A mode-localized MEMS electrical potential sensor based on three electrically coupled resonators

We report a new class of MEMS resonant potential sensor based on the mode localization effect using a 3-degree-of-freedom (DoF) electrically weakly coupled resonator system. As opposed to previously reported electrically coupled 2DoF mode-localized resonant sensors, it can be shown in theory that the 3DoF structure has an improved sensitivity without sacrificing signal transduction, in addition to a reduced nonideal effect with regard to the vibration amplitudes and the motional currents. Experimentally, it has also been shown that several orders of magnitude higher sensitivity can be achieved compared to frequency shift and 2DoF mode-localized sensor. In the best case, we are able to demonstrate over 4 orders of magnitude improvement in sensitivity compared to frequency shift as an output signal. Compared to current state-of-the art 2DoF mode-localized sensor, the highest sensitivity improvement is over 123 times. An estimation of the noise floor of the sensor is 614μV/ √ Hz for potential sensing, or an equivalent 57.6e/ √ Hz for charge sensing, and a dynamic range of 66.3 dB can be achieved. Furthermore, two different approaches for detection were investigated, perturbing the stiffness in the form of either an axial electrostatic force or a change in electrostatic spring. We were able to demonstrate that the approach of changing electrostatic spring is more sensitive than its counterpart.


Introduction
The detection of electrical potential is of significant interest in surface potential distribution characterizations (Nonnenmacher et al., 1991) and biological (Sinensky and Belcher, 2007) and chemical analysis (Gao and Cai, 2009).Electrometers (Lee et al., 2008) are another application for potential sensing devices, which can be employed for particulate matter detection (Jaramillo et al., 2013).MEMS resonant devices have been widely used for these applications with the advantage of high resolution and a large dynamic range.
Recently, mode-localized MEMS resonant sensors have emerged as an alternative resonant sensing scheme (Thiruvenkatanathan et al., 2009;Zhao et al., 2015b), in which the mode shape of a weakly coupled resonator system changes subject to an external stiffness perturbation caused by the electrical potential change.Orders of magnitude improvement in sensitivity for electrometers (Thiruvenkatanathan et al., 2010a;Zhang et al., 2016a) have already been reported.Furthermore, mode-localized sensors exhibit better common-mode rejection capability (Thiruvenkatanathan et al., 2010b).Previously, mode-localized sensors were implemented with two resonators weakly coupled electrically (Thiruvenkatanathan et al., 2011) or mechanically (Spletzer et al., 2006;Zhang et al., 2016b), with the electrical coupling element offering advantages of tunability of the sensitivity (Manav et al., 2014).
In this paper, we demonstrate an alternative approach for potential sensing applications using a 3-degree-of-freedom (3DoF) structure.The structure has already been reported elsewhere (Zhao et al., 2016); however, the advantages of the 3DoF design have not been discussed in full detail, as only improvements in sensitivity have been shown.In this paper, we focus on the design considerations of the 3DoF structure, which we believe also have other advantages -for instance, the alleviation of the electrical nonlinear driving force, as well as the nonideal sensing current.Furthermore, in terms of potential sensing applications, two sensing methods exist: (i) modulating the electrostatic spring and (ii) directly applying an axial electrostatic force.However, the two sensing methods were not directly compared previously.In this work, we are able to demonstrate that, by modulating the electrostatic spring, an improvement in sensitivity can be observed.
The paper is arranged as follows: in Sect.2, the advantages of the 3DoF structure design are discussed; in Sect.3, the two potential sensing schemes by applying a DC potential to different ports are discussed; in Sect.4, experimental results are presented and the paper is concluded in Sect. 5.

Advantages of the 3DoF structure with electrical coupling
The schematic of the 3DoF mode-localized MEMS resonant potential sensor is shown in Fig. 1.Each resonator has four suspension beams acting as springs and a relatively large proof mass to reduce the effect of fabrication tolerances to the mass.In addition, resonators 1 and 3 have a tether structure that is capable of translating an axial electrostatic force to the suspension beams.Electrical coupling was chosen due to the ability to tune the coupling strength and thus the sensitivity of the sensor to external perturbations.A DC voltage V bias was applied to resonators 1 and 3, while resonator 2 was connected to ground.The voltage difference created electrostatic springs to couple the resonators to its neighbouring counterparts.An AC drive voltage is applied on the drive electrode, generating the actuation force.Details of the design and its fabrication process were reported in Zhao et al. (2016).However, due to the limited scope of the previous work, the advantages of the design were not discussed in full detail; these are presented in the following sections.

