SIGNAL ENHANCEMENT IN MEM RESONANT SENSORS USINGPARAMETRIC SUPPRESSION

James M. L. Miller, Nicholas E. Bousse, Dongsuk D. Shin, Hyun-Keun Kwon,and Thomas W. Kenny

Department of Mechanical Engineering, Stanford University, Stanford, California, USA

ABSTRACTWe use parametric suppression for signal amplifi-

cation in a phase-modulated microelectromechanical(MEM) charge detector. Simultaneously driving andparametrically pumping a MEM resonator with the ap-propriate relative phase increases the phase slope nearthe resonant frequency. We use this to increase thesensitivity of a MEM electrometer more than tenfold.

KEYWORDSparametric amplification, electrometer, MEM res-

onator, perturbation theory, phase modulation

INTRODUCTIONParametric enhancement is widely used for signal

amplification of MEM resonant sensors, such as atomicforce microscopes (AFMs) [1, 2, 3, 4], gyroscopes [5, 6,7, 8], and magnetometers [9]. MEM resonators are of-ten parametrically pumped by modulating the springconstant of a mode at twice its natural vibration fre-quency. Parametric pumping is phase-dependent, incontrast with mechanical pumping and the other phase-independent pumping strategies [10, 11, 12]. A para-metric pump amplifies one quadrature of motion, whilesuppressing the other quadrature. This can be used toamplify signals in an amplitude-modulated (AM) con-figuration, provided that the signal frequency is closeto the resonant frequency and the signal is in-phasewith the pump [9]. The AM parametric enhancementstrategy will amplify signals with minimal added noiseand can improve the signal-to-noise ratio (SNR) of themeasurement until the output signal is dominated bythe thermomechanical noise, saturating at a thermo-mechanical SNR improvement of

√2 at the onset of

parametric oscillations [1, 9].We demonstrate an alternate strategy for using para-

metric pumping to amplify signals in resonant sensors:phase-modulated (PM) parametric suppression. Theresonator phase lag behind an external drive signal hasa frequency-dependence that can be measured to infershifts in the resonant frequency. This sensing strategyworks by applying a drive signal of constant amplitudeand frequency at resonance, and measuring the signalof interest via the phase shift that accompanies thecorresponding resonant frequency shift [13]. Paramet-ric enhancement increases the vibration amplitude butmakes the phase slope less steep than the case with-out pumping, as is shown in Fig. 1. This contrastswith phase-independent feedback, where the increasein vibration amplitude is accompanied by an increasein phase slope [14, 15]. Parametric suppression makesthe phase slope at resonance more steep than the case

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)/X

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Increasing pump

1

Figure 1: (a) Amplitude and (b) phase of a driven me-chanical mode subjected to parametric enhancement,corresponding to ψ = −π/4 in Eq. 1. The legendspecifies the normalized pump values, λ/λthresh, whereλthresh is the parametric resonance threshold. Thedashed line corresponds to the λ = 0 solution in Eqs.2 and 3. The amplitude curves are normalized to theresponse in Eq. 2 at resonance, X0.

without pumping, as is shown in Fig. 2, so a PM res-onant sensor will experience a larger shift in phase forthe same shift in resonant frequency. While parametricsuppression is clearly unsuitable for signal amplifica-tion in AM resonant sensors, it will amplify signals insensing topologies that detect signals via a shift in theresonant frequency. We illustrate this amplificationstrategy by using parametric suppression to improvethe signal of a PM resonant charge detector.

MODELThe dynamics of a single degree-of-freedom reso-

nant system subjected to a direct drive and a para-metric pump can be represented by:

ẍ+ γẋ+ ω20x+ λ cos(2ωt)x = f cos(ωt+ ψ), (1)

978-1-5386-8104-6/19/$31.00 ©2019 IEEE 881

T3P.008

Transducers 2019 - EUROSENSORS XXXIIIBerlin, GERMANY, 23-27 June 2019

where x is the displacement, ω0 =√k/m is the natural

frequency, γ = ω0/Q is the damping rate, λ is theparametric pump strength, f is the direct drive force,ψ is the relative phase between the direct drive andthe parametric pump, k is the spring constant, m isthe effective mass, and Q is the inverse measure ofdissipation known as the quality factor. The drivensystem without a pump (λ = 0) has a closed-formsolution for the amplitude (X) and phase (φ) of theresponse:

