Imaging vibrations of locally gated, electromechanical few layer graphene resonators with a moving vacuum enclosure
Heng Lu, Chen Yang, Ye Tian, Jun Lu, Fanqi Xu, FengNan Chen, Yan Ying, Kevin G. Schädler, Chinhua Wang, Frank H. L. Koppens, Antoine Reserbat-Plantey, Joel Moser
IImaging vibrations of locally gated, electromechanical few layergraphene resonators with a moving vacuum enclosure
Heng Lu, Chen Yang, Ye Tian, Jun Lu, Fanqi Xu,FengNan Chen, Yan Ying, Chinhua Wang, and Joel Moser ∗ School of Optoelectronic Science and Engineering & CollaborativeInnovation Center of Suzhou Nano Science and Technology,Soochow University, Suzhou 215006, People’s Republic of China andKey Lab of Advanced Optical Manufacturing Technologies of Jiangsu Province& Key Lab of Modern Optical Technologies of Education Ministry of China,Soochow University, Suzhou 215006, People’s Republic of China
Kevin G. Sch¨adler, Frank H. L. Koppens, and Antoine Reserbat-Plantey † ICFO–Institut de Ciencies Fotoniques,The Barcelona Institute of Science and Technology,08860 Castelldefels Barcelona, Spain a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n bstract Imaging the vibrations of nanomechanical resonators means measuring their flexural modeshapes from the dependence of their frequency response on in-plane position. Applied to two-dimensional resonators, this technique provides a wealth of information on the mechanical proper-ties of atomically-thin membranes. We present a simple and robust system to image the vibrationsof few layer graphene (FLG) resonators at room temperature and in vacuum with an in-planedisplacement precision of ≈ . µ m. It consists of a sturdy vacuum enclosure mounted on athree-axis micropositioning stage and designed for free space optical measurements of vibrations.The system is equipped with ultra-flexible radio frequency waveguides to electrically actuate res-onators. With it we characterize the lowest frequency mode of a FLG resonator by measuring itsfrequency response as a function of position on the membrane. The resonator is suspended overa nanofabricated local gate electrode acting both as a mirror and as a capacitor plate to actuatevibrations at radio frequencies. From these measurements, we estimate the ratio of thermal expan-sion coefficient to thermal conductivity of the membrane, and we measure the effective mass of thelowest frequency mode. We complement our study with a globally gated resonator and image itsfirst three vibration modes. There, we find that folds in the membrane locally suppress vibrations. ∗ [email protected] † [email protected] . INTRODUCTION Imaging the flexural vibrations of two-dimensional (2-D) nanomechanical resonators isan important task. These resonators, made of atomically-thin membranes of graphene [1–7]and various thin materials such as few layer transition metal dichalcogenides [8–12], of-fer the opportunity to study the physics of vibrational modes in regimes where extremelysmall mass, low bending rigidity, large stretching rigidity and large aspect ratio combineto give rise to a wealth of mechanical behaviors [13]. In its simplest form, imaging vibra-tions means measuring their time averaged, resonant amplitude as a function of positionon the membrane. Driven vibrations of graphene resonators were imaged with an AtomicForce Microscope (AFM) [14] and with an optical interferometry setup [3]. Both driven andthermal vibrations of resonators based on graphene [15, 16], MoS [10], black phosphorus[17], and hexagonal boron nitride [18] were measured using a similar optical interferometrytechnique. These measurements advanced our understanding of 2-D resonators in severalimportant ways. They made it possible to identify modes in the vibration spectrum unam-biguously, including degenerate modes that are otherwise difficult to detect [3, 15]. Theyrevealed the impact of unevenly distributed stress and mass on the mode shape [14]. Theyalso proved to be exquisitely sensitive to mechanical anisotropies, such as the anisotropy ofYoung’s modulus that stems from the crystal structure of the membrane [17]. All these inter-esting measurements may benefit nanomechanical sensing applications, including spatiallyresolved nanomechanical mass spectroscopy [19]. Vibration imaging may also complementother techniques including AFM, Raman spectroscopy and photoluminescence where theseare used to inspect 2-D materials for defects, impurities and grain boundaries.With all its merits, imaging vibrations of 2-D resonators remains a challenging task. Dif-ficulties come in part from the necessity to measure