Interactions between directly- and parametrically-driven vibration modes in a micromechanical resonator

The interactions between parametrically- and directly-driven vibration modes of a clamped-clamped beam resonator are studied. An integrated piezoelectric transducer is used for direct and parametric excitation. First, the parametric amplification and oscillation of a single mode are analyzed by the power and phase dependence below and above the threshold for parametric oscillation. Then, the motion of a parametrically-driven mode is detected by the induced change in resonance frequency in another mode of the same resonator. The resonance frequency shift is the result of the nonlinear coupling between the modes by the displacement-induced tension in the beam. These nonlinear modal interactions result in the quadratic relation between the resonance frequency of one mode and the amplitude of another mode. The amplitude of a parametrically-oscillating mode depends on the square root of the pump frequency. Combining these dependencies yields a linear relation between the resonance frequency of the directly-driven mode and the frequency of the parametrically-oscillating mode.

Figure 1

Measurement setup. (a) False-colored scanning electron micrograph of a SiN beam with the piezo actuator (white arrow) on top (scale bar is 20 $\mu$m). (b) Optical deflection setup is used to detect motion in air and the vacuum. The piezoactive AlN layer is depicted in red. The piezo actuator and photodiode are connected to a digital signal processor. (c), (d) Typical frequency responses of the first and second modes (amplitude $A$), respectively, in the vacuum. The inset in (c) shows the frequency response in air, of the resonator with length 750 $\mu$m, with a resonance frequency of 98 kHz. The response of a damped-driven harmonic oscillator is fitted through the responses to obtain Q factors and resonance frequencies.

Figure 2

Characterization of the parametric amplification in air. (a) Q-factor enhancement is proportional to the parametric pump voltage. (b) Measured gain vs phase relation; the solid blue line represents Eq. (2), with fit parameter $k_p/k_t=0.26$.

Figure 3

Parametric oscillations of the first flexural mode in the vacuum. (a) Frequency spectra at three pump voltages; the parametric oscillation becomes visible when $V_{\mathrm{pump}}>$ 85 mV. (b) Parametric tongue, showing frequency responses when the resonator is driven directly ($V_{\mathrm{direct}}=5$ mV) and parametrically past the instability threshold. Color indicates the amplitude of oscillation. (c) Hysteresis between the forward (red) and reverse sweep (green) when driving parametrically ($V_{\mathrm{pump}}=95$ mV). The blue dashed line shows the square root dependence of the amplitude $A$ on the frequency $f$. (d) Phase dependence of the parametric oscillations at $V_{\mathrm{pump}}=$ 95 mV. The color indicates the amplitude of oscillation.

Figure 4

Interactions between a directly- and parametrically-driven mode. (a) Frequency responses of the second mode while varying pump frequency of the first mode. Color scale indicates the amplitude of the second mode. The linear dependence of $f_{\mathrm{R}},1$ on $f_{\mathrm{pump}},2$ is observed as explained in the text. (b) Reversed experiment: frequency responses of the first mode for varying the pump frequency of the second mode.