X(ω) =f√

(ω20 − ω2)2 + ω20γ2, (2)

φ(ω) = tan

(γω

ω2 − ω20

). (3)

To derive the response in the presence of a para-metric pump, we first decompose the system responseinto two slowly-varying quadratures, a(t) and b(t):

x(t) = a(t) cos(ωt) + b(t) sin(ωt), (4)

which must satisfy the following equation of constraint:

0 = ȧ(t) cos(ωt) + ḃ(t) sin(ωt). (5)

We use the method of averaging to obtain the equa-tions for a(t) and b(t), accurate to first order [16, 17]:

ȧ = −ωb2− γa

2+ω20b

2ω− λb

4ω+f sin(ψ)

2ω, (6)

ḃ =ωa

2− γb

2− ω

20a

2ω− λa

4ω+f cos(ψ)

2ω, (7)

The steady-state response can be obtained by settingȧ = 0 and ḃ = 0 in Eqs. 6 and 7, and numericallysolving for a(ω) and b(ω) for a given resonant fre-quency, damping rate, pump strength, drive strength,and phase between the pump and drive. The twoquadratures can finally be converted to an amplitudeand phase using X = a2 + b2 and tan(φ) = b/a.

Figures 1 and 2 plot the solutions to Eqs. 6 and7 for increasing values of λ and a relative phase thatyields maximum parametric enhancement (ψ = −π/4)and maximum parametric suppression (ψ = π/4), re-spectively. Beyond the onset of parametric resonance,λthresh, the quadrature of motion in-phase with thepump parametrically resonates, and additional nonlin-earities must be incorporated into Eq. 1 to bound theresponse [18, 19]. Fig. 2(b) shows that for parametricsuppression, the change in phase for a small change infrequency near the resonant frequency can be amplifiedwith increasing pump strength.

EXPERIMENT AND DISCUSSIONWe use parametric suppression for signal amplifi-

cation in the MEM torsional electrometer depicted inFig. 3. Micromechanical electrometers have reachedsingle electron sensitivities, exceeding the sensitivitiesof commercial charge detectors by several orders ofmagnitude [20, 21, 22]. Our electrometer is fabricated

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0

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X(ω

)/X

0

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ω/ω0 − 1 (ppm)

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Figure 2: (a) Amplitude and (b) phase of a drivenmechanical mode subjected to parametric suppression.

within a wafer-scale encapsulation process that pro-duces high-Q devices in a hermetic vacuum-sealed en-vironment [23]. The device is anchored at two points,and resonates via a torsional mode that modifies thecapacitance to the sense electrodes. When a bias volt-age is applied between the vibrating element and thesense electrodes, this induces a motional current thatcan be amplified to measure the displacement. A se-ries of internal electrodes are etched out of the vibrat-ing element to increase the sensitivity of the capacitivereadout. We use two sets of these internal electrodesto differentially transduce the motional current. Thisenables us to easily resolve the thermomechanical dis-placement noise in Fig. 4 using a bias voltage of only4 V, comparable to the available voltages in handheldconsumer electronic devices.

A mechanical resonator can be used to measurean external signal, such as a nearby distribution ofcharges, via a change in the resonant frequency of oneof its mechanical modes. In our device, as the voltagedifference between the top left electrode and suspendedelement increases, the resonant frequency decreases viaelectrostatic softening. This is demonstrated in Fig.4, showing that the peak thermomechanical noise fre-quency decreases with increasing voltage difference.

The device responses in Fig. 5 confirm the modelplotted in Figs. 1 and 2. We use the top right electrodeto sweep an external constant drive across resonancewhile simultaneously applying a parametric pump of

882

Vd + Vp

TIA +-

A VoutVb

TIAB

Vs

1

Figure 3: The schematic of the fabricated resonator.The biased device is forced and pumped with the topright electrode, and differentially sensed via two tran-simpedance amplifiers (TIAs) using the bottom left andright electrodes. A square wave voltage is applied to thetop left electrode to slowly vary the spring constant.