vibrations in a controlled environment.Measuring the mechanical response of thin membranes while minimizing their damping ratesrequires keeping them in vacuum. This immediately places technical constraints on the mea-suring equipment. Piezo linear actuators [20, 21] were used in some imaging experiments,where they either moved the vacuum enclosure [15] or moved a microscope objective withrespect to the vacuum enclosure [16], while a high-precision motorized stage was used inother experiments [10, 17, 18]. Adding to the complexity of imaging vibrations, radio fre-quency electrical signals are sometimes supplied to the resonator to drive vibrations while3he position of the resonator is changing. Implementing this driving technique is difficultbecause radio frequency cables are stiff and hinder the motion of piezoelectric actuators andmotorized stages.Here we demonstrate vibration imaging of electrically driven, few layer graphene (FLG)resonators using a moving vacuum enclosure. The enclosure is mounted on a three-axismicropositioning stage with a measured in-plane displacement precision of ≈ . µ m andis equipped with homemade, ultra-flexible radio frequency waveguides. Our design makesuse of the large load capacity of the stage and the high sensitivity of the manual adjustersattached to it. The adjusters are driven by inexpensive stepper motors connected to themby simple gears and rubber belts. The benefits of our system are its sturdy design thatprotects it against acoustic vibrations, its submicrometer displacement precision unaffectedby a rather heavy load, and its capability of delivering radio frequency signals to a movingresonator. We employ our system to image vibrations of two FLG resonators. The firstresonator is suspended over a nanofabricated local gate electrode acting both as a mirrorfor optical detection and as a capacitor plate to actuate vibrations at radio frequencies. Wemeasure the hardening of the spring constant of the lowest frequency mode of the resonatoras a function of absorbed optical power. From these measurements we estimate the ratioof thermal expansion coefficient to thermal conductivity of the membrane, which plays animportant role in thermal transport across the resonator. In addition, imaging the vibrationsof this mode allows us to measure its effective mass, which is important for quantitative massand force sensing applications based on 2-D resonators. Our measurements combine threeinteresting features which, to the best of our knowledge, have not been reported thus far ina single device: they demonstrate (i) vibration imaging of a lowest frequency mode which(ii) resonates above 60 MHz using (iii) a local metal gate to enhance optical readout. Thesecond resonator has a fold in the membrane that is not visible in an optical microscope. Weimage the first three vibration modes and show that the fold locally suppresses vibrations.These measurements demonstrate that our system can be used to identify mesoscopic defectsin the resonator and study their impact on the mechanical response.4 I. EXPERIMENTAL SETUP
The mechanical part of our system is robust and easy to operate. Our enclosure is shapedas a cylinder with a diameter of 120 mm and a depth of 75 mm, and is made of stainlesssteel (Fig. 1a, b). There is a clear advantage to such a sturdy design: with a total weightof ≈ . − mbar using a small size turbomolecularpump connected to a one-meter long KF16 flexible bellow. The enclosure is mounted on athree-axis micromechanical linear stage (Newport M-562-XYZ). Each axis is connected to amanual adjuster (Newport DS-4F) with a specified fine sensitivity of 20 nm. Each adjusteris driven by a stepper motor (Makeblock 42BYG) whose shaft is connected to the adjusterusing gears and a rubber belt. In vibration imaging experiments where a laser beam isused to measure the response of a resonator, our system guarantees that only the positionof the resonator is varied while the light path remains unchanged, so the incident opticalpower and the shape of the focused beam are unaffected by the imaging process. Using ananofabricated calibration sample, we measure the precision of in-plane displacement of theenclosure to be ≈ . µ m (Supplementary Material, Section I).Our optical setup, shown in Fig. 2a, is similar to those used to detect vibrations of 2-Dresonators (see e.g. Ref. [1]). Its design originates from the setup presented in Refs. [22,23]. Briefly, we employ a Helium-Neon laser emitting at a wavelength λ ≈
