Abstract
Nonlinearity in macroscopic mechanical systems may lead to abundant phenomena for fundamental studies and potential applications. However, it is difficult to generate nonlinearity due to the fact that macroscopic mechanical systems follow Hooke’s law and respond linearly to external force, unless strong drive is used. Here we propose and experimentally realize high cubic nonlinear response in a macroscopic mechanical system by exploring the anharmonicity in chemical bonding interactions. We demonstrate the high tunability of nonlinear response by precisely controlling the chemical bonding interaction, and realize, at the singlebond limit, a cubic elastic constant of 1 × 10^{20} N m^{−3}. This enables us to observe the resonator’s vibrational bistates transitions driven by the weak Brownian thermal noise at 6 K. This method can be flexibly applied to a variety of mechanical systems to improve nonlinear responses, and can be used, with further improvements, to explore macroscopic quantum mechanics.
Introduction
Nonlinearity in micro and nanomechanical systems has been used to study processes such as fluctuationenhanced dynamics^{1,2,3,4,5}, synchronization^{6,7}, mode mixing^{8}, noise control^{9,10}, signal amplification^{11,12} and logic devices^{13,14,15}. Its dynamics can be modelled by a driven Duffing oscillator equation as follows:
Here m, ω_{0}, Q and are the mass, resonance frequency, quality factor and linear spring constant, respectively. And αx^{3} is the Duffing nonlinearity with α the Duffing constant, that is, the cubic elastic constant. Under weak drive F_{drive}, the nonlinear response is negligible due to its cubic dependence on the amplitude x of the resonator, and so the resonator behaves like a simple harmonic oscillator. This is the wellknown Hooke’s law of elasticity^{16}. On the other hand, nontrivial dynamics of the resonator emerges when the drive is strong enough. A famous example is the occurrence of the driven Duffing biastability when the drive strength reaches a certain threshold F_{c} (ref. 17), with the corresponding threshold power P_{c}.
Since F_{c} is inversely proportional to as , the larger is α, the weaker driving force is required for the system to reach nonlinear regime. One benefit of low driving force is low driving noise, while in those micro and nanomechanical systems reported in the literature, driving noise is far beyond the system’s intrinsic Brownian thermal noise even at room temperature^{1,2,3,4,5,9,11}. On the other hand, , so increasing α can lower the threshold power, which is favourable in some scenarios^{13,14,15}. In practice, since the system’s fundamental parameters m, ω_{0} and Q are limited by various factors, such as working bandwidth, material or fabrication, a universal way to increase and to tune α independently is significant.
What is more demanding is from quantum science, where strong nonlinearity can make quantum effects emerge from a classical harmonic resonator^{18}. Such quantum nonlinearity is still elusive in macroscale mechanical systems due to the naturally weak nonlinear response. Generally speaking, the emergence of quantum behaviour requires quantum nonlinear strength to overcome the system’s decoherence^{19,20,21,22,23,24}, and since the quantum fluctuation x_{zpf} is usually extremely tiny for a macroscale resonator, generating ultrastrong nonlinear response α is of paramount importance.
In this work we demonstrate a system with a macroscopic mechanical resonator coupled to a single chemical bond, where the anharmonicity of the chemical bond deformation potential induces a giant nonlinear response of the resonator and can be tuned using external force. When driving to the nonlinear bistates regime, stochastic transitions between bistates are observed, which are demonstrated to be induced by the intrinsic Brownian thermal noise of the resonator.
