Abstract
Understanding and controlling modal coupling in micro/nanomechanical devices is integral to the design of highaccuracy timing references and inertial sensors. However, insight into specific physical mechanisms underlying modal coupling, and the ability to tune such interactions is limited. Here, we demonstrate that tuneable mode coupling can be achieved in capacitive microelectromechanical devices with dynamic electrostatic fields enabling strong coupling between otherwise uncoupled modes. A vacuumsealed microelectromechanical silicon ring resonator is employed in this work, with relevance to the gyroscopic lateral modes of vibration. It is shown that a parametric pumping scheme can be implemented through capacitive electrodes surrounding the device that allows for the mode coupling strength to be dynamically tuned, as well as allowing greater flexibility in the control of the coupling stiffness. Electrostatic pump based sideband coupling is demonstrated, and compared to conventional strainmediated sideband operations. Electrostatic coupling is shown to be very efficient, enabling strong, tunable dynamical coupling.
Introduction
The interactions of micro and nanomechanical resonators with various physical fields have been researched for several decades and have been engineered into specific device formats for applications to timing and frequency control^{1,2}, sensing^{3,4,5,6}, as well as more fundamental studies in alternative approaches to information processing^{7,8,9}, and quantum science^{10,11,12,13}. Several recent studies have also focused on fundamental emergent behaviors in coupled micro/nanomechanical systems^{14,15,16,17} and the mutual coupling between two distinct mechanical resonators or modes^{18,19,20,21,22,23,24,25,26,27}. By regarding one of the coupled mechanical modes as a phonon cavity, optomechanicslike dynamical operations, such as cooling (dynamical coupling)^{22,28,29,30,31}, amplification^{22,31,32}, squeezing^{33,34}, coherent manipulation^{28,35,36}, and phonon lasing^{30,37}, have been demonstrated in micro and nanomechanical systems. Mode coupling mechanisms based on physical mechanical linkages^{28}, dielectric coupling^{21,35}, tensioninduced parametric coupling^{18,23,29}, and internal resonance^{20,25} have been previously investigated. However, physical insight is often limited due to the limited experimental control of mode coupling in such systems. Often, manufacturing tolerances or small asymmetries can have a huge impact, and the ability to define modal coupling parameters by design is still limited, often even elusive. Static electrostatic tuning in such systems is possible but provides for only weak tunability, and the ability to define and exercise good control on modal coupling in micro and nanomechanical resonators remains a key challenge limiting practical applications as well.
In this paper, we demonstrate that dynamic capacitive tuning can enable significant tunability of modal coupling in microelectromechanical devices. The intrinsic modal coupling in such systems can be tuned by the nonlinear fields associated with parallel plate transducers. Such fields can be dynamically modulated enabling a further knob on the tunability of system response. A vacuumsealed micromachined gyroscopic ring resonator is employed as the experimental testbed in this work. Such a system demonstrates linear hybrid coupling between the gyroscopic neardegenerate modes due to configurational/structural asymmetries arising from manufacturing tolerances or material properties, and misalignment between the principal axis and the detection electrode. We demonstrate that, akin to the tensioninduced stiffness hardening mechanisms^{18,22,23}, electrostatically induced stiffness softening can also provide a nonlinear parametric coupling term. While the static linear and nonlinear interactions generated by the electrostatic field have been previously studied^{38,39}, capacitive nonlinear parametric coupling between normal modes in a single resonator has not been presented previously. Here, it is shown that apart from the tensioninduced parametric coupling, this electrostatic nonlinear parametric coupling may reside, or even dominate, in capacitively transduced devices^{30,31}. Dynamical mode coupling between gyroscopic modes is implemented by applying a reddetuned pump adjusted based on the system parameters and the builtin intrinsic coupling. This capacitive device is demonstrated to be a couplingabundant multiplemode system, which used to be very difficult to construct due to the difficulty of combining different types of mode coupling mechanisms in one system^{40}. Strong dynamical coupling has significant practical applications to tuning the response of micro and nanoelectromechanical devices, such as modelocalized sensors^{41,42}, Coriolis gyroscopes^{43,44}, wireless filter resonators^{45}, and many other devices^{16,32}.
Results
Capacitive symmetric electromechanical resonator
A capacitive symmetric microelectromechanical ring resonator is used as the experimental testbed in this work^{44}. The resonator is constructed of 45 equispaced nested rings. Adjacent rings are interconnected with interleaved spokes. The resonator is supported by a central anchor. The diameter and the thickness of the resonator are 720 and 40 µm, respectively. The width of the rings and spokes is 3 \(\upmu\)m. The resonator is surrounded by 16 capacitive electrodes, employed to actuate, sense, and tune the inplane response. The capacitive gap \({d}_{0}\) is designed to be 1.5 µm wide. The device is fabricated using highly doped Ptype (111) singlecrystal silicon with a resistivity of 1–3 \({\rm{m}}\Omega \, {\rm{cm}}\), and encapsulated using the “Episeal” process^{46}, enabling a stable \(< 1\) Pa vacuum environment. The capacitive electrodes introduce a nonlinear electrostatic field surrounding the mechanical resonator, which produces inhomogeneous force terms, stiffness modification terms, and nonlinear mode interactions among multiple modes. Combined with the intrinsic properties of the bare mechanical resonators, the capacitive features allows for abundant manipulations of the system dynamics, the most interesting aspect studied here is the parametric pumpinduced dynamical operations.