633 nm as amonochromatic light source. The output of the laser is filtered with a single mode fiberto obtain a clean fundamental transverse Gaussian mode. The combination of a polarizingbeam splitter and a quarter-wave plate ensures that light incident on the resonator andreflected light have orthogonal polarizations so the photodetector mostly collects reflectedlight. Incident light is focused with and reflected light is collected by a long working distance5bjective (Mitutoyo M Plan Apo 100X) with a numerical aperture NA = 0 . w of the waist of the focused laser beam using a modified version of the knife edgetechnique (Supplementary Material, Section II). We find w ≈ . µ m, which is consistentwith the input beam parameters. III. RESULTS AND DISCUSSION
With the beam optimally focused, we demonstrate that our system can image the flexuralvibrations of 2-D resonators based on suspended membranes of few layer graphene (FLG).We present vibration imaging data obtained with two devices. The first device is a locallygated FLG resonator. It is characterized by a nanofabricated gate electrode made of evap-orated gold over which FLG is suspended. The gate electrode serves as a highly reflectivemirror for optical detection, and also forms a capacitor with FLG to actuate vibrations atradio frequencies. We use this device to measure the ratio of thermal expansion coefficientto thermal conductivity of FLG. We also use it to measure the effective mass of the funda-mental mode of vibration. The second device is a globally gated FLG resonator. It consistsof FLG suspended over a doped silicon substrate. We use this device to demonstrate thatour system can be employed to detect the presence of folds in the thin membrane that areotherwise invisible in an optical microscope. The advantage of a local gate made of gold overa global silicon gate is the higher reflectance of the mirror combined with the possibility ofactuating individual resonators within an array of devices. To the best of our knowledge, theuse of a local gate both for optical detection and capacitive actuation has not been reportedthus far.We first consider our locally gated resonator. It is fabricated by exfoliating FLG andtransferring it onto a prefabricated substrate using the viscoelastic transfer method [24].The substrate is thermal silicon oxide grown on highly resistive silicon. It is patternedwith source and drain electrodes to contact FLG (Fig. 3a), a 3 µ m diameter, cylindricalcavity etched in the oxide, and a local gate electrode nanofabricated at the bottom of thecavity (Fig. 3a, b). The distance between FLG and the gate is nominally 250 nm (Fig. 3c).We estimate that our FLG is composed of N L = 8 graphene layers from measurements of6ptical power reflected by the oxidized silicon substrate with and without supported FLG,away from the cavity. We obtain N L by comparing the ratio of these measured powers tocalculations based on the transfer matrix method [25, 26], see Fig. 3d and SupplementaryMaterial, Section III. Additional reflected power measurements made on the gold electrodesconfirm this result. The resonator is actuated electrically by applying an oscillating voltage V ac superimposed on a dc offset V dc between FLG and the gate (Fig. 2b). This resultsin an electrostatic force of amplitude V dc V ac | d C d z | , where the third factor is the derivativewith respect to flexural displacement of the capacitance C between FLG and the gate.We favor electrostatic drive over optical drive [1] because the former allows actuating veryhigh frequency vibrations without the inconvenience of dissipation caused by photothermaleffects. To supply radio frequency signals for this actuation without impeding the motion ofthe micropositioning stage, we use a homemade waveguide consisting of a copper microstrippatterned on a 15 cm long ribbon cut from a thin Kapton film (Fig. 2c). Our waveguide isultra-flexible, its insertion loss is smaller than 1 dB up to at least 3 GHz, and its scatteringparameters are insensitive to bending and twisting of the waveguide. The resonator is keptat room temperature in a vacuum of ≈ − mbar. We detect the flexural vibrations ofthe resonator using a standard technique [1, 22, 23]. Briefly, the resonator is placed in anoptical standing wave from which it absorbs energy. Vibrations render absorbed energytime-dependent, resulting in modulations of the reflected power. Correspondingly, the meansquare voltage (cid:104) V (cid:105) at the output of the photodetector reads (cid:104) V (cid:105) ≈ (cid:18) G × T × P inc (cid:12)(cid:12)(cid:12) d R d z (cid:12)(cid:12)(cid:12) z M (cid:19) (cid:104) z (cid:105) + (cid:104) δV (cid:105) , (1)where G , in