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A force sensor based on three weakly coupled resonators with ultrahigh sensitivity


Sensors and Actuators A 232 (2015) 151–162

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Sensors and Actuators A: Physical
journal homepage: www.elsevier.com/locate/sna



A force sensor based on three weakly coupled resonators with ultrahigh sensitivity
Chun Zhao a,? , Graham S. Wood a , Jianbing Xie b , Honglong Chang b , Suan Hui Pu a,c , Michael Kraft d,??
a

Nano Research Group, School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, UK MOE Key Laboratory of Micro and Nano System for Aerospace, Northwestern Polytechnical University, 127 Youyi West Road, Xi’an 710072, Shaanxi, China University of Southampton Malaysia Campus, Nusajaya, 79200 Johor, Malaysia d University of Liege, Monte?ore Institute, Grande Traverse 10, 4000 Liege, Belgium
b c

a r t i c l e

i n f o

a b s t r a c t
A proof-of-concept force sensor based on three degree-of-freedom (DoF) weakly coupled resonators was fabricated using a silicon-on-insulator (SOI) process and electrically tested in 20 ?Torr vacuum. Compared to the conventional single resonator force sensor with frequency shift as output, by measuring the amplitude ratio of two of the three resonators, the measured force sensitivity of the 3DoF sensor was 4.9 × 106 /N, which was improved by two orders magnitude. A bias stiffness perturbation was applied to avoid mode aliasing effect and improve the linearity of the sensor. The noise ?oor of the amplitude ratio output of the sensor was theoretically analyzed for the ?rst time, using the transfer function model of the 3DoF weakly coupled resonator system. It was shown based on measurement results that the output noise was mainly due to the thermal–electrical noise of the interface electronics. The output noise spectral density was measured, and agreed well with theoretical estimations. The noise ?oor of the force sensor output was estimated to be approximately 1.39nN for an assumed 10 Hz bandwidth of the output signal, resulting in a dynamic range of 74.8 dB. ? 2015 Elsevier B.V. All rights reserved.

Article history: Received 16 December 2014 Received in revised form 13 May 2015 Accepted 15 May 2015 Available online 2 June 2015 Keywords: MEMS Force sensor Resonant sensor Force sensitivity Thermal noise

1. Introduction For the last couple of decades, emerging micro- and nano-scale devices enabled the measurement of forces in the region of pN to ?N. Measurement of the forces in this range plays important roles in many different areas, including surface characterization [1], contact potential difference measurement [2], study of biomechanics [3] and cell mechanobiology [4], inertial sensing [5], manipulation of microscale objects [6] and magnetometer for electronic compass [7], among many others. Among these miniature force sensors, resonant sensing devices are attractive to researchers due to its quasi-digital output signal and high accuracies [8]. The conventional approach employs a single degree-of-freedom (DoF) resonator; when an external force is exerted on the resonator, the stiffness changes while the mass remains the same, leading to a frequency shift [9]. The challenge

? Corresponding author. Tel.: +44 7730043696. ?? Corresponding author. E-mail address: cz1y10@ecs.soton.ac.uk (C. Zhao). URL: http://www.ecs.soton.ac.uk/people/cz1y10 (C. Zhao). http://dx.doi.org/10.1016/j.sna.2015.05.011 0924-4247/? 2015 Elsevier B.V. All rights reserved.

to improve the performance of the force sensor, aiming to sense smaller forces motivates research in alternative sensing paradigms. One promising approach, which couples two identical resonators with a spring much weaker than that of the resonators, is to form a 2DoF system [10]. This approach utilizes a mode localization effect which was ?rst described in solid-state physics by Anderson [11]. When a small perturbation is applied on one of the resonators, the mode shapes of the system change. It was demonstrated that by measuring the eigenstates shift caused by mode localization, orders of magnitude improvement in sensitivity of mass change was observed [10]. Various groups demonstrated that orders of magnitude enhancement in sensitivity of stiffness change [12–15] and force [16] could be achieved using this approach. Another advantage of this type of device is its intrinsic common mode rejection [17]. The force to be measured can be applied to a resonator in different directions, depending on the application: one way is to apply a vertical force or force gradient to the tip of a horizontal cantilever, as demonstrated in [18]. This approach is widely used for atomic force microscopy (AFM) due to its simplicity. However, for non-contact AFM applications, when the gradient of the Van der Waals force exceeds the stiffness of the cantilever, snap-down

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C. Zhao et al. / Sensors and Actuators A 232 (2015) 151–162

instability occurs [19], which is analogous to the pull-in effect occurring in parallel plate actuation. Hence, a large stiffness for the vibrating structure is required for some applications, which, in turn, deteriorates the force sensitivity and resolution of the sensor. To reach maximum stability while not compromising the sensitivity, an alternative method is to apply the force along the length of a beam [20,21]. Due to a relatively high longitudinal stiffness of a beam, the instability is alleviated [21]. In this work, a novel proof-of-concept force sensor consisting of three resonators with enhanced force sensitivity is presented. The reason for using a 3DoF resonator system is that, the third resonator located in between two identical resonators reduces the energy propagation due to its absorption of energy, thus increases the energy attenuation along the chain. Consequently, it enhances the mode localization when a structural disorder is present. It has been demonstrated in both theory and by measurement results [22,23] that a 3DoF weakly coupled resonator sensor can exhibit enhanced sensitivity compared to existing 2DoF mode-localized sensors. The device was fabricated using a silicon-on-insulator process, and tested electrically. The external force was a quasi-static electrostatic force applied along the direction of the beam length, which avoided the potential instability mentioned above. The 3DoF force sensor utilized the mode shape change due to a stiffness perturbation introduced by an external force. The vibration amplitude ratio of two resonators at one mode of interest was used to measure the mode shape change. Two orders of magnitude improvement in sensitivity, compared to 1DoF resonator sensors with frequency shift as an output signal, was observed from the measurement. In addition to sensitivity, resolution and dynamic range of the force sensing device is also discussed.