The linear and nonlinear modecoupling mechanisms are investigated in this system. Moreover, those couplings can be manipulated dynamically using a parametric pumping scheme. The experimental setup is shown in Fig. 1a; all the experiments are conducted at room temperature. The inplane mechanical modes involved in this study are illustrated in Fig. 1b, c, involving two pairs of neardegenerate order2 and order3 modes, II1, II2, III1, and III2, with resonant angular frequencies \({\omega}_{{\rm{II}}{\hbox{}}1}=2\pi\,\times\,134,209\ {\rm{Hz}}\), \({\omega}_{{\rm{II}{\hbox{}2}}}=2\pi \,\times \,134,253\ {\rm{Hz}}\), \({\omega }_{{\rm{III}{\hbox{}1}}}=2\pi\, \times\, 166,498\ {\rm{Hz}}\), and \({\omega }_{{\rm{III}{\hbox{}2}}}= 2\pi\, \times\, 166,949\ {\rm{Hz}}\). The damping rates of the modes of the same order are identical, with values of \({\gamma }_{{\rm{II}}}\,\approx\, 2\pi \, \times 1.05\ {\rm{Hz}}\) and \({\gamma }_{{\rm{III}}}\approx 2\pi \, \times 2.51\ {\rm{Hz}}\). The electrodes are marked with numbers anticlockwise from 1 to 16 (Fig. 1a). The drive signals \(\pm {\!}{V}_{{\rm{d}}}\cos \, ({\omega }_{{\rm{d}}}t)\) are applied on electrodes 3 and 7 to actuate order2 modes. The pump signal \({V}_{{\rm{p}}}\cos \, ({\omega }_{{\rm{p}}}t)\) is applied on electrodes 1 and 5. A directcurrent (DC) voltage \({V}_{0}\) of 30 V is applied on the resonator body. The mechanical motion is transduced by the detection electrodes along drive and pump axes. The current signal is introduced to a lockin amplifier following further amplification stages. A tuning voltage \({V}_{{\rm{t}}1}\) can be superposed on drive electrode 7. Another tuning voltage \({V}_{{\rm{t}}2}\) can be applied on offaxis electrode 6.
Hybrid state coupling and dynamical manipulation
Order2 modes II1 and II2 are a pair of widely used degenerate modes^{47,48}. The antinodal axes (principal axes) of the order2 normal modes have an angular interval of \(4{5}^{\circ }\) (Fig. 2a). This system can be equivalent to a twodegreeoffreedom lumped parameter system in Cartesian coordinates (Fig. 2b). The angle in this equivalent system is the double of that in the real system depicted in Fig. 2a (see Supplementary Note 1). Axes \(x{\hbox{}}o{\hbox{}} y\) are defined along the directions in which mechanical motions are probed, the drive and pump are applied along \(x\) and \(y\) directions, respectively. Coordinates \({x}_{\omega}{\hbox{}}o{\hbox{}}{y}_{\omega }\) are defined along the principal axes of normal modes II1 (\({x}_{\omega }\) direction) and II2 (\({y}_{\omega }\) direction).
If the tuning voltage \({V}_{{\rm{t}}1}\) is applied along \(x\) direction, the resonant frequencies of modes II1 and II2 will disperse (Fig. 2d). An avoided crossing is obviously illustrated when \({V}_{{\rm{t}}1}\) is around 3.5 V, which indicates that the hybrid states observed from \(x{\hbox{}}o{\hbox{}}y\) coordinates H1 (\(x\) direction) and H2 (\(y\) direction) can be regarded as two coupled resonators, as depicted in Fig. 2c. This is also revealed by the nondiagonal equations of motion in \(x{\hbox{}}o{\hbox{}}y\) coordinates (see Supplementary Note 2). Those hybrid states are combinations of normal modes II1 and II2. The strength of the avoided crossing indicates the coupling rate of the two hybrid states. As shown in Fig. 2e, the \({V}_{{\rm{t}}1}\) tuning process can be simulated (see Supplementary Note 2). This coupling originates from two factors. The first one is the frequency difference between the normal modes II1 and II2 (\(\Delta \omega ={\omega }_{{\rm{II}{\hbox{}}{2}}}{\omega }_{{\rm{II}{\hbox{}}{1}}}\, \ne \, 0\)), and the second one is the misalignment (denoted as \(\theta\) in the equivalent Cartesian coordinates, as shown in Fig. 2b) of the principal axes with electrode axes, which are both due to manufacturing tolerances. The initial \(\theta\) of the tested resonator is estimated to be about \(1{8}^{\circ }\). In the setup reference system, the real misalignment is \(\theta /2\) (Fig. 2a). The mode shapes of the order2 modes in the \({V}_{{\rm{t}}1}\) tuning process can be obtained by calculating \(\theta\) at different values of \({V}_{{\rm{t}}1}\), which are selectively depicted in Fig. 2f. Both \(\Delta \omega\) and \(\theta\) can be changed by \({V}_{{\rm{t}}1}\) or \({V}_{{\rm{t}}2}\) (see Supplementary Note 2). A given set of \({V}_{{\rm{t}}1}\) and \({V}_{{\rm{t}}2}\) can determine a specific group of \(\Delta \omega\) and \(\theta\).