units of V/W, is the product of the responsivity and the transimpedance gain ofthe photodetector, T is the transmittance of the reflected light path, P inc is the optical powerincident on the resonator, z vib is the amplitude of vibrations in the flexural direction, δV b isthe amplitude of fluctuations of the measurement background, and (cid:104)·(cid:105) averages over time.The quantity | d R d z | z M is the derivative of the reflectance R of the whole device consisting ofthe resonator and the reflective gate, at a distance z M between the resonator and the gate.The local gate acts as highly reflective mirror that optimizes the transduction of (cid:104) z (cid:105) into (cid:104) V (cid:105) . We calculate | d R d z | ≈ × − /nm for our device at λ = 633 nm, which is abouttwice the value calculated with a regular silicon gate [25]. To study the frequency response (cid:104) z (cid:105) ( f ) of the resonator, we sweep the drive frequency f of V ac and measure (cid:104) V (cid:105) with7 spectrum analyzer. Figure 3e displays (cid:104) V (cid:105) as a function of f and V dc for the lowestfrequency mode we can resolve. The response shifts to higher frequencies as | V dc | increasesmostly because the electrostatic force ∝ | d C d z | V tensions the membrane as it pulls it towardsthe gate, making the resonant frequency of the mode tunable.With the resonator positioned near the center of the focused beam, we characterize thelowest frequency resonance in the spectrum of (cid:104) V (cid:105) ( f ) and its dependence on P inc . Wehave verified that, on resonance, the electromechanical signal represented by the root meansquare voltage ¯ V = ( (cid:104) V (cid:105) − (cid:104) δV (cid:105) ) / increases linearly with V ac within the range used inthis work while the resonant frequency does not shift, indicating that the resonator is drivenin a regime where the restoring force is linear in displacement. Figure 4a shows (cid:104) V (cid:105) / ( f )measured at V ac = 12 . rms and V dc = 5 V for P inc ranging from 20 µ W to 205 µ W. Hereas well, we have verified that the peak value of ¯ V ( f ) increases linearly with P inc . Overall,¯ V can be linearly amplified either by increasing the vibrational amplitude electrostaticallywith V ac or by increasing the optical readout with a larger probe power P inc . The lineshapeof ¯ V ( f ) is Lorentzian, which indicates that the resonator behaves as a damped harmonicoscillator with susceptibility χ ( f ) = 14 π f − f − i f f /Q , (2)where Q is the spectral quality factor, f is the resonant frequency of the vibrational mode,and ¯ V ( f ) ∝ | χ ( f ) | . Figure 4b shows that Q is low and does not change within the rangeof P inc , while f increases with P inc (Fig. 4c). In the absence of nondissipative spectralbroadening processes, Q is inversely proportional to the rate at which energy stored in aresonator gets dissipated in a thermal bath. In low dimensional resonators based on 2-Dmaterials and on nanotubes, Q at room temperature is always found to lie between 10and ≈
100 [1, 27], which is surprisingly low given the high crystallinity of the resonators.Proposed mechanisms to explain such low Q ’s include losses within the clamping area [28]and spectral broadening due to nonlinear coupling between the mode of interest and alarge number of thermally activated modes [29, 30]. In turn, the increase of f reveals ahardening of the spring constant of the resonator. The latter may be due to vibrationsresponding to photothermal forces with a delay [5, 31–33]. However, because Q does notappreciably change with P inc , the hardening of the spring constant is more likely to becaused by absorptive heating accompanied by a contraction of the membrane [15, 34]. In8his case, it is interesting to relate the change ∆ f induced by a change ∆ P inc to the thermalexpansion coefficient α and to the thermal conductivity κ of the membrane. For this weborrow a result from Ref. [35], namely | ∆ f / ∆ P abs | = | αf η/ (4 π(cid:15)κh ) | , with P abs the powerabsorbed by the membrane, (cid:15) the strain within the membrane, h = N L × . × − m thethickness of the membrane, and η ≈ P inc into P abs using the absorbance A = P abs /P inc of our FLG suspended over the gate electrode. We measure A ≈ . (cid:15) ≈ × − from f by calculating the elastic energy of a disk-shaped membrane [36, 37] andderiving from it the spring constant of the fundamental mode (Supplementary Material,Section IV). We make the simplifying assumption that the electrostatic force is uniform overthe membrane. We also assume a Young’s modulus of 10 Pa