Dispersive and Dissipative Coupling in a Micromechanical Resonator

Mar 9, 2015 - A micromechanical resonator embedded with a nanomechanical resonator is developed whose dynamics can be captured by the coupled-Van ...
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Letter pubs.acs.org/NanoLett

Dispersive and Dissipative Coupling in a Micromechanical Resonator Embedded with a Nanomechanical Resonator I. Mahboob,* N. Perrissin, K. Nishiguchi, D. Hatanaka, Y. Okazaki, A. Fujiwara, and H. Yamaguchi NTT Basic Research Laboratories, NTT Corporation, Atsugi-shi, Kanagawa 243-0198, Japan S Supporting Information *

ABSTRACT: A micromechanical resonator embedded with a nanomechanical resonator is developed whose dynamics can be captured by the coupled-Van der Pol−Duffing equations. Activating the nanomechanical resonator can dispersively shift the micromechanical resonance by more than 100 times its bandwidth and concurrently increase its energy dissipation rate to the point where it can even be deactivated. The coupledVan der Pol−Duffing equations also suggest the possibility of self-oscillations. In the limit of strong excitation for the nanomechanical resonator, the dissipation in the micromechanical resonator can not only be reduced, resulting in a quality factor of >3× 106, it can even be eliminated entirely resulting in the micromechanical resonator spontaneously vibrating. KEYWORDS: Nanomechanical resonator, coupled resonators, nonlinear coupling, Van der Pol resonator

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The electromechanical system was fabricated from a GaAs/ AlGaAs high electron mobility transistor heterostructure sustaining a two-dimensional electron gas (2DEG) 90 nm below the surface; the fabrication procedure and the layer structure are detailed elsewhere.13 The electromechanical system consists of a micromechanical resonator embedded with a nanomechanical resonator with lengths, widths, and thickness of 200, 60, 1.3 μm and 50, 8, 1.3 μm, respectively. The coupling between the two resonators is expected to be dispersive, as previously pioneered between different modes in a single resonator,14−21 where the motion of one resonator creates tension that modifies the restoring force and hence the resonance frequency of the other resonator. In order to maximize this coupling between the two distinct resonators, a finite element analysis was carried out to identify the position of maximum strain on the micromechanical resonator that as expected occurred at the clamping points and consequently the nanomechanical resonator was defined at this optimized location as shown in Figure 1a. Both resonators in the electromechanical system can be independently probed via the piezoelectric transducers formed from the Au electrodes located on the surface and the 2DEG confined in a shallow mesa.13 Application of AC bias to one of the piezotransducers can activate the resonance of the associated mechanical element that can be detected via the motion induced piezovoltage in the other piezotransducer on the same mechanical element that is then further amplified via an on-chip Si-nanotransistor as detailed elsewhere.22 Probing

lectromechanical systems consisting of a mechanically compliable element embedded in an electrical transduction circuit enable the spectral purity of the mechanical resonators to be exploited for sensors,1 mechanical nonlinearities to be accessed for information processing applications,2 and even the quantum mechanical properties of the mechanics to be studied.3 The resonance dynamics of the mechanical element can be captured by the driven damped harmonic equation of motion in which the energy dissipation rate is proportional to the velocity of the mechanical element and is quantified by the term γ.4 More recently and primarily carbon-based mechanical systems have emerged in the guise of carbon nanotubes, graphene, and diamond, which exhibit energy dissipation that cannot be captured by γ alone.5−8 Instead the complete dissipation characteristics can only be accounted for with the inclusion of an additional nonlinear dissipation term η that is proportional to the mechanical element’s displacement squared. Apart from the underlying microscopic nature of this effect being unclear, the ability to harness this phenomenon has also remained elusive. Although III−V semiconductor-based mechanical resonators have universally exhibited linear dissipation, here a structurally interconnected system is developed in which a nanomechanical resonator is embedded within a micromechanical resonator. Previously identical mechanical resonators that are coupled via a structural overhang have been investigated9−12 but in the present approach, the large size mismatch between the mechanical resonators results in the emergence of nonlinear dissipation that not only can be completely controlled it can even be exploited to eliminate the velocity proportional dissipation entirely resulting in the micromechanical resonator spontaneously vibrating. © XXXX American Chemical Society

