Parametrically Ampliﬁed Z-Axis Lorentz Force Magnetometer

702 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 20, NO. 3, JUNE 2011

Parametrically Amplified Z-AxisLorentz Force Magnetometer

Matthew J. Thompson, Member, IEEE, and David A. Horsley, Member, IEEE

Abstract—This paper describes a microelectromechanical sys-tems (MEMS) Lorentz force navigation magnetometer whose fab-rication is fully compatible with existing foundry processes usedto manufacture MEMS inertial sensors. Operating at 1 atm, themagnetometer is shown to have a Brownian-noise-limited fieldresolution of 87 nT/

√Hz and a corresponding angular resolution

of 0.7◦/√

Hz. To achieve Brownian-noise-limited performance at1 atm, parametric amplification was performed which amplifiesthe Lorentz force 40% more than the Brownian noise force andincreases the field sensitivity of the device up to 82.5 folds. Here,the parametric signal is also used to perform electromechanicalamplitude modulation to mitigate capacitive feedthrough of thedrive signal to the sensing electronics. [2010-0229]

Index Terms—Force measurement, magnetometers, micro-electromechanical devices, navigation, parametric amplifiers.

I. INTRODUCTION

A. Magnetometers

THIS PAPER describes a microelectromechanical systems(MEMS) Lorentz force navigation magnetometer whose

fabrication is fully compatible with existing foundry processesused to manufacture MEMS inertial sensors [1]. The currenttrend in consumer electronics is to add more functionality andproduce multipurpose devices (e.g., cell phones that are alsopersonal digital assistants, cameras, MP3 players, GPS naviga-tion, and Internet terminals). Part of this trend, the inclusionof an electronic compass (magnetometer) to facilitate naviga-tion and location-based services, is expected to dramaticallyincrease magnetometer sales, with 8.7 million units sold in2008, expected to reach 540 million units in 2013 [2]. A typ-ical magnetometer used in consumer electronics applicationssuch as cell phones, the AKM Semiconductor AK8973, is athree-axis Hall effect sensor with a resolution of 976 nT [3].Magnetoresistive magnetometers are more sensitive than Halleffect sensors, and several companies including Honeywell arecompeting for the market. Honeywell’s three-axis HMC5843magnetoresistive sensor has a 700-nT resolution [4]. The earth’smagnetic field ranges between 10 and 100 μT depending on the

Manuscript received July 29, 2010; revised March 1, 2011; acceptedMarch 4, 2011. Date of publication May 11, 2011; date of current versionJune 2, 2011. This work was supported by National Science FoundationCAREER Award ARRA CMMI-084379. Subject Editor H. Zappe.

M. J. Thompson was with the University of California at Davis, Davis, CA95616 USA. He is now with InvenSense, Sunnyvale, CA 94089 USA (e-mail:[email protected]).

D. A. Horsley is with the University of California at Davis, Davis, CA 95616USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JMEMS.2011.2140355

TABLE ILORENTZ FORCE MAGNETOMETERS

location. Using 10 μT as a conservative estimate, the AK8973sensor is therefore able to resolve the orientation angle of theearth’s field to 7.9◦, and the HMC5843 can resolve 5.7◦.

Both silicon Hall effect sensors and MEMS Lorentz forcemagnetometers have advantages over magnetoresistive sensorsbecause they are free from magnetic hysteresis and require nospecialized magnetic materials. However, the field sensitivity ofsilicon Hall devices is poor and, in electronic compass applica-tions, is often enhanced using magnetic flux concentrators (asin the AK8973), eliminating these advantages. MEMS Lorentzforce sensors have the additional advantage of easy integrationwith other MEMS inertial sensors. Inertial measurement units(IMUs) require the integration of accelerometers with eithergyroscopes or magnetometers or both. Here, we study a MEMSmagnetometer fabricated using a very simple foundry process(SOIMUMPs, MEMSCAP Inc.) that is similar to that used forthe fabrication of MEMS inertial sensors so that a single chipIMU is conceivable.

A summary of earlier MEMS Lorentz force magnetometersis presented in Table I. Devices which measure the field com-ponent perpendicular to the chip surface (z-axis field) haveachieved resolutions of as low as 70 nT/

√Hz but both re-

quired a vacuum package and utilized a relatively complicatedfabrication process with electrically isolated metal traces onthe surface of the MEMS device [5]. Here, we examine theperformance of a magnetometer where the moving structure iscomposed of silicon, requiring no additional electrical isolation.To avoid the need for vacuum packaging, parametric amplifica-tion is applied to the device to enhance its performance whenoperating at 1 atm, resulting in over an order of magnitudeimprovement in resolution relative to earlier results obtained onan identical device operated as a linear resonator [6].

