Nanomechanical vibrational response from electrical mixing measurements

 View Online  Export Citation RESEARCH ARTICLE | NOVEMBER 13 2023 Nanomechanical vibrational response from electrical mixing measurements C. Samanta  ; D. A. Czaplewski ; S. L. De Bonis; C. B. Møller ; R. Tormo-Queralt ; C. S. Miller ; Y. Jin ; F. Pistolesi ; A. Bachtold Appl. Phys. Lett. 123, 203502 (2023) https://doi.org/10.1063/5.0184931  CHORUS Articles You May Be Interested In Steady-state transition of buckled nano graphite sheets in vibration processes Appl. Phys. Lett. (March 2025) Nanomechanical resonant structures in single-crystal diamond Appl. Phys. Lett. (September 2013) Mode-dependent scaling of nonlinearity and linear dynamic range in a NEMS resonator Appl. Phys. Lett. (August 2024) 16 October 2025 13:10:55 Nanomechanical vibrational response from electrical mixing measurements Cite as: Appl. Phys. Lett. 123, 203502 (2023); doi: 10.1063/5.0184931 Submitted: 28 June 2023 .Accepted: 30 October 2023 . Published Online: 13 November 2023 C. Samanta, 1,a) D. A. Czaplewski, 2 S. L. De Bonis, 1 C. B. Møller, 1 R. Tormo-Queralt, 1 C. S. Miller, 2 Y. Jin, 3 F. Pistolesi, 4 and A. Bachtold 1 AFFILIATIONS 1 ICFO - Institut De Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain 2 Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439, USA 3 C2N, CNRS, Universit e Paris-Saclay, Palaiseau, France 4 Universite de Bordeaux, CNRS, LOMA, UMR 5798, F-33400 Talence, France a) Author to whom correspondence should be addressed: [email protected] ABSTRACT Driven nanomechanical resonators based on low-dimensional materials are routinely and efficiently detected with electrical mixing measurements. However, the measured signal is a non-trivial combination of the mechanical eigenmode displacement and an electrical contribution, which makes the extraction of the driven mechanical response challenging. Here, we report a simple yet reliable method to extract solely the driven mechanical vibrations by eliminating the contribution of pure electrical origin. This enables us to measure the spectral mechanical response as well as the driven quadratures of motion. This method is crucial for nanomechanical vibrations in the nonlinear regime, since the shape of the mechanical response depends on the physics at work. We further show how to calibrate the measured signal into units of displacement. Our method marks a key step forward in the study of nanoelectromechanical resonators based on low-dimensional materials in the nonlinear regime. V C2023 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http:// creativecommons.org/licenses/by/4.0/).https://doi.org/10.1063/5.0184931 Nanomechanical resonators 1 are exquisite sensors of mass adsorption 2–4 and external forces. 5–8 These sensing capabilities enable advances in different research fields, such as mass spectrometry, 9 surface science, 10–12 heat transport, 13,14 in situ nanofabrication, 15 magnetic resonance imaging, 16–18 scanning probe microscopy, 19–21 nanomagnetism, 22–25 and probing viscosity in liquids. 26 Many of these studies are carried out with mechanical resonators based on lowdimensional materials, such as carbon nanotubes, 27,28 because of their tiny mass. However, the detection of motion becomes increasingly difficult as resonators get smaller. The electrical detection of resonators based on low-dimensional materials is usually realized with a mixing-based method, 28,29 where the vibrations are driven near resonance frequency and detected at a low frequency within the RC bandwidth of the circuit. The electrical mixing detection has been applied to resonators based on carbon nanotubes, 6,10,12,28,30–47 graphene, 35,48–59 transition metal dichalcogenides (TMDs), 60–65 and semiconducting nanowires. 66–74 Different variants of the mixing method were developed by applying either two signals 28 on the device or one signal that is amplitude 49 or frequency 33 modulated. The transduction from displacement into current can be based on capacitive 28 or piezo-resistive measurements. 