Received: November 18, 2014 Revised: February 5, 2015

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DOI: 10.1021/nl5044264 Nano Lett. XXXX, XXX, XXX−XXX

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Figure 1. (a) A false color electron micrograph of the electromechanical system consisting of a nanomechanical resonator embedded into the left clamping point of the micromechanical resonator as detailed in the raw inset. The 2DEG is confined in a 150 nm thick shallow mesa (blue) and the piezoelectric transducers are formed via the 150 nm thick Au electrodes (orange) located above. The positions of the piezoelectric transducers are also defined with reference to a finite element analysis that identifies the location of maximum strain from the mechanical motion. Also shown is a simplified measurement schematic where the ac signals are injected and extracted from the electromechanical system via superconducting niobium coaxial cables where the output signals are first amplified on-chip in Si-nanotransistors (blue triangles) and subsequently demodulated in a conventional homodyne measurement setup. (b) The frequency response of the microresonator as a function actuation amplitude ranging from 0.1 to 0.8 mVrms reveals a negative Duffing constant where the inset shows the corresponding mode shape extracted from a finite element calculation. (c) The frequency response of the nanoresonator as a function actuation amplitude ranging from 5 to 15 mVrms reveals a positive Duffing constant where the inset shows the corresponding mode shape extracted from a finite element calculation. Note both on-chip nanotransistors are biased far from their optimal points as drain currents approaching a microampere gave rise to resistive heating and thus only modest gains of 10−16 dB are achieved.

Figure 2. (a−d) The microresonator’s frequency response when harmonically probed with an amplitude of 100 μVrms while the nanoresonator is simultaneously harmonically pumped from 5, 10, 15, and 20 mVrms, respectively. Both the probe and pump frequencies have been normalized to the micro- and nanomechanical resonances, respectively, corresponding to 0 Hz. (e,f) The microresonator’s frequency shift and quality factor extracted from the above measurements clearly correlate with nanoresonator’s frequency response where the frequency shift and energy dissipation are maximized when the nanoresonator vibrates with its largest amplitude.

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when the nanoresonator is more strongly pumped; this would not only yield increased dissipation in the microresonator but its resonance would also migrate to lower frequencies as observed in this instance.4,24 However, a similar series of measurements in which the microresonator’s third mode is probed while the nanoresonator is pumped not only reveals the enhanced dissipation but crucially this resonance migrates to higher frequencies thus ruling out the enhanced dissipation originating from heating as detailed in Supporting Information. In order to understand the physics of this phenomenon, the electromechanical system is modeled via the following coupled equations of motion