respectively. The longitudinal stiffness of the suspension beam is given by [25]: Klong = Etw L (2)

where w and L are the width in the x-axis and the length of the suspension beam. To applied forces in the negative y-direction, the tether and the suspension beams act similarly to two springs in parallel [24]. Ideally, the tether does not absorb any force applied in the y-axis, so that all the forces can be measured by the resonator. For our design, the shortest effective length of the tether is 60 ?m, resulting in a maximum stiffness of Ktether = 538 N/m. Whereas in the y-axis, suspension beams 1 and 3 are in series, therefore the effective longitudinal stiffness is Klong = 2.48 × 104 N/m. This indicates that more than 97.9% of the force applied is absorbed by the suspension beams, with less than 2.1% of the force exerting on the tether. Hence, we are able to assume that the entire electrostatic force is transmitted to the resonators for measurement. When two different DC voltages are applied to the resonator and the electrode below, an electrostatic force is generated in the negative y-axis pulling the resonator. Due to the relatively large length of the electrode in the x-axis of 160 ?m compared to the air gap of 4.5 ?m, the fringe ?eld can be neglected. Assuming small displacements in the y-axis, the tensile force for the resonator T in terms of voltage difference V between the resonator and the electrode, cross-sectional area of electrode Ae , air gap de and dielectric constant of vacuum ε0 is given by [25]: T= ε0 Ae
2 2de

V2

(3)

2. Theory 2.1. Force sensing mechanism To understand the behaviour of the 3DoF resonator force sensor, the system is modelled as a lumped parameter block diagram as shown in Fig. 1a. A schematic drawing of the left resonator to which the force to be measured is applied, is shown in Fig. 1, and a SEM image of our proof-of-concept chip is shown in Fig. 1c. A tether structure [24] was used in our design to allow the transmission of an axial electrostatic force to the suspension beams of the left resonator. In addition, it is also used to impede the movement of the electrode attached to the bottom of the suspension beams when the resonator is vibrating, so that the electrostatic force is kept as constant as possible. Therefore, the tether was made wide in the x-axis (170 ?m), but thin in the y-axis (5 ?m) in our design. The design ensures that the tether has a high mechanical stiffness in the x-direction. In addition, when the displacement of the resonator in the direction of vibration is small compared to the length of the beam, the movement of the resonator in the y-axis is negligible. Consequently, the tether ef?ciently constraints the movement of the electrode attached to the suspension beams, and thus it can also be regarded as a ?xed end for the two suspension beams attached. In the y-axis, the tether, which is a cantilever beam in essence, has a stiffness of [25]: Etw3 t
3 4Lt

For an applied force in the y-axis, the two identical suspension beams (beams 3 and 4 in Fig. 1), are in parallel. Hence the tensile force T is evenly distributed to the two suspension beams. Furthermore, the suspension beams 1 and 3 are in series, so are suspension beams 2 and 4. Therefore, the tensile force applied on each suspension beam equals to T/2. The suspension beams have one end ?xed, while the other end moves perpendicular with respect to the beam length. Given the displacement functions along the axis of the beam for these boundary conditions [26], the stiffness of each suspension beam under weak axial tensile force T/2 is given by [27]: Kbeam = Etw3 0.6T + L L3 (4)

Moreover, due to the high longitudinal stiffness of the suspension beams, the elongation of the beams are trivial compared to the beam length L. For a tensile force of 1 ?N, the resulting elongation of the beams is less than 0.1 nm, which is negligible compared to the beam length of 300 ?m; the strain change is therefore neglected. The stiffness perturbation introduced by the tensile force, normalized to the effective stiffness of the resonator K, is therefore: Kforce 2.4T = LK K (5)

Ktether =

(1)

With the coupling voltage Vc applied, suppose d is the air gap between parallel plates and A, Acf are the cross-sectional area of the actuation parallel plate and the comb ?nger overlap, respectively. Neglecting the intrinsic tension introduced during fabrication process, the effective stiffness is given by [15]: K = 4 × Kbeam ? Kelec =
2 ε0 (A + 6Acf )Vc 4Etw3 ? 3 3 L d

where E, t, wt , Lt are the Young’s modulus, the thickness of the device, the width in the y-axis and effective length of tether,

(6)

C. Zhao et al. / Sensors and Actuators A 232 (2015) 151–162

153

Fig. 1. Figures showing: (a) block diagram of a 3DoF resonator sensing device [15]; (b) a detailed schematic diagram of the left resonator to which the force is applied; and (c) SEM image of the fabricated 3DoF resonator force sensor.

2.2. Amplitude ratio In the model of the 3DoF force sensor, as shown in Fig. 1a, each resonator consists of a mass, spring and damper, and is coupled to its neighbouring resonator through springs (Kc1 and Kc2 ). Suppose the mass of all resonators and their corresponding coupling spring stiffness are identical, i.e., M1 = M2 = M3 = M and Kc1 = Kc2 = Kc , while the spring stiffness of the resonators are asymmetrical with a quasi-static stiffness perturbation of K, thus K3 = K + K, K < 0 and K1 = K, and the stiffness of the resonator in / K. the middle is different to the other two resonators with K2 = The damping coef?cients are included due to its constraints on the value of K (as will be shown in Section 2.3), and it will be used in later sections for noise considerations. Further, assuming all springs are linear, and no movement in the y and z-axis, the motion in the x-direction of the resonators can be described by the equations of motion of the system (Eqs. (A.1)–(A.3) in Appendix A).

Now, consider the case where the coupled resonator system is only driven by F1 (s) = F1 and F2 (s) = F3 (s) = 0, the frequencies of the in-phase and out-of-phase modes can be calculated, and are given by [23]: ωip ≈ 1 1 K + Kc + ( K ? ? ? M 2 1 1 K + Kc + ( K ? ? + M 2 K 2 + ?2 ) (7)

ωop ≈ where ?=

K 2 + ?2 )

(8)

2 2Kc K2 ? K + Kc

(9)

Let s = jω, and substitute the out-of-phase mode frequency (Eq. (8)) into the equations of the transfer functions of |H11 (jω)| (Eq. (A.12)) and |H31 (jω)| (Eq. (A.17)), for the system driven only by F1 ,

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C. Zhao et al. / Sensors and Actuators A 232 (2015) 151–162 Table 1 Values used for the simulations to demonstrate mode aliasing. Component
3(

the amplitude ratios of the out-of-phase modes can be approximated as: X1 (jωop ) ≈ ? X3 (jωop ) where
3 2( 3

K/K ) + 4 ? 2

2

Values 0.489 MH 0.254 fF 84.8 aF ?19.07 fF 8.77 M (a) 0 fF (b) ?0.17 pF

Mechanical model equivalent M K K2 /K = 3 K/Kc = ?75, Q = 5000

K/K )