and a Poisson’s ratio of 0.165[38]. Combining ∆ f / ∆ P inc , A and (cid:15) , we find | α/κ | ≈ × − m/W. This is a reasonableestimate, considering for example | α/κ | ≈ × − m/W with α ≈ − × − K − fromsuspended singe layer graphene [34] and κ ≈ − K − from pyrolytic graphite [39],both at room temperature.We now measure the response of the resonator as a function of its in-plane position withrespect to the beam. Data shown in Figs. 4d-f are measured with V ac = 12 . rms , V dc =5 V and P inc = 110 µ W. Figure 4d shows the same resonance as in Fig. 4a measured at variousfixed positions along the x direction, with the center of the beam at x . Correspondingly,Fig. 4e shows Q and Fig. 4f shows f as a function of x − x . Figures 4d-f are strikinglysimilar to Figs. 4a-c: Q does not depend on position while f increases as the resonatorapproaches x and decreases as it moves out of the beam. As the resonator moves withrespect to the position of the beam, it samples the intensity of the beam in a similar way tothat of a reflective structure in a knife edge measurement (Supplementary Material, SectionII). Importantly, the dependence of f on position means that vibration imaging requiresmeasuring the full resonance at each position on the resonator, as we do next.We present spatially resolved amplitude measurements of the lowest frequency mode inFigs. 4g-i. Figure 4g shows the peak (resonant) value of the time averaged electrical power (cid:104) V (cid:105) /
50 dissipated across the input impedance of the spectrum analyzer, expressed in dB m ,as a function of x and y . Figure 4h shows the peak value of ¯ V on a linear scale. The latter9s proportional to the mean square of the resonant vibrational amplitude, see Eq. (1), henceit is proportional to the potential energy of the mode. The background (cid:104) δV (cid:105) is measuredin the cavity at f but with the drive frequency shifted up and far away from resonance.Measurements are made at V dc = 5 V with V ac = 0 . rms and P inc = 110 µ W. At such low V ac , we find that the peak value of ¯ V is sizeable only in a central area away from the edgeof the cavity (highlighted by the dashed circle in Fig. 4h). Within this area, the measuredpeak frequency of ¯ V ( f ) is almost uniform and defines the resonant frequency f of the mode(Fig. 4i).We use the vibration imaging data shown in Figs. 4g-i to measure the effective mass m eff of the mode. The latter is related to the potential energy U as U ≡ mπa (2 πf ) (cid:90) (cid:90) z ( f , x, y )d S = 12 m eff (2 πf ) z , (3)where m is the geometrical mass and a is the radius of the membrane, respectively, z vib ( f , x, y ) is the resonant amplitude at position ( x, y ), d S is an elementary area onthe membrane and z max is the largest value of z vib ( f , x, y ) over the membrane. Replacingthe integral in Eq. (3) with a discrete sum yields m eff m = (cid:80) i,j ¯ V ( f , x i , y j ) N ¯ V , (4)where x i and y j are discrete coordinates over the cavity, N is the number of pixels withinthe area of the cavity in Figs. 4g-i, and ¯ V is the largest value of ¯ V ( f , x i , y j ) over themembrane. While Figs. 4g-i represent the convolution of the focused beam with the vibrationmode shape (instead of the mode shape alone), we verified numerically that m eff calculatedfrom the convolution and m eff calculated from the mode shape agree within 5% for ourmeasured radius w ≈ . µ m of the waist of the focused laser beam. Equation (4) yields m eff /m = 0 . ± .
01. This estimate agrees well with the value of 0.27 calculated for a diskshaped graphene membrane without bending rigidity and subjected to electrostatic pressure[40]. It shows that the assumption of negligible bending rigidity compared to stretchingrigidity is still a valid one for FLG.We complement our study with vibration imaging measurements performed on a seconddevice, showing the effect of a mesoscopic defect in FLG on the mechanical response. Herethe resonator consists of FLG suspended over silicon oxide grown on doped silicon. The sili-con substrate serves as a global gate electrode. We show the mechanical resonance spectrum10f the resonator and its dependence on V dc in Supplementary Material, Section V. Figure 5ais a scanning electron microscope image of the device, which reveals the presence of a fold inFLG near the bottom edge of the cavity. This fold cannot be seen in an optical microscopewith a 100 × magnification objective. Figures 5b-d display the resonant value of (cid:104) V (cid:105) asa function of in-plane displacements x and y for the first three vibrational modes we areable