75 Methods were also implemented to measure thermal vibrations 39 and ring-downs 43,76 at temperatures down to below 0.1 K. The fundamental detection limit was theoretically investigated in Ref. 77. Despite this large amount of work, the measurement of the spectral response of nanomechanical vibrations to a driving force—the most common method to study mechanical resonators 1 —remains to be demonstrated with the mixing detection. Here, we report on a simple, yet reliable, method to measure the spectral mechanical response to a driving force using the mixing method with two signals applied to the device. By properly tuning the phase of the measured signal, we are able to separate the signal of the mechanical vibrations from the signal of pure electrical origin inherent to the mixing method. Moreover, we use the pure electrical contribution as a resource to calibrate the displacement. The ability to measure the spectral mechanical response is a key step for exploring the rich physics of nonlinear phenomena. We produce nanotube mechanical resonators by growing nanotubes using chemical vapor deposition on prepatterned electrodes. The Appl. Phys. Lett. 123, 203502 (2023); doi: 10.1063/5.0184931 123, 203502-1 V CAuthor(s) 2023 Applied Physics Letters ARTICLE pubs.aip.org/aip/apl 16 October 2025 13:10:55 nanotube is suspended ’150 nm above a gate electrode and connected between two metal electrodes 6,78 [Fig. 1(a)]. We clean the nanotube surface from contamination molecules by applying a large current through the device under vacuum at low temperature. 79 We detect the vibrations of the nanotube resonator by capacitively driving it with an oscillating voltage Vac gcos xton the gate electrode, applying the voltage Vac scos ððxþdxÞtþueÞon the source electrode, and measuring the current at frequency dx from the drain electrode with a lock-in amplifier 6 [Fig. 1(a)]whereueis the phase difference between the two oscillating voltages. We set dx within the bandwidth of the circuit, and we sweep xthrough the mechanical frequency xm(dx xm). All the measurements are carried out with the device in the single-electron tunneling regime 80–84 at the temperature T¼6K. To detect the vibrations, the nanotube has to behave as a transistor such that the conductance Gdepends on the charge Qin the nanotube. The application of Vac gcos xtmodulatesthechargethroughtwo terms dQ¼CgdVgþdCgVg. The first term has a pure electrical origin, while the second term is proportional to the driven vibration displacement dzvia dCg¼C0 gdz,whereC0 gis the spatial derivative of the capacitance. The application of Vac scos ððxþdxÞtþueÞenables one to mix down the modulation of Ginto a current oscillation at the frequency dx within the circuit bandwidth via Ohm’slawI¼GVs.The mixing intertwines the two terms of the charge modulation. The downside of the mixing method is that the measured current is not directly proportional to the driven vibration displacement, making the extraction of the driven mechanical response complicated. To illustrate this, we consider the amplitude of the current AI¼affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðXzþCgVac g=C0 gVdc gÞ2þY2 z q, where we express the vibration displacement and the measured current as z¼Azcosðxtþ/zÞ¼Xzcos xtþYzsin xt;(1) I¼AIcosðdxtþ/IÞ¼XIcos dxtþYIsin dxt(2) and a¼ð@G=@VgÞVac sVdc gC0 g=2Cgis a constant that depends on the transconductance ð@G=@VgÞ. In the limit where the displacement zis much smaller than CgVac g=C0 gVdc g, the response of AIconsists of a signal proportional to the quadrature Xztogether with a large, frequencyindependent background that has a pure electrical origin, see Fig. 1(b). In the opposite limit, the responses of AIand Azbecome proportional to each other [Fig. 1(c)]. This shows that while the shape of the mechanical response remains the same in these two limits, the measured shape of the spectral response of AIis completely different. In practice, the measured response of AIin the linear regime is usually compared with the predictions based on a Lorentzian line shape using several free parameters. 