By regarding mode II2 as a phonon cavity, a dynamical sideband coupling operation can be realized based on the above structural coupling by applying a reddetuned parametric pump. In this implementation, \({V}_{{\rm{t}}1}\) and \({V}_{{\rm{t}}2}\) are tuned to make \(\theta \approx 3{6}^{\circ }\) and \(\Delta \omega \approx 2\pi \times 39.7\) Hz. An alternating current signal \({V}_{{\rm{p}}}\cos {\omega }_{{\rm{p}}}t\) applied along \(y\) axis will periodically change the stiffness of hybrid state H2. The dynamics of the normal modes are given by the equations of motion
where \({x}_{{\rm{II}}{\hbox{}}{\mathrm{j}}}\) and \({F}_{j}\)\((j=1,2)\) are the displacement and drive force amplitude (normalized by mass) of mode II\(j\), respectively. \({\Delta }_{{\rm{p}}}\) is the pump produced by signal \({V}_{{\rm{p}}}\cos ({\omega }_{{\rm{p}}}t)\),
where \(\kappa ={A}_{{\rm{p}}}{\epsilon }_{0}/({d}_{0}^{3}{m}_{{\rm{II}}})\), \({A}_{{\rm{p}}}\) is the area of pump electrodes, \({\epsilon }_{0}\) is the permittivity of vacuum, and \({m}_{{\rm{II}}}\) denotes the effective mass of mode II1 or II2. There is a DC term, a firstorder harmonic term, and a secondorder harmonic term in the pump. The DC term in the pump will tune the hybrid coupling condition, thus affecting resonant frequencies (see Supplementary Fig. 6). The resonant frequencies of order2 modes are functions of \({V}_{{\rm{p}}}\) and are denoted as \({\omega }_{{\rm{II}{{\hbox{}}}}\square ,{{\rm{V}}}_{{\rm{p}}}}\) (\(\square =1,2\)). When a 3V pump is applied, \({\omega }_{{\rm{II}{\hbox{}}{1,3V}}}\) (\({\omega }_{{\rm{II}{\hbox{}}{2,3V}}}\)) is slightly shifted from the initial \({\omega }_{{\rm{II}{\hbox{}}{1,0V}}}\) (\({\omega }_{{\rm{II}{\hbox{}}{2,0V}}}\)) by 0.33 Hz (0.60 Hz), as shown in Fig. 3f.
The amplitude of the firstorder term is \(4{V}_{0}/{V}_{{\rm{p}}}\) times that of the secondorder term. The firstorder term is dominant in this implementation, since \({V}_{0}\) is much larger than \({V}_{{\rm{p}}}\). The secondorder term itself and its interaction with the firstorder term only slightly contribute to higher order (\(\ge {\!}2\)) coupling. In the following interpretation parts, the DC and secondorder harmonic terms are not considered. \({\Gamma }_{1}=2\kappa {V}_{0}{V}_{{\rm{p}}}{\sin }^{2}\theta\) and \({\Gamma }_{2}=2\kappa {V}_{0}{V}_{{\rm{p}}}{\cos }^{2}\theta\) are defined as intramodal coupling terms, and \(\Lambda =2\kappa {V}_{0}{V}_{{\rm{p}}}\cos \theta \sin \theta\) is defined as an intermodal coupling term.
When simultaneously applying drive and reddetuned pump with frequencies \({\omega }_{{\rm{d}}}\approx {\omega }_{{\rm{II}{{{\hbox{}}}1}}}\) and \({\omega }_{{\rm{p}}}\approx \Delta \omega\), the firstorder sideband dynamical coupling process is illustrated in Fig. 3a. An idler spectrum line near \({\omega }_{{\rm{II}{{{\hbox{}}}2}}}\) is generated due to an upconversion (\({\Lambda }_{+}\)) process, in which the antiStokes sideband 3 with frequency \({\omega}_{{\rm{II}{{{\hbox{}}}{1}}}}+{\omega }_{{\rm{p}}}\) is produced in the dynamics of mode II2 during the mixing of the intermodal coupling pump with the displacement of mode II1. Meanwhile, the mixing of mode II2 displacement with the intermodal coupling pump causes a downconversion (\({\Lambda }_{}\)) process, which produces a Stokes sideband 2 with frequency \({\omega }_{{\rm{II}{{{\hbox{}}}2}}}{\omega }_{{\rm{p}}}\) in the dynamics of mode II1. Sideband 2 has a phase delay relative to the external actuation tone. For the reddetuned pump, sideband 2 is in antiphase, which results in dynamic backaction cooling and avoided crossing. The offresonance sideband 1 (4) is produced by the down (up)conversion process of mode II1 (II2), which will not affect the dynamical coupling. In this implementation, sidebands are generated near the mechanical modes rather than the pumps in cavity optomechanics systems^{49}, because the eigenfrequencies of the mechanical modes are much higher than the frequency difference here.
If the pump frequency is changed to \({\omega }_{{\rm{p}}}\approx \Delta \omega /2\), the secondorder sideband coupling process is illustrated in Fig. 3b. Wave mixing processes of (\({\Gamma }_{1+}\times {\Lambda }_{+}\)) and (\({\Lambda }_{+}\times {\Gamma }_{2+}\)) of mode II1 will generate sidebands 7 and 8 in dynamics of mode II2, respectively. Similarly, wave mixing processes of (\({\Gamma }_{2}\times {\Lambda }_{}\)) and (\({\Lambda }_{}\times {\Gamma }_{1}\)) of mode II2 will generate sidebands 9 and 10 in dynamics of mode II1. Offresonance sidebands 5 and 6 act as the intermediary points for those processes. Other offresonance sidebands 11, 12, 13, 14 are also generated by first and secondorder mixing processes, which would not contribute to the dynamical coupling.
Sidebands 9 and 10 will produce back actions on mode II1. However, the effects of sidebands 9 and 10 on mode II1 are opposite to each other. Based on the secondorder coupling strength model in Supplementary Eq. (58), back actions caused by sidebands 9 and 10 will completely cancel each other when \(\theta =\arctan \sqrt{(3{\omega }_{{\rm{II}}{{\hbox{}}}1}+{\omega }_{{\rm{II}}{{\hbox{}}}2})/({\omega }_{{\rm{II}}{{\hbox{}}}1}+3{\omega }_{{\rm{II}}{{\hbox{}}}2})}\approx 4{5}^{\circ }\). In this condition, the secondorder dynamical coupling is invisible, as shown in Supplementary Fig. 3a.