Sensitivity improvement
As reported in Zhao et al. (2015b), the sensitivity to stiffness changes of a 3DoF mode-localized resonant sensor can be expressed by, assuming linear springs, K 2 > 2K and K/K c > 10: where K, K 2 and K c denote the stiffness of the suspension beam of resonator 1 (and 3), resonator 2 and the coupling spring, respectively.Moreover, the sensitivity of a 2DoF mode-localized resonant sensor can be expressed by Thiruvenkatanathan (2010): For identical K/K c , the sensitivity of the 3DoF modelocalized resonant sensor can be enhanced by a factor of . Over 2 orders of magnitude improvement has already been demonstrated (Zhao et al., 2016).

Sensitivity improvement without sacrificing signal transduction
The electrostatic coupling K c for a parallel plate configuration as shown in Fig. 1 can be expressed by Thiruvenkatanathan (2010): where ε 0 , A and d are the dielectric constant of vacuum, cross-sectional area and the air gap between the plates, respectively.V is the potential difference between the two plates, which is the determining factor of the coupling strength for a given design.Therefore, for a 2DoF modelocalized resonant sensor, decreasing V , and thus K c , is beneficial for sensitivity enhancement.On the other hand, a high V is often desirable due to the required motional current level for a reasonable signal-to-noise ratio in the readout circuit.This design contradiction for choosing an optimal V can be solved by adopting a 3DoF resonant sensor configuration.An additional third parameter, the effective spring constant of the middle resonator K 2 , can be altered to maintain or even improve the sensitivity without sacrificing the readout signal level.As shown in Eq. ( 1), increasing K 2 can improve the sensitivity.

Electrostatic nonlinearity reduction
For an ideal 3DoF mode-localized resonant sensor with identical resonators 1 and 3 and negligible damping, there are three fundamental modes of vibration (Nguyen, 1999): in the first mode, all three resonators vibrate in phase with each other; in the second mode, resonators 1 and 3 vibrate out of phase, whereas resonator 2 is statutory; in the third mode, each resonator vibrates out of phase with its neighbours, and resonators 1 and 3 are in phase.The second mode, which is referred to as out-of-phase mode, is the focus of this study.The balanced and perturbed mode shapes of the out-of-phase mode of a 3DoF resonant structure are illustrated in Fig. 2.
When a stiffness perturbation is introduced, resonator 2 starts to vibrate due to mode localization.However, in the case of weak coupling, the amplitude of resonator 2 is orders of magnitude lower than the resonator with highest amplitude (e.g.resonator 1) (Zhao et al., 2015a).This can also be seen qualitatively from Fig. 2.
Consider an abstract model of the drive electrode, resonator 1 and resonator 2 as shown in Fig. 3.Only resonator 1 is considered because, under normal operating conditions, resonator 1 has a higher amplitude than resonator 3, meaning that it is more susceptible to nonlinear effects, as will be shown in Sect. 4.
Assuming v ac V bias and neglecting nonlinear terms with orders higher than 3, the total electrostatic force exerted on resonator 1 can be approximated by For a 3DoF mode-localized sensor with X 2 X 1 , or quasistatic motion of resonator 2, the second-order nonlinear term of the electrostatic actuation (i.e. between drive electrode  and resonator 1) cancels out that of the electrostatic coupling (i.e. between resonators 1 and 2), therefore rendering the total second-order electrostatic nonlinearity negligible.
Thus the overall nonlinearity is reduced.This is often desirable for resonator design, leaving only the third-order electrostatic nonlinearity which, in turn, can be used to eliminate the third-order mechanical nonlinearity intrinsic to the vibrating beams (Shao et al., 2008).

Reduction of nonideal sense current components
Due to the amplitude detecting method used in a modelocalized sensor, it is important to obtain a measure of the linear motion of the resonators with high accuracy.From the structure design perspective, it is desirable to use differential sensing to cancel out the common-mode second-order nonlinearity while doubling the magnitude of the first-order term.This is shown in Fig. 4. In addition to the second-order nonlinear term of the motional current, another commonmode nonideal current component, the feed-through current (Lee and Seshia, 2009), can also be cancelled out.

