Frequency stability improvement for piezoresistive micromechanical oscillators via synchronization

Micro and nano mechanical resonators have been widely studied as a potential alternative to conventional quartz oscillators due to their advantages such as easier to miniaturize and to be integrated with electronics.^1,2 As frequency control or time reference devices, micromechanical oscillators have to own a better frequency stability. However, their performances are often degraded by large displacement instabilities due to scaling effect and nonlinearity,^3–5 frequency noise imposed by thermomechanical noise^6–8 and intrinsic frequency fluctuations.⁹ According to the comprehensive literature review by Sansa et al.,⁹ there still has several orders of magnitude over the thermo-mechanical noise limit to improve the frequency stability for micro/nanomechanical oscillators.

Synchronization phenomenon first discovered in Huygens’ clock shows that the rhythms of oscillating objects can be adjusted via an interaction¹⁰ and accordingly synchronization is considered to be an effective technique to overcome the frequency dispersing. In recent few years, special efforts have been invested in the reduction of phase noise and the improvement frequency stability in micro/nanomechanical oscillators via synchronization. An improvement of the frequency stability is observed when two micromechanical oscillators are synchronized.¹¹ The phase noise can be significantly reduced when two anharmonic nanomechanical oscillator in the phase synchronized state.¹² In the previous work,¹³ one micromechanical oscillator is synchronized to an external force and the phase noise spectrum is given analytically which implies that the largest phase noise reduction occurs in the center of the Shapiro step. Different from the oscillators reported in those studies, piezoresistive micromechanical oscillators are proposed recently because of their better electrical transduction efficiency but the induced joule heating is believed to affect their stability of frequency output. In this work, the synchronization phenomenon in a piezoresistive micromechanical clamped-clamped (C-C) beam oscillator is investigated and the observed improvement of frequency stability is studied.

The piezoresistive oscillator is built up on a micromechanical C-C beam resonator that is fabricated by a standard Silicon-On-Insulator (SOI) process. Fig. 1 shows a microscopic picture of the micro-resonator and a schematic drawing of the oscillator circuit.⁴ The dimensions of the resonator are $478μm$ long, $10μm$ wide and $10μm$ thick respectively. It is electrostatically actuated in its principle flexural mode. The high impedance of resonators at resonance makes other transduction methods (such as piezoresistive and piezoelectric) popular except using the capacitive transduction to detect the mechanical resonance. Here, the differential piezoresistive sensing method is used to detect the transverse displacement.¹⁴ After amplification, the resulting signal is filtered and phase shifted by a prescribed amount of $ϕ0$⁠. Then the square wave signal with a fixed amplitude V_ac2 from the comparator is applied on the comb-drive electrode to electrostatically drive the resonator. The feedback time-varying driving force compensates the energy dissipation and in the absence of any external perturbations, this closed loop feedback sustains the beams vibration at an amplitude A₀ and a frequency $Ω0$⁠. To entrain the oscillator into synchronized motion, an external harmonic signal of amplitude V_ac1 and frequency $Ωs$ is applied on the other comb-drive electrode as a perturbation force to the beam oscillator. A frequency counter (Agilient 53230A) is used to log the frequency output of the piezoresistive oscillator.

The equation of motion for the forced self-sustained nonlinear micromechanical beam’s oscillation is well described using the Duffing equation.

Where, $m,μ,k1,k3$ and F_act are the effective mass, damping coefficient, linear mechanical stiffness, cubic mechanical stiffness and the actuation force, respectively. The micro-beam is actuated electrostatically and the driving force F_act combining the self-sustaining force with the synchronization perturbation can be expressed as follow.

Where, $∂C1∂x$ and $∂C2∂x$ are capacitance gradients,V_dc1 and V_dc2 are DC driving voltages, $ϕt=ϕ+ϕ0$ is the instantaneous oscillation phase of the coordinate x(t). In our experiment, the AC driving voltage is much smaller than DC voltage. The capacitance gradient can be expanded in Taylor’s series and accordingly the driving force can be simplified as follow.

Where k_1e,k_3e,F₀,F_s are respectively the linear electric spring constant, nonlinear electric spring constant, the self-sustaining force and the external perturbation. The main resonance region is discussed here and the parametric terms are neglected. Redefining time units and normalizing the Eq. 1 by the spring constant k(k = k₁ −k_1e) leads to the following equation of the oscillation.

Where, $Q=kmγ,β=k3−k3ek,f0=F0k,fs=Fsk,Ωs′=Ωsmk,τ=kmt$⁠. Here Q is the quality factor, the cubic-term coefficient $β$ is positive (negative) depending on the hardening (k₃) and softening (k_3e) nonlinearities. The self-sustaining force f₀ reaches the maximal when $ϕ0$ is $π/2$⁠. The sign function is expanded and simplified as follow after neglecting the higher order harmonics.

For the self-excited oscillation without external perturbation (f_s = 0), the frequency $Ω0$ and amplitude A₀ can be expressed as follows by using the method applied by Antonio.⁵ 

When a harmonic perturbation is applied (⁠$fs≠0$⁠), there exists a synchronization region $[Ω0−12Ωc,Ω0+12Ωc]$ within which the frequency of mechanical oscillator $Ω0′$ locks to synchronizing frequency $Ωs$ and the synchronization bandwidth $Ωc$ is derived as follow.

This synchronization region of the oscillator synchronized to an external force can also be obtained similarly using averaging method.¹³ Fig. 2 shows the synchronization region at various perturbation intensities (V_ac1) predicted using Eq. 7. It is clear that the synchronization bandwidth is 170.5Hz at V_ac1 of 20mV and increases as the harmonic perturbation intensity increases. In the verifying experiments, the beam micro-resonator is tested in a vacuum chamber with the air pressure less than 0.01Pa. The resonator is driven into oscillation and a harmonic signal from a RF source (Agilient 33250A) is applied to perturb the oscillation. The free-running self-oscillation frequency is 202400Hz. Both the frequency outputs from the signal generator and the piezoresistive oscillator are logged simultaneously using the dual channel frequency counter. The frequency ratio $Ωs/Ω0′$ is obtained and plotted against $Ωs$ as shown in Fig. 3.