Here, we monitor the frequency response along the \(x\) axis at \({\omega }_{{\rm{d}}}\), and the results with different pump frequencies are shown in Fig. 3c. The pump strength is maintained at \({V}_{{\rm{p}}}=3\) V. The vertical dotted lines indicate the pumpon resonant frequencies of the modes II1 and II2. First level normalmode splitting can be observed when \({\omega }_{{\rm{p}}}\approx \Delta \omega\), which indicates strong firstorder dynamical mode coupling. The avoided crossings induced by the very strong firstorder dynamical coupling shift the lower branches of the resonance peaks, which makes the secondorder avoided crossings take place at a pump frequency higher than \(\Delta \omega /2\), as shown in Fig. 3c, d. However, it should be noted that the secondorder dynamical coupling still takes place at the pump frequency of exactly \(\Delta \omega /2\). The observed secondorder avoided crossings higher than \(\Delta \omega /2\) are simultaneously affected by both the first and the secondorder coupling. Some slices of response curves near and between the first and secondorder coupling are also provided in Fig. 3e to better illustrate the avoided crossing evolution process. The Stückelberg interferometry pattern is also observed at the bottom of the Fig. 3c where the pump frequency is low^{50}. In this system, the intramodal coupling terms are responsible for the Stückelberg interferometry phenomenon, and the intermodal coupling term leads to the firstorder dynamical coupling. Both intraand intermodal coupling terms are necessary for higherorder dynamical couplings. The experimental data can be reproduced by solving Eqs. (1), (2), as shown in Fig. 3d (see Supplementary Note 3).
The dynamical coupling strength is controllable. Fig. 3f shows the pump strength \({V}_{{\rm{p}}}\) dependence of the firstorder coupling rate \({g}_{1}\), which is described by the frequency split. The firstorder coupling rate is given by \({g}_{1}=\sqrt{{\kappa }^{2}{V}_{0}^{2}{V}_{{\rm{p}}}^{2}{\sin }^{2}(2\theta )/(4{\omega }_{{\rm{II}{{{\hbox{}}}1}}}{\omega }_{{\rm{II}{{{\hbox{}}}2}}}){\gamma }_{{\rm{II}}}^{2}}\). The theoretical values of mode splitting are depicted by dotdashed lines in Fig. 3f. In this study, firstorder coupling rate of more than \(2\pi \times 30\) Hz can be obtained, which exceeds the damping rate (\(2\pi \times 1.05\) Hz) by more than 28 times. Besides, when \({V}_{{\rm{p}}}\) reaches values around 6 V, an intermediate avoided crossing is observed, which originates from the interaction of the two firstorder idler sideband. If \({V}_{{\rm{p}}}\) exceeds this value, the frequency split will decrease. The maximum coupling rate of this study is restricted by the limited value of \(\Delta \omega\). It is noteworthy that the dynamical coupling strengths are both \({V}_{{\rm{p}}}\) and \(\theta\) dependent. Adjusting \(\theta\) provides a new degree of freedom to control dynamical coupling.
The pump strength dependence of the firstorder coupling strength can be reproduced by solving Eqs. (1), (2) (Fig. 3g). There is an overall trend of dispersing to lower frequency for the peaks in Fig. 3f, which is caused by the DC term in the pump \({\Delta }_{{\rm{p}}}\). The experiment and simulation results for another implementation of \(\theta \approx 4{5}^{\circ }\) and \(\Delta \omega \approx 2\pi \times 40.7\) Hz are also provided in Supplementary Fig. 3.
Electrostatic nonlinear coupling and dynamical manipulation
In this setup, the ring resonator is dominated by stiffnesssoftening electrostatic nonlinearity because of the narrow capacitive gap^{51}, which is confirmed by the \({V}_{0}\)dependence of the nonlinear response (see Supplementary Note 4). Dopinginduced material nonlinearity^{52} and tensioninduced mechanical nonlinearity^{51} may also reside, but they are relatively weak compared to the electrostatic one when \({V}_{0}\) is set to be 30 V.
It was previously demonstrated that the mechanical modes in clamped–clamped beam resonators can be parametrically coupled to each other due to tensioninduced mechanical nonlinearity, because the displacement of one mode produces a beam extension thus modifying the other mode’s resonant frequency^{18,22,23}. Here, we demonstrate that the origin of electrostatic nonlinearity can also produce intermodal parametric coupling. The electrostatic nonlinear coupling between order2 and order3 modes is experimentally and theoretically described.
Modes II1 (II2) and III1 are simultaneously actuated by drive and pump electrodes, respectively. Modes II are actuated in linear region with a drive signal amplitude of 2 mV. Mode III1 is actuated in the stiffnesssoftening Duffing nonlinear condition with a drive signal amplitude of 10 mV. The frequency responses of modes II are recorded, and a dispersive parametric coupling is observed. If the drive frequency of mode III1 is changed from low to high, the dispersion of modes II is shown in Fig. 4d. If the drive frequency of mode III1 is changed from high to low, the dispersion of modes II is shown in Fig. 4e, and the shifts of modes II1 and II2 when mode III1 is at resonance are 5.9 Hz and 6.2 Hz, respectively. The nonlinear bifurcation of mode III1 is revealed by the frequency dispersion of modes II. When mode III1 is actuated, the resonant frequencies of modes II will disperse to lower values. This phenomenon is opposite to that of the mechanical nonlinear parametric coupling^{18,22,23}, which should make modes II disperse to higher resonance frequencies.