the micromechanical resonator (μ) in this configuration reveals the fundamental mode ωμ/2π = 151 kHz with a quality factor Qμ ≈ 290 000 as shown in Figure 1b and similarly the nanomechanical resonator (n) exhibits its fundamental mode ωn/2π = 2.07 MHz with a quality factor Qn ≈ 100 000 as shown in Figure 1c. With increasing actuation voltage, resulting in larger motional amplitudes, both resonators develop the wellknown Duffing nonlinearity β where the restoring force experienced by the mechanical elements exhibits a term that is proportional to its amplitude cubed. The micromechanical resonator sustains a negative Duffing nonlinearity, a consequence of it being slightly buckled at rest, which results in its resonance migrating to lower frequencies namely it experiences beam softening. On the other hand, the nanoresonator exhibits a positive Duffing nonlinearity that results in its resonance migrating to higher frequencies, namely, it experiences beam stiffening.23 The onset of the positive Duffing nonlinearity in the nanoresonator can be used to calibrate its displacement that yields a responsivity of 320 pA/nm.24 In order to evaluate the coupling between the resonators, the microresonator is probed while simultaneously the nanoresonator is harmonically pumped. The results of this measurement shown in Figure 2a reveal that the nanoresonator’s frequency response is captured by the dispersion of the micromechanical resonance that arises from the tension generated in the microresonator from the nanoresonator’s motion and is proportional to its amplitude squared.14,19 The coupling strength between the two resonators can be extracted from the above responsivity and the corresponding frequency shift that yields ∼0.1 Hz/nm2. Although this value is modest, it should be noted that it is limited to the nanoresonator’s linear response regime. However, if the nanoresonator is more strongly pumped, the larger resultant displacement activates its Duffing nonlinearity and the corresponding tension generated in the microresonator is amplified thus resulting in the coupling between the resonators being enhanced as shown in Figure 2b−d. Indeed increasing the pump amplitude by only a factor of 4 results in the microresonator frequency dispersion being enhanced by 1 order of magnitude as shown in Figure 2e. For context, application of dc bias to the nanoresonator can also create tension via the piezoelectric effect that can cause the micromechanical resonance to disperse13 and it yields a 3 Hz frequency shift on application of 1 V (see Supporting Information). In contrast, the above results detailed in Figure 2d,e reveal that a pump amplitude of 20 mVrms yields a 40 Hz frequency shift, or in other words this motional-induced coupling is 3 orders of magnitude stronger. It should be noted that this coupling is reversible and can even be observed when the nanoresonator is probed while the microresonator is pumped as detailed in Supporting Information. Unexpectedly, pumping the nanoresonator not only causes the microresonator’s frequency to disperse but it also increases its dissipation, namely, the quality factor Qμ = ωμ/γμ is reduced. The increased dissipation also correlates with the nanoresonator’s frequency response and is maximized at it largest displacement as shown in Figure 2f. Indeed pumping the nanoresonator to only 35 mVrms not only shifts the microresonator’s frequency by more than 100 times its intrinsic bandwidth (0.52 Hz) but it also increases its dissipation to the point where it is completely deactivated as detailed in Supporting Information. A plausible explanation for this surprising observation could arise from heating generated in the electromechanical system

mμX″μ + (mμγμ + ηX n2)X′μ + (kμ + βμXμ2 + ΓX n2)Xμ = Fμ(t ) mnX n″ + (mnγn + ηXμ2)X n′ + (kn + βnX n2 + ΓXμ2)X n = Fn(t )

with resonators of mass mi where i = μ or n, the second term describes the total dissipation,5−7 the third term describes the restoring potential with the spring constant ki = miωi2, the Duffing nonlinearity, and the nonlinear coupling Γ between the micro and nanoresonator14,18 where the electromechanical system is harmonically pumped/probed with force Fi. Unlike the coupling between modes in a single resonator, which is captured by Γ alone, the dispersive and dissipative coupling between the micro- and nanoresonators is encapsulated in both the restoring potential, that is, Γ, and the dissipation, that is, η, respectively, where the latter is the well-known nonlinearity of the Van der Pol oscillator. The coupled-Van der Pol−Duffing equations can be numerically solved in the rotating frame approximation by expressing the position of both resonators as Xi = ai(t)cos(ωit) + bi(t)sin(ωit) where ai and bi are the slowly varying envelopes of the in-phase and quadrature components and Fi = Λicos((ωi + δi)t) where Λi is proportional to the pump/probe voltage and δi is the detuning about the resonances (see Supporting Information). The results of this analysis shown in Figure 3 reveal that the coupled-Van der Pol−Duffing equations can reproduce the experimental responses shown in Figure 2 where the microresonator’s frequency dispersion arises from Γ and its enhanced energy dissipation can be attributed to η. These results also indicate the sign of Γ changes from negative when a node from the microresonator’s vibration coincides with the nanoresonator as shown in Figure 1b to positive when an antinode from the microresonator’s third mode coincides with the nanoresonator as shown in Supporting Information, which reflects the different strain profile across the nanoresonator. Experimental observations of nonlinear dissipation in mechanical resonators have only recently emerged and have predominantly been limited to carbon-based systems where its signatures only became apparent in the fundamental mode as it underwent sufficiently large amplitude vibrations to activate η.5−8 However, in this limit this effect also competes with the Duffing nonlinearity thus making transparent observations of the nonlinear dissipation impossible. In the present case, the microresonator is always weakly probed (100 μVrms) so as not to activate βμ as the nanoresonator is strongly pumped concurrently. This allows the quality factor and hence the dissipation to be readily extracted from the microresonator’s linear spectral response thus permitting unambiguous observations of its nonlinear dissipation.4,24 Further, these results also indicate that the nonlinear dissipation is not a material C