Potential sensing methods
Using two different approaches, we investigated the application of the device to detect a DC electrical potential applied to either "port 1" or "port 2", as illustrated in Fig. 1a.When a DC potential is applied to port 1, an axial electrostatic force change is created, modulating the stress in the suspension beams and thus leading to a stiffness perturba-tion of resonator 1.Alternatively, when a DC potential is applied to port 2, a change in the electrostatic spring, instead of an electrostatic force, modulates the spring softening effect compared to the case in which no potential is applied; this is equivalent to introducing a stiffness perturbation to resonator 3.

Potential detection using port 1
The stiffness perturbation of resonator 1, K 1 , as a function of an applied potential V 1 to port 1, can be expressed as (Zhao et al., 2015b) where ε 0 is the dielectric constant of free space.Typically, to initially avoid mode aliasing, a negative stiffness perturbation, K p < 0, created by a constant voltage V p on port 2 is introduced (Zhao et al., 2015b).With an applied K p , the where , M is the effective mass of each of all three resonators, K is the stiffness of resonators 1 and 3, K 2 is the stiffness of resonator 2 and is the electrostatic coupling stiffness between neighbouring resonators.The positive and negative sign is for the out-of-phase and in-phase mode, respectively.
From Eq. ( 6), it can be derived that the out-of-phase mode has a more significant response subject to a stiffness perturbation.Also, assuming V 1 V bias and weak coupling, K c K, we can find an expression of the sensitivity for frequency shift as an output signal, S f,1 , with respect to V 1 : It can be seen that Eq. ( 7) is similar to a conventional single DoF resonant sensor with frequency shift as an output signal (Schmidt and Howe, 1987), allowing a direct comparison to using amplitude ratio as an output signal.If a mode-localized sensing approach is used, the linearized sensitivity of amplitude ratio as an output with respect to the potential, S AR , can be calculated based on the assumption of weak coupling as elaborated in Zhao et al. (2016), and V 1 V bias : where L is the length of the suspension beams, and A i and d i are the overlapping cross-sectional area and the gap of the parallel plate for the ith potential port (i = 1 and 2), respectively; A and d are the cross-sectional area and the gap of the electrodes for the electrostatic coupling, respectively.It can be seen that the improvement in sensitivity is 2γ .For weak coupling K c < K/10 < K 2 /20, the improvement is at least 2 orders of magnitude (Zhao et al., 2016).

Potential detection using port 2
For a potential applied to port 2, V 2 V bias , cancelling the common term to both resonators 1 and 3 proportional to V 2 bias , the stiffness perturbation of resonator 3 can be approximated as a linear function of V 2 : For | K 3 | |K p |, the in-phase mode frequency has a stronger response (Zhao et al., 2015a).The sensitivity for the in-phase mode frequency shift, as well as the amplitude ratio as an output signal, can be approximated by It should be noted that the length of the suspension beams is 350 µm, while the capacitive gap is 4.5 µm.Consequently, for the dimensions of this device, K 1 is around 2 orders of magnitude lower than K 3 .Therefore, applying the potential to port 2 should induce a more significant stiffness perturbation, hence a higher output signal.

Device description
The device was fabricated by a single-mask SOI-based process with details described in Chang et al. (2011), Xie et al. (2013) and Hao et al. (2016), which achieved a good antistiction capability through selective roughening on the bottom side of the device layer using the notching effect.The design of the device is elaborated elsewhere (Zhao et al., 2016).Some key parameters are L = 350 µm, The SEM image of the fabricated device is shown in Fig. 5a, with zoom-in images showing port 1 (Fig. 5b) and port 2 (Fig. 5c).In the experiment, the input impedance is dominated by the capacitances, approximately 7fF and 15fF for port 1 and port 2, respectively.