The observed parametric mode coupling can be explained by the model that two modes sharing one polarized capacitor. It was previously shown that a mechanical resonator can be coupled to a microwave cavity using a capacitor^{10,53,54}, and the stiffness of the mechanical resonator is modified by varying the bias voltage on the capacitor. In this model, the resonant frequency of mode II is modified by the capacitance gap variation induced by displacement of mode III. When two modes are actuated simultaneously, their displacements are superposed (Fig. 4a–c). Displacement of heavily actuated mode III will cause a electrostatic stiffness variation for mode II, thus modifying its resonant frequency. The capacitive parametric coupling can be described by following equations of motion (see Supplementary Note 4):
where \({x}_{{\rm{II}}}\) and \({\omega }_{{\rm{II}}}\) denote the displacement and resonant angular frequency of mode II1 or II2, and \({x}_{{\rm{III}}}\) and \({\omega }_{{\rm{III}}}\) denote the displacement and resonant angular frequency of one order3 mode (mode III1 in this case). \({F}_{\square }\) and \({\omega }_{{\mathrm{d}}\hbox{}\square }\) (\(\square =\) II, III) are the amplitudes and frequencies of the external forces acting on the corresponding modes. Parameters \({\alpha }_{\square }\), \({\beta }_{\square }\), \({\nu }_{\square }\) are provided by Supplementary Eqs. (70–72). The key factor for the observed modeII frequency shift is the \(3{\nu }_{{\rm{II}}}{x}_{{\rm{III}}}^{2}{x}_{{\rm{II}}}\) term in expanded Eq. (4). The displacement square of modeIII scaled by the thirdorder nonlinearity coefficient \({\nu }_{{\rm{II}}}\) will directly influence the effective stiffness of modeII. Eqs. (4), (5) are solved using multiplescale analysis, and the results are provided in Supplementary Eqs. (75, 76).
The dispersive frequency shifts of modes II caused by actuation of mode III1 can be simulated based on Eqs. (4), (5) (see Supplementary Fig. 9). The frequency shift of mode II \({\hat{\sigma }}_{{\rm{II}}}\) caused by displacement of mode III and that of mode III \({\hat{\sigma }}_{{\rm{III}}}\) caused by displacement of mode II are given by
where \({f}_{\square }={F}_{\square }/({d}_{0}{\gamma }_{{\rm{II}}}^{2})\). The frequency shift direction of modeII (the sign of \({\hat{\sigma }}_{{\rm{II}}}\)) is determined by the sign of the thirdorder nonlinearity coefficient \({\nu }_{{\rm{II}}}\), which is negative for this resonator dominated by electrostatic nonlinearity. Thus, frequency of modeII will shift downward.
As long as the associated modes can simultaneously modulate the response of a common capacitive transducer, those modes are coupled. Thus, this electrostatic nonlinear mode coupling is very common in capacitive micro or nanomechanical resonators. The coupling strength is significantly impacted by the characteristics of the shared capacitor.
It has been illustrated that the structural asymmetry will transform normal modes II1 and II2 into mechanically coupled hybrid states H1 and H2. The hybrid states are electrostatically coupled to order3 modes, an antiStokes pump can transform this electrostatic coupling into tunable strong dynamical coupling. In this case, order3 modes are regarded as phonon cavities. The drive signal (\({V}_{{\rm{d}}}=2\) mV) sweeps from below mode II1 to above mode II2, and the pump signal (\({V}_{{\rm{p}}}=3\) V) with frequency from below \({\omega }_{{\rm{III}}{\hbox{}}1}{\omega }_{{\rm{II}}{\hbox{}}2}\) to above \({\omega }_{{\rm{III}}{\hbox{}}2}{\omega }_{{\rm{II}}{\hbox{}}1}\) are simultaneously applied. The frequency responses of modes II1 and II2 are measured, as shown in Fig. 4f, in which a skewed “#” configuration with four avoided crossings is illustrated. When the pump frequency is kept \({\omega }_{{\rm{III}}{\hbox{}}1}{\omega }_{{\rm{II}}{\hbox{}}1}\), the dynamical coupling strength of modes II1 and III1 depicted by the frequency split of mode II1 will increase if pump voltage is increased (Fig. 4h). The pump voltage dependence of coupling strength of modes II2 and III1 is also obtained (Fig. 4j). Moreover, avoided crossings can be observed between the higher splitting branch of II1 and mode II2 in Fig. 4h, and between the lower splitting branch of II2 and mode II1 in Fig. 4j, which are produced by the structural hybrid coupling of H1 and H2.
The skewed “#” coupling configuration are simulated by separately modeling the sequential dynamical coupling of III1 to II1 and II2 and that of III2 to II1 and II2, which can be described similarly by the following equations of motion:
The subscript III refers to mode III1 or III2. The detailed derivation of those equations of motion and the parameters \({\alpha }_{j}\), \({\beta }_{j}\), \({\lambda }_{j}\), \({\Lambda }_{j}\), and \({g}_{j}\) (\(j=1,2,3\)) are provided in Supplementary Note 5. By solving Eqs. (8)–(10), we can simulate the observed avoided crossings, as shown in Fig. 4g. The \({V}_{{\rm{p}}}\) dependence of coupling strengths depicted in Fig. 4h, j can also be simulated by setting \({\omega }_{{\rm{p}}}={\omega }_{{\rm{III}}{\hbox{}}1}{\omega }_{{\rm{II}}{\hbox{}}1}\) and \({\omega }_{{\rm{III}}{\hbox{}}1}{\omega }_{{\rm{II}}{\hbox{}}2}\), as shown in Fig. 4i, k, respectively.
By showing the abundant coupling phenomena between order2 and order3 modes, we demonstrate that this electrostatic mechanical system has the potential to couple even more modes. If the dynamical couplings are implemented simultaneously, a classical analog of multiplelevel system could be constructed, which would enable abundant varieties of interesting manipulations^{40}.