+j

3

Q

(10)

L C C2 Cc R C

3

=11,174

=

K (K2 ? K + Kc ) 2K = 2 ? Kc

(11)

K=0 K/K = ?1.5 × 10?3

With a tensile force applied on resonator 1, the stiffness of resonator 1 is increased by a positive Kforce . When the tensile force is weak, so that Kforce K, this is equivalent to break the symmetry of the resonator system with a negative stiffness perturbation ? Kforce acting on resonator 3. Therefore, we are able to gauge an external force applied along the beam length of the resonator, by measuring the amplitude ratio change resulting from a stiffness change caused by the force. 2.3. Mode aliasing effect Damping has the effect that it lowers the quality factor of the vibration modes, therefore limiting the bandwidth of the modes. If the in-phase and out-of-phase modes overlap, this effect is termed mode aliasing. Hence, for a given bandwidth f3dB of the modes, the two main modes having a frequency difference of f do not alias, the following antialiasing condition has to be satis?ed: f>2× f3dB (12)

preventing the observer from identifying the out-of-phase mode. On the contrary, when the anti-aliasing condition is satis?ed for K/K = ?1.5 × 10?3 , the out-of-phase mode can be distinguished, and the amplitude ratio can be measured. Hence, it is important that the anti-aliasing condition Eq. (12) is satis?ed for all input conditions. 2.4. Nonlinearity of amplitude ratio It can be seen from the amplitude ratio expression (Eq. (10)) that the amplitude ratio is a nonlinear function of the normalized stiffness perturbation K/K. Mathematically, it can be further deduced that for large | K/K|, the amplitude ratio approaches a linearized scale function of K/K: X1 (jωop ) ≈ X3 (jωop )
3

K

K

(14)

With the frequencies of the in-phase and out-of-phase modes given by Eqs. (7) and (8), the frequency difference can be approximated as: f = fop ? fip ≈ 1 2 K M K 2K
2

+

1
3

2

(13)

It can be seen from Eq. (10) that two nonlinearity errors contribute to the total nonlinearity error: (a) a nonlinearity from the ?rst term occurring even without damping and (b) a nonlinearity due to damping (term j 3 /Q). By calculating the nonlinearity errors separately and superimposing, we are able to estimate the total nonlinearity error, , as: =
1

It can be seen from Eq. (13) that the frequency difference is a function of the stiffness perturbation K. Hence, small stiffness perturbations K can result in a frequency difference f violating the condition of Eq. (12); and thus strong mode aliasing would occur. This is illustrated by a simulation using an equivalent RLC electrical circuit model of the 3DoF weakly coupled resonator system [28]. The values used in the simulation are chosen to be close to the device design, and are listed in Table 1. It can be seen from Fig. 2 that for K = 0 strong mode aliasing occurs, which stops the sensor from functioning properly by

+

2



K
3

2

K

+

1 2

1 K Q K

2

(15)

where 1 is the nonlinearity error from the ?rst term and 2 is the nonlinearity error from the second term. To verify this estimation, a simulation using the equivalent RLC electrical circuit model is run using the values listed in Table 2. The stiffness perturbations used in the simulations complied with the anti-aliasing condition. The results shown in Fig. 3 showed good agreement between the theoretical estimations and the simulated results of the nonlinearity error. It can also be seen from the ?gure that the nonlinearity

Fig. 2. Simulated frequency responses of resonators 1 and 3 using an equivalent electrical RLC network model [28] with different stiffness perturbations: (a) 3 =11,174, K/K = 0 and (b) 3 =11,174, K/K = ?1.5 × 10?3 . The theoretically calculated frequency difference and the 3 dB bandwidth of the modes are also shown in the ?gure. Strong mode aliasing can be seen in (a), as the in-phase and the out-of-phase modes merged; this is due to the frequency difference violating the anti-aliasing condition (Eq. (12)). The mode aliasing effect reduces to a negligible level when the anti-aliasing condition is satis?ed, and the two modes can be distinguished in (b).

C. Zhao et al. / Sensors and Actuators A 232 (2015) 151–162 Table 2 Values used for the simulations to verify the nonlinearity error estimation Component L C C2 R Cc Values 0.489 MH 0.254 fF 84.8 aF 8.77 M (a) ?12.72 fF (b) ?19.07 fF Mechanical model equivalent M K K2 /K = 3 Q = 5000 K/Kc = ?50, K/Kc = ?75,
3 3

155

= 4950 =11,174

error diminished as the value of the negative K/K decreased. Hence, in order to improve the linearity of the amplitude ratio, it is desired to have a high stiffness perturbation.

2.5. Bias point To ensure that any tensile force applied will not result in severe mode aliasing, a negative bias stiffness perturbation Kbias < Kmax < 0 can be applied to resonator 3; this is depicted in Fig. 4. With this bias perturbation, for Kforce > 0 resulting from a tensile force, the total stiffness perturbation K satis?es K = Kbias ? Kforce < Kmax . In addition, the negative bias stiffness perturbation also makes K = Kbias ? Kforce < ? Kforce , hence, the nonlinearity of the amplitude ratio is also decreased; this is also shown in Fig. 4. To introduce the negative stiffness perturbation bias, we applied a DC voltage on the electrode on the right, hence lowering the effective stiffness of resonator 3. Once an appropriate bias stiffness perturbation (which will be discussed in Section 4.2) is introduced, the mode aliasing effect and nonlinearity can be made negligible. Therefore Eq. (10) can be linearized as Eq. (14). Combining Eqs. (5) and (14) and neglecting the nonlinearity, the change in amplitude ratio is approximately linear with the tensile force, T: X1 (jωop ) ≈ X3 (jωop )
3

Fig. 4. Demonstration of the bias point concept. The amplitude ratio as a function of the stiffness perturbation is based on an analytical model described in the previous sections. The black curve shows the actual curve of amplitude ratio, the red dotted line shows the linearized scale function and the grey area illustrates the region with severe mode aliasing. It demonstrates that with a bias point outside the region with strong mode aliasing, the working region for tensile forces will not suffer from the same effect. It also shows that improved linearity can be achieved through deploying a bias point. (For interpretation of the references to colour in this ?gure legend, the reader is referred to the web version of this article.)