1057-7157/$26.00 © 2011 IEEE

THOMPSON AND HORSLEY: PARAMETRICALLY AMPLIFIED Z-AXIS LORENTZ FORCE MAGNETOMETER 703

Fig. 1. SEM image of the magnetometer with three folded flexures used asLorentz force actuators (Dr+, Dr−), linear comb drives used for electrostaticforcing (Fb+, F b−), and differential parallel-plate capacitors used for bothmotion sensing and parametric excitation (Cs+, Cs−).

B. Parametric Amplification

Parametric amplification increases the force-to-displacementtransduction of a resonant sensor by artificially increasing thesensor’s quality factor, typically by modulating the device’sspring constant at twice the natural frequency. Several groupshave experimentally shown parametric amplification on MEMSdevices such as cantilevers [7], infrared sensors [8], mixers [9],and mass sensors [10]. The original motivation for investigatingparametric amplification in micromechanical devices was toimprove the signal-to-noise ratio (SNR) in a device limitedby measurement noise [7]. A more recent theoretical analysisdemonstrated that parametric amplification carries no benefitfor sensors which are already limited by thermomechanicalnoise [11]. However, many practical MEMS sensors operatingin air are limited by electronic, rather than thermomechanical,noise. For these sensors, we demonstrate that parametric ampli-fication can increase the SNR to the thermomechanical-noise-limited value.

II. DESIGN

The magnetometer, shown in Fig. 1 and detailed in Table II,is a MEMS force sensor where an external magnetic fieldinteracts with current-carrying flexures to induce Lorentz forceon a movable MEMS shuttle. The magnetic field is detectedby measuring the shuttle motion using differential parallel-platecapacitors whose capacitance is proportional to the position ofthe shuttle. The device is fabricated in the SOIMUMPs foundryprocess [12]. Because an SOIMUMPs MEMS structure is con-structed from a single layer of heavily doped silicon, isolationof individual capacitive pick-off electrodes is impossible, andeach moving electrode has two opposing fixed electrodes thatare electrically connected, the first with a small gap g1 and thesecond having a larger gap g2, as shown in Fig. 2. As shownin the following, making g2 larger than g1 avoids significantlydegrading the displacement sensitivity.

A z-axis (out-of-plane) magnetic field interacts with ay-axis current in the MEMS flexures to create an x-axis force.

TABLE IIDESIGN PARAMETERS

Fig. 2. SEM image showing the parallel-plate capacitors used for motionsensing and parametric amplification. Each moving plate is bordered by twofixed pick-off plates that are electrically connected.

The Lorentz force applied to the MEMS shuttle when thecurrent-carrying beams of the flexures are in a magnetic field isshown in

FL(t) = �leff i�y × B(t) (1)

where leff is the effective length of the flexure, i is the current,and B is the z-axis magnetic field. An effective length (leff)is calculated for complicated flexure geometries. For a foldedflexure, the effective length is equivalent to the length of thelongest beam. Folded flexures are advantageous in comparisonto clamped–clamped beams because they minimize stress (andparasitic motion) caused from constrained expansion due toohmic heating [6] and reduce sensitivity to packaging stress.

To increase the field sensitivity, the flexure current is modu-lated at the mechanical natural frequency of the MEMS struc-ture ωn = 2πfn. The result is that the field-induced motion isamplified by the quality factor of the resonator Q. For devicesoperating near atmospheric pressure, Q is dominated by viscousair damping between the capacitive plates, and a tradeoff exists


between Q and the amount of capacitance used for motionsensing.

A. Design for Parametric Amplification

Parametric amplification allows a resonator’s Q to be arti-ficially increased to a desired level. In electrostatically drivenMEMS devices, parametric excitation can be achieved by us-ing a voltage-controlled electrostatic spring to modulate thedevice’s spring constant. In the magnetometer, this parametricinput is created using the same parallel-plate capacitors used formotion sensing. Both a dc bias Vb and a small-signal ac voltageat a frequency 2ω are applied across both capacitors Cs+and Cs−

V (t) = Vb − vac sin(2ωt). (2)

Because vac is used to generate the parametric excitation signal,it is referred to here as the parametric input or pump voltage.

The voltage V (t) results in a negative electrostatic springstiffness with components at both dc and 2ω

ke ≈ ke,0 + Δk sin(2ωt) (3)

where

Δk = −∂2C

∂x2

∣∣∣∣x=0

2Vbvac (4a)

ke,0 =∂2C

∂x2

∣∣∣∣x=0

(V 2

b + v2ac/2

)(4b)

∂2C

∂x2

∣∣∣∣x=0

= − 2ε0A

g31

(1 +

g31

g32

). (5)

Here, ε0 denotes the permittivity of air, A is the total area of theparallel-plate electrodes, g1 is the small gap, and g2 is the largegap of the neighboring pick-off electrodes.