Discussion
In this study, we report significant progress in modeling and manipulating the structural asymmetry and misalignment induced mode coupling between a pair of degenerate wineglass modes in ring resonators. We also discover an electrostatic fieldimposed nonlinear parametric coupling effect among different modes in a single resonator. These modal coupling effects in capacitively transduced mechanical systems can be tuned very significantly through the parametric pumping scheme. Electrostaticpumpbased manipulations show consequent advantages with respect to tensionmediated manipulations. Depending on the topology of the resonators, the effect of tensioninduced manipulations can vary. For instance, its effect can be significant for clamped–clamped or thin film resonators, but negligible for centrally anchored or bulkfabricationprocess resonators, such as the ring resonator in this work, if the same tension is applied. In comparison, the electrostaticmediated pumpbased manipulation removes such topology restrictions for capacitively transduced resonators, hence making it applicable for a wider range of resonator designs.
Recently, significant breakthroughs in enhancing quality factor (\(Q\)) of mechanical resonators have been made^{55,56,57,58}. Though some of those resonators are dielectric, the proposed dissipation mitigating techniques could inspire conductive high\(Q\) mechanical resonators. An interesting direction for further research is to combine the electrostaticmediated parametric coupling and dynamical manipulations illustrated here in such conductive high\(Q\) mechanical resonators, which may enable purely phononbased quantum sideband manipulations at the macroscopic scale^{10,11}. Some interesting dynamical tunable coupling experiments have been implemented in capacitive mechanical systems^{30,31}. However, these effects were attributed to tensioninduced parametric interaction. Here, we demonstrate that electrostatic fieldinduced parametric coupling should also reside in these systems, and may at least partly contribute to the observed results.
Electrostatic mechanical resonators are widely used for sensing applications, and their susceptibility to mode coupling has often been not fully assessed in published studies. This paper shows that mode coupling should be taken into account when designing sensors, as the associated interactions may greatly alter the properties of the mechanical devices. More interestingly, electrostatic dynamical sideband coupling may be very useful in terms of enhancing sensor performance, such as manipulating energy transfer between dynamically coupled modes to improve or decrease \(Q\) factor of a specific mode^{59}, greatly speeding up mode switching for Coriolis gyroscopes by replacing the stopdecayactuate process with coherent mode manipulation^{60}, thus calibrating the bias error without affecting the bandwidth, dynamically tuning the frequency split of gyroscopes, and providing tunable coupling for modelocalized sensors.
Methods
Experiment and simulation setup
The die embedding the vacuumsealed MEMS resonators is electrically packaged in a ceramic leadless chip carrier. The device is operated in an ambient room temperature environment. The bias voltages are generated by a low noise voltage source (Keysight B2961A). The drive and pump signals are provided by a twochannel lockin amplifier (Zurich Instruments HF2LI). The response motion of the resonator is detected by a capacitance–voltage converting scheme that is based on charge amplifier, and measured by the lockin amplifier. The simulation codes are based on Python 3.7 with NumPy and Matplotlib packages.
Data availability
The authors declare that all data supporting the findings of this study are included in the paper and its supplementary information files, and are available on request from the corresponding authors. The source data underlying Figs. 1b, 2d, f, 3c, e, f, and 4d, e, f, h, j and Supplementary Figs. 2c, 3a, c, e, 4c–i, 5, 6, 7, and 8 are provided as a Source Data file, which is also available in figshare (https://doi.org/10.6084/m9.figshare.8397908).
Code availability
The authors declare that all codes supporting the findings of this study are included in the paper and its supplementary information files, and are available on request from the corresponding authors. The simulation codes underlying Figs. 2e, 3d, g, and 4g, i, k and Supplementary Figs. 2d, e, 3b, d, f, 4c–i, 6, and 9 are available in figshare (https://doi.org/10.6084/m9.figshare.8397908).
References
 1.
Nguyen, C. T. C. MEMS technology for timing and frequency control. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54, 251–270 (2007).
 2.
Ng, E. et al. The long path from MEMS resonators to timing products. In Proceedings of 28th IEEE International Conference on Micro Electro Mechanical Systems. 1–2 (IEEE, 2015).
 3.
Roy, S. K., Sauer, V. T. K., WestwoodBachman, J. N., Venkatasubramanian, A. & Hiebert, W. K. Improving mechanical sensor performance through larger damping. Science 360, eaar5220 (2018).
 4.
Moser, J. et al. Ultrasensitive force detection with a nanotube mechanical resonator. Nat. Nanotechnol. 8, 493–496 (2013).
 5.
Hanay, M. S. et al. Singleprotein nanomechanical mass spectrometry in real time. Nat. Nanotechnol. 7, 602–608 (2012).
 6.
Unterreithmeier, Q. P., Weig, E. M. & Kotthaus, J. P. Universal transduction scheme for nanomechanical systems based on dielectric forces. Nature 458, 1001–1004 (2009).
 7.
Hafiz, M. A. A., Kosuru, L. & Younis, M. I. Microelectromechanical reprogrammable logic device. Nat. Commun. 7, 11137 (2016).
 8.
Hatanaka, D., Mahboob, I., Onomitsu, K. & Yamaguchi, H. Phonon waveguides for electromechanical circuits. Nat. Nanotechnol. 9, 520–524 (2014).
 9.
Mahboob, I. & Yamaguchi, H. Bit storage and bit flip operations in an electromechanical oscillator. Nat. Nanotechnol. 3, 275–279 (2008).
 10.
OckeloenKorppi, C. F. et al. Stabilized entanglement of massive mechanical oscillators. Nature 556, 478–482 (2018).
 11.
Riedinger, R. et al. Remote quantum entanglement between two micromechanical oscillators. Nature 556, 473–477 (2018).
 12.
