Piezoresistive Sensing of a Dielectrically Actuated Silicon Bar Resonator

This paper reports on a dielectrically actuated and piezoresistively sensed 4.41 GHz silicon bar resonator with an electromechanical Q (Qem) of 8180. The 2-port piezoresistive transconductance measurement performed provides a promising alternative to capacitive measurement at high frequencies, where nominal and feed-through capacitance often dominates the output signal. The electromechanical f·Q product of the silicon-based resonator is 3.6×10 s, the highest reported to date for a silicon resonator using a direct 2-port measurement. INTRODUCTION Silicon-based high frequency electromechanical resonators offer low power, small footprint, high-Q solutions for a variety of applications including microprocessing, RF communications, and sensor networks. In recent years, much of MEMS resonator research has focused on different mechanisms for driving and sensing acoustic resonance at frequencies exceeding 1 GHz. Promising transduction methods include solid dielectric capacitive transduction [1,2], piezoelectric transduction [3,4], and air-gap capacitive actuation with piezoresistive sensing [5]. The authors previously demonstrated the internal dielectric transduction of a 4.51 GHz silicon bar resonator [2]. The 3 and 9 longitudinal-mode harmonics of the device were excited and detected capacitively with dielectric films sandwiched inside the resonator body at displacement nodes (Figure 1). The 9 harmonic resonance at 4.51 GHz with a mechanical quality factor of 11,200, demonstrated a 9.8 dB improvement in signal strength over the 3 harmonic at 1.5 GHz. The resulting mechanical f·Q product of 5.1×10 s is the highest measured to date in silicon. The analytical and experimental study of internal dielectric transduction performed in [2,6] indicates improved transduction efficiency with increasing frequency, enabling mechanical resonance at previously unattainable frequencies. A three-port scalar mixer measurement [7] was used to characterize the internal dielectrically transduced resonator. However, the large nominal and feed-through capacitance inherent in high-frequency capacitive sensing makes such resonators impractical in most integrated CMOS applications. In this work, the 9 harmonic of a longitudinal mode silicon bar resonator is detected in a two port measurement, coupling the benefits of internal dielectric capacitive actuation with the piezoresistive sensing demonstrated by van Beek et al [5]. THEORY Capacitive Actuation and Sensing Internal dielectric transduction employs capacitive drive and sense to excite and detect acoustic resonance. In [2], a longitudinal-mode bar resonator is driven and sensed electrostatically with thin vertical dielectric layers, as shown in Figure 1. The resonator body is biased to VDC, and a harmonic excitation of amplitude vin is applied to the drive electrode at the resonant frequency. The amplitude of vibrations of the n harmonic resonance is given by ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = 2 sin 2 sin 2 2 2 2 0 g k d k g k d k g L Y n v V Q U n n n n in DC f π ε (1) suspension beams anchor anchor A A’ dielectric film A A’ Figure 1: Schematic of internal dielectric transduced longitudinalmode bar resonator. The dielectric films (yellow) are incorporated into the resonant mode shape. A cross section of the bar along AA’ is illustrated on the right. The 9th harmonic longitudinal mode is shown, with dielectric films for driving resonance positioned at displacement nodes. where Q is the quality factor, and Y and ρ are the Young’s modulus and mass density of the resonator, respectively. Here, f ε is the dielectric permittivity, g is the dielectric thickness, d is the position of the dielectric along the bar, A is the transduction area, and L n kn π = is the resonance wave number. The 2-port motional impedance out in X i v R ≡ for the n harmonic of the resonator is given by ( ) ( ) 2 sin cos 2 2 2 4