When the flexure current is modulated i(t) = i cos(ωt + ϕ),the parametrically excited magnetometer is described by theforced Mathieu equation

mx +mωn

Qx + [keff + Δk sin(2ωt)] x = FL(t) cos(ωt + ϕ)

(6)

where FL(t) is the Lorentz force amplitude from (1), x and mare the displacement and mass of the shuttle, respectively, andkeff = k + ke,0 is the effective spring stiffness resulting fromthe mechanical flexures and the dc component of the electro-static spring (4b). Parametric amplification is phase sensitive[7], and the maximum amplification occurs when the harmonicforcing input is in phase with the pump input, requiring ϕ =π/2. The Mathieu equation can be solved using a perturbationmethod by separating the solution into two time scales, namely,stretched time (ωt) and slow time (η), as shown by [13]

x(t) = xc(η) cos(ωt) + xs(η) sin(ωt) (7)

where xc(η) and xs(η) are slowly time varying and can becalculated explicitly

dxc

dη= − xc

keff

Q

(1 − QΔk

2keff

)+ xs(keff − mω2) + FL(t)

dxs

dη= − xs

keff

Q

(1 +

QΔk

2keff

)− xc(keff − mω2). (8)

Assuming that the system is forced at its natural frequencyω = ωn, the steady-state solution to (8) is given by

x(t) = xc,ss(t) cos(ωnt),

xc,ss(t) =Q

keff

(1 − QΔk

2keff

)−1

FL(t). (9)

In a linear resonator, Δk = 0, and the force sensitivity isQ/keff . With parametric amplification, Δk is arbitrarily set bythe designer using (4a) to obtain a desired additional amplifica-tion Gp, termed “parametric gain” and shown in

Gp =(

1 − QΔk

2keff

)−1

. (10)

The effective quality factor is defined as the product of themechanical quality factor and the parametric gain, i.e., Qeff =GpQ. Using this definition, the sensor’s frequency response canbe approximated with a second-order linear system model

X(ω)FL(ω)

=1

keff

11 − (ω/ωn)2 + jQ−1

eff ω/ωn

. (11)

The sinusoidal flexure current at ωn modulates the Lorentzforce so that the force spectrum is centered around ωn. Hence,the displacement amplitude resulting from a dc magnetic fieldis found by evaluating (11) at ω = ωn

|X(ωn)| =Qeff

keff|FL(ωn)| =

Qeff

keffleff iB. (12)

For ac magnetic signals, the half-width of the resonance peakin (11) ωn/2Qeff determines the 3-dB bandwidth of the sensor.

B. Drive and Sense Electronics

The capacitive sensing and driving circuit used to evaluatethe device is shown in Fig. 3. To create the current for theLorentz force actuators, a voltage modulated at the naturalfrequency (fn) is applied to the top of the flexure (Dr+), andthe inverse is applied to the bottom of the flexure (Dr−). Thisdifferential voltage creates not only the required current butalso a zero-voltage node where the flexures join the shuttle.The parametric signal (2fn) is applied to the MEMS shuttle byadding a common-mode voltage signal to both the top (Dr+)and the bottom (Dr−) of the flexure.

The motion is detected using displacement-dependentdifferential parallel-plate capacitors (Cs+, Cs−). The fixedelectrodes of the sensing capacitors are connected to high-input-impedance JFET buffers, selected due to their low inputbias current (< 1 pA) and low input capacitance. A dc voltageVb is applied to the fixed electrodes through a bias resistor


Fig. 3. Electrical circuit used to drive the magnetometer and detect shuttlemotion.

whose resistance is large enough that, for motion at frequenciesclose to ωn, the charge on each sense capacitor is approximatelyconstant. The displacement-induced voltage on the fixed elec-trode of each sense capacitor is then given by

v(t) =1

CT

dC

dx

∣∣∣∣x=0

V (t)x(t) (13)

where V (t) is the voltage across the sensing capacitor from (2),C is the sense capacitance, and CT is the total capacitance toground at the gate of the JFET. The displacement sensitivity forthe differential capacitance is

dC

dx

∣∣∣∣x=0

=2ε0A

g21

(1 − g2

1

g22

). (14)

Equation (14) shows that the displacement sensitivity is reducedby the factor (g1/g2)2. In the design presented here, this factoramounts to approximately 12%. Further increasing g2 wouldslightly improve the displacement sensitivity but at the cost ofa larger MEMS structure.

Mixing occurs between the 2ωn component of the voltagesignal (2) and the displacement oscillations centered around ωn

(9) so that the output voltage (13) can be reexpressed as

v(t) =xc,ss

CT

dC

dx

(Vb cos(ωnt) +

vac

2sin(ωnt)

−vac

2sin(3ωnt)

). (15)

Based on (15), the displacement signal can be detected ateither ωn or 3ωn. Because the bias voltage Vb is larger thanvac, the signal at ωn is larger than the 3ωn signal. However,measurements at 3ωn have the advantage that they are free fromfeedthrough that arises from capacitive coupling of the currentdrive signal into the sensing pick-offs.