Møller, C. B. et al. Quantum backactionevading measurement of motion in a negative mass reference frame. Nature 547, 191–195 (2017).
 13.
Wollman, E. E. et al. Quantum squeezing of motion in a mechanical resonator. Science 349, 952–955 (2015).
 14.
Ganesan, A., Do, C. & Seshia, A. A. Phononic frequency comb via intrinsic threewave mixing. Phys. Rev. Lett. 118, 033903 (2017).
 15.
Matheny, M. H. et al. Phase synchronization of two anharmonic nanomechanical oscillators. Phys. Rev. Lett. 112, 014101 (2014).
 16.
Agrawal, D. K., Woodhouse, J. & Seshia, A. A. Observation of locked phase dynamics and enhanced frequency stability in synchronized micromechanical oscillators. Phys. Rev. Lett. 111, 084101 (2013).
 17.
Shim, S. B., Imboden, M. & Mohanty, P. Synchronized oscillation in coupled nanomechanical oscillators. Science 316, 95–99 (2007).
 18.
Westra, H. J. R., Poot, M., van der Zant, H. S. J. & Venstra, W. J. Nonlinear modal interactions in clampedclamped mechanical resonators. Phys. Rev. Lett. 105, 117205 (2010).
 19.
Eichler, A., del Álamo Ruiz, M., Plaza, J. A. & Bachtold, A. Strong coupling between mechanical modes in a nanotube resonator. Phys. Rev. Lett. 109, 025503 (2012).
 20.
Antonio, D., Zanette, D. H. & López, D. Frequency stabilization in nonlinear micromechanical oscillators. Nat. Commun. 3, 806 (2012).
 21.
Faust, T. et al. Nonadiabatic dynamics of two strongly coupled nanomechanical resonator modes. Phys. Rev. Lett. 109, 037205 (2012).
 22.
Mahboob, I., Nishiguchi, K., Okamoto, H. & Yamaguchi, H. Phononcavity electromechanics. Nat. Phys. 8, 387–392 (2012).
 23.
Matheny, M. H., Villanueva, L. G., Karabalin, R. B., Sader, J. E. & Roukes, M. L. Nonlinear modecoupling in nanomechanical systems. Nano Lett. 13, 1622–1626 (2013).
 24.
Mahboob, I. et al. Dispersive and dissipative coupling in a micromechanical resonator embedded with a nanomechanical resonator. Nano Lett. 15, 2312–2317 (2015).
 25.
Chen, C., Zanette, D. H., Czaplewski, D. A., Shaw, S. & López, D. Direct observation of coherent energy transfer in nonlinear micromechanical oscillators. Nat. Commun. 8, 15523 (2017).
 26.
Seitner, M. J., Abdi, M., Ridolfo, A., Hartmann, M. J. & Weig, E. M. Parametric oscillation, frequency mixing, and injection locking of strongly coupled nanomechanical resonator modes. Phys. Rev. Lett. 118, 254301 (2017).
 27.
Luo, G. et al. Strong indirect coupling between graphenebased mechanical resonators via a phonon cavity. Nat. Commun. 9, 383 (2018).
 28.
Okamoto, H. et al. Coherent phonon manipulation in coupled mechanical resonators. Nat. Phys. 9, 598–598 (2013).
 29.
Liu, C. H., Kim, I. S. & Lauhon, L. J. Optical control of mechanical modecoupling within a MoS_{2} resonator in the strongcoupling regime. Nano Lett. 15, 6727–6731 (2015).
 30.
De Alba, R. et al. Tunable phononcavity coupling in graphene membranes. Nat. Nanotechnol. 11, 741–746 (2016).
 31.
Mathew, J. P., Patel, R. N., Borah, A., Vijay, R. & Deshmukh, M. M. Dynamical strong coupling and parametric amplification of mechanical modes of graphene drums. Nat. Nanotechnol. 11, 747–751 (2016).
 32.
Sun, F., Dong, X., Zou, J., Dykman, M. I. & Chan, H. B. Correlated anomalous phase diffusion of coupled phononic modes in a sidebanddriven resonator. Nat. Commun. 7, 12694 (2016).
 33.
Mahboob, I., Okamoto, H., Onomitsu, K. & Yamaguchi, H. Twomode thermalnoise squeezing in an electromechanical resonator. Phys. Rev. Lett. 113, 167203 (2014).
 34.
Patil, Y. S., Chakram, S., Chang, L. & Vengalattore, M. Thermomechanical twomode squeezing in an ultrahigh\(Q\) membrane resonator. Phys. Rev. Lett. 115, 017202 (2015).
 35.
Faust, T., Rieger, J., Seitner, M. J., Kotthaus, J. P. & Weig, E. M. Coherent control of a classical nanomechanical twolevel system. Nat. Phys. 9, 485–488 (2012).
 36.
Zhu, D. et al. Coherent phonon Rabi oscillations with a highfrequency carbon nanotube phonon cavity. Nano Lett. 17, 915–921 (2017).
 37.
Mahboob, I., Nishiguchi, K., Fujiwara, A. & Yamaguchi, H. Phonon lasing in an electromechanical resonator. Phys. Rev. Lett. 110, 127202 (2013).
 38.
Truitt, P. A., Hertzberg, J. B., Altunkaya, E. & Schwab, K. C. Linear and nonlinear coupling between transverse modes of a nanomechanical resonator. J. Appl. Phys. 114, 114307 (2013).
 39.
Buks, E. & Roukes, M. L. Electrically tunable collective response in a coupled micromechanical array. J. Microelectromech. Syst. 11, 802–807 (2002).
 40.
Okamoto, H. et al. A strongly coupled \(\Lambda\)type micromechanical system. Appl. Phys. Lett. 108, 227402 (2016).
 41.