Results
Theoretical model and density functional calculations
Our system consists of a macroscopic mechanical resonator tightened to an anchor via a chemical bonding structure shown schematically in Fig. 1a. Strong nonlinearity is achieved when the resonator moves along xdirection by deforming the chemical bond. This is because, although the resonator alone follows the elasticity theory with linear dynamics, the response of the chemical bond is highly anharmonic. This is illustrated by the chemical bond’s energy curve U_{chem}(x) (Fig. 1b) obtained by density functional theory calculations (see Methods for details). For simplicity, x=0 is set at the minimum of U_{chem}(x) (Fig. 1b). By applying on the chemical bond an external control force F, the resonator’s equilibrium position x_{eq} can be tuned, and along with this the spring constant for sufficient small vibration is modified by Δk=∂^{2}U_{chem}/∂x^{2}. As shown in Fig. 1c, when the resonator’s equilibrium position is far away from the atom contact, the chemical bonding interaction is weak. By tuning the control force, that is, shifting the resonator’s equilibrium position towards x=0, the strength of Δk first reaches a local maximum and then drops to zero at the point where the chemical bonding attraction reaches maximum. The Duffing constant from chemical bonding, that is, α=(1/6)∂^{4}U_{chem}/∂x^{4}, changes differently (Fig. 1d): its strength first reaches a local maximum, then drops to zero before approaching the maximum attraction point, and finally changes its sign and increases in strength. At the maximum attraction point, α=2 × 10^{22} N m^{−3} is reached, while Δk is zero.
Figure 1e,f shows the threshold driving force F_{c} and the corresponding power P_{c} as a function of the macroscopicity of the resonator (here characterized by the resonator’s mass) with/without using the chemical bonding forceinduced nonlinearity. With specific basic mechanical parameters, that is, resonance frequency ω_{0}, quality factor Q and mass m, introducing chemical bonding force can reduce F_{c} as well as P_{c} significantly. The term proportional to x^{3} in the expansion of chemical bonding potential U_{chem} and higherorder terms beyond the Duffing nonlinearity are also presented in our system, but are insignificant for bistates dynamics studied in current experiments.
Experimental realization of the system
Now we demonstrate the above idea by using a macroscopic doubly clamped beam whose fundamental vibrational mode couples to a goldatom contact, as shown in Fig. 2a. The atom contact is made by electric current migrations (see Methods) on a nanobridge, which anchors the beam to a stiff electrode (Fig. 2b). The dimensions of beams is l × w × t=50 × 1.5 × 0.51 μm. The mass of the beam is ∼0.2 ng. The vibrations of the beam deform the gold–gold bonds of the contact along x direction. The beam has a typical intrinsic frequency around ω_{0}/2π=1.6 MHz, with measured quality factor Q ranging from 1,000 to over 3,000, depending on specific device. The device is placed in an ultrahigh vacuum chamber and has an environment temperature of 6 K.
We carried out the experiments (see Methods) with several devices. The measured beam’s resonance frequency and the corresponding spring constant are plotted as functions of control force F in Fig. 3a–c for devices A to C correspondingly. Device A is in noncontact regime while B and C are in contact regime but with different atom structure in the contact. In typical contact regime devices, the frequency varies by about 1 MHz when the beam is pulled out to noncontact regime. Figure 3d–f shows the corresponding electron conductance results measured simultaneously as those for Fig. 3a–c. In the noncontact regime, the conductance due to quantum tunnelling shows a strong dependence on the control force and a value as low as ∼300 kΩ^{−1} is reached, which confirms that a shortrange chemical bonding force has been significantly involved^{25}, while in the contact regime, quantized conductance is observed and is insensitive to control force. Such a ballistic conductance is resulted from small number of metallic bonds^{26}.
From the dependence of the spring constant k on the applied static force F, we can obtain the Duffing constant as
where ξ is the shape factor with value the order of 1 and depends on the definition of the mode shape normalization of the device (Supplementary Note 1). To reliably obtain the Duffing constant from the data with noise, we have smoothed the α using a running average of five data points. Figure 3g–i plots the measured Duffing constant α as a function of control force F. In the singlebond case, the maximum strength of the Duffing constant achieved in our experiments is (1.1±0.2) × 10^{20} N m^{−3}, with an enhancement of six orders relative to the estimated intrinsic nonlinearity due to the elongation of the beam (α_{0}≈1 × 10^{14} N m^{−3}), and is many orders of magnitude larger in strength than those reported previously^{11,12,27,28,29,30,31} (see Supplementary Table 1 for comparison to other systems). High tunability is easily reached by control force. In practical, both noncontact and contact regimes can be used depending on detailed applications. In the noncontact regime, jumptocontact leads to system’s instability (see Supplementary Note 1 for detail), while in contact regime such instability is naturally avoided. The frequency response of the device was studied by increasing the drive force, and the results are plotted in Fig. 3j–m, with Fig. 3j from device A, Fig. 3k,l from device B corresponding to regions I and II in Fig. 3h, respectively, and Fig. 3m from device C. The hysteresis responses are due to Duffing induced bistates^{17}, with their directions agreeing with the signs of the nonlinear coefficients measured in Fig. 3g–i.