85 These can include the resonance frequency, the mechanical linewidth, the offset CgVac g=C0 gVdc g, the constant a,and the phase difference ue. This analysis usually becomes unreliable in the nonlinear regime, since the mechanical response can take a large range of different line shapes. Therefore, a method to directly extract the mechanical response without any fitting procedure is highly desirable. We demonstrate a method to separate the signal of the mechanical vibrations from the signal of pure electrical origin. The idea is to put all the current signal of pure electrical origin in the quadrature XI such that XI/XzþCgVac g=C0 gVdc gand YI/Yz.Thisisachievedby manipulating the data after the measurements with a rotation of the angle /I[Eq. (2)]intheplaneðXI;YIÞ(see supplementary material section, Sec. I). To illustrate this method, we proceed with the response of the two quadratures XIand YIof the current directly acquired from the lock-in amplifier [Figs. 2(a) and 2(b)]. The two responses cannot be described by the usual functional forms of driven linear oscillators. We then compute the background offset of YIby incrementing the rotation phase /Iby d/Ifrom 0 to 2p[Fig. 2(c)]. When this background offset in YIis zero, all the current signal of pure electrical origin is in XIand can be subtracted from the data. The resulting quadrature responses have now the familiar functional form of linear oscillators [Figs. 2(e) and 2(f)], and the spectral response of the displacement is well described by a Lorentzian [Fig. 3(a)]. We use the subtracted background current Xe Iof pure electrical origin to calibrate the displacement of the nanotube resonator in units of meters [Fig. 3(a)]. This background current is given by Xe I¼ð@G=@VgÞVac sVac g=2; we verify that it depends linearly on Vac g [Fig. 2(d)]. The two quadratures then read Xz¼CgVac g C0 gVdc g XIXe I Xe I ;Yz¼CgVac g C0 gVdc g YI Xe I :(3) The calibration of the displacement is subject to the uncertainty in the estimation of Cg=C0 g(see below). This method enables us to obtain the mechanical response in the nonlinear regime, see Fig. 3(c). While the response can be described by FIG. 1. (a) Schematics of the measured device. The nanotube is suspended over a gate electrode and electrically connected to two metal electrodes. Two oscillating voltage signals are applied to the device. The current Iis measured with a RLC resonator and a low-temperature amplifier. 6 (b) Calculated response of a underdamped harmonic oscillator electromechanically driven by a capacitive force FðxÞ/Vac gðxÞexpected from the mixing method in the limit where the mechanical displacement is much smaller than CgVac g=C0 gVdc g. (c) Same as panel b but in the opposite limit. Applied Physics Letters ARTICLE pubs.aip.org/aip/apl Appl. Phys. Lett. 123, 203502 (2023); doi: 10.1063/5.0184931 123, 203502-2 V CAuthor(s) 2023 16 October 2025 13:10:55 the usual Duffing equation without nonlinear damping [red line in Fig. 3(c)], a detailed study of this device where vibrations are coupled to single-electron indicates that the nonlinear mechanical response to a driving force is significantly modified by thermal fluctuations (see Ref. 87). This results in a decreasing responsivity for an increasing drive and the absence of hysteresis when the driving frequency is swept back and forth. 87 When taking into account the effect of thermal fluctuations, the Duffing constant is 1:91031 m 2 s 2 . Note that while the thermal fluctuations modify the response of the driven vibrations, they are barely visible in the measurement shown in Fig. 3(c) as they are averaged out with the lockin amplifier. The current Xe Iof pure electrical origin also enables quantifying the mass of the mechanical mode in a way that is simple and reliable. In Fig. 3(b), we compute the force response of the displacement amplitude at resonance frequency xmin the linear regime using Az¼Cg C0 g Vac g Vdc g YI Xe I ;F¼bC0 gVdc gVac g;(4) where YIcorresponds to the current amplitude at resonance frequency after having separated the signal of pure electrical origin. The constant bcan be different from one for electron transport in the singleelectron tunneling regime [Eq. (5) and supplementary material, Sec. II]. The mass mis determined from the slope of the forcedisplacement response using Az¼ðQm=mx2 mÞFwith Q m the quality factor. The slope depends on the current terms YIand Xe Imeasured from the lock-in amplifier but is independent of Vac g;Vac s,and @G=@Vgthat enter the prefactor ain the current-displacement conversion and whose values could be somewhat altered by the amplification chain and the losses along the coaxial cables. We determine m¼4:561:5 ag from the mass measured at different Vdc gvalues [Fig. 4(b)]. This value is consistent with the length of the suspended nanotube measured by scanning electron