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Figure 3. (a,b) The frequency response of the microresonator extracted by numerically solving the coupled-Van der Pol−Duffing equations when the nanoresonator is simultaneously pumped with Λn = 2 and 8 × 105 corresponding to 5 and 20 mVrms as shown in Figure 2a,d with Λμ = 700, mn/mμ ≈ 0.013, βμ = − 1, βn = 2 ± 1, Γ = −14 ± 6 × 1013, and η = 25 ± 15 × 106. (c,d) The microresonator’s frequency shift and quality factor respectively for all pump amplitudes can also be extracted from the above numerical solutions which faithfully reproduce the experimental responses depicted in Figures 2e,f.

Figure 4. (a) The microresonator’s frequency response exhibits a negative Duffing linearity when actuated at 500 μVrms with the nanoresonator deactivated. (b) Activating the nanoresonator on resonance with an amplitude of 35 mVrms can deactivate the microresonator’s nonlinearity that can only be reactivated with a response similar to that shown in (a) if the microresonator is actuated with an amplitude of >2.5 mVrms. (c) The microresonator’s amplitude following a 100 μVrms on-resonance 2.5 s pulse as a function of the nanoresonator’s pump amplitude (V n). The microresonator’s amplitude decay (averaged between 102−103 times) reveals its ring down time τμ, which can be reduced by almost 2 orders of magnitude when the nanoresonator is pumped with an amplitude of only 25 mVrms.

dependent phenomenon as the present device is composed from a III−V semiconductor. Although the origin of the microresonator’s frequency dispersion can be explained in terms of the strain generated from the nanoresonator’s motion that creates a softening in its restoring potential in a mechanism analogous to the Duffing term βμ,14 the underlying nature of the microresonator’s enhanced energy dissipation is less clear.5−8,25 However, because the enhanced dissipation only emerges as more phonons are activated in the nanoresonator by virtue of its larger displacements, it would suggest a geometric origin where the large phonon ensemble in the nanoresonator at the microresonator’s clamping point acts as conduit to channel phonons away from the microresonator into its bulk environment. Consequently, placing the nanoresonator in the center of the microresonator away from its clamping points in principle should neutralize the nonlinear dissipation and is the subject of future study. In the meantime, the potential to exploit the nonlinear dissipation in the present system is available in contrast to earlier studies.5−8 For example, as the size of mechanical resonators has been reduced, their dynamic range over which they exhibit a linear response has also reduced as consequence of their Duffing nonlinearity being activated more easily.14,26 However, since the dissipation can be increased in the microresonator via the nanoresonator, the onset to nonlinearity consequently occurs at higher drives thus resulting in an enhanced dynamic range. For instance, the linear microresonance shown in Figure 4b with nanoresonator simultaneously activated requires the microresonator to be excited at least five times more strongly in order to replicate the nonlinear response shown in Figure 4a where the nanoresonator is

inactive. In fact, because the microresonance can be nearly deactivated via the nanoresonator as shown in Supporting Information Figure S3b, the dynamic range can actually be increased by several orders of magnitude. A second example concerns the quality factors of mechanical resonators where their increase27 has resulted in narrow bandwidths that has inhibited high speed operation.4,24 However, again as the microresonator’s dissipation can be dynamically engineered via η this limitation can be bypassed. Indeed as shown in Figure 4c, the microresonator’s response time can be decreased from nearly 0.5 to 0.01 s corresponding to Qμ ≈ 5000 when the nanoresonator is simultaneously activated. Tantalisingly, the coupled Van der pol equations also offer the potential to reduce energy dissipation in the electromechanical system if the sign of η can be tuned. To that end, both the nanoresonator and the microresonator are strongly pumped and probed as shown in Figure 5a. As expected, both resonators exhibit an asymmetric frequency response characteristic of their Duffing nonlinearities. However, as the nanoresonator is pumped beyond its nonlinear resonance, the D