Observation of thermally activated bistates dynamics
In the following we present a demonstration of a dynamical effect of the strong nonlinear response. Thermal fluctuation effects in nonlinear regime can lead to various complex dynamics, and have been studied in various systems, with an example being the centre of mass motion of optical trapped particles^{32}. However, such a thermal nonlinear regime is exclusive in widely studied micro and nanomechanical systems. A wellknown phenomenon is noiseactivated bistates transitions in a driven Duffing oscillator^{33}. As a consequence of lacking strong enough nonlinear response, this phenomenon has only been previously observed by introducing strong artificial noise that far above the thermal fluctuation^{1,2,3,4,5,9}. With the successful realization of strong nonlinear response here, we have observed in our system the bistates transitions activated by the Brownian thermal noise even at cryogenic temperatures.
In the process, we chose a device (device D) of a relative high quality factor Q=3,100, and tuned the control force to a point with the Duffing constant approximately −1 × 10^{17} N m^{−3}. We avoided using the maximum absolute Duffing constant, in which case the corresponding bistates amplitude would be too small and that would make the motion sensing the noise. We drove the system to the bistable state regime and observed the switching behaviour by recording the amplitude of vibrations of the beam, as shown by the green trajectory in Fig. 4a.
To verify that such switchings are indeed from the Brownian thermal noise of the beam, we introduced an amplitude modulation to the driving signal of the form F_{drive}(t)=F_{drive}+δF_{drive} cos(Ωt), with Ω=0.5 Hz and δF_{drive}=0.18 pN. The amplitudes of the vibration of the beam show periodic switchings (Fig. 4a, purple trajectory) instead of random switchings. Figure 4b shows the corresponding power density, S_{mod}(Ω) with modulations and S_{noise}(Ω) without modulations, from which the signal to noise ratio (SNR) is calculated with SNR(Ω)=S_{mod}(Ω)/S_{noise}(Ω). By using the standard stochastic resonance theory (Methods), we estimated the total force noise as =(3.6±0.6) × 10^{−16} N Hz^{−1/2}. It agrees nicely with the resonator’s Brownian thermal noise, estimated^{34} by =4mω_{0}k_{B}T/Q as 3.3 × 10^{−16} N Hz^{−1/2}. In the limit of Ω approaching zero, the modulation becomes a perturbative force signal δF_{driv}ecos(ωt) added to the driving force with the same phase. In the rotating frame of the driving signal, the bistates dynamics is described by an overdamped double well whose shape is tuned by δF_{drive} (ref. 33), and becomes sensitive to the tuning near the bifurcation point. By applying a weak force δF_{drive}=2 fN that is only several times larger than the resonator’s Brownian thermal noise force, significant changes in the statistic distribution are observed, as shown in Fig. 4c. The sensitivity of such a response to the external force is limited by the total force noise . It is noted that, in our device, in addition to the thermal noise, the electronic noise from the circuit also contributes to the total force noise, but with a weak power density of . We estimated that is 11 dB below the level, corresponding to an electronic noise temperature of 500 mK, and is mainly limited by our roomtemperature detection circuit (see Supplementary Table 2 for detailed data).
Discussion
In conclusion, we have demonstrated a highly controllable nonlinearity in a macroscopic mechanical system, with its fluctuation dynamics dominated by Brownian thermal noise. The universal existence and the small scale of the chemical bonding force make our method applicable to current widely used micro and nanomechanical systems in improving their nonlinear responses. With further improvements, nonlinearityinduced quantum behaviours in macroscopic mechanics^{19,20,21,22,23,24} are foreseeable in the type of systems described here.