Frequency response
The electrical characterization of the device was performed under a pressure of 20 µTorr to improve the quality factor of the resonators; a Q factor of 6221 was achieved.To set the initial operating point of the sensor so that the mode aliasing effect can be avoided (Zhao et al., 2015b), a constant DC voltage V p = 4.15 V was applied to port 2 and maintained during the experiment.An AC signal with a peak-peak amplitude of 20 mV was used to drive the resonators into oscillation for a bias voltage of V bias = 30 V. The motional currents of resonators 1 and 3 were converted to voltage and further amplified using an interface electronics board and recorded on an oscilloscope.The drive frequency was swept from 14 924 to 14 934 Hz manually to obtain the frequency response curve shown in Fig. 6a.
It can be seen from the frequency response that the resonator can be regarded as working in the linear regime at the out-of-phase mode frequency.Using a transimpedance gain of 6.6 M , voltage gain of 200 V V −1 and transduction factor of 6.05 × 10 −8 VF m −1 , the vibrational amplitude can be estimated as 115 nm.

Potential sensitivity characterization
AC voltages were applied to the drive electrode to actuate the 3DoF system at the desired mode frequency in the linear region for different DC bias voltages.
In the first experiment, potential port 2 was maintained at the aforementioned constant DC voltage V p for a particular V bias .We then applied varying DC potentials (both positive and negative) to port 1 and measured the averaged amplitudes of resonators 1 and 3 using an oscilloscope to calculate the amplitude ratio.The same experiment was repeated for V bias = 30, 34.5 and 40 V.The measured amplitude ratio as a function of the potential applied is plotted in Fig. 7a.Also shown is the out-of-phase mode frequency shift, normalized to the frequency when V 1 = 0, as a function of a potential applied to port 1 for V bias = 30 V. The in-phase mode frequency shift was orders of magnitude less pronounced.Therefore it is not shown.
For the second experiment, we connected port 1 to ground, while applying a DC potential to port 2. The difference between the potential applied and the aforementioned DC voltage V p is equivalent to the potential to be sensed.The amplitude ratio with respect to the potential to be sensed is plotted in Fig. 7b.The more significant in-phase mode frequency  shift for V bias = 30, normalized to that of V 2 = 0, is also plotted in Fig. 7b.
For a potential within a range of −5 V< V < 5 V (where V V bias can be considered as valid), we can extract a linear sensitivity, as given in Table 1.It can be clearly seen that when using the sensor as a mode-localized sensor, using amplitude ratio as the output signal (with sensitivity S AR ), there is at least 4 orders of magnitude improvement compared to frequency shift as an output (with sensitivity S f ).This is valid for both ports 1 and 2. Comparing S AR,1 to S AR,2 , it is also apparent that S AR,2 has over 2 orders of magnitude improvement in sensitivity for any V bias .This suggests that using port 2 is superior in terms of potential sensitivity.Another observation is that the smaller V bias , the higher the sensitivity.This is also predicted in theory by Eqs. ( 8) and ( 11).This will be helpful for biomedical applications where a low voltage is preferred.However, it should be pointed out that even though further reduction of the bias voltage is beneficial for the sensitivity, it also results in a reduction of full scale and thus dynamic range, due to the mode aliasing effect (Zhao et al., 2015b).For instance, when port 2 was used, the full scale was reduced from 7.3 to 4 V when V bias was decreased from 40 to 30 V. Nevertheless, the maximum sensitivity improvement compared to a current state-of-the-art potential sensor (Zhang et al., 2016a), which has been employed as an electrometer having a linear sensitivity of 0.083/V , is 123.6 times.
Using the noise estimation approach described in Zhao et al. (2015b), the noise floor of the amplitude ratio was regarded as white and derived to be 6.30 × 10 −3 /

√
Hz for V bias = 30 V. For port 1, the noise floor for the potential detection is approximately 73.3 mV/

√
Hz.If port 2 is used, the performance is superior.The noise floor for the potential sensor is estimated as 614 µV/

√
Hz.If a measurement bandwidth of 10 Hz is assumed (after Lassagne et al., 2008), a dynamic range of 66.3 dB can be achieved.In addition, the linear fit R 2 value for V bias = 30 V for full scale is 0.999, suggesting a very linear response of the sensor.The dominant source of noise is the interface electronics (Zhao et al., 2015b), which should be optimized to improve the performance of the sensor further.