73.08 73.09 73.1 73.11 73.12

ω (2π kHz)

5

10

15

20

25

30

35

ASD

(µV

Hz−

1/2)

Vs-V

b=0 V

Vs-V

b=5 V

Vs-V

b=8 V

1

Figure 4: The thermomechanical noise amplitude spec-tral density (ASD) with varying electrode bias.

increasing amplitude at twice the drive frequency. InFig. 5(a-b), the pump is in-phase with the drivenquadrature of motion, so increasing the pump strengthincreases the amplitude but reduces the phase slope.In Fig. 5(c-d), the pump is orthogonal to the drivequadrature, which suppresses the amplitude at reso-nance while increasing the phase slope.

We employ parametric suppression to increase de-vice charge sensitivity via the slope detection tech-nique [13]. We measure ω0(t0) of the torsional modeby grounding the top left electrode, and then applyto the top right electrode a drive and pump at ω0(t0)and 2ω0(t0), respectively. Changes in the top left elec-trode voltage will now shift the resonant frequency toa new value, ω0(t), which in turn shifts the phase lagof the displacement behind the drive. In Fig. 6, we ap-ply a square wave voltage to the top left electrode tomodulate ω0(t) at well below the resonant frequency;this is the signal of interest. In Fig. 6(a), increas-ing the parametric enhancement causes the phase sig-

0 25 50 75 100 125

ω − ω0 (2π Hz)

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t(m

V)

(a)

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30 mV

34 mV

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t(m

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0 mV 20 mV30 mV

34 mV

0 25 50 75 100125

ω − ω0 (2π Hz)

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φ(◦)

(d)

1

Figure 5: (a, c) The amplitude-frequency curves forparametric enhancement and suppression, respectively,showing an increase (decrease) in the amplitude at res-onance with increasing pump strength. (b, d) Thephase-frequency curves, showing a decrease (increase)in phase-slope with increasing pump. The curves areeach shifted forward by 10 Hz to better illustrate thedynamics with pumping.

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1

Figure 6: Charge detection using a drive (at ω0) andpump (at 2ω0) applied to the top right electrode anda 50 mV amplitude, 300 mHz frequency square wavevoltage applied to the top left electrode. The pumpamplitudes (in order of increasing shading) are 0 mV,20 mV, and 30 mV. Parametric oscillations occur atapproximately 31 mV. Increasing parametric (a) en-hancement, and (b) suppression reduces (increases) thephase amplitude for a given square wave amplitude.

nal to decrease, even though the amplitude at reso-nance increases. In Fig. 6(b), increasing the para-metric suppression causes the phase signal to increase,even though the amplitude at resonance decreases. Atpump amplitudes near threshold, the sensitivity is in-creased tenfold while the sensor bandwidth decreasesbelow the charge modulation rate.

Parametric suppression can be leveraged to am-plify the response of PM sensors that detect signalsvia a shift in the resonant frequency, such as AFMs,electrometers, and mass sensors. However, parametricsuppression will simultaneously amplify the intrinsicfrequency noise [24]. The SNR will saturate once theerror in the phase detection is negligible compared tothe amplified intrinsic frequency noise.

AcknowledgmentsFabrication was performed in the nano@Stanford

labs, which are supported by the National Science Foun-dation (NSF) as part of the National Nanotechnol-ogy Coordinated Infrastructure under award ECCS-

883

1542152, with support from the Defense Advanced Re-search Projects Agency Precise Robust Inertial Guid-ance for Munitions (PRIGM) Program, managed byRon Polcawich and Robert Lutwak. J.M.L.M. is sup-ported by the National Defense Science and Engineer-ing Graduate (NDSEG) Fellowship and the E.K. Pot-ter Stanford Graduate Fellowship. J.M.L.M. is grate-ful to Guillermo Villanueva, Steven Shaw, and DanielRugar for helpful discussions.

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Email: jmlm@stanford.edu

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