microscopy and assuming a 1.5 nm radius single-wall nanotube. The uncertainty in the mass measurement and the displacement calibration comes from the uncertainty in the estimation of the nanotube-gate separation dand the mass fluctuations in Fig. 4(b).The separation d¼150 620 nm measured by atomic force microscopy FIG. 2. (a) and (b) Spectral response of the current quadratures XIand YIto the driven capacitive force. The two blue boxes indicate the YIvalues used to compute the background offset Ye I. (c) Estimated background offset Ye Ifrom the data in a and b by incrementing the phase /Iin Eq. (1) by d/I. (d) Background current Xe I with pure electrical origin with d/Iset so that Ye I¼0. The red line is a linear fit of the data. (e) and (f) Spectral response of the displacement quadratures Xzand Yz to the driven capacitive force after having subtracted Xe Ifrom XI. The data are compared to the quadratures expected for a linear oscillator (red lines). FIG. 3. (a) Spectral response of the displacement amplitude Azto the driven capacitive force in the linear regime after having subtracted Xe Ifrom XI. The data are compared to a Lorentzian peak (red line). (b) Force response of the displacement amplitude Azat the mechanical resonance frequency. The red line is a linear fit of the data. The force is multiplied by the quality factor, since the latter varies when increasing the driving force. 35 (c) Nonlinear mechanical response of the displacement amplitude Azto the driven capacitive force. The spectral response is extracted using the method described in Fig. 2. The red line is the fit to the usual Duffing equation. 86 The Duffing constant is much larger than that in panel (a) due to the coupling of vibrations and single-electron tunneling, see Ref. 87; concomitantly, the resonance frequency is lower. Applied Physics Letters ARTICLE pubs.aip.org/aip/apl Appl. Phys. Lett. 123, 203502 (2023); doi: 10.1063/5.0184931 123, 203502-3 V CAuthor(s) 2023 16 October 2025 13:10:55 enters in the estimation of C0 g¼Cg=dln ð2d=rÞin Eqs. (3) and (4) when considering the capacitance between a tube with radius rseparated from a plate by the distance d. We estimate Cg¼e=DVg¼9:7 aF from the separation DVgin gate voltage between two conductance peaks associated with single-electron tunneling [Fig. 4(a)]. This capacitance is consistent with Cg¼12:9 aF obtained from the device geometry measured by scanning electron microscopy and atomic force microscopy. The fluctuations of min Fig. 4(b) are partly due to the error in the estimation of the average charge occupation f, which varies between 0 and 1 when sweeping Vdc gthrough the conductance peaks [Fig. 4(a)], since fenters in the prefactor bof the driven force in Eq. (4) as b¼1Cg CRþfð1fÞCg CR e2=CR kBT(5) in the incoherent single-electron tunneling regime. Here, CRis the total capacitance of the single-electron transistor and varies gradually from 19.9 to 26.5 aF when sweeping Vdc gover multiple conductance peaks. The fluctuations of mare also attributed to the slow increase in the contamination on the nanotube surface at 6 K; the three largest m values in Fig. 4(b) are obtained from force-displacement measurements carried out one month after the first measurements. We now discuss the limits of our method to measure the mechanical response. The method is expected to be reliable for any device as long as the current contributions with mechanical, and pure electrical origins are both well resolved. This means that the signal of both contributions should be large compared to the electrical noise floor of the measurement circuit. This is the case in our measurements, see Figs. 2 and 3. The noise from the electrical circuit at 6 K is given by the Johnson–Nyquist noise of the 7.5 kXimpedance of the RLC resonator. 6 Using the current noise SI¼210 fA =ffiffiffiffiffiffi Hz pat 6 K, the conversion abetween current and displacement, and the 1.5 Hz measurement bandwidth, we get a 22 pm standard deviation in the displacement noise, which is consistent with the noise observed in Fig. 3(a). The contribution to the measured signal from the thermal mechanical fluctuations is significantly smaller than that from the electrical noise. The vibrations can be driven to large amplitudes, but it is better to have displacements much smaller than the resonator-gate separation to avoid nonlinearity in the detection. In this way, the capacitance Cgand its derivative C0 gremain constant resulting in a linear relation between the measured quadratures XIXe Iand YIand the displacement quadratures Xzand Yzin Eq. (3). The nonlinearity in the detection is related to Cg=C0 g¼ðddzÞln ð2ðddzÞ=rÞin our device. For large vibrations, the nonlinearity in the detection has to be taken into account when estimating Az.InFig. 