1 Introduction

Micro-electro-mechanical system (MEMS) filter has received significant attention due to their fascinating functional application in robotics, sensors, energy harvester, signal processing system in loudspeaker and micro-communication (Bijari et al. 2012; Chivukula and Rhoads 2010; Hore et al. 2016; Lang et al. 2011; Saha et al. 2011; Wang et al. 2010). A low-pass or high-pass filter is an electronic filter that passes signals with a frequency lower or higher than a certain cutoff frequency. And capacitor takes a key factor of the frequency cutoff in the filter (Saha et al. 2011; Wang et al. 2010). Electrostatically MEMS capacitor not only has the advantage with mechanical dynamic motion and electrical storing for their frequency response (Baginsky et al. 2013; Galayko and Basset 2011; Tvedt et al. 2010), but also has advantage with compact size, simple geometry, low power consumption and ease integration in micro fabrication. Many researchers have reported that MEMS capacitor through electrostatic actuation is highly dependent on parameters, such as actuation forces, squeeze film damping effects, large structural deformation, and intermolecular surface forces (Erbes et al. 2016; Shirazi et al. 2011; Kambali and Pandey 2016; Langfelder et al. 2011; Yang et al. 2010; Zhang et al. 2014). And the behavior of many kinds structural designed filter has also been reported in the literature (Basu and Bhattacharyya 2011; Caruntu and Knecht 2015; Dong et al. 2014; Xu and Cheng 2015; Zhang et al. 2015). Basu and Bhattacharyya 2011 have demonstrated a 2 μm thick polysilicon flexural mode clamped–clamped beam device at a frequency of 10 MHz in vacuum ambient. Xu and Chen2015 have reported in silicon extensional mode device vibrating at 2.18 MHz using square plates operating in the bulk mode of vibration. However, all the researches were focused on the extremely high frequency in resonator. Almost no group has carried out the research about the variable MEMS capacitor with beam and plate structural application in high-pass RC filter, especially the frequency around 1 kHz. We have taken the research about a suspended fixed-end thin SiO2 film beam and plate as high-pass RC filter for the frequency range 300–2500 Hz, and we also found an asymmetric offset voltage electrical response because of the coupling of the electrostatic force and dynamic motion of the electrode plate in the variable MEMS capacitor.

2 Design and modeling

The suspended fixed-end variable beam and plate capacitor shown in Fig. 1a, b was designed with bulk silicon micromachined technology. The proof-mass movable 5 μm thickness SiO2 film was supported by two fixed-end SiO2 suspension beams. The top surface of the SiO2 film was sputtered Cu by megnetron sputtering (PVD 75 Pro Line Kurt J. Lesker Company) for 5 min and the bottom fixed square plate on the silicon substrate was patterned with chromium (Cr) 1 min and gold (Au) 2 min by megnetron sputtering equipment as well. Both the suspended SiO2 film plate and the bottom gold square plate formed a variable capacitor. When a rectangular wave signal imposed onto this variable capacitor shown in Fig. 1c, the movable plate motions due to the changeable electrostatic force interaction with the bottom gold fixed plate. When a variable capacitor and output test resistor is connected to a circuit with direct current (DC) source, two processes related the capacitor, which are called “charging” and “discharging” the capacitor, will happen due to the energy storing of the capacitor in its electric field. And both plates get the equal and opposite charges, and an increasing potential difference is created while the capacitor is charging. Once the voltage at the terminals of the capacitor and the output resistor is equal to the power supply voltage V in , the capacitor is fully charged and the transient current stops flowing. Notice that the transient currents decay and the time in the designed circuit depend on the resistance (R out ) and capacitance (C).