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Figure 5. (a−c) The microresonator’s frequency response when probed with an amplitude of 500, 10, and 2 μVrms, respectively, while the nanoresonator is simultaneously pumped at 35 mVrms. The microresonator exhibits a nontrivial response in the boxed region in (a) that is probed in greater detail in (b,c) respectively. As the nanoresonator is pumped beyond its resonance, the microresonator exhibits a quality factor of more than three million (d), a positive Duffing nonlinearity (e), and self-oscillations (f) corresponding to the solid lines in (c).

existence of such a mode is found in either the micro- or nanoresonator. Further, solving the coupled-Van der Pol− Duffing equations with a large pump excitation namely Λn ≫ 106 fails to reproduce the experimental observations. Indeed, even freely varying all the parameters in the numerical simulations cannot reproduce the self-oscillation phenomenon indicating the incompleteness of the coupled-Van der Pol− Duffing model for describing this system. In addition, parasitic coupling between possible harmonics of the strongly pumped nanoresonator and the microresonator was also considered as the source of this phenomenon. However, this was ruled out with the absence of such harmonics in a detailed examination of both the micro- and nanoresonator’s spectral response from ωμ/2π to 10 MHz. In order to test the robustness of this phenomenon, the electromechanical system was even thermally cycled which as expected resulted in the micro and nanoresonator’s frequencies shifting slightly, however, the selfoscillation phenomenon remained as shown in Supporting Information. Although the underlying microscopic nature of both the nonlinear dissipation and the self-oscillation phenomena remains unclear, these results unequivocally demonstrate the rich dynamics available to this simple coupled mechanical system. Indeed, as more complex variations of this system are developed in the vein of integrated electrical circuits, it will inevitably result in nonlinear phenomena such as selfoscillation, fractals, and chaos emerging and dominating their dynamical behavior.30,31 Consequently, the further study of such coupled systems is essential in elucidating the microscopic origins of these effects that if harnessed could lead to the creation of a new class nonlinear devices. An electromechanical system in which a microresonator is embedded with a nanoresonator exhibits both dispersive and

microresonator’s response becomes complex as highlighted in the box in Figure 5a. This region is probed again with a weaker excitation while the pump excitation remains unchanged as shown in Figure 5b. Again, as the nanoresonator is pumped across its Duffing response, the microresonator exhibits a nontrivial response. The region enclosed in the box in Figure 5b is further probed with an even weaker excitation as shown in Figure 5c where three distinct regimes for the microresonator’s frequency response are identified and detailed in Figure 5d−f corresponding to the solid lines in Figure 5c. Specifically, as the pump frequency is increased, the microresonator’s quality factor first increases beyond 3 × 106 as shown in Figure 5d after which it develops an asymmetric response characteristic of the Duffing nonlinearity but now with a positive polarity (see Figure 5e) in contrast to the negative Duffing nonlinearity exhibited by the microresonator when probed in isolation (see Figure 1b). Beyond this point, the microresonator exhibits highly unstable large amplitude oscillations that are characteristic of self-oscillation. Indeed, if the microresonator’s output noise spectra are measured instead without application of any probe excitation (i.e., Λμ = 0) while the nanoresonator is pumped in the same spectral window, the microresonator still spontaneously oscillates, which suggests that its intrinsic dissipation γμ has been eliminated via η (see Supporting Information). A simple explanation for the observed self-oscillations could arise from the nanoresonator’s excitation acting as an antiStokes parametric pump for a hitherto unobserved mechanical mode corresponding to the frequency ωn − ωμ. In this configuration, phonons can be generated in both the fundamental mode of the microresonator and this unknown mode, and at sufficiently large pump amplitudes it will result in both self-oscillating.28,29 However, no evidence for the E