3(c), the nonlinearity in the detection modifies Azby about 1% at the resonance. In summary, we show how to measure the spectral mechanical response using electrical mixing measurements. Our method enables us to calibrate the displacement in meters. Another asset of this method is the determination of the mass of the measured mechanical eigenmode, which is a key parameter of the mechanical resonators when used in sensor applications. This work opens the possibility to quantitatively study nanoelectromechanical resonators in the nonlinear regime, where interesting mesoscopic phenomena emerge. 1 The driven nonlinear mechanical response can be described by a large range of different shapes depending on, for example, the strength of the thermal noise, the presence of frequency noise, the nonlinear resonant coupling to a high-frequency mode, and whether the restoring force is described by only the cubic nonlinear term in the displacement or also higher order terms. Our method enables us to determine such driven response spectra in an unambiguous way from electrical mixing measurements. Another important aspect of the method is the calibration of the displacement, which enables quantifying the strength of nonlinear forces by determining for instance the Duffing constant from the measurement of the resonance frequency shift as a function of displacement. 1 See the supplementary material for theoretical description of the two-source mixing technique and the methodology used to derive the driving force in the single electron tunneling domain. We acknowledges ERC Advanced Grant No. 692876 and MICINN Grant Nos. RTI2018-097953-B-I00 and PID2021122813OB-I00. Work performed at the Center for Nanoscale Materials, a U.S. Department of Energy Office of Science User Facility, was supported by the U.S. DOE, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. We also acknowledge AGAUR (Grant No. 2017SGR1664), the Fondo Europeo de Desarrollo, the Spanish Ministry of Economy and Competitiveness through Quantum CCAA, Nos. TED2021129654B-I00, EUR2022-134050, and CEX2019-000910-S (MCIN/ AEI/10.13039/501100011033), MCIN with funding from European Union NextGenerationEU (PRTR-C17.I1) and Generalitat de Catalunya, CERCA, Fundacio Cellex, Fundacio Mir-Puig, the French Agence Nationale de la Recherche (Grant No. SINPHOCOM ANR-19-CE47-0012), Marie Skłodowska-Curie (Grant No. 101023289). AUTHOR DECLARATIONS Conflict of Interest The authors have no conflicts to disclose. Author Contributions Chandan Samanta: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Writing –original draft (equal). David Czaplewski: Conceptualization (equal); Data curation (equal); Formal analysis (supporting); Funding acquisition (equal); Methodology (equal); Writing –original draft (equal). Sergio Lucio FIG. 4. (a) Electrical conductance of the nanotube device as a function of gate voltage. (b) Mass of the eigenmode measured at different gate voltage values. The red dashed line indicates the average mass of 4.5 ag. Applied Physics Letters ARTICLE pubs.aip.org/aip/apl Appl. Phys. Lett. 123, 203502 (2023); doi: 10.1063/5.0184931 123, 203502-4 V CAuthor(s) 2023 16 October 2025 13:10:55 De Bonis: Formal analysis (supporting); Funding acquisition (equal); Methodology (supporting); Writing –original draft (equal); Writing – review & editing (equal). Christoffer Møller: Formal analysis (supporting); Methodology (supporting); Writing –original draft (equal); Writing –review & editing (equal). Roger Tormo-Queralt: Formal analysis (supporting); Writing –original draft (equal); Writing – review & editing (equal). C. S. Miller: Formal analysis (supporting); Methodology (equal); Writing –review & editing (equal). Yong Jin: Methodology (equal). Fabio Pistolesi: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Writing –original draft (equal). 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