Methods
Theoretical description of the system
Density functional theory calculations were used to estimate qualitatively the chemical bonding force (see Supplementary Note 1 for detailed description). Owing to stability considerations, a small structure of two goldatom clusters was adopted in the calculation (see Supplementary Fig. 1 for data). By changing only the relative position x of the left cluster relative to the right one while otherwise keeping the relative positions of all the gold atoms fixed, we obtained a chemical bonding energy function U_{chem}(x), from which the nonlinearity is calculated as the fourthorder derivative. Other structures of clusters are also considered (see Supplementary Fig. 2 for data). Owing to the existing systematic error in performing density functional theory calculations, the secondorder derivative of the total energy shows slightly an oscillatory behaviour at large distance range. We smoothed the calculated data by fitting it to the function . In reality, however, the relaxation of gold atoms can occur, which may modify the energy–displacement curve U_{chem}(x). We have considered this effect and find out that the results do not change significantly. To estimate the longrange forces, we considered analytically the van der Waals attraction and the electrostatic force^{25} for the geometric model of two gold cylinders close to each other, similar to the case of our nanobridge structure and in modelling nonlinear dynamics, we have dropped the quadratic nonlinearity in the chemical interaction (see Supplementary Fig. 3 for data). To estimate the intrinsic macroscopic mechanical property, we modelled the macroscopic resonator as a doubly clamped beam. The lowest vibrational mode has a weak intrinsic nonlinearity due to elongation. In calculating the threshold drive force F_{c} and the corresponding power P_{c} in Fig. 1e,f, we took the mechanical quality factor 3,000 and the ratio between the beam’s thickness and length t/l≥0.005. We also estimated the electrostatic forceinduced nonlinear response based on the same doubly clamped beam (see Supplementary Fig. 4 for data).
Fabrication of the device
The device was fabricated on a commercial silicononinsulator wafer using nanolithography (see Supplementary Note 2 for detailed descriptions). The nanobridge connecting the beam and the stiff electrode is about 200 nm in length and 80 nm in diameter. Focusion beam was adopted to narrow it down to <50 nm. Once such a device, that is, the doubly clamped beam with a suspended nanobridge, was successfully fabricated, it was placed in the ultrahigh vacuum environment of the cryogenic system, and then the bridge was electromigrated to form an atomic point contact^{35}. After the atom contact was produced, the beam was disconnected from the stiff electrode and its lowest vibrational mode was measured. The position of the equilibrium of the beam is controlled by applying a d.c. current through the beam so that the atom interaction can be tuned.
Device characterization and electronic noise
The resonant frequency (2π)^{−1}ω of the device under a control force F is measured directly when the device is working in the linear regime under a weak drive. The spring constant k is calculated as k=mω^{2}. Here the effective mass m for the first vibrational mode of the beam is estimated by using finite element simulation, which is widely used for similar systems^{36}. The electric conductance is measured by using G=dI(dV)^{−1}.
The Brownian thermal noise on the beam is calculated to be 3.3 × 10^{−16} N Hz^{−1/2}. On the basis of this and the calculated mass, the measured resonator frequency and quality factor, we estimate the thermal motion amplitude noise on resonance as 1.0 × 10^{−13} m Hz^{−1/2} (refs 34, 36), which cannot be measured directly due to the sensitivity limitation of our roomtemperature measurement electrics. We modelled the mechanical resonator as LCR elements following standard procedure^{37} (see Supplementary Note 3 for detailed descriptions), and analysed the noise in the equivalence electric circuit (see Supplementary Fig. 5 for circuit diagram). The separation of the atom contact can be controlled with a precision of about 1 pm without any feedback control for thousands of seconds (see Supplementary Fig. 6 for the data), and once jumptocontact occurred, a strong enough control force is used to pulled the tip out again and device’s main nonlinear character is commonly preserved (see Supplementary Fig. 7 for data). The intrinsic nonlinearity is measured from the frequency response to the driving strength in a standard way based on the Duffing nonlinear model^{17}(see Supplementary Fig. 8 for data). To do this, we have pulled the beam far apart from the stiff electrode with contact separation larger than 30 nm, so the interaction due to the atom contact is negligible. A 6T magnetic field is used in our experiments so that control force can be generated using a small current. This decreases the system total quality factor from its intrinsic value (see Supplementary Fig. 9 for data). We also applied a voltage bias <200 mV between the beam and the stiff electrode to compensate the contact potential of the atom contact so that the contribution of electrostatic force in our experiments is minimized (see Supplementary Fig. 10 for data).
