Simultaneous electrical and mechanical resonance drive for large signal amplification of micro resonators

Achieving large signal-noise ratio using low levels of excitation signal is key requirement for practical applications of micro and nano electromechanical resonators. In this work, we introduce the double electromechanical resonance drive concept to achieve an order-of-magnitude dynamic signal amplification in micro resonators. The concept relies on simultaneously activating the micro-resonator mechanical and electrical resonance frequencies. We report an input voltage amplification up to 15 times for a micro-resonator when its electrical resonance is tuned to match the mechanical resonance that leads to dynamic signal amplification in air (Quality factor enhancement). Furthermore, using a multi-frequency excitation technique, input voltage and vibrational amplification of up to 30 times were shown for the same micro-resonator while relaxing the need to match its mechanical and electrical resonances.

Micro/nano-electromechanical system (M/NEMS) devices have been used in many applications, such as high-frequency switches, 1 sensors, 2-9 micro-mirrors, 10 and RF amplifiers. 11 However, due to their high input voltage requirement, MEMS devices are still not employed to their potential. 12 To amplify the output signal of MEMS resonant devices several approaches have been utilized, including driving them around their mechanical resonance frequency, optimizing designs [13][14][15] by increasing the MEMS structure surface area, reducing its stiffness, or narrowing the gap between the stationary and movable electrodes. However, these methods were not effective to boost the device response while reducing their operating voltage to a level compatible with complementary metal-oxide semiconductor (CMOS) technology. Moreover, most of these methods increase squeeze film damping 16 and the risk of electrode stiction. 17 Other research have focused on utilizing parametric nonlinear resonance to increase MEMS dynamic deflection and enhance the output voltage. [18][19][20] However, parametric resonance activation requires complex actuation techniques to modulate the stiffness and strict low damping conditions; and hence is limited for specific applications.
In this work, we report the amplification of the response of a MEMS resonator at resonance by simultaneously activating its electrical and mechanical resonances (double resonance actuation). Moreover, a multi-frequency signal [21][22][23] was utilized to trigger the double resonance in MEMS resonators with a mechanical and electrical natural frequency mismatch.
Toward this, we fabricated a clamped-clamped microbeam resonator, as shown in Fig. 1(a). The microbeam is fabricated on a silicon wafer coated with 500 nm of thermally grown silicon dioxide (SiO 2 ) layer. The lower electrode is formed by pattering the Cr/Au layer that is sputtered on the wafer surface. The microbeam is composed of a 4. We characterized the mechanical and electrical properties of the MEMS resonator by studying the frequency response near the mechanical and electrical resonances. The mechanical resonance frequency of the fundamental mode was measured to be around 123 kHz using white noise signal excitation. 21 The MEMS device is electrically modeled as a nominal capacitance C e , formed by the deformable MEMS microbeam and the substrate beneath it, connected in parallel to a series branch of motional resistance, motional capacitance, and motional inductance, donated by R m , C m and L m , respectively, Fig. 1(b). We create a resonant R e L e C e circuit drive by connecting the MEMS device to a variable external inductance L e , Fig. 1(c). The circuit's external resistance R e is the equivalent parasitic resistance from the wires and the internal resistance of the inductor L e . In our experiments, we varied L e to control the series R e L e C e resonance frequency and the voltage gain across the MEMS capacitor C e . The frequency sweep for electrical resonance characterization was conducted at ambient pressure and the input voltage was kept low so that the effect of C m , R m and L m can be minimized during this characterization step.
The frequency response for characterizing the pure electrical resonance is conducted by an impedance analyzer, as shown in Fig. 2. We identified electrical resonance by one of the two methods: (i) by monitoring the amplitude of the circuit conductance G with respect to the frequency, Fig. 2(a), where the corresponding frequency at the peak conductance value represents the electrical resonance, and (ii) by monitoring the voltage across the MEMS capacitor C e , where the maximum voltage is achieved at the electrical resonance frequency as shown in Fig. 2(b). The increase in the conductance is due to the circuit reactance X e going to zero at resonance, as shown in (1) and (2): where f is the excitation frequency. By solving the external R e L e C e resonance circuit characteristic equation (3), the steady state voltage across the MEMS capacitor C e can be obtained using equation (4): where Q is the charge stored in the MEMS capacitor C e , V in is the input voltage and |V c | is the amplitude of the voltage across the MEMS device. The electrostatic force in the MEMS resonator is a function of the square of the voltage across the movable and stationary electrodes, and is given by: where F e is the electrostatic force per unit length, ε is the air permittivity, d is the gap between the microbeam and the substrate beneath, x is the location along the length of the microbeam, t is time and w(x,t) is the out-of-plane deflection of the microbeam in the direction of the substrate. Next, we compare the response of the MEMS device with simultaneous activation of electrical and mechanical resonances to that actuated using conventional mechanical resonance alone. Fig. 3(a) shows the response of the device operated at atmospheric pressure, with V DC = 30V, and for various AC voltages. We note that for this experiment we have disconnected L e . Fig. 3(b) shows the response of the double resonance driven circuit at the same pressure but with V DC = 10V. In order to activate double resonance, the electrical resonance frequency was tuned to be near the mechanical resonance frequency by using L e = 4 mH. To compare the two actuation methods, we find the product V AC V DC that results in similar maximum vibrational amplitudes. For instance, to achieve a response of 0.7 µm, the required product of V AC V DC without electrical resonance is 750 V 2 compared to only 50 V 2 when both resonances are simultaneously activated. Thus, as the DC component of the signal remains unamplified, we show an effective 15 times voltage amplification of the AC actuation signal by driving the MEMS resonator with double resonance drive. We note that the system resonance frequency is slightly less in Fig. 3 for the case of the double resonance drive compared to mechanical resonance alone. This softening behavior is an indication of the strong nonlinearities due to the electrical and mechanical resonance coupling similar to those achieved by high voltage excitation. 24 However, when a double resonance MEMS is used as a sensor, the MEMS electrical resonance frequency is almost fixed and any change in the overall system resonance frequency is due to the effect of the measured entity on the mechanical resonance frequency.