Therefore, the sensitivity of the force sensor can be approximated as: S3DoF =

?(Amplitude ratio) 2.4 3 ≈ KL ?T

(17)

With the bias, the lower limit of the dynamic range of the quasistatic force is only limited by the noise ?oor of the sensor and the interface electronics. Hence, we shall discuss the noise in the following section. 2.6. Noise Assuming the noise of the sensing device is Gaussian and the noise of resonator 1 and 3 are not correlated, the output noise power

2.4 3 T Kforce = K KL

(16)

Fig. 3. Simulated amplitude ratios compared to the linearized scale function given by Eq. (14). The total nonlinearity errors were also calculated and are plotted for (a) 3 = 4950 and (b) 3 =11,174. The total theoretical nonlinearity error was estimated using Eq. (15). The nonlinearity errors determined by simulation match the theoretical predictions well.

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C. Zhao et al. / Sensors and Actuators A 232 (2015) 151–162

Suppose C1 = C2 = C3 = C, the noise power of the displacement of resonator 1 is:
2 = X1

2kB TC

ωop + ω ωop ? ω 2 2 2 (H11 + H12 + H13 )dω

(23)

Fig. 5. Block diagram of a 3DoF resonator sensing device.

of the 3DoF sensor, equivalent to the variance of the amplitude ratio |X1 /X3 |, can be derived according to [29]: X1 X3
2

Assume a quality factor of Q = 2 ×104 in vacuum, which is a conservative estimation compared to other similar devices [12,33] and setting |K/Kc | = 200, K2 /K = 2, 3 can be calculated to be 40,000 using Eq. (11). The sensor, without any force applied, has a bias of | Kbias /K| = 5/ 3 . As shown in Appendix B, it can be demonstrated that near the out-of-phase mode, |H12 |2 and |H13 |2 are both negligible compared to |H11 |2 . Therefore Eq. (23) can be simpli?ed as:
2 ≈ X1

=
noise

2

X1 X3
2

2kB TC

ωop + ω ωop ? ω 2 H11 (jω)dω

(24)

= =

X1 X3 X1 X3

(X1 ) X1
2 Xn, 1 2 X1

2

+
2 Xn, 3 2 X3

(X3 ) X3

2

(18)

2

+

Based on the integrals derived in [34], we are able to estimate the spectral density of the thermal–mechanical noise displacement of resonator 1, X1 (jωop ) 2 , in close vicinity of the out-of-phase mode, assuming ω ωop , as: X1 (jωop )
2



where 2 (f) is the variance of function f, which by de?nition equals 2 (i = 1, 3) is the noise power of the ith resto the noise power; Xn,i onator. Hence the signal-to-noise ratio (SNR) is: X1 SNR = X3
2

8kB TCQ 2 K2

(25)

Furthermore, for ω = ωop /(2Q), the mechanical SNR of resonator 1 within the 3 dB bandwidth of the out-of-phase mode can be evaluated as: SNRm,1 ≈
2 (jω )K X1 op eff

X1 / X3

2

=
noise

2 Xn, 1 2 X1

+

2 Xn, 3 2 X3

?1

(19)

kB T

(26)

SNR1 × SNR3 = SNR1 + SNR3 It can be seen from Eq. (19) that the output SNR increases as the SNR of resonator 1 and/or 3 improves. The noise power of resonator 1 and 3 is dominated by two parts, mechanical–thermal noise of the resonators and the electricalthermal noise of the interface electronics [30]. Therefore, the SNR of resonator 1 and 3 can be written as: SNRi = SNRm,i × SNRe,i , SNRm,i + SNRe,i i = 1 or 3 (20)

In a similar manner, the noise displacement power spectral density at the out-of-phase mode X3 (jωop ) 2 and the mechanical SNR of resonator 3 can be approximated by: X3 (jωop )
2



2 (jω ) 8kB TCQ 2 X1 op 2 (jω ) K2 X3 op

(27)

SNRm,3

≈ ≈

2 (jω )K 2 X3 op eff X1 (jωop )

2 kB T
2 (jω )K X1 op eff

2 (jω ) X3 op

(28)

kB T

where SNRm,i and SNRe,i are the mechanical and electrical SNR of the ith resonator, respectively. 2.6.1. Mechanical SNR To theoretically calculate the mechanical noise, a transfer function model [15] of the 3DoF resonator sensor was used. The block diagram of the model is shown in Fig. 5. Considering three mechanical–thermal noise inputs Fn,r (r = 1–3) (see Fig. 5), the noise power in terms of displacement near the outof-phase mode of the rth resonator Xmn,i (i = 1 to 3) can be evaluated as [31]:
2 Xmn,i

2.6.2. Electrical SNR For a standard transimpedance ampli?er, the input-referred current noise power spectral density can be expressed as [31]:
2 2 = ina + in

Rm + Rf Rm Rf

2 2 ena +

4kB T Rf

2

(29)

where ina , ena , Rm and Rf are the current noise, voltage noise spectral density of the op-amp, equivalent motional resistance of the resonator and feedback resistance, respectively. Given the sensing transduction factor ?s [25] of the device and the 3 dB bandwidth of the out-of-phase mode f3dB , the electrical SNR of resonator 1 and 3 within the 3 dB bandwidth can therefore be calculated as: SNRe,i = Xi2 ?2 s , 2 in,i f3dB i = 1 or 3 (30)

1 = 2

ωop + ω ωop ? ω

3 2 2 Fn,r Hir dω r =1

(21)

where Hir is the transfer function from rth input to ith output, which is derived in Appendix A, and the power spectral density of the thermal driving force is given by [32,30]:
2 = 4kB TC r , Fn,r

r = 1, 2, 3

(22)

where kB , T and Cr are the Boltzmann constant, ambient temperature and damping coef?cient of rth resonator, respectively.

As will be shown in the experimental results, for a biased 3DoF resonator sensor, within the 3 dB bandwidth, for resonator 1 with larger vibration amplitude, the mechanical noise from the resonators is the dominant noise source, whereas outside of the bandwidth, the total noise was mainly attributed to the electronic noise. But for resonator 3 having a smaller vibration amplitude, the electrical noise dominated. The ultimate limit of the output noise power was imposed by electrical noise from the interface electronics.