to measure. While these results are qualitatively consistent with the shapes of a first,second and third mode, we observe the presence of a node in the vicinity of the fold. Thefold presumably causes a local stiffening of FLG [41] which pins vibration modes and forcesthem to a low amplitude state. Our measurements show that folds have a strong impacton the mechanical response of 2-D resonators. As with membranes with free standing edges[10] and membranes with inhomogeneous strain and mass distributions [14], membranes withfolds may have mechanical properties that may not be found in uniform membranes. Ourmeasurement system is well suited to investigate these properties as it is noninvasive and,unlike scanning electron imaging, does not contaminate the surface of resonators. IV. CONCLUSION
Our simple system composed of a vacuum enclosure mounted on a three-axis microp-ositioning stage is well suited to measure the mechanical response of few layer grapheneelectromechanical resonators and to image their vibrations. From the hardening of thespring constant of the lowest frequency mode in response to increased incident power, weestimate the ratio of thermal expansion coefficient α to thermal conductivity κ of the mem-brane. Doing so requires either a strong temperature gradient induced by the beam acrossthe membrane or a resonator with a low spring constant, both of which are more likely tobe obtained with resonators larger than our 3 µ m diameter device [5, 15]. We image theshape of the lowest frequency mode and measure its effective mass, which is important forquantitative sensing applications based on those devices. We also image vibration modes inthe presence of a fold in FLG, and show that the fold strongly affects the mechanical re-sponse of the resonator. Built-in calibration of in-plane displacement is possible by mappingthe reflectance of the cavity, which can be done by averaging the measurement backgroundaway from resonance in between two resonant measurements. Our system may be used incombination with a small size, dry cryostat that would replace our heavy vacuum enclosure11nd with the objective outside the cryostat. Measuring the dependence of f on temper-ature would yield α which, combined with f ( P inc ), would yield κ [35]. Planning for suchexperiments, we have experimentally verified that our homemade waveguides remain flexibleand that their insertion loss remains low at cryogenic temperatures by bending 4 of themwith a piezopositioner at 800 mK. If precision and accuracy on the nanometer scale are notneeded, our system may offer an alternative to systems based on piezo positioners which arefragile, have a small load capacity, and often come at a prohibitive cost to experimentalistson a budget. ACKNOWLEDGMENTS
J. Moser is grateful to Yin Zhang, Warner J. Venstra and Alexander Eichler for help-ful discussions. This work was supported by the National Natural Science Foundation ofChina (grant numbers 61674112 and 62074107), the International Cooperation and Ex-change of the National Natural Science Foundation of China NSFC-STINT (grant number61811530020), Key Projects of Natural Science Research in JiangSu Universities (grant num-ber 16KJA140001), the project of the Priority Academic Program Development (PAPD) ofJiangsu Higher Education Institutions, and the Opening Fund of State Key Laboratory ofNonlinear Mechanics in Beijing. [1] J. S. Bunch, A. M. van der Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M.Parpia, H. G. Craighead, and P. L. McEuen, Science , 490 (2007).[2] C. Chen, S. Rosenblatt, K. I. Bolotin, W. Kalb, P. Kim, I. Kymissis, H. L. Stormer, T. F.Heinz, and J. Hone, Nat. Nanotech. , 861 (2009).[3] R. A. Barton, B. Ilic, A. M. van der Zande, W. S. Whitney, P. L. McEuen, J. M. Parpia, andH. G. Craighead, Nano Lett. , 1232 (2011).[4] A. Reserbat-Plantey, L. Marty, O. Arcizet, N. Bendiab, and V. Bouchiat, Nat. Nanotech. ,151 (2012).[5] R. A. Barton, I. R. Storch, V. P. Adiga, R. Sakakibara, B. R. Cipriany, B. Ilic, S.-P. Wang,P. Ong, P. L. McEuen, J. M. Parpia, and H. G. Craighead, Nano Lett. , 4681 (2012).
