Amplitude stabilization of micromechanical oscillators using engineered nonlinearity

Micromechanical oscillators provide periodic output signals for clocks and sensors by vibrating in a single mechanical mode. The mode is conventionally excited into self-sustained oscillations and stabilized with an external electronic feedback loop. A paradigm is emerging for sustaining vibrations by coupling the mechanical mode with internal degrees of freedom, such as photons, electrons, or auxiliary mechanical modes. An open question in these hybrid vibrational systems is the corresponding internal sources of nonlinearity that can stabilize the oscillations, and their impact on oscillator performance. Here, we delineate two kinds of amplitude-stabilization mechanisms in micromechanical oscillators, geometric nonlinear damping and repulsive contact, and show that these mechanisms can coexist in the same device and their interplay and resonance frequency stability can be tuned in situ by adjusting the feedback strength. An auxiliary source of viscous dissipation and nonlinear dissipation accompanies the repulsive contact, which stabilizes the amplitude during sidewall collisions. The onset of self-sustained oscillations yields distinct spectral-temporal signatures that can be used to identify the amplitude stabilization nonlinearities.

Figure 1

Characterization of the nonlinear electromechanical resonators. (a) A depiction of amplitude stabilization in a micromechanical cantilever via nonlinear dissipation in a high-stress connecting element. (b) An analogous depiction to (a) for stabilization via repulsive contact with an adjacent electrode. (c) A simplified schematic of the micromechanical resonator, showing the vibration modeshape, electrical biasing, and displacement detection scheme. The device is voltage biased at $V_b$, and a current over-bias is applied between the two anchors to flow a direct current ($I_{\mathrm{dc}}$) through the resonator. The motion is read out piezoresistively ($V_{\mathrm{pzr}}$), or capacitively. An external harmonic drive ($V_{\mathrm{ac}}$) signal can be alternately applied to an adjacent electrode to characterize the forced response. [(d),(e)] The amplitude and phase of the forced response vs frequency detuning $\mathrm{\Delta }\omega =\omega -{\omega }_0$ for a repulsive-contact-stabilized resonator, Device D ($L=50\mu \mathrm{m}$, $g=700\mathrm{nm}$). The harmonic drive voltage varies from $V_{\mathrm{ac}}=100\mathrm{mV}$ to $V_{\mathrm{ac}}=370\mathrm{mV}$. [(f),(g)] The corresponding theoretical amplitude and phase curves of the device measured in [(d),(e)], using the nonlinear model for the amplitude and phase in Eqs. (4) and (5), respectively.

Figure 2

The self-oscillation dynamics for three regimes of amplitude stabilization nonlinearities. (a) The measured self-oscillation waveform for the nonlinear-damping-stabilized oscillator, Device A ($L=6\mu \mathrm{m}$, $g=2\mu \mathrm{m}$), vs time after switching on the direct current, as measured using the piezoresistive readout ($x_{\mathrm{pzr}}$). The exponential ring-up envelope from Eq. (20) (solid-black curves) and the steady-state amplitude from Eq. (21) (dashed-black curves) are overlaid. (b) The same measurement as (a), except measured using the capacitive readout ($x_{\mathrm{cap}}$). (c) The oscillation waveform at various points along the self-oscillation envelope, as measured using the piezoresistive readout. (d) A surface plot showing the Fast Fourier transforms near the resonance frequency during self-oscillations, with a 50-ms bin time. The black line plots the measured centroid frequency with a 5-ms bin time. The feedback strength is characterized by $\lambda =I_{\mathrm{dc}}/I_{\mathrm{thresh}}$, where $I_{\mathrm{thresh}}$ is the threshold current for self-oscillations. (e) The amplitude spectral density (ASD) of the steady-state oscillations, extracted from the piezoresistive output for a 1000-ms bin time. The fundamental peak and readout harmonics are labeled. [(f)–(j)] The same measurement as in [(a)–(e)], except for the intermediate-nonlinearity-stabilized oscillator, Device B ($L=6\mu \mathrm{m}$, $g=700\mathrm{nm}$). [(k)–(o)] The same measurement as in [(a)–(e)], except for the repulsive-contact-stabilized oscillator, Device C ($L=20\mu \mathrm{m}$, $g=700\mathrm{nm}$). The feedback strength is $\lambda =1.017$ for the nonlinear-damping-stabilized oscillator in [(a)–(e)], $\lambda =1.221$ for the intermediate-nonlinearity-stabilized oscillator in [(f)–(j)], and $\lambda =1.080$ for the repulsive-contact-stabilized oscillator in [(k)–(o)].