C. Zhao et al. / Sensors and Actuators A 232 (2015) 151–162 Table 3 Design parameters of the device. Parameter Device layer thickness Suspension beam lengths (L) (resonators 1, 2 and 3) Suspension beam width (w) (resonators 1 and 3) Suspension beam width (w2 ) (resonator 2) Tether length (Lt ) Tether width (wt ) Air gaps (d = dc = db ) Air gaps (de ) Cross-sectional area (A = Ac = Ab ) Cross-sectional area (Ae ) Cross-sectional area (Acf ) Sensing transduction factor (?s ) (12 V coupling voltage) Actuation transduction factor (?t ) (12 V coupling voltage) Proof mass (M) Resonant frequency (single resonator) Design value 30 300 4 5 170 5 3.5 4.5 360 × 22 160 × 22 70 × 22 4.01 × 10?8 6.87 × 10?8 6.87 × 10?9 13.24 Unit ?m ?m ?m ?m ?m ?m ?m ?m (?m)2 (?m)2 (?m)2 A/(m rad/s) A/(m rad/s) kg kHz

157

dc are the cross-sectional area and the air gap between resonators, the coupling strength is given by [25]: Kc = ?
2 ε0 Ac Vc 3 db

(31)

Similarly, given that Ab and db are the cross-sectional area and the air gap between the electrode on the right and resonator 3, the bias stiffness Kbias is given by: Kbias = ? ε0 Ab (Vc ? Vb )2
3 db

(32)

It should be noted that the same con?guration was used for resonator 3. Therefore, when the electrode for resonator 3 was grounded as demonstrated in Fig. 6, and Ve < 0 applied for perturbation on resonator 1, using Eq. (3), the effective perturbation force can be calculated as: T = =
2] ε0 Ae [(Vc ? Ve )2 ? Vc 2 2de 2 ? 2V V ) ε0 Ae (Ve c e 2 2de

(33)

The resulting perturbation is therefore: 3. Experiment 3.1. Device description To demonstrate the concept of a 3DoF force sensor, a device was fabricated using a single mask silicon on insulator (SOI) process [35] with a structural layer of 30 ?m thickness. The fabrication process for the device is described in detail elsewhere [15]. The design of this chip is different from our previous work [15]. In this device, the beam width was 4 ?m (compared to 5 ?m, which was the stated minimum width for good yield). In addition, the length of the beams were 300 ?m, the resulting aspect ratio of the beam was 75, higher than the previous device of 70. This demonstrates the potential capability of the process to fabricate compliant beams. With these dimensions, the spring constant of the resonators was weaker for the device tested in this work, which is desirable for sensitivity improvement. Moreover, the air gap was reduced to 3.5 ?m to achieve higher actuation and sensing transduction factors. With smaller air gaps, the DC voltage required to achieve a certain coupling strength is also lowered. One downside of smaller air gap is the increased electrostatic nonlinearity, which contributes to a the nonlinear sensitivity, as discussed later. The design parameters of the fabricated device are listed in Table 3.
2 ? 2V V ) 2.4 T 1.2ε0 Ae (Ve Kforce c e = = 2 K LK KLd e

(34)

3.2. Measurement methodology To electrically test the chip, the chip was mounted on a chip carrier and wire bonded to the contacts. The chip carrier was then inserted into a socket on a printed circuit board. The circuit board was placed into a customized vacuum chamber with electrical feedthroughs. The ambient pressure was 20 ?Torr ensuring minimum air damping loss, so a high quality factor could be obtained. Three DC voltages were used in the experiment: (a) a ?xed coupling voltage of Vc = 12 V was applied to resonators 1 and 3, while resonator 2 was grounded, hence the resonators were electrostatically coupled; (b) a variable voltage Vb , the value of which will be discussed later, was used to bias the 3DoF sensor to an appropriate operating point; (c) a variable voltage Ve < 0 was used to apply a tensile force on resonator 1. With the voltages applied, suppose Ac and

Motional currents were used to measure the motion of resonators 1 and 3. With both resonators vibrating at the same frequency, the ratio of the motional currents equals to the amplitude ratio. Standard TIAs (AD8065, Analog Devices Inc.) with feedback resistance of 6.6M were used to convert and amplify the differential motional currents to differential voltage signals, which were further ampli?ed by subsequent instrumentation ampli?ers (INAs) (AD8421, Analog Devices Inc.) with a differential gain of 100. The sub-nano ampere motional currents from the resonators were ampli?ed to voltages at a measurable level of hundreds of millivolts, whereas the common mode signals such as the feedthrough signals, were suppressed to the sub-millivolt range. A two-channel oscilloscope (DSO6032A, Agilent Technologies) was used for measuring the voltage amplitudes of the resonators simultaneously. By manually altering the frequency of the drive signal, which was generated from a signal generator with variable frequency function, in 0.01 Hz steps, two distinct peaks in the amplitudes could be found, i.e., the in-phase and out-of-phase modes. The out-of-phase mode was used in our measurement for high sensitivity, which was identi?ed by the phase difference between the resonators. Then, the applied frequency was maintained at the out-of-phase mode frequency for the oscilloscope to measure the amplitudes in over 500 cycles. The oscilloscope computed the mean value of the amplitudes of both resonators, which were then used to calculate the amplitude ratios. Additionally, the mode frequencies were recorded as displayed by the signal generator. It is worth noting here that the third mode was neglected in the analysis due to the fact that in the experiment this mode could not be detected as the amplitudes of resonators 1 and 3 were below the noise level. 4. Results and discussion 4.1.
3

and offset values extraction

Before proceeding to demonstrate the force sensor, 3 and stiffness offset values were extracted due to their importance in analysing the experimental results [15]. The 3 dB bandwidth of the out-of-phase mode was found to be 0.48 Hz, the quality factor was 28,653 (as shown in Fig. 9). While

158

C. Zhao et al. / Sensors and Actuators A 232 (2015) 151–162

Fig. 6. Test con?guration of the prototype 3DoF resonator sensing device.