Equivalent charge detection
For the non-contact potential detection approach described here, the sensor can also be employed as an electrometer; the DC potential applied can be directly translated into a charge injection, with Q = V C as the governing equation.For port 1, the motion of the resonator along the y axis direction (direction illustrated in Fig. 1) can be neglected.Therefore the parallel plate capacitance can be regarded as a constant 7fF.A resolution of 3205e/

√
Hz is estimated with V bias = 30 V.
On the other hand, due to the vibrating motion of resonator 3 along the x axis, the capacitance is not of constant value.However, due to the low vibrating amplitude (less than 12.1 nm) within the operating range for V bias = 30 V, the relative variation of the charge due to the motion is less than 0.27 % for a gap of 4.5 µm.If the charge variations due to the motion are neglected, the equivalent charge detection resolution is estimated to be 57.6e/

√
Hz with C = 15fF.This is an improvement of 2.5 times compared the state-of-the-art MEMS mode-localized electrometers at room temperature, which has a resolution of 147e/

Conclusions
In this paper, we have demonstrated the design of a 3DoF weakly coupled resonant sensor for potential sensing applications, which could also be extended to an electrometer.We have presented the design advantages for the 3DoF structure in theory, including sensitivity improvement, electrical nonlinearity and nonideal current reduction.We have shown that, by using the device as a mode-localized sensor, the sensitivity can be improved by over 4 orders of magnitude, com-pared to conventional frequency shift as an output.In addition, we have also compared the sensitivity of the modelocalized sensor for different bias voltages.We demonstrated that the lower the bias voltage, the higher the sensitivity.Finally, we have shown two viable methods for sensing an electrical potential.The more sensitive approach is by applying the potential to port 2, where a change in electrostatic spring is used to perturb the stiffness.The best sensitivity improvement compared to the state-of-the-art mode-localized sensor is 123 times.If employed as an electrometer, the best resolution can also be improved by 2.5 times compared to the state of the art.

Figure 1 .
Figure 1.(a) Schematics of the 3DoF mode-localized potential sensor, showing three resonators coupled electrostatically with their neighbours.In addition, electrostatic springs and forces caused by electrical potential differences are also shown; (b) the linearized spring-mass-damper model of the 3DoF mode-localized potential sensor.

Figure 2 .
Figure2.Illustration of the mode shapes of the out-of-phase mode of a representative 3DoF resonant structure, simulated using Coven-torWare FEM tool: (top) without perturbations and (bottom) with stiffness perturbations.The perturbation in the bottom panel is a positive stiffness perturbation applied to resonator 1.It should be pointed out that the same mode shape is generated given an identical negative stiffness perturbation applied to resonator 3.

Figure 3 .
Figure3.Abstract model of the drive electrode, resonator 1 and resonator 2 illustrated as parallel plates.X 1 and X 2 are the displacement of resonator 1 and resonator 2, respectively.The crosssectional area A and gap d are supposed to be identical for all electrodes considered.

Figure 4 .
Figure 4.An SEM image of one pair of comb fingers attached to the proof mass of resonator 1 for vibration motion sensing.(As shown in Fig.5c, another pair of comb fingers with identical configuration is also attached to the proof mass of resonator 3.) The top and bottom sets of the comb fingers on the proof mass are reversely configured relative to the stationary sense electrodes 1 and 2; this enables differential sensing.Therefore, a reduction in nonideal sense current components is achieved by common-mode rejection.
sensing mechanism in response to a stiffness perturbation caused by the potential change is explained below.Assuming | K 1 | |K p |, based on a transfer function model of the 3DoF weakly coupled resonators device described in Zhao et al. (2015b), the mode frequencies of interest can be calculated as

Figure 5 .
Figure 5. Scanning electron micrograph (SEM) showing (a) the 3DoF MEMS mode-localized resonant sensor; (b) the structure for potential detection at potential port 1; (c) the structure for potential detection at potential port 2.

Figure 6 .
Figure 6.The responses of the 3DoF mode-localized resonant sensor with a driving voltage of 20 mV and V bias = 30 V: (a) the frequency response of the sensor (the voltage amplitude is the peak-peak amplitude); (b) measured steady-state time domain responses of the resonators 1 and 3 at the out-of-phase mode frequency.

Figure 7 .
Figure 7. Graphs showing (a) measured amplitude ratio as a function of potential applied to port 1 for different V bias , with inset showing measured normalized out-of-phase mode frequency shift for V bias = 30 V; (b) measured amplitude ratio as a function of potential applied to port 2 for different V bias , with inset showing measured normalized in-phase mode frequency shift for V bias = 30 V.

Table 1 .
Summary of potential sensitivity.
NA: not available.