In our experiment, there are mainly three sources of electronic noise. The first one is the Johnson–Nyquist noise , with T room temperature and R the circuit’s resistant, which produces a force noise with power density . The second one is the current leakage from the input of the voltage preamplifier . We model the preamplifier backaction by a current source similar to that described in ref. 38. In doing this, we assume that there is no correlation between voltage imprecision and the backaction (ref. 39), so leads to a force noise with power density . Another one is from the radio frequency driving signal, which generates a phase noise and works as an equivalent force noise on the beam with power density . We estimate the total electronicinduced force noise as , and in the experiments, =0.9 × 10^{−16} N Hz^{−1/2}, with power density about 11 dB below the .
Measurement of the total force noise on the beam
The dynamics of our system is modelled by the standard Duffing oscillator of equation of motion
where the dissipation rate γ=mω_{0}/Q, F_{drive}(t)=F_{drive}+δF_{drive} cos(Ωt) is the amplitudemodulated driving force and F_{noise}(t) is the total noise with power density . The minimum force that drives the system into the nonlinear regime where bifurcation occurs is F_{c}, with the corresponding vibration frequency being (2π)^{−1}ω_{c}, and the amplitude x_{c} can be calculated from equation (3)^{17}. Near the nonlinear bifurcation point, we transform this equation to an overdamped one by following the standard procedure^{33}. For the case of modulation frequency Ω being much smaller than the decay rate ω_{0}/Q of the system, the SNR is related to the system’s noise power as^{40}
with x_{m} the half of the vibration amplitude of the bistable states, δω the nonlinearity frequencyinduced shift from linear resonance peak and γ_{k} the measured random switching rate without modulation. So from the measured SNR we obtained .
Additional information
How to cite this article: Huang, P. et al. Generating giant and tunable nonlinearity in a macroscopic mechanical resonator from a single chemical bond. Nat. Commun. 7:11517 doi: 10.1038/ncomms11517 (2016).
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Acknowledgements
We thank L. Jiang for many stimulating discussions and comments and for his help in improving the manuscript. We thank Y.X. Liu, Z.Q. Yin, L. Tian C.P. Sun, and M. Lukin for helpful discussions. We thank Yiqun Wang from Suzhou Institute of NanoTech and NanoBionics for nanofabricating supports and Guizhou Provincial Key laboratory of Computational NanoMaterial Science for calculation supports. This work was supported by the 973 Program (Grant No. 2013CB921800), the NNSFC (Grant Nos. 11227901, 91021005, 11104262, 31470835, 21233007, 21303175, 21322305, 11374305 and 11274299), and the ‘Strategic Priority Research Program (B)’ of the CAS (Grant Nos. XDB01030400 and 01020000).
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J.D. supervised the experiments; J.D. and P.H. proposed the idea and designed the experimental proposal; P.H., J.Z., L.Z., W.D. and F.X. prepared the experimental setup; P.H., J.Z., L.Z., S.L. and C.J. performed the experiments; J.Z. fabricated the device; D.H. and X.Z. carried out the theoretical calculation; M.C. carried out the finite element simulation; P.H., C.D., F.X. and J.D. wrote the paper; all authors analysed the data, discussed the results and commented on the manuscript.
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Supplementary Figures 110, Supplementary Tables 12, Supplementary Notes 13 and Supplementary References (PDF 6735 kb)
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Huang, P., Zhou, J., Zhang, L. et al. Generating giant and tunable nonlinearity in a macroscopic mechanical resonator from a single chemical bond. Nat Commun 7, 11517 (2016). https://doi.org/10.1038/ncomms11517
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DOI: https://doi.org/10.1038/ncomms11517
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