ensuring the mode aliasing effect was negligible and with Ve kept at 0 V, bias voltage Vb was altered to change K/K. The amplitude ratios were recorded for different K/K, and the amplitude ratio curve was ?tted to Eq. (10), as shown in Fig. 7. The extracted values of 3 =29,119 and the offset in normalized stiffness offset = 5.16 × 10?4 . Compared to the theoretically calculated value of 3 =39,557 from the designed dimensions, the relative error is approximately 26%, this is due to the variances introduced during the fabrication process. 4.2. Bias point selection A bias stiffness perturbation Kbias was intentionally introduced in the experiment, in order to avoid the mode aliasing effect. This was achieved by applying a ?xed bias voltage Vb , as shown in Fig. 6. To reduce the mode aliasing effect, the anti-aliasing condition, Eq. (12), should be satis?ed. A mode frequency measurement was carried out to ?nd the range of the bias voltage Vb that satis?es Eq. (12). The results are shown in Fig. 8. Since the 3 dB bandwidth of the in-phase and

out-of-phase modes were 0.48 Hz from the measurement, the minimum frequency difference that satis?es Eq. (12) was 0.96 Hz, which was marked with a blue line in Fig. 8. Therefore, a bias voltage of Vb ≤ 0.5 V satis?ed the anti-aliasing condition. Moreover, as mentioned in Section 2.4, a negative Kbias with larger magnitude, therefore, a lower Vb (refer to Eq. (32)), is desired for better linearity. However, as shown from Eq. (18), a larger | K| leads to a larger |X1 /X3 |, hence leading to larger noise in the amplitude ratio. Therefore, to balance the trade-off, Vb = 0.4 V was used for perturbation. The corresponding normalized stiffness perturbation Kbias /K and amplitude ratio were 1.91 × 10?4 and 5.75, respectively. 4.3. Force measurement To demonstrate the functionality of the proof-of-concept force sensing device, electrostatic forces along the beam length were

Fig. 7. Measured amplitude ratios (in red dots) were ?tted to Eq. (10) to extract 3 and offset value in normalized stiffness perturbation. The ?tted curve is shown in black. The extracted 3 =29,119 and offset = 5.16 × 10?4 . (For interpretation of the references to colour in this ?gure legend, the reader is referred to the web version of this article.)

Fig. 8. Measured (red) and theoretically calculated (black) frequency difference as a function of the bias voltage Vb . The theoretical frequency differences were calculated using Eqs. (7) and (8) with 3 , offset value extracted and the designed dimensions. 2f3dB = 0.96 Hz is marked with a blue line in the ?gure. Measured frequency differences match well with theoretical calculated values. It can also be seen that for bias voltages smaller than 0.5 V, the anti-aliasing condition was satis?ed. (For interpretation of the references to colour in this ?gure legend, the reader is referred to the web version of this article.)

C. Zhao et al. / Sensors and Actuators A 232 (2015) 151–162 Table 4 Sensitivity comparison with state-of-the-art resonant force sensors. Reference Type Sensitivity expression
?( f/f ) ?T

159

Sensitivity (/N)

[5]

1DoF resonator with differential sensing and leverage 2DoF resonant sensor 3DoF resonator sensor

8995

[16] Our work

?(Eigenstates shift) ?T ?(Amplitude ratio) ?T

1478 4.9 × 106

A comparison of sensitivity to other state-of-the-art resonant force sensors are listed in Table 4. It can be seen that signi?cant improvement in sensitivity of at least two to three orders of magnitude was achieved. 4.4. Force resolution and dynamic range Since any motion caused by mechanical noise (SNR given by Eqs. (26) and (28)) went through the same ampli?cation stages on the printed circuit board, the output mechanical SNR of the ith resonator is therefore: SNRm,i = Vi2 Keff kB T (2ωop ?s Rf GINA )2 , i = 1 or 3 (35)

Fig. 9. Measured frequency response of resonator 1 and 3 under two different perturbation conditions: (a) Vb = 0.4 V and Ve = 0 V, shown in solid lines; (b) Vb = 0.4 V and Ve = ?28.5 V, shown in dotted lines. The quality factor was calculated to be 28,653.

created by applying Ve to the electrode for resonator 1. With Ve < 0 applied, a tensile force was exerted on resonator 1, therefore decreasing K. Hence the frequency difference f increased and the mode aliasing effect could be neglected, as shown in Fig. 9. It can also be seen from Fig. 9 that negligible spring nonlinearity was present; therefore the assumption of linear springs can be regarded as valid. Varying Ve , we were able to measure the amplitude ratios. Using Eq. (33), the effective tensile forces applied were calculated. Hence, we can obtain the theoretical amplitude ratio using Eqs. (10) and (34). Fig. 10 shows the measured amplitude ratios and linearized scale function, given by Eq. (16), together with the nonlinearity error. It can be seen from Fig. 10 that the measured amplitude ratio matched well with the linearized scale function, with a nonlinearity error smaller than 10% for all the data points. The linear force sensitivity was found to be 4.9 × 106 /N. The theoretical force sensitivity is calculated to be 6.6 × 106 /N. The relative error compared to theoretical prediction is ?26%, which is attributed to fabrication tolerances.

where Vi is the rms-value of the output voltage of the ith resonator and GINA are the differential gain of the instrumentation ampli?ers. From Eqs. (29) and (30), the electrical SNR at the output can be computed as: Vi2 , SNRe,i = √ 2 ( 2in,i Rf GINA ) i = 1 or 3 (36)

The noise spectral density was measured using a two channel dynamic signal analyser (35670A by Agilent Technologies) without any driving signals applied, while Vc = 12 V, Vb = 0.4 V and Ve = 0 V were retained. Averaging of 50 measurement results were used to reduce the measurement variation, hence the peak caused by the mechanical noise could be found. The theoretical noise was calculated using Eqs. (35) and (36), together with the equations in Section 2.6. It can be seen from Fig. 11 that the measurement results and theoretical predictions agreed well. Therefore we were able to evaluate the noise power based on the theoretical noise. Assuming an ambient temperature of 290 K, using the designed value of the sensing transduction factor, ?s from Table 3, Rf = 6.6 M

Fig. 10. Measured amplitude ratios and the linearized scale function with respect to the applied tensile force. The measured amplitude ratios matched well with the linearized scale function, with nonlinearity error smaller than 10% for all the data points. The force sensitivity is found to be 4.9 × 106 /N.

Fig. 11. Output voltage noise spectral density of resonator 1 and 3 compared to the theoretically estimated noise density. The measured noise ?oor agreed well with theoretical calculations.