Figure 3

The self-oscillation dynamics and stability for the nonlinear-damping-stabilized oscillator, Device A ($L=6\mu \mathrm{m}$, $g=2\mu \mathrm{m}$), for decreasing feedback strength. (a) The measured mean (red curve) and $\pm 2$ standard deviations (black curves) of the amplitude ($r$), using a 50-ms bin time and $\tau =500\mu \mathrm{s}$ averaging time. (b) The corresponding measured mean (red curve) and $\pm 2$ standard deviations (black curves) of the resonance frequency (${\omega }_0/2\pi$), as measured from the interpolated zero crossings, using a 50-ms bin time and $\tau =500\mu \mathrm{s}$ averaging time. (c) The extracted amplitude Allan deviation (${\sigma }_r$), for the lower (red curves) and upper (dashed-green curves) amplitude envelope. The light curves show the amplitude Allan deviation during the initial wall collision period, corresponding to the light red box in (a). The dark curves show the amplitude Allan deviation during the subsequent nonlinear-damping-stabilized period, corresponding to the dark red box in (a). The feedback strength for [(a)–(d)] is $\lambda =I_{\mathrm{dc}}/I_{\mathrm{thresh}}=1.021$, where $I_{\mathrm{thresh}}$ is the threshold current for self-oscillations. (d) The extracted resonance frequency Allan deviation (${\sigma }_{{\omega }_0}$). The light curves show the frequency Allan deviation during the initial wall collision period, corresponding to the light red box in (b). The dark curves show the frequency Allan deviation during the subsequent nonlinear-damping-stabilized period, corresponding to the dark red box in (b). $\tau$ corresponds to the integration time. [(e)–(h)] The same measurements as [(a)–(d)], except at a reduced feedback strength $\lambda =1.011$. The Allan deviations are only extracted from one time window, corresponding to the dark red boxes in [(e)–(f)]. [(i)–(l)] The same measurements as [(e)–(h)], except at a further reduced feedback strength $\lambda =1.006$.

Figure 4

The self-oscillation dynamics and stability for the repulsive-contact-stabilized oscillator, Device C ($L=20\mu \mathrm{m}$, $g=700\mathrm{nm}$), for decreasing feedback strength. (a) The measured self-oscillation waveform vs time after switching on the direct current, as measured using the piezoresistive readout ($x_{\mathrm{pzr}}$), overlaid for decreasing feedback strength (light to dark red envelopes) beyond threshold. The feedback strength is characterized by $\lambda =I_{\mathrm{dc}}/I_{\mathrm{thresh}}$, where $I_{\mathrm{thresh}}$ is the threshold current for self-oscillations. The corresponding measured $\pm 2$ amplitude standard deviation curves lie on top of the mean amplitude for this plotting range. (b) The corresponding measured mean (red curve) and $\pm 2$ standard deviations (gray curves) of the resonance frequency (${\omega }_0/2\pi$), as measured from the interpolated zero crossings, using a 50 ms bin time and $\tau =500\mu \mathrm{s}$ averaging time, for feedback strengths $\lambda =1.106$, $\lambda =1.025$, and $\lambda =1.005$ (light- to dark-shade curves). (c) The extracted amplitude Allan deviation (${\sigma }_r$), for the lower (red curves) and upper (dashed-green curves) amplitude envelope, for decreasing feedback strength (light to dark shade curves), extracted from the steady-state vibrations in (a). (d) The extracted resonance frequency Allan deviation (${\sigma }_{{\omega }_0}$), for decreasing feedback strength (light- to dark-red curves), extracted from the steady-state vibrations in (a). (e) The steady-state self-oscillation frequency (${\omega }_0/2\pi$) for a longer time period after the initial transient interval, for decreasing feedback strength (light- to dark-red curves), as measured using a frequency counter. (f) The resonance frequency Allan deviation (${\sigma }_{{\omega }_0}$) corresponding to the time window from 2 to 8 minutes in (e) for decreasing feedback strength (light to dark red curves).

Figure 5

Measuring the contact dissipation during repulsive contact. [(a),(b)] The measured amplitude and phase curves (gray circles) with the overlaid simulations (blue curves), including neither contact viscous damping nor contact nonlinear damping. [(c),(d)] The measured amplitude and phase curves with the overlaid simulations, including contact viscous damping but not including contact nonlinear damping. [(e),(f)] The measured amplitude and phase curves with the overlaid simulations, including both contact viscous damping and contact nonlinear damping.

Figure 6

A close-up view of the measured amplitude-frequency curves (gray circles) and the overlaid simulations (blue curves), with contact viscous damping and contact nonlinear damping.