While double resonance activation was simple to achieve for the experimental results shown in Fig. 3(b), due to the proximity of the systems' mechanical and electrical resonance frequencies, this might not be the case for general MEMS devices. To overcome this limitation, we introduce a multi-frequency excitation signal composed of a beating signal with two frequency components: f 1 and f 2 . Due to the quadratic voltage term shown in Equ. (5), the frequency spectrum of the resulting electrostatic force include 23 2f 1 , 2f 2 , ( f 1 +f 2 ), and ( f 1 -f 2 ) frequency components. Therefore, for double resonance activation, either f 1 or f 2 is selected to be around the electrical resonance frequency while at least one of the forcing spectral components is made to be equal to the mechanical resonance frequency. To demonstrate this concept, we show in Fig. 4, the response of the MEMS resonator to a multi-frequency input signal with electrical resonance frequency at 308 kHz and a mechanical resonance frequency at 123 kHz (a mismatch of 185 kHz). We note that this experiment was also conducted under atmospheric pressure. In Fig. 4(a), the input signal has a fixed frequency component, f 1 = 308 kHz at different V AC1 values, near the electrical resonance frequency, while the second frequency component f 2 , was swept such that ( f 1 -f 2 ) is near the mechanical resonance (123 kHz). This resulted in voltage amplification of V AC1 across the MEMS resonator (capacitor) due to the electrical resonance and overall forcing amplification, hence, higher amplitude of motion. In contrast, when both f 1 and f 2 are far from the electrical resonance frequency, significantly higher input voltages are required to achieve comparable results, as shown in Fig. 4(b). For instance, to achieve an amplitude of 0.36 µm, the product of V AC1 V AC2 is 1176 V 2 while the required V AC1 V AC2 with double resonance drive is about 39 V 2 . Thus, Fig. 4(b) demonstrates a voltage amplification gain of ∼30 through double resonance excitation. This amplification is almost twice the amplification obtained in Fig. 3(b) by matching the resonator electrical resonance to its mechanical resonance using a larger inductor. We attribute this higher gain to the flexibility of using a smaller external inductor, and hence less parasitic resistance, when matching the two frequencies is not required. Finally, to demonstrate the increase in the quality factor of the system, we compare the response of the MEMS device with and FIG. 4. The frequency response of the MEMS device for: (a) Mixed frequency excitation away from the electrical resonance frequency: f 1 is chosen far away from the electrical and mechanical resonance frequency (80 kHz) with V AC2 =42 V while f 2 was swept such that f 1 +f 2 is around the mechanical resonance frequency. (b) Double resonance excitation: f 1 is fixed at the electrical resonance (308 kHz) with multiple values of V AC1 , V AC2 = 6.5 V and f 2 is swept such that |f 1 -f 2 | is around the mechanical resonance. We show that to achieve similar deflection in (a), significantly more voltage is required. (c) Double resonance (blue circle, V AC1 = 3V, V AC2 =6.5V, f 1 =308 kHz, f 2 = 170 to 210 kHz, f effective =f 1 -f 2 ) versus traditional mechanical resonance (red triangle, V AC1 =3V, V AC2 = 7 V, f 1 =80 kHz, f 2 = 20 to 60 kHz, f effective = f 1 +f 2 ). Here, f effective is a frequency near the MEMS mechanical resonance frequency. The figure shows the increase in the resonator quality factor using double resonance. without double resonance activation, Fig. 4(c). Each case utilizes a multi-frequency signal excitation such that f 1 and f 2 produce a forcing signal that has a frequency-spectrum component equal to the mechanical resonance frequency. A significant increase in the quality factor of the MEMS resonator and a vibrational amplitude amplification of ∼30 times was found when the system was driven using double resonance drive.
In conclusion, we introduced a mean of signal amplification in electrostatic MEMS resonators by utilizing electrical resonance. We demonstrated two different schemes to utilize this signal amplification to improve the dynamic response of MEMS resonators. The approach was demonstrated first by tuning the electrical resonance frequency to coincide with the mechanical resonance frequency, using a variable external inductor, which is viable when the two resonances are proximate. We have also shown a more generic approach to activate electrical and mechanical resonances simultaneously by actuating the device using a multi-frequency signal. In this case, one of the signal's frequency components was near the electrical resonance. The other component was chosen such that at least one of the forcing spectral components (such as f 1 +f 2 or f 1 -f 2 ) matches the MEMS mechanical resonance. In both the cases, a high amplitude response was recorded. An increase in the quality factor of the resonator response was also shown. We note that the activation of simultaneous electrical and mechanical resonances does not require any changes in the design of MEMS devices. However, the electrical resonance frequency and its corresponding voltage gain may differ between similar devices due to parasitic capacitance and resistance variation. More precise fabrication and tuning of external electrical components (inductor, capacitor) can be used to alleviate this issue. Nonetheless, the simultaneous electrical and mechanical resonance activation scheme may alleviate the need for CMOS incompatible high AC voltage source or amplifiers for