C. Sensor Noise

Noise in the sensor output arises from two sources, namely,Brownian and electronic noise. Brownian noise is a white noiseforce that arises from thermally induced molecular vibrationdissipated through the dominant damping mechanism in theMEMS structure. The Brownian noise force is given by

FB =√

4kbTKmωn

Q[N/

√Hz] (16)

where kb is Boltzmann’s constant and TK is the temperature indegrees Kelvin.

The major source of electronic noise is current noise inthe bias resistor. For measurements conducted near ωn, theelectronic noise can be converted into an equivalent noise forceusing (9) and (13)

FV =k

Qeff

CT

Vb

(dC

dx

)−1

vn [N/√

Hz] (17)

where vn is the input-referred voltage noise density of the JFETvoltage follower.

Assuming ideal noiseless electronics, the minimum field res-olution of the sensor is limited by Brownian noise. To achievethis limit with practical electronics requires that FB > FV .Similarly, the resolution is said to be limited by electronic noisewhen FV > FB . The SNR in a narrow bandwidth Δf aroundfn is given by the ratio of the signal power in the Lorentzforce from (1) to the total noise power found by combiningthe Brownian force from (16) and the electronic noise forcefrom (17)

SNR =FL√

Δf√

F 2B + F 2

V

. (18)

Note that the Brownian noise density (16) scales inverselywith

√Q (where Q is the true quality factor), and parametric

amplification has no impact on FB . However, since FV isinversely proportional with Qeff , parametric amplification isable to decrease the contribution of electronic noise to theoverall SNR, allowing Brownian-noise-limited performance tobe achieved at atmospheric pressure. In addition, as demon-strated in the following, the phase-sensitive nature of parametricamplification conveys an additional

√2 improvement to the

SNR of Brownian-noise-limited sensors due to the fact that onlythe in-phase component of the Brownian noise is parametricallyamplified.

If the sensor is Brownian noise limited, the resolution ofthe sensor is calculated by making the SNR equal to one in(18) and solving for B in FL from (1). Using the values fromTable II and a peak current of 2.67 mA, the Brownian-noise-limited resolution of this sensor is 87 nT/

√Hz, including the

factor of√

2 resulting from parametric amplification. Using10 μT as a conservative estimate of the earth’s field, the sensorhas an angular resolution of 0.7◦/

√Hz.

III. EXPERIMENTAL RESULTS

Unless otherwise stated, experiments were conducted withVb = 18.6 V and a 2.67-mA peak flexure current. AC mag-netic fields were applied by placing the sensor on top of anelectromagnet, while dc magnetic fields were generated witha permanent magnet.

A. Sensor Calibration

Initial experiments were performed to calibrate thedisplacement-to-voltage relationship of the capacitive sensingcircuitry as given in (13) by comparing the sensor outputvoltage to displacement data obtained from a laser Dopplervibrometer (LDV). The magnetometer frequency response wasmeasured in a B = 110 mT dc magnetic field by applying a


Fig. 4. Shuttle displacement measured with (Sensor Output) capacitive sens-ing and LDV. Measurements were collected at 133-μA flexure current in a110-mT dc magnetic field. The (Sensor FT) capacitive feedthrough was mea-sured with the same flexure current but with no magnetic field.

133-μA swept-sine flexure current from 7.5 to 9 kHz. Themeasurements, shown in Fig. 4, were fit with a second-ordertransfer function (11), yielding fn = 8.2 kHz and Q = 48.8.The experiment was also repeated with B = 0 to measure thecapacitive feedthrough present in the output signal. Based onthe modeled value for dC/dx = 20 fF/μm (14), the calibrationmeasurements showed a total capacitance of CT = 12.5 pFat the JFET inputs. This capacitance is dominated by thecapacitance of the large bond pads on the MEMS device.

B. Parametric Amplification of the Lorentz Force

Testing was conducted with varying levels of parametric am-plification using an electromagnet to apply sinusoidal magnetictest signals with frequencies from 1 to 10 Hz in 1-Hz incrementswith 40-μT amplitude. In the power spectrum of the capacitivesensing voltage, shown in Fig. 5, the ten test signals appearas sidebands around the drive frequency (fn = 8.2 kHz). Ca-pacitive feedthrough of the excitation signal at fn corrupts themeasurement at dc field but has no effect on the ac test signals.The full spectrum is shown for the case with no parametricexcitation, while only the amplitude of the ten test signals isplotted as the pump voltage is increased from 0 to 8.5 V. Tomeasure the increase in mechanical transduction, a second-order transfer function was fit to each frequency response withQeff used as a fitting parameter [14]. The parametric gain Gp isshown in Fig. 6 as a function of the applied parametric voltageand agrees well with the analytical model from (10). At thehighest pump voltage (vac = 8.5 V), the sensitivity is increased82.5 folds, conferring an effective Qeff of 4026. The bandwidthof the device decreases in proportion to the increase of Qeff ,with the original 3-dB bandwidth of 85 Hz shrinking to ∼1 Hzfor a parametric gain of 82.5.