Fig. 1
figure 1

5 μm thick suspended fixed-end SiO2 film variable capacitive high-pass filter: a SEM of a 500 × 500 μm2 plate and 1 mm × 50 μm supporting beam; b SEM of a 1 × 2 array, note the dimension of them are not the same; c physical model of MEMS capacitive filter; d electrical equivalent model of the variable capacitive RC filter

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When the capacitive filter is subjected with a rectangular wave, the arrival of the falling edge of the input waveform causes the variable capacitor to reverse charge giving a negative output spike, then as the square wave input changes during the output spike changes from a positive value to a negative value. As shown in Fig. 1c, the initial separation between the two plates, d 0 , is used as the characteristic length throughout of this paper. The variable x is the dimensionless displacement of the movable plate from the equilibrium. Parameter C(t) is the variable capacitor, C p represents the parasitic capacitance related to the variable capacitor, C in represents the input capacitance of the voltage measurement circuit which is directly related with the measured voltage. Since the effect of C p can be lumped into C(t), we assume that C p  = 0 is without loss of generality. The charge on C(t) shown in Fig. 1d is given by

$$Q = C(t)(V_{in} - V_{out} ),$$
(1)

the derivative on both side of equation is as follows,

$$\dot{Q} = \dot{C}(t)(V_{in} - V_{out} ) - C(t)\dot{V}_{out} .$$
(2)

It knows that the charge over C in is given by

$$Q_{1} = C_{in} V_{out} .$$
(3)

Based on the Kirchhoff current law in the electrical circuit, we have

$$\dot{Q} = \dot{Q}_{1} + \frac{{V_{out} }}{{R_{out} }}.$$
(4)

Therefore, we get

$$\dot{C}(t)(V_{in} - V_{out} ) - C(t)\dot{V}_{out} = C_{in} \dot{V}_{out} + \frac{{V_{out} }}{{R_{out} }}.$$
(5)

Rearranging the terms in Eq. (5), we finally get

$$R_{out} [C_{in} + C(t)]\dot{V}_{out} + [R_{out} \dot{C}(t) + 1]V_{out} = V_{in} R_{out} \dot{C}(t)$$
(6)

which is

$$\dot{V}_{out} = \frac{{\dot{C}(t)(V_{in} - V_{out} ) - \frac{{V_{out} }}{{R_{out} }}}}{{C_{in} + C(t)}}.$$
(7)

The parasitic capacitance C in has negligible effect if the capacitive impedance is much larger than R out . The microcantilever and the suspended plate of the MEMS variable capacitive filter are considered as a spring-mass system. The model of the equivalent mass m, spring stiffness k, damping factor b has been extensively explored and modeled (Saha et al. 2011; Galayko and Basset 2011). The electrostatic force between two parallel plates is derived from the following equation,

$$F_{e} = - \frac{\partial U}{\partial d}$$
(8)

where d is the separation of two parallel plates, U is the potential energy given by

$$U = \frac{{Q^{2} }}{2C(x)}.$$
(9)

The capacitance between two plates can be calculated using the equation,

$$C(x,\theta ) = \iint\limits_{A} {\frac{\varepsilon dA}{d}}.$$
(10)

The movable plate of the variable capacitor is subjected to the nonlinear electrostatic force F e which is driven by rectangular wave signal, the suspension force from linear springs and the damping force from the air. We follow Newton’s second law,

$$m\ddot{x} + c_{1} \dot{x} + kx = F_{e} .$$
(11)

where c 1 is the viscous damping coefficient added to represent the damping loss. The mechanical and electrical effects are coupling with each other through the motion of the movable plate in the variable capacitive high-pass filter system.

3 Device fabrication

Figure 2 illustrates the fabrication process flow of the variable MEMS capacitor using bulk silicon micromachining technology: (1) a thin SiO2 was plasma enhanced chemical vapor deposition (PECVD) on a piece of silicon, and then thick photoresist sacrificial layer was coated on this specific SiO2 (see Fig. 2b). (2) A mask was used during the lithography exposure for the specific structure (see Fig. 2c). (3) Cr and Au was megnetron sputtering deposited (see Fig. 2d), and after this the photoresist sacrificial layer on top of the silicon was removed by carefully flushing with acetone and DI water (see Fig. 2e). (4) A SiO2 layer with a thickness of 5 µm on the top side and 1 µm SiO2 film on the back side of a wafer was PECVD grown as well, respectively (see Fig. 2f). (5) A thick photoresist sacrificial layer was coated with a thickness of 15 µm on both side of the wafer for lithography on both side of the water (see Fig. 3g). (6) 5 µm deep SiO2 and 1 µm SiO2 were etched away using Inductively Coupled Plasma (ICP) 20 min (see Fig. 3h). Note that a few thickness of silicon etched away. (7) KOH solution was used as bulk wet etching for 48 h (see Fig. 2i). (8) Metallization was carried out by Cr/Au (30/300 nm) which were sputter on the top of the suspended plate and cantilever (see Fig. 2j). (9) Finally, the photoresist sacrificial layer on top of the silicon was removed carefully by flushing with acetone and DI water. (10) Both the two components Fig. 2e and j were boding together to form the variable capacitor (see Fig. 2k).