It is obvious that there is a synchronization region in which the frequency of external perturbation equals that of the piezoresistive oscillator. The synchronization bandwidths are respectively 185.7Hz, 262.1Hz and 328.8Hz for V_ac1 of 20mV, 25mV and 30mV, which are close to the analytical predictions shown in Fig. 2 and confirm that the synchronization bandwidth can be tuned by the intensity of harmonic perturbation (or the signal to perturbation ratio SPR=$𝑓𝑠/𝑓0∝Vac1/Vac2$⁠). By sweeping up and down using different perturbation intensities, the Arnold tongue is figured out by connecting the edges with dashed lines.

In the synchronization region, a series of experiments on frequency stability are performed at various frequency detunings $δΩ$ (⁠$δΩ=Ωs−Ω0$⁠) and a fixed perturbation intensity (V_ac1=10mV). Allan deviations of frequency outputs from the piezoresistive oscillator are plotted in Fig. 4 comparing with that of the free running oscillator and the synchronizing RF source. It clearly shows that the frequency stability of the piezoresistive oscillator is enhanced as compared with that of the oscillator before synchronization, but there is no much difference in the frequency stability for different frequency detunings in the synchronization region, which indicates that the intensity of harmonic perturbation has greater effect on the frequency stability than the frequency of perturbation.

To further confirm this, a series of experiments on frequency stability are performed at various intensities of harmonic perturbation V_ac1 from 10mV to 50mV, a fixed self-sustaining amplitude V_ac2=400mV and a fixed frequency of harmonic perturbation located in the center of synchronization region. Allan deviation of the oscillator frequency output is calculated as a function of integration time $τA$ from 100ms to 100s as shown in Fig. 5. It is clearly shown that the frequency stability can be improved nearly 10 folds at $τA$=2s. For a linearly increasing SPR, we can find the frequency stability can be enhanced gradually with slow growth. Intuitively, a bigger SPR brings a better synchronization state which leads to better frequency stability. Interestingly, the frequency stability is not linearly improved.

In order to better understand these phenomena of improving frequency stability via synchronization, a theoretical investigation is conducted. A model of micromechanical oscillator synchronized to an external force in a noisy environment has been put forward by Shoshani¹³ and the phase noise spectrum is given. Starting from this point, a normalized equation of our closed loop oscillator affected by additive noise $ξ1$⁠, multiplicative noise $ξ2$ and a synchronization perturbation is written as follow.

Using the averaging method,¹³ the power spectral density of the random phase fluctuation can be derived as follow.

Where $ω$ is the modulation frequency, $Ωc$ is obtained in deterministic situation and D is the non-dimensional noise intensity (⁠$D=Dd/2πf0$⁠). Noted that D_d is defined by the auto-correlation function $ξ(t)ξ(t+Δt)=2Ddδ(Δt)$⁠. Allan variance is a function of the power spectral density¹⁵ and is given as follow.

Where, $τA$ is the time interval, and $Ω0$ is the angular frequency of free-running oscillator. The synchronization bandwidth $Ωc$ and the frequency detuning $δΩ$ discussed here are all positive real. Assuming $Ωc>δΩ$⁠, i.e. the oscillator operates in the synchronized state, and Allan deviation can be obtained

Where $ΔΩ=Ωc2−δΩ2$⁠. According to Eq. 7 and Eq. 11, it is clear that the Allan deviation is a function of the synchronizing perturbation intensity (or SPR) and the frequency detuning. The dimensional noise intensity $Dd≈9.93×10−4$ is proportional to the noise spectra density which can be obtained from the spectrum analyzer. The calculated Allan deviation $σA$ of the piezoresistive oscillator of different frequency detunings and perturbation intensities are shown in Fig. 6a and Fig. 6b. It should be noted that the additive noise and multiplicative noise are used to give a general description of the noise in the whole closed loop system rather than specific sources, and the data from spectrum analyzer already takes the noise from the electrical devices into account. From the experiments results and analytical results given above, we can find that a high-precision perturbation brings a good performance of frequency stability of this synchronized micromechanical oscillator. Besides, the perturbation is assumed to be a pure source in our model which might account for the one order of magnitude difference between the experimental and analytical results in Fig. 4, Fig. 5 and Fig. 6. Nevertheless, the trends that the frequency stability of the synchronized oscillator stays relatively stable in the synchronization region and can be further improved by a larger perturbation intensity agree well with the trends of the experimental data in Fig. 4 and Fig. 5.

To conclude, a piezoresistive oscillator is built up on a C-C micromechanical beam resonator and the improvement of its frequency stability is experimentally demonstrated via synchronization phenomenon. The effects of external perturbation detuning and intensity are studied experimentally and a theoretical model is built up to analyze the noise polluted oscillator and the explicit analytical formula of Allan deviation successfully explains experimental observations. The parameters such as nonlinearities (mechanical hardening and electrostatic softening), quality factor, signal to perturbation ratio and noise intensity are all influencing the final frequency stability of the oscillator. In this work, the frequency $Ω0$ of our piezoresistive oscillator equals to that of the external force $Ωs$ once they are synchronized and their frequency ratio $η$ is 1. However, one should be noted that sub-harmonic (e.g. $η=3$⁠) or super-harmonic (e.g. $η=13$⁠) synchronization phenomena can be observed as well in micro/nanomechanical oscillator.¹⁶