C. SNR of Parametric Amplification

Experiments were conducted to demonstrate parametric am-plification of the Brownian noise force and the resulting SNR

Fig. 5. Sensor response at various pump voltages to 40-μT magnetic fieldsignals. The tone at 8.2 kHz is capacitive feedthrough. With vac = 0, thesensitivity is constant over the 30-Hz measurement span, whereas at vac =8.5 V, the 3-dB bandwidth is ∼1 Hz.

Fig. 6. Parametric gain versus the amplitude of the pump input at 2fn. Thetheoretical curve was generated using the model from (10).

of the sensor. The Lorentz force from (1) was set to FL =39 pN, the Brownian noise force density from (16) was FB =173 fN/

√Hz, and the electronic noise density was measured at

vn = 50 nV/√

Hz, making the equivalent electronic noise forcefrom (17) FV = 1.13 pN/

√Hz. Before parametric amplifica-

tion, the electronic noise force was larger than the Browniannoise force and dominated the SNR, which was equal to 35 in aΔf = 1 Hz bandwidth. With the ten magnetic test tones appliedto the sensor, the parametric voltage was set to give a parametricgain of 20. The sensor output spectrum was then measured withthe current drive turned off so that the parametric pump and theBrownian noise force were the only inputs to the sensor. Themeasured output spectra are shown in Fig. 7. The fit obtainedfrom the Lorentz force test tones shows an amplification that isa factor of

√2 greater than the fit obtained from the Brownian

noise force for the same parametric pump, resulting in an SNRof 320. This result can be explained by a phenomenon callednoise squeezing [7]. Noise squeezing occurs because paramet-ric amplification is phase sensitive, with the in-phase signal


Fig. 7. Measured spectrum and second-order frequency-response fit for exper-iments with the parametric excitation amplitude fixed at 8.3 V. Measurementswere performed (filled squares) with the drive signal on to measure the Lorentzforce response and (filled circles) with the drive signal off to measure theBrownian noise response.

being amplified and the in-quadrature signal being attenuated.The Lorentz current drive is phase locked with the parametricsignal to achieve maximum amplification (Gp). However, thepower in the Brownian noise is equally distributed among thein-phase and in-quadrature components, with each having anrms amplitude equal to 1/

√2 of the total noise level. Parametric

amplification only amplifies the in-phase component of thenoise, while the quadrature component is attenuated. If Gp issufficiently large, the total noise is dominated by the in-phasenoise, and the effective noise gain is Gp/

√2. We note that the

resulting√

2 improvement in the SNR is identical to that whichmay be obtained using other phase-sensitive detection methods.

D. Noise Squeezing

To demonstrate noise squeezing, a lock-in amplifier wasused to demodulate and record the in-phase and in-quadratureoutputs of the sensor both with and without parametric ampli-fication. For each data set, the 8.2-kHz flexure current signalwas used as the reference frequency for the lock-in amplifier,and 10 000 samples of the demodulated output sampled at100 Hz are shown in Fig. 8. When the device is not beingparametrically amplified, the in-phase and in-quadrature com-ponents of the noise are equal in amplitude, and electronicnoise dominates the response. When the device is parametri-cally excited, the output noise distribution becomes elliptical,demonstrating noise squeezing. The elliptical shape is causedby the phase-sensitive amplification of the Brownian noiseforce. The in-phase component of the Brownian noise forceis amplified, while the quadrature component is attenuated.However, the attenuation of the quadrature component is notdemonstrated here because the electronic noise dominates overthe attenuated quadrature component of the Brownian noise.

E. Sensing at 3fn

The detection of dc magnetic fields is corrupted by thedrive-to-sense capacitive feedthrough, which results in the large

Fig. 8. Time history of the amplitude and phase of the demodulated sensoroutput with no excitation and with an 8.4-V parametric excitation amplitude. Alock-in amplifier was used to demodulate the 8.2-kHz output signal, and 10 000samples were collected at a 100-Hz sample rate.