6] J. P. Mathew, R. N. Patel, A. Borah, R. Vijay, and M. M. Deshmukh, Nat. Nanotech. ,747 (2016).[7] Z.-Z. Zhang, X.-X. Song, G. Luo, Z.-J. Su, K.-L. Wang, G. Cao, H.-O. Li, M. Xiao, G.-C.Guo, L. Tian, G. W. Deng, and G.-P. Guo, Proc. Natl. Acad. Sci. U. S. A. , 5582 (2020).[8] S. Sengupta, H. S. Solanki, V. Singh, S. Dhara, and M. M. Deshmukh, Phys. Rev. B ,155432 (2010).[9] A. Castellanos-Gomez, R. van Leeuwen, M. Buscema, H. S. J. van der Zant, G. A. Steele,and W. J. Venstra, Adv. Mater. , 6719 (2013).[10] Z. Wang, J. Lee, K. He, J. Shan, and P. X.-L. Feng, Sci. Rep. , 3919 (2014).[11] N. Morell, A. Reserbat-Plantey, I. Tsioutsios, K. G. Sch¨adler, F. Dubin, F. H. L. Koppens,and A. Bachtold, Nano Lett. , 5102 (2016).[12] S. J. Cartamil-Bueno, P. G. Steeneken, F. D. Tichelaar, E. Navarro-Moratalla, W. J. Venstra,R. van Leeuwen, E. Coronado, H. S. J. van der Zant, G. A. Steele, and A. Castellanos-Gomez,Nano Res. , 2842 (2015).[13] M. I. Dykman, Fluctuating Nonlinear Oscillators , 1st ed. (Oxford University Press, Oxford,U. K., 2012).[14] D. Garcia-Sanchez, A. M. van der Zande, A. San Paulo, B. Lassagne, P. L. McEuen, andA. Bachtold, Nano Lett. , 1399 (2008).[15] D. Davidovikj, J. J. Slim, S. J. Cartamil-Bueno, H. S. J. van der Zant, P. G. Steeneken, andW. J. Venstra, Nano Lett. , 2768 (2016).[16] A. Reserbat-Plantey, K. G. Sch¨adler, L. Gaudreau, G. Navickaite, J. G¨uttinger, D. E. Chang,C. Toninelli, A. Bachtold, and F. H. L. Koppens, Nat. Comm. , 10218 (2016).[17] Z. Wang, H. Jia, X.-Q. Zheng, R. Yang, G. J. Ye, X. H. Chen, and P. X.-L. Feng, Nano Lett. , 5394 (2016).[18] X.-Q. Zheng, J. Lee, and P. X.-L. Feng, Microsystems & Nanoengineering , 17038 (2017).[19] M. S. Hanay, S. I. Kelber, C. D. O’Connell, P. Mulvaney, J. E. Sader, and M. L. Roukes,Nat. Nanotech. , 339 (2015).[20] S.-T. Ho and S.-J. Jan, Prec. Eng. , 285 (2016).[21] Y.-T. Liu and B.-J. Li, Prec. Eng. , 118 (2016).[22] D. W. Carr and H. G. Craighead, J. Vac. Sci. Technol. B , 2760 (1997).[23] D. Karabacak, T. Kouh, and K. L. Ekinci, J. Appl. Phys. , 124309 (2005).
24] A. Castellanos-Gomez, M. Buscema, R. Molenaar, V. Singh, L. Janssen, H. S. J. van der Zant,and G. A. Steele, 2-D Mater. , 011002 (2014).[25] S. Roddaro, P. Pingue, V. Piazza, V. Pellegrini, and F. Beltram, Nano Lett. , 2707 (2007).[26] F. Chen, C. Yang, W. Mao, H. Lu, K. G. Sch¨adler, A. Reserbat-Plantey, J. Osmond, G. Cao,X. Li, C. Wang, Y. Yan, and J. Moser, 2D Mater. , 011003 (2018).[27] V. Sazonova, Y. Yaish, H. ¨Ust¨unel, D. Roundy, T. A. Arias, and P. L. McEuen, Nature ,284 (2004).[28] J. Rieger, A. Isacsson, M. J. Seitner, J. P. Kotthaus, and E. M. Weig, Nat. Commun. , 3345(2014).[29] A. W. Barnard, V. Sazonova, A. M. van der Zande, and P. L. McEuen, Proc. Natl. Acad.Sci. U. S. A. , 19093 (2012).[30] Y. Zhang and M. I. Dykman, Phys. Rev. B , 165419 (2015).[31] C. H¨ohberger Metzger and K. Karrai, Nature , 1002 (2004).[32] C. Metzger, I. Favero, A. Ortlieb, and K. Karrai, Phys. Rev. B , 035309 (2008).[33] S. Zaitsev, A. K. Pandey, O. Shtempluck, and E. Buks, Phys. Rev. E , 046605 (2011).[34] D. Yoon, Y.-W. Son, and H. Cheong, Nano Lett. , 3227 (2011).[35] N. Morell, S. Tepsic, A. Reserbat-Plantey, A. Cepellotti, M. Manca, I. Epstein, A. Isacsson,X. Marie, F. Mauri, and A. Bachtold, Nano Lett. , 3143 (2019).[36] S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells , 2nd ed. (McGraw-Hill,New York, NY, 1987).[37] Y. Zhang, Sci. China-Phys. Mech. Astron. , 624602 (2016).[38] O. L. Blakslee, D. G. Proctor, E. J. Seldin, G. B. Spence, and T. Weng, J. Appl. Phys. ,3373 (1970).[39] Y. S. Touloukian, Thermal Conductivity : Nonmetallic Solids , 1st ed. (IFI/Plenum, New York,NY, 1970).[40] P. Weber, J. G¨uttinger, I. Tsioutsios, D. E. Chang, and A. Bachtold, Nano Lett. , 2854(2014).[41] J. Moser and A. Bachtold, Appl. Phys. Lett. , 173506 (2009). IG. 1. Enclosure to image vibrations in vacuum. (a) Assembled enclosure with a window (blueshaded disk) for free space optical measurements, a KF port on the side for SMA connectors andanother KF port on the back side for pumping. (b) Exploded view. The red shaded square repre-sents the substrate hosting the resonators. It is glued on a printed circuited board (lightly coloreddisk) that is attached to a copper holder (brown shaded disk with a shallow recess). (c) Crosssection showing the holder (brown), the board (yellow), the substrate (red) and SMA connectorsattached to the back side of the board and connected to the front side of it. IG. 2. Vibration measurement setup. (a) Optical setup. NDF: neutral density filter. SMF: singlemode fiber. λ/
2: half-wave plate. PBS: polarizing beam splitter. λ/
4: quarter-wave plate. XYZ:micro-positioner. SA: spectrum analyzer. PD: photodetector. (b) Electrical actuation scheme.FW: flexible waveguide. The gate electrode is connected to a semi-rigid radio-frequency cable insidethe enclosure, which is connected to FW outside of the enclosure via a hermetic feed-through. (c)Front and back side of FW consisting of a 15 cm long copper microstrip patterned on Kapton andterminated with SMA connectors.