160 Table 5 Theoretical noise evaluation of the 3DoF sensor. Noise type Mechanical noise (resonator 1) Mechanical noise (resonator 3) Electrical noise (resonator 1) Electrical noise (resonator 3) Amplitude ratio noise Measured signal power 0.53 (V2 ) 1.60 × 10?2 (V2 ) 0.53 (V2 ) 1.60 × 10?2 (V2 ) 33.11

C. Zhao et al. / Sensors and Actuators A 232 (2015) 151–162

4.5. Nonlinearity
Evaluated SNR (dB) 84.80 84.80 87.18 71.98 71.72 Evaluated noise power 1.76 × 10?9 (V2 ) 5.30 × 10?11 (V2 ) 1.02 × 10?9 (V2 ) 1.02 × 10?9 (V2 ) 2.23 × 10?6

and GINA = 100 as designed, when Vb = 0.4 V and Ve = 0 V, resulting in an amplitude ratio |X1 /X3 | = 5.75, the SNRs can be calculated from the noise power within 3 dB bandwidth (f3dB = 0.48 Hz) using the measured output signal. The evaluated SNR and noise power are listed in Table 5. It can be seen from Table 5 that the electrical noise of resonator 3 (the resonator with smaller amplitude) ultimately sets the noise ?oor of the amplitude ratio. Due to the fact that the thermal–electrical noise can be regarded as uniformly distributed in a wide frequency span, as a consequence, the amplitude ratio noise can also be regarded as white noise. Therefore, from Table 5, we can evaluate the minimum resolvable force of the sensor near the bias point as: T
min

From the measurement results, it can be noticed that the nonlinearity error of the 3DoF device started off decreasing in value as the amplitude ratio increased, as shown in Fig. 12, which agreed with the theoretical prediction in Section 2.4. However, the linearity of the 3DoF sensor tended to deteriorate as the amplitude ratio increased, as shown in Figs. 10 and 12. It should be noticed that this nonlinearity was found to be insigni?cant for the device in [15]. One possible reason for this is that the air gap between the resonators was 3.5?m in this design, smaller compared to 4.5?m in [15]. For example, when amplitude of resonator 1 is signi?cantly higher than resonator 3 (larger than 30 times), the nonlinearity of Kc1 becomes larger than that of Kc2 , making the assumption of Kc1 = Kc2 invalid for larger amplitude ratios. 5. Summary and future work In this work, a proof-of-concept force sensing device consisting of three weakly coupled resonators with enhanced sensitivity is reported. Two orders of magnitude improvement in sensitivity compared to current state-of-the-art resonant force sensors was observed. A noise ?oor of the output signal, i.e. for 10 Hz bandwidth of the output signal, 1.39 nN could be demonstrated. Currently the measurement method requires mode frequency searching, which makes real-time measurement of a fast changing force impossible. Hence, only quasi-static stiffness and force perturbations were used as inputs. Future work will include the design of a self-oscillating loop that is capable of locking to a particular mode of interest. This would enable the measurement of the dynamic inputs. Furthermore, since the resonators were coupled by electrostatic forces, in a future study we can completely switch off the coupling voltages to decouple the resonators. This would enable a better comparison between the 3DoF and the 1DoF resonator sensor in the future. Appendix A. Transfer function derivation The motion of the three resonators can be described by three differential equations: M x¨1 + C1 x˙1 + (K + Kc )x1 ? Kc x2 = F1 M x¨2 + C2 x˙2 + (K + 2Kc )x2 ? Kc x1 ? Kc x3 = F2 M x¨3 + C3 x˙3 + (K + K + Kc )x3 ? Kc x2 = F3 (A.1) (A.2) (A.3)

= =

Amplitude ratio min Force sensitivity 2.23 × 10?6 /0.48 4.9 × 106 √ N/ Hz (37)

√ = 4.40 × 10?10 N/ Hz where Amplitude ratio min is the evaluated noise power spectral density of the amplitude ratio, hence, the frequency bandwidth in Eq. (37) is the bandwidth of the output voltage signals. To estimate the dynamic range of the 3DoF sensor, a bandwidth of 10 Hz of the output voltage signal was supposed; for this assumption, the minimum detectable DC force is 1.39 nN. For a maximum force of 7.6 ?N in the experiment, a dynamic range of approximately 74.8 dB can be achieved.

where xi , Fi denote the displacement of the proof mass with respect to a ?xed frame and external force on the mass of the ith resonator (i = 1, 2, 3), respectively. After performing a Laplace transformation and rearranging, we obtain: H1 (s)X1 (s) = Kc X2 (s) + F1 (s) H2 (s)X2 (s) = Kc X1 (s) + Kc X3 (s) + F2 (s) H3 (s)X3 (s) = Kc X2 (s) + F3 (s) where the transfer functions are de?ned as: H1 (s) ≡ Ms2 + C1 s + (K + Kc )
Fig. 12. Measured amplitude ratio and the linearized scale function (Eq. (14)) as a function of normalized stiffness perturbation. Nonlinearity error was also calculated and shown in the ?gure. Nonlinearity error decreased in value as the amplitude ratio increased.

(A.4) (A.5) (A.6)

(A.7) (A.8) K) (A.9)

H2 (s) ≡ Ms2 + C2 s + (K2 + 2Kc ) H3 (s) ≡ Ms2 + C3 s + (K + Kc +

C. Zhao et al. / Sensors and Actuators A 232 (2015) 151–162

161

Let s = jω, we are able to obtain the matrix form of the forced response in terms of angular frequency ω:

References
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?

? ? ? ? ? X2 (jω) ? = H ? F2 (jω) ?
X3 (jω) where F3 (jω)

X1 (jω)

?

?

F1 (jω)

?

(A.10)

? ?

H11 (jω)

H12 (jω) H22 (jω) H32 (jω)

H13 (jω)

? ?
(A.11)

H = ? H21 (jω) H31 (jω)

H23 (jω) ? H33 (jω)

By applying Mason’s rule [36] to the block diagram shown in Fig. 5, we are able to obtain the following: H11 (jω) = H22 (jω) = H33 (jω) =
2 H2 (jω)H3 (jω) ? Kc D(jω)

(A.12) (A.13)

H1 (jω)H3 (jω) D(jω)
2 H1 (jω)H2 (jω) ? Kc D(jω)

(A.14) (A.15) (A.16)

H12 (jω) = H21 (jω) = H23 (jω) = H32 (jω) = H13 (jω) = H31 (jω) = where

H3 (jω)Kc D(jω) H1 (jω)Kc D(jω)
2 Kc D(jω)

(A.17)

2 D(jω) = H1 (jω)H2 (jω)H3 (jω) ? [H1 (jω) + H3 (jω)] Kc

(A.18)

Appendix B. Transfer function ratio approximations of out-of-phase mode We make the following assumptions as in the text: a quality factor of Q = 2 ×104 , |K/Kc | = 200, K2 /K = 2, 3 =40,000, the sensor is perturbed by | K/K| = 5/ 3 . Near the out-of-phase mode frequency: ω≈ 1 1 K + Kc + ( K ? ? + M 2 K 2 + ?2 ) (B.1)

2 /H 2 | near the out-of-phase mode Since H13 = H31 , the ratio |H11 13 frequency can be approximated using Eq. (10): 2 2 H11 2 H13 2

≈ ?