Fig. 9. Sensor output at fn (8.2 kHz) and at 3fn (24.6 kHz).

peak at 8.2 kHz in Figs. 5, 7, and 9. One method to bypassfeedthrough is to drive and sense at different frequencies,through a technique commonly known as electromechanicalamplitude modulation. As shown in (15), the parametric pumpsignal applied to the MEMS shuttle causes the motion signalat fn to be mixed with the parametric signal at 2fn, creatingtwo additional signals, one centered around a carrier at fn anda second centered around a carrier at 3fn. The signal at fn isin quadrature with the original sense signal at fn and has asmaller amplitude in proportion to 1/2(vac/Vb), as describedin (15). The combination of the two signals at fn adds 2.6%to the final amplitude and a 12◦ phase shift. The responseof the magnetometer to ten 40-μT magnetic tones is shownat fn (8.2 kHz) and at 3fn (24.6 kHz) in Fig. 9. The figureshows a tone at 3fn that is 50% smaller than the peak due tocapacitive feedthrough at fn. However, the amplitude of the 3fn

tone is still 40 times larger than the amplitude expected due tothe ambient 60-μT dc magnetic field that was present duringthe experiment. This experiment demonstrates the presence ofresidual motion not caused by the Lorentz force at the drive


Fig. 10. Transfer characteristic measured with and without parametric am-plification to demonstrate linear response. Each data point was generated bysubjecting the sensor to ten magnetic field test tones and fitting the outputspectrum with a second-order frequency-response model.

frequency. The source of this residual motion is discussed inthe following.

F. Sensitivity and Linearity

The magnetic field transfer characteristic of the device,measured in the range of 10–180 μT, is shown in Fig. 10.Measurements were collected without parametric amplification,with 65-fold parametric amplification sensing at fn, and with65-fold parametric amplification at 3fn. If the device is notparametrically amplified, the field sensitivity is 0.16 V/T. Thefield sensitivities of the device for a parametric gain of 65 are11.4 and 2.8 V/T when measured at fn and 3fn, respectively.

The linearity of the transfer characteristic demonstrates that,although parametric amplification is a nonlinear phenomenon,the response to the Lorentz force remains linear. The dominantnonlinearity in the device comes from the motion detectioncapacitors. This nonlinearity is exploited to achieve parametricamplification, but it also limits the sensor’s linear range. Sincethe onset of nonlinearity is determined by displacement, themaximum field input for linear response depends on the para-metric gain. The experimental displacement is plotted versusthe displacement predicted from a linear model in Fig. 11. Themeasured nonlinearity is less than 1.5% for displacements up to46 nm (approximately 0.5% of the nominal 10-μm capacitiveair gap), which corresponds to a field input of 68 μT at aparametric gain of 65.

G. Sources of Residual Motion

The residual motion observed at fn could be caused byseveral different mechanisms. The two most likely causes arethermal expansion of the current-carrying flexures or electro-static actuation due to fabrication mismatches in the flexureresistances and capacitive air gaps. Table III shows a list of theresistances of the Lorentz force flexures, and Fig. 12 shows theequivalent circuit model. Although the device is driven sym-

Fig. 11. Experimental displacement measured with a 1-Hz magnetic fieldinput versus the displacement predicted from a linear model. Deviation fromthe linear model is observed at displacements larger than 46 nm.

TABLE IIIRESISTANCE VALUES

metrically, small fabrication variations result in asymmetries inthe resistance of the various flexures, causing a small residualvoltage at fn on the MEMS shuttle. With a 2-V differentialdrive level, the residual voltage on the shuttle is on the order ofa few millivolts. This residual voltage creates a net electrostaticforce at fn if either (or both) (a) the residual voltage is unequalon the left and right halves of the shuttle which form Cs+ andCs− or (b) the shuttle is slightly offset so that the capacitivegaps of Cs+ and Cs− are not exactly equal. In case (a),the electrostatic force resulting from this small voltage is onthe order of (dC/dx)Vbvr, where vr denotes the differencein residual voltage at fn on the left and right halves of theshuttle. Although the residual voltage is small, the relativelylarge dc bias voltage (Vb = 18.6 V) used for capacitive sensingmeasurements results in electrostatic force that is nonnegligiblein comparison to the relatively tiny Brownian noise force.For the electrostatic force to be less than the Brownian force,the residual voltage on the shuttle must be less than 63 μV.Since three flexures were used in the present design, the taskof nulling the residual voltage is complicated, as it requiresindividual adjustment of the six voltages driving the flexurecurrents. When a single current-carrying flexure is used, thetask is relatively easy: Adjusting the current-driving voltagesto V1 = 0.985 sin(ωnt) and V2 = −1.15 sin(ωnt) maintains a


Fig. 12. Resistance model of the MEMS magnetometer. R1a−c represent thetop Lorentz force actuators, and R2a−c represent the bottom Lorentz forceactuators, while RSL and RSR are the resistances of the left and right halvesof the shuttle.