Fig. 2
figure 2

Fabrication process flow of SiO2 film fixed-end MEMS variable capacitor

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Fig. 3
figure 3

MEMS variable capacitor with the increasing sweeping frequency: a capacitance; b loss of the coefficient

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4 Measurement and discussion

Figure 3 illustrates the electrical characteristic performance of the bulk silicon fabricated MEMS variable capacitor using E4980A Precision LCR. The 1 × 2 array has 540 pF compared with 500 pF for a single capacitor at frequency 1.2 kHz shown in Fig. 3a. Figure 3b depicts the single cavity has 2 times loss coefficient than the 1 × 2 array. The reason to explain this phenomenon is that the electrostatic force on the variable 1 × 2 array capacitor is much stronger than the single device.

The function generator and an oscilloscope were used to do the experimental test shown in Fig. 4a. The function generator was as an input rectangular wave power source, and the oscilloscope was used to measure the output voltage on the resistors 220 kΩ. Compared with the commercial static capacitor, we found that the stability voltage of capacitor is slightly above zero in MEMS variable device. The coupling effect of the electrostatic force could be used to explain them. When compared with the single device in Fig. 4a and the 1 × 2 array in Fig. 4b, we found that the parallel connection of the 1 × 2 array capacitors with greater value than single capacitor caused better electrical performance. The resonance frequency of the high-pass filter was observed around 1 kHz for both devices (shown in Fig. 4d).

Fig. 4
figure 4

Measured electrical performance of the variable capacitive high-pass filter: a results using single capacitor; b results using 1 × 2 array capacitor; c results using commercial static capacitor; d the peak output voltage with increasing frequency

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5 Numerical simulation

The first mode of MEMS variable capacitor is shown in Fig. 5a and has been obtained from a finite element analysis (FEA) simulation performed with COMSOL Multiphysics. The simulation with approximately 16072 3D solid elements resulted in a 1015.3 Hz expected resonance frequency. The numerical simulation to the parameters such as displacement of the movable plate and the output voltage were shown in Fig. 5b. The parameters used in the computation are given in Table 1. The output voltage on the resistor and capacitor are the red curve and blue curve, respectively. When the MEMS variable capacitor is charging, the voltage across the capacitor increases but the output voltage across the resistor decreases. We found that the displacement of the movable plate was motioned around 16.6 μm position. We also found that the trends of the output voltage across the resistor with the numerical results are quite similar with experimental test results shown in Fig. 4a. Keep in mind that we assumed the squeeze film air damping a constant coefficient in the numerical modeling.

Fig. 5
figure 5

Simulation results of the single variable capacitive filter: a FEM of the beam and plate; b numerical simulation results of the plate motion and the output voltage

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Table 1 Simulation parameters of the fixed-end variable capacitive high-pass filter
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6 Conclusion

In this study, we successfully fabricated a suspended fixed-end 5 μm thin SiO2 film beam and plate using bulk silicon two-sided alignment micromachining technology. We have built a mathematical model which both includes a high-pass filter electrical system and variable capacitors driven by mechanical structural deformation. We found that the room temperature experiment of the designed MEMS high-pass filter can convert a rectangular wave input signal into high frequency spikes as its output. We also used both finite element calculating software COMSOL and numerical simulation MATLAB to evaluate the motion and the output voltage, and we demonstrated the room temperature measurement and the simulation modeling had a good agreement. We found that the parallel connection of the 1 × 2 array capacitors caused stronger asymmetric offset voltage electrical response performance than single capacitor due to the nonlinear dynamic of the variable electrostatic force with the motioned plate.