2( 3

K/K ) + 4 ? 2

3(

K/K )

+j

3

Q

= 30.96

(B.2)

2 /H 2 | near the out-of-phase mode Now consider the ratio |H12 13 frequency: 2 H12 2 H13



H3 (jωop ) Kc

2

(B.3)

Let s = jω in H3 in Eq. (A.9) and substitute into Eq. (B.2):
2 H12 2 H13



K Kc

+

Kc K + Kc +j K2 ? K + Kc QK c

2

= 7.51 × 10?4

(B.4)

2 |/|H 2 | = 2.43 × 10?5 . So we are able to conclude Therefore, |H12 11 2 that |H12 | and |H13 |2 are both negligible compared to |H11 |2 .

162

C. Zhao et al. / Sensors and Actuators A 232 (2015) 151–162 2010, respectively. He is currently an Associate Professor at the Micro and Nano Electromechanical Systems Laboratory, NPU. His research interests include MEMS inertial sensors, micromachining process, etc. Honglong Chang received the B.S., M.S., and Ph.D. degrees in mechanical engineering from Northwestern Polytechnical University (NPU), Xi’an, China, in 1999, 2002, and 2005, respectively. From 2011 to 2012, he was a Visiting Associate (Faculty) with the Micromachining Laboratory, California Institute of Technology, Pasadena, CA, USA. He is currently a Professor with the Micro and Nano Electromechanical Systems Laboratory, NPU, where he is also the Department Head of the Department of Microsystem Engineering. He has authored over 27 international journal papers and over 20 international conference papers in microelectromechanical systems (MEMS). His research interests include MEMS inertial sensors and micro?uidics. Suan Hui Pu received the M.Eng. degree in mechanical engineering from Imperial College London, London, U.K. in 2006 and the Ph.D. degree in electrical and electronic engineering from Imperial College London, London, U.K. in 2010. He is currently an Assistant Professor at the University of Southampton Malaysia Campus and a visiting academic in the Nano Research Group, Electronics and Computer Science, University of Southampton, U.K. From 2010 to 2012, he was a Product Engineer with In?neon Technologies (Kulim) Sdn. Bhd., working on yield enhancement for a bipolar-CMOS-DMOS process technology. His current research interests include MEMS/NEMS sensors and actuators, graphene, nanocrystalline graphite and wearable technology. He has published more than 16 peer-reviewed papers in international journals and conferences. Dr. Pu is currently serving as a technical committee member for IEEE Electronics Packaging Technology Conference (EPTC). He is a reviewer for IEEE Journal of Microelectromechanical Systems and IEEE Transactions on Components, Packaging and Manufacturing Technology. Michael Kraft received the Dipl.-Ing. degree in electrical and electronics engineering from the Friedrich Alexander Universitat Erlangen Nurnberg, Erlangen, Germany, in 1993, and the Ph.D. degree from Coventry University, Coventry, U.K., in 1997. He is currently a Professor for Micro- and Nanosystems at the University of Liege, Belgium. Before joining the University of Liege, from October 2012 to December 2014, he worked at the Fraunhofer Institute for Microelectronic Circuits and Systems in Duisburg, Germany, heading the Department of Micro- and Nanosystems with a focus on fully integrated microsensors and biohybrid systems. Concurrently, he held the Professorial Chair (W3) of Integrated Micro- and Nanosystems at the University of Duisburg-Essen. From 1999 to 2012, he was an academic with the School of Electronics and Computer Science, University of Southampton, Southampton, U.K, where he acted as the Director with the Southampton Nanofabrication Center, Southampton. He was with the Berkeley Sensors and Actuator Center, University of California, Berkeley, CA, USA, working on integrated microelectromechanical systems (MEMS) gyroscopes. He has focused on novel microand nanofabrication techniques, microsensors, and actuators and their interface circuits, particularly for capacitive sensors. His current research interests include MEMS and nanotechnology ranging from process development to system integration of MEMS and nanodevices. He is the author or co-author over 160 peer-reviewed journal and conference papers. He has contributed to three textbooks on MEMS and an edited MEMS for Aerospace and Automotive Applications. Dr. Kraft serves on several steering and technical committees of international conferences, such as the IEEE Sensors, Eurosensors, and the Micromechanics and Microsystems Europe Workshop.

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Biographies
Chun Zhao received the B.Eng. degree in measurement & control technology and instrument from the Huazhong University of Science and Technology, Wuhan, China in 2009, and M.Sc. degree in analog and digital IC design from Imperial College London, London, U.K. in 2011. He is currently working towards the Ph.D. degree in microelectronics in the Nano Research Group, School of Electronics and Computer Science, University of Southampton, Southampton, U.K. His current research interests include micro-electromechanical systems (MEMS), miniature sensing devices, micro-resonators and interface circuit for sensors design. Graham S. Wood received the M.Eng. degree in electronics and electrical engineering and the M.Sc. by Research degree in microelectronics from the University of Edinburgh, Edinburgh, U.K., in 2008 and 2011, respectively. He is currently working towards the Ph.D. degree in microelectronics in the Nano Research Group, School of Electronics and Computer Science, University of Southampton, Southampton, U.K. From June 2008 until April 2010, he was a Research Associate with the Scottish Microelectronics Center, Institute for Integrated Micro and Nano Systems, School of Engineering, The University of Edinburgh, where he conducted research concerning the actuation and sensing of SiC microelectromechanical systems (MEMS) resonators for high-frequency RF applications. His current research interests include the use of mode localization in electrically coupled MEMS resonators for sensor applications. Jianbing Xie received his B.S., M.S. and Ph.D. degrees in mechanical engineering from Northwestern Polytechnical University (NPU), Xi’an, China in 2003, 2006 and


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