Spletzer, M., Raman, A., Wu, A. Q., Xu, X. & Reifenberger, R. Ultrasensitive mass sensing using mode localization in coupled microcantilevers. Appl. Phys. Lett. 88, 254899 (2006).
 42.
Thiruvenkatanathan, P., Yan, J., Woodhouse, J., Aziz, A. & Seshia, A. A. Ultrasensitive modelocalized mass sensor with electrically tunable parametric sensitivity. Appl. Phys. Lett. 96, 083562 (2010).
 43.
Ayazi, F. & Najafi, K. A HARPSS polysilicon vibrating ring gyroscope. J. Microelectromech. Syst. 10, 169–179 (2001).
 44.
Ahn, C. H. et al. Encapsulated high frequency (235 kHz), high\(Q\) (100 k) disk resonator gyroscope with electrostatic parametric pump. Appl. Phys. Lett. 105, 243504 (2014).
 45.
Nguyen, C. T. C. Frequencyselective MEMS for miniaturized lowpower communication device. IEEE Trans. Microw. Theory Tech. 47, 1486–1503 (1999).
 46.
Yang, Y., Ng, E. J., Chen, Y., Flader, I. B. & Kenny, T. W. A unified Episeal process for fabrication of highstability microelectromechanical devices. J. Microelectromech. Syst. 25, 489–497 (2016).
 47.
Zhou, X. et al. Mitigating thermoelastic dissipation of flexural micromechanical resonators by decoupling resonant frequency from thermal relaxation rate. Phys. Rev. Appl. 8, 064033 (2017).
 48.
Nitzan, S. H. et al. Selfinduced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Sci. Rep. 5, 9036 (2015).
 49.
Kippenberg, T. J. & Vahala, K. J. Cavity optomechanics: backaction at the mesoscale. Science 321, 1172–1176 (2008).
 50.
Ivakhnenko, O. V., Shevchenko, S. N. & Nori, F. Simulating quantum dynamical phenomena using classical oscillators: LandauZenerStückelbergMajorana interferometry, latching modulation, and motional averaging. Sci. Rep. 8, 12218 (2018).
 51.
Lifshitz, R., Cross, M. C. Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators (ed Schuster, H. G.) 1–52 (Wiley VCH Verlag GmbH & Co. KGaA, 2008).
 52.
Yang, Y. et al. Nonlinearity of degenerately doped bulkmode silicon MEMS resonators. J. Microelectromech. Syst. 25, 859–869 (2016).
 53.
Rocheleau, T. et al. Preparation and detection of a mechanical resonator near the ground state of motion. Nature 463, 72–75 (2010).
 54.
Brown, K. R. et al. Passive cooling of a micromechanical oscillator with a resonant electric circuit. Phys. Rev. Lett. 99, 137205 (2007).
 55.
Ghadimi, A. H. et al. Elastic strain engineering for ultralow mechanical dissipation. Science 360, 764–768 (2018).
 56.
Tsaturyan, Y., Barg, A., Polzik, E. S. & Schliesser, A. Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution. Nat. Nanotechnol. 12, 776–783 (2017).
 57.
Güttinger, J. et al. Energydependent path of dissipation in nanomechanical resonators. Nat. Nanotechnol. 12, 631–636 (2017).
 58.
Moser, J., Eichler, A., Güttinger, J., Dykman, M. I. & Bachtold, A. Nanotube mechanical resonators with quality factors of up to 5 million. Nat. Nanotechnol. 9, 1007–1011 (2014).
 59.
Venstra, W. J., Westra, H. J. R. & van der Zant, H. S. J. \(Q\)factor control of a microcantilever by mechanical sideband excitation. Appl. Phys. Lett 99, 013112 (2011).
 60.
Okamoto, H., Mahboob, I., Onomitsu, K. & Yamaguchi, H. Rapid switching in high\(Q\) mechanical resonators. Appl. Phys. Lett. 105, 083114 (2014).
Acknowledgements
We thank Sijun Du, Milind Pandit, Malar Chellasivalingam, and Atif Aziz for helpful discussions and assisting with training on laboratory equipment. This work is partly supported by the National Key R&D Program of China (NKPs) (2018YFB2002304) and the National Natural Science Foundation of China (NSFC) (51905539, 51575521 and 51705527). Experimental devices are designed and fabricated in the nano@Stanford labs, which are supported by the NSF as part of the National Nanotechnology Coordinated Infrastructure under award ECCS1542152, with support from the Defense Advanced Research Projects Agency Precise Robust Inertial Guidance for Munitions (PRIGM) Program, managed by Dr. Robert Lutwak and Dr. Ron Polcawich, and the NSF under grant number CMMI1662464. This work is primarily supported by the UK Natural Environment Research Council under grant number NE/N012097/1.
Author information
Affiliations
Contributions
X.Z. conceived the idea and designed the research under the guidance of A.A.S. and assistance of C.Z. The measurements and data analyses were performed by X.Z. and C.Z. with the assistance of J.S. and G.S. The test circuitry was developed by D.X. and X.Z. The theoretical works were done by X.Z., supervised by A.A.S., D.X. and X.W. The device was designed and fabricated by D.D.G., Y.C., I.F. and T.W.K. The manuscript was written by X.Z. and A.A.S. All authors contributed to manuscript preparation. The project was planned by A.A.S.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer Review Information Nature Communications thanks G. P. Guo, Zenghui Wang and other anonymous reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Zhou, X., Zhao, C., Xiao, D. et al. Dynamic modulation of modal coupling in microelectromechanical gyroscopic ring resonators. Nat Commun 10, 4980 (2019). https://doi.org/10.1038/s41467019127960
Received:
Accepted:
Published:
Further reading

Single particles as resonators for thermomechanical analysis
Nature Communications (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.