2-V differential amplitude while reducing the residual signalat v7 from 1.4 mV to 35 μV, below the threshold required toreduce the parasitic motion to the level of Brownian motion.

Another approach to reduce parasitic motion is to use activecancellation. Here, electrostatic force generated with linearcomb drive actuators is used to cancel the parasitic motion byapplying an equal and opposite force using feedforward control.To study cancellation of electrostatic motion, a 1-V ac voltageat fn (over 100× larger than the residual voltage present inthe magnetic field experiments described earlier) was appliedto the shuttle, and the resulting displacement was monitoredwith the LDV. As described previously, an initial offset of theshuttle position makes the force generated by Cs+ and Cs−unbalanced. From the experiments, the offset was determinedto be approximately 150 nm. Feedforward compensation wasperformed by first using a dc electrostatic force to null the resid-ual motion by centering the shuttle. This approach succeeded inreducing the displacement amplitude at fn from 42 to 2.26 nm.Further suppression was achieved by applying an ac voltage atfn to the electrostatic comb drive actuators and adjusting thephase and amplitude of this feedforward signal to minimize themotion of the shuttle. Fig. 13 shows the measured displace-ment at fn as a function of the feedforward signal amplitudeand phase. The displacement amplitude at fn was reduced to30 pm, a 63-dB improvement over the original 42-nm ampli-tude. This result suggests that this method could be successfullyused to suppress the parasitic motion observed at 3fn in Fig. 9to a level approximately 10× above the Brownian noise floor.

Various approaches for suppressing parasitic motion are pos-sible. Kyynáráinen et al. [5] used a more complicated fabri-cation process where an extra oxide layer electrically separatesthe Lorentz current and the detection capacitance, thereby elim-inating parasitic motion. The tradeoff is between a more costlyfabrication without parasitic motion or a cheaper fabricationand improved signal processing to suppress parasitic motion.

IV. CONCLUSION

Parametric amplification was shown to improve the perfor-mance of a Lorentz force magnetometer in three ways. First,the force-to-displacement sensitivity was increased by a factorof up to 85. Second, the force sensitivity of the MEMS structurewas increased until the sensor became Brownian noise limited,rather than electronic noise limited. Third, the Lorentz force

Fig. 13. Feedforward cancellation of residual motion due to electrostaticforce at fn = 8.2 kHz. The displacement amplitude at fn is plotted versusthe amplitude of a feedforward voltage applied to electrostatic comb driveactuators. The experiment was conducted by injecting a 1-V 8.2-kHz signalonto the MEMS shuttle to generate residual motion due to parasitic electrostaticforce, after which the dc voltage on the comb drives was adjusted to minimizethe displacement at fn by centering the shuttle. Finally, the feedforward signalat 8.2 kHz was adjusted to null the remaining displacement.

sensitivity was increased 40% more than the Brownian forcesensitivity due to noise squeezing. The parametric signal can beused to mitigate capacitive feedthrough by separating the driveand the sense frequencies. For a drive current of 2.67 mA and aminimum parametric gain of 10 folds, the sensor has a magneticfield resolution of 87 nT/

√Hz. Using 10 μT as a conservative

estimate of the earth’s field, the sensor has an angular resolutionof 0.7◦/

√Hz.

A parasitic motion at fn, corrupting dc magnetic field sens-ing, is present because the current-carrying flexures and themotion sensing capacitors are electrically coupled. A methodto use electrostatic force to null the parasitic motion wasdemonstrated. Further study is required to mitigate the parasiticmotion from this magnetometer before it can be used as acompass in a navigation system.

REFERENCES

[1] M. J. Thompson and D. A. Horsley, “Lorentz force MEMS magnetome-ter,” in Proc. Solid-State Sens. Actuators, Microsyst. Workshop, HiltonHead, SC, 2010, pp. 44–48.

[2] J. Bouchaud, Silicon Magnetic Sensor Market Offers Attractive Growth.El Segundo, CA: ISuppli, 2008, p. 1.

[3] AK8973 3-Axis Electronic Compass, Asahi Kasei Microdevices Semi-cond. Inc., Tokyo, Japan, 2008, pp. 1–31. [Online]. Available:http://www.asahi-kasei.co.jp/akm/en/product/ak8973nbs/ak8973nbs.html

[4] HMC5843 3-Axis Digital Compass IC, Honeywell Inc., Morristown, NJ,2009, pp. 1–19. [Online]. Available: www.honeywell.com

[5] J. Kyynäräinen, J. Saarilahti, H. Kattelus, A. Kärkkäinen, T. Meinander,A. Oja, P. Pekko, H. Seppa, M. Suhonen, and H. Kuisma, “A 3D microme-chanical compass,” Sens. Actuators A, Phys., vol. 142, no. 2, pp. 561–568,Apr. 2008.