10 0 10 V dc [V] f [ M H z ] width [ m] -300-200-100050 d e p t h [ n m ] N L R F L G = R n o F L G (a) 5 m (b) 1 m (e)(c)(d) FIG. 3. Cavity, gate electrode, FLG thickness and tuning of the resonant frequency. (a) Opticalmicroscopy image showing FLG and source (S) and drain (D) electrodes. (b) AFM image of thegate electrode at the bottom of the cavity, and (c) cross section along the blue dashed line. (d)Calculated ratio of reflectances R FLG /R noFLG as a function of the number of layers N L . R FLG is thereflectance of the structure composed of FLG on a 500 nm thick slab of SiO on Si at λ = 633 nm,and R noFLG is the reflectance without FLG. The blue shaded area is the uncertainty related tooxide thickness measurements. (e) (cid:104) V (cid:105) as a function of f and V dc for the lowest frequency mode. V ac = 0 . rms , P inc = 110 µ W. Orange side of the color bar: 5 × − V . x [ m] y [ m ] -100 -80 -60 0 1 2 3 4 5 x [ m] y [ m ] f [MHz] h V pd i = [ V r m s ] ! -4 -2 0 2 x ! x [ m] Q -4 -2 0 2 x ! x [ m] f [ M H z ]
63 64 65 f [MHz] h V pd i = [ V r m s ] !
10 100 P inc [ W] Q
10 100 P inc [ W] f [ M H z ] x [ m] y [ m ] ! [V rms] [MHz](i) FIG. 4. Effect of the beam on the response of the resonator and vibration imaging. (a) Root meansquare of the voltage at the output of the photodetector, (cid:104) V (cid:105) / as a function of drive frequency f at V ac = 12 . rms and V dc = 5 V with the resonator near the center of the beam. From blue tored: P inc = 20, 45, 70, 135, 160, and 205 µ W. (b) Quality factors Q and (c) resonant frequency f ,measured from a series of resonances partially shown in (a), as a function of P inc . (d) (cid:104) V (cid:105) / as afunction of f as the resonator is moved along x into the beam at x ( V ac = 12 . rms , V dc = 5 Vand P inc = 110 µ W). (e) Q and (f) f as a function of in-plane displacement x − x taken fromresonances partially shown in (d). (g) Peak (resonant) value of the time averaged electrical power (cid:104) V (cid:105) /
50 as a function of x and y . (h) Peak value of the electromechanical signal ¯ V , shown here ona linear scale. V dc = 5 V, V ac = 0 . rms and P inc = 110 µ W. (i) Peak frequency of ¯ V ( f ). Datain (g)-(i) were obtained after optimizing the collection efficiency of the photodetector compared todata in (a)-(f). x [ m] y [ m ] -100 -85 -70 0 2 4 x [ m] y [ m ] -100 -90 -800 2 4 x [ m] y [ m ] -100 -90 -80(c) (d)(a) (b)[dBm] [dBm][dBm] FIG. 5. Effect of a fold on a the mode shapes of the globally gated resonator. (a) Scanning electronmicroscope image of the device. The arrows point to a fold in FLG (red shading) near the bottomedge of the cavity. (b-d) (cid:104) V (cid:105) /
50 as a function of in-plane displacements x and y for the lowestfrequency mode near 75 MHz (a), for the second mode near 100 MHz (b) and for the third modenear 150 MHz (d). V ac = 7 .
07 mV rms is used in (b) and (c) and V ac = 22 .