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(14) Westra, H. J. R.; Poot, M.; van der Zant, H. S. J.; Venstra, W. J. Phys. Rev. Lett. 2010, 105, 117205. (15) Gaidarzhy, A.; Dorignac, J.; Zolfagharkhani, G.; Imboden, M.; Mohanty, P. Appl. Phys. Lett. 2011, 98, 264106. (16) Westra, H. J. R.; Karabacak, D. M.; Brongersma, S. H.; CregoCalama, M.; van der Zant, H. S. J.; Venstra, W. J. Phys. Rev. B 2011, 84, 134305. (17) Castellanos-Gomez, A.; Meerwaldt, H. B.; Venstra, W. J.; van der Zant, H. S. J.; Steele, G. A. Phys. Rev. B 2012, 86, 041402. (18) Lulla, K. J.; Cousins, R. B.; Venkatesan, A.; Patton, M. J.; Armour, A. D.; Mellor, C. J.; Owers-Bradley, J. R. New J. Phys. 2012, 14, 113040. (19) Matheny, M. H.; Villanueva, L. G.; Karabalin, R. B.; Sader, J. E.; Roukes, M. L. Nano Lett. 2013, 13, 1622−1626. (20) Truitt, P.; Hertzberg, J.; Altunkaya, E.; Schwab, K. J. Appl. Phys. 2013, 114, 114307. (21) Dorignac, J.; Gaidarzhy, A.; Mohanty, P. J. Appl. Phys. 2008, 104, 073532. (22) Mahboob, I.; Flurin, E.; Nishiguchi, K.; Fujiwara, A.; Yamaguchi, H. Appl. Phys. Lett. 2010, 97, 253105. (23) Nayfeh, A. H., Mook, D. T. Nonlinear Oscillations; Wiley: New York, 1995. (24) Ekinci, K. L.; Roukes, M. L. Rev. Sci. Instrum. 2005, 76, 061101. (25) Imboden, M.; Mohanty, P. Phys. Rep. 2014, 534, 89−146. (26) Kozinsky, I.; Postma, H. W. C.; Bargatin, I.; Roukes, M. L. Appl. Phys. Lett. 2006, 88, 253101. (27) Verbridge, S. S.; Craighead, H. G.; Parpia, J. M. Appl. Phys. Lett. 2008, 92, 013112. (28) Mahboob, I.; Nishiguchi, K.; Okamoto, H.; Yamaguchi, H. Nat. Phys. 2012, 8, 387−392. (29) Kippenberg, T. J.; Vahala, K. J. Science 2008, 321, 1172−1176. (30) Kao, Y. H.; Wang, C. S. Phys. Rev. E 1993, 48, 2514−2520. (31) Venkatesan, A.; Lakshmanan, M. Phys. Rev. E 1997, 56, 6321− 6330.

dissipative coupling. The dispersive coupling between the resonators enables the microresonance to be tuned by more than 100 times its bandwidth where its polarity can also be controlled by selecting different modes in the microresonator. On the other hand, the dissipative coupling enables the microresonator’s dissipation rate to be either increased resulting in a larger dynamic range and a wider bandwidth or reduced yielding a quality factor of millions and the ability to tune the sign of its Duffing nonlinearity. Indeed, the dissipation in the microresonator can even be eliminated entirely by the nanoresonator resulting in it spontaneously vibrating. This simple coupled electromechanical system paves the way toward a new class of actively engineered mechanics where novel nonlinear phenomena can both be activated and harnessed.

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ASSOCIATED CONTENT

S Supporting Information *

Experimental measurements further detailing the nonlinear dissipation and self-oscillation in the electromechanical system are shown in Figures S1−S7. The expansion of the coupled-Van der Pol−Duffing equations in the rotating frame approximation used to generate the numerical solutions are also described. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful to S. Miyashita for growing the GaAs/ AlGaAs sample. This work was partly supported by JSPS KAKENHI Grant 23241046.

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REFERENCES

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DOI: 10.1021/nl5044264 Nano Lett. XXXX, XXX, XXX−XXX