[6] M. J. Thompson and D. A. Horsley, “Resonant MEMS magnetometerwith capacitive read-out,” in Proc. IEEE Sensors 2009, Christchurch,New Zealand, Oct. 2009, pp. 992–995.

[7] D. Rugar and P. Grutter, “Mechanical parametric amplification and ther-momechanical noise squeezing,” Phys. Rev. Lett., vol. 67, no. 6, pp. 699–702, Aug. 1991.


[8] T. Ono, H. Wakamatsu, and M. Esashi, “Parametrically amplified thermalresonant sensor with pseudo-cooling effect,” J. Micromech. Microeng.,vol. 15, no. 12, pp. 2282–2288, Dec. 2005.

[9] M. Koskenvuori and I. Tittonen, “Improvement of the conversion per-formance of a resonating multimode microelectromechanical mixer-filterthrough parametric amplification,” IEEE Electron Device Lett., vol. 28,no. 11, pp. 970–972, Nov. 2007.

[10] W. H. Zhang and K. L. Turner, “Application of parametric resonance am-plification in a single-crystal silicon micro-oscillator based mass sensor,”Sens. Actuators A, Phys., vol. 122, no. 1, pp. 23–30, Jul. 2005.

[11] A. N. Cleland, “Thermomechanical noise limits on parametric sens-ing with nanomechanical resonators,” New J. Phys., vol. 7, pp. 2–16,Nov. 2005.

[12] K. Miller, A. Cowen, G. Hames, B. Hardy, SOIMUMPs Design Rules,4th ed. Research Triangle Park, NC, MEMSCAP Inc., 2004,pp. 1–28. [Online]. Avialable: http://www.memscap.com/mumps/nc-smumps.refs.html

[13] W. H. Zhang, R. Baskaran, and K. L. Turner, “Effect of cubic nonlinearityon auto-parametrically amplified resonant MEMS mass sensor,” Sens.Actuators A, Phys., vol. 102, no. 1/2, pp. 139–150, Dec. 2002.

[14] M. J. Thompson and D. A. Horsley, “Parametrically amplified MEMSmagnetometer,” in Proc. 15th Int. Conf. Solid-State Sens., Actuators,Microsyst. (TRANSDUCERS), Denver, CO, 2009, pp. 1194–1198.

[15] H. Emmerich and M. Schofthaler, “Magnetic field measurements with anovel surface micromachined magnetic-field sensor,” IEEE Trans. Elec-tron Devices, vol. 47, no. 5, pp. 972–977, May 2000.

[16] A. L. Herrera-May, P. J. García-Ramírez, L. A. Aguilera-Cortés,J. Martínez-Castillo, A. Sauceda-Carvajal, L. Garca-González, andE. Figueras-Costa, “A resonant magnetic field microsensor with highquality factor at atmospheric pressure,” J. Micromech. Microeng., vol. 19,no. 1, pp. 1–11, Jan. 2009.

[17] Z. Kádár, A. Bossche, P. M. Sarro, and J. R. Mollinger, “Magnetic-fieldmeasurements using an integrated resonant magnetic-field sensor,” Sens.Actuators A, Phys., vol. 70, no. 3, pp. 225–232, Oct. 1998.

Matthew J. Thompson (M’09) received the B.E. de-gree (with honors) from The University of Auckland,Auckland, New Zealand, in 2001, the M.S. degreefrom the University of Nevada, Reno, in 2003, andthe Ph.D. degree from the University of California,Davis, in 2010.

Following receipt of the B.E. degree, he was aMachine Designer for Compac Sorting Equipment.After receiving the M.S. degree, he became anInstructor in the Department of Mechanical Engi-neering, University of Nevada, and was a Research

Associate with the Nevada Terawatt facility. In the summer of 2009, he waswith The University of Newcastle, Newcastle upon Tyne, U.K., studying thenonlinear dynamics and control of parametrically excited gyroscopes. He iscurrently with InvenSense, Sunnyvale, CA.

David A. Horsley (M’97) received the B.S., M.S.,and Ph.D. degrees in mechanical engineering fromthe University of California, Berkeley, in 1992, 1994,and 1998, respectively.

He held research and development positions atDiCon Fiberoptics, Hewlett-Packard Laboratories,and Onix Microsystems. He is currently an Asso-ciate Professor in the Department of Mechanicaland Aerospace Engineering, University of Califor-nia, Davis (UC Davis), and has been a Codirectorof the Berkeley Sensor and Actuator Center since

2005. His research interests include microfabricated sensors and actuatorswith applications in optical microelectromechanical systems, communication,displays, and biological sensors.

Prof. Horsley was a recipient of a National Science Foundation CAREERAward and the UC Davis College of Engineering’s Outstanding Junior FacultyAward.

Parametrically Ampliﬁed Z-Axis Lorentz Force Magnetometer

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