Frequency unlocking-based MEMS bifurcation sensors

Abstract

MEMS resonators exhibit rich dynamic behaviors under the internal resonance regime. In this work, we present a novel MEMS bifurcation sensor that exploits frequency unlocking due to a 1:3 internal resonance between two electrostatically coupled micro-resonators. The proposed detection mechanism allows the sensor to operate in binary (digital) and analog modes, depending on whether the sensor merely detects a significant jump event in the peak frequency upon unlocking or measures the shift in the peak frequency after unlocking and uses it in conjunction with a calibration curve to estimate the corresponding change in stimulus. We validate the success of this sensor paradigm by experimentally demonstrating charge detection. High charge resolutions are achieved in binary mode, up to 0.137 fC, and in analog mode, up to 0.01 fC. The proposed binary sensor enables extraordinarily high detection resolutions due to the excellent frequency stability under internal resonance and the high signal-to-noise ratio of the shift in peak frequency. Our findings offer new opportunities for high-performance ultrasensitive sensors.

Keywords: Electrical and electronic engineering; Engineering.

Fig. 1. Characterization of the uncoupled resonators.

a A diagram of the measurement circuit in an open-loop configuration. b, c The frequency-response curves of uncoupled resonators R1 and R3, respectively. d, e The linear responses of R1 and R3, respectively, where the insets are the corresponding primary mode shapes. The measured quality factors of the two resonators are $Q_1=36296$ and $Q_3=11541$, respectively

Fig. 2. Frequency locking and unlocking behaviors under a 1:3 IR.

a The frequency-response curves of resonator R1 under bias DC voltage $V_{dc}=20V$, electrothermal voltage $V_{th}=23.7V$, coupling voltage $V_C=8V$ and varied excitation amplitudes. Inset: the frequency-response curve of resonator R3 due to IR. b A schematic of the 1:3 internal resonance between two coupled resonators. c The peak frequency versus excitation amplitude $V_{ac}$ of resonator R1, extracted from a. d The frequency-response curves of resonator R1 under $V_{th}=23.8V$. The dependence of the measured peak frequency-shift $\mathrm{\Delta }f$ of resonator R1 from locking to unlocking, at an excitation amplitude of 10 mV above the critical value, on the (e) initial frequency offset $\sigma$ (blue circles) between the natural frequency and the locked frequency of R1 and (d) coupling voltage $V_C$ between two resonators (reddish-brown circles). The theoretical predictions of Eq. (13) are shown as solid green lines

Fig. 3. Detection mechanism for unlocking-based bifurcation sensors.

a The binary mode exploits the qualitative difference between the peak frequency before and after unlocking, reporting On-state for target stimuli above the threshold and OFF-state for stimuli below the threshold. b The analog mode exploits the peak frequency shift after unlocking

Fig. 4. Charge detection in binary mode.

a A schematic diagram of the charge detection. b The peak frequency before (OFF) and after (ON) the coupling voltage reduction (electron injection) of 0.01 V. c Allan deviation of the locked peak frequency of resonator R1

Fig. 5. Charge detection in analog mode.

a Measured peak frequencies of R1 after unlocking vs. the coupling voltage. b The real-time detection ladder diagram as the coupling voltage varies in steps of 0.1 V. The markers represent the tracked real-time peak frequency by exploiting a PLL, and the solid line denotes the mean value in one test. Inset: zoom on a detection ladder. c The time series of a specific peak frequency over 5 minutes. The colored markers represent the average frequency over 200 adjacent frequency samples. d Allan deviation of the peak frequency of R1 after unlocking