigated in detail. Thedependence of the oscillation amplitude on pump fre-quency and phase difference are in agreement with the-ory. In this work, we demonstrate that the parametricoscillation of one mode induces a change in the resonancefrequency of the other vibrational modes. This frequencychange is proportional to the pump frequency, as is shownfor the first and second mode. The sensitivity of the res-onance shift of the second mode on the pump frequencyof the first mode is found to be 1.4 Hz/Hz. When theexperiment is reversed, i.e. the oscillating second modeis detected by a shift in resonance frequency of the firstmode, the sensitivity is 79 mHz/Hz.
Acknowledgments
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