Transducers'07 - Volume 1 Paper

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TRANSDUCERS & EUROSENSORS ’07

The 14th International Conference on Solid-State Sensors, Actuators and Microsystems, Lyon, France, June 10-14, 2007

DIAMAGNETICALLY LEVITATED MEMS ACCELEROMETERS

D. Garmire

1

, H. Choo

1

, R. Kant

2

, S. Govindjee

3

, C. H. Séquin

1

, R. S. Muller

1

, J. Demmel

1

Berkeley Sensor & Actuator Center, Berkeley, CA, USA

2

Stanford University, Palo Alto, CA, USA

3

ETH Zürich, Institut für Mechanische Systeme, Zürich, Switzerland

Abstract:

We introduce the theory and a proof-of-concept design for MEMS-based, diamagnetically-

levitated accelerometers. The theory includes an equation for determining the diamagnetic force above a

checkerboard configuration of magnets. We demonstrate both electronic probing and a rapid MEMS-

based interferometer technique for position sensing of the proof mass. Through a proof-of-concept

design, we show electrostatic-measurement sensitivity achieving 34 μg at a 0.1 V sense signal and

interferometer-measurement sensitivity achieving 6 μg for in-plane vibrations at 5 Hz. We conclude by

outlining batch-fabrication steps to produce levitated accelerometers.

Keywords:

accelerometer, diamagnetism, levitation, fast phase-shifting interferometer

1. INTRODUCTION

This research presents the theory and

verification for a MEMS-based, diamagnetically

levitated proof mass to improve the low-frequency

performance of accelerometers for applications

such as the study of earthquakes and structural

health monitoring. We have built a levitated proof-

mass device and obtained experimental results that

show electrostatic-measurement sensitivity of 34

μg at a 0.1 V sense signal and interferometer-

measurement

sensitivity

of

6

μg.

The

measurements are targeting 5 Hz vibrations. An

ADXL 203, a commercial MEMS accelerometer,

achieves 285 μg for the same vibrations. Previous

approaches to levitating a proof mass in a MEMS

process use electrostatic active control and suffer

from stiction [1]. In our accelerometer,

diamagnetism is employed to passively levitate a

proof mass containing pyrolytic graphite [2]. We

measure the position of the proof mass using off-

chip circuitry that differentially senses the change

in capacitance across a pair of comb drives. The

comb drives are fabricated in an SOI process [3].

2. EQUATIONS OF MOTION

We show that a lightly damped, non-stiff

suspension gives more precise results when

measuring low-frequency vibrations than does the

stiff suspensions that are typically used in MEMS

accelerometers. Diamagnetic levitation provides

the means to create these optimized suspensions.

The 1D mechanical theory for accelerator

response is formulated in Eq. 1. Calling

x

the

target displacement, the acceleration

)

(

t

x

is:

(

)

(

)

(

)

(

1

)

(

t

z

t

Kz

t

z

D

M

t

x

−

+

−

=

,

(1)

where

y

(

t

) is the proof-mass position,

z

(

t

) =

y

(

t

)-

x

(

t

) is the measured displacement, and

M

,

D

, and

K

are the mass, damping, and stiffness of the

system. We simulate the response of this system to

a 10 μg sinusoidal excitation over a frequency

range from 0.1 to 10 Hz (Fig. 1).

The assumed

noise of the position detector has a standard

deviation of 1 nm when sampling at 500 Hz.

10

-1

10

0

10

1

10

-8

10

-7

10

-6

10

-5

Frequency of Vibration

Standard Deviation of Error

Simulated

Accuracy

of

Suspensions

K/M=100000, D/M=100

K/M=100, D/M=100

K/M=100, D/M=10

K/M=0, D/M=0

Figure 1: The accuracy of accelerometers using

four different suspensions is measured by finding

the standard deviation of the suspension response

compared to a sinusoidal input at each frequency.

(Hz)

(g)

Isolated Monitoring

1/

¥

Hz tren

d

1/Hz tren

d

2

n

d

derivative error

Typical Suspension

Levitated Suspensions

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TRANSDUCERS & EUROSENSORS ’07

The 14th International Conference on Solid-State Sensors, Actuators and Microsystems, Lyon, France, June 10-14, 2007

3. DIAMAGNETIC FORCE

Using Fourier analysis and Taylor-series

expansions applied to the diamagnetic governing

equations [2] of a square plate of graphite above a

checkerboard configuration of magnets (Fig. 2),

we derive Eq. 2 which approximates the plate-

levitation height

z

when the proof-mass weight

balances the diamagnetic force (Fig. 3):

(

)

A

z

h

z

w

h

B

F

gA

h

graphite

load

total

̧

¹

·

̈

©

§

+

≈

+

2

0

5

16

235

.

0

μ

π

χ

ρ

, (2)

where

h

total

is the total plate thickness,

ȡ

is the

plate density (2300 kg/m

3

),

g

is 9.8 m/s

2

,

A

is the

plate area (8mm × 8mm),

F

load

is the vertical load

applied to the plate

,

Ȥ

is graphite’s magnetic

susceptibility (450×10

-6

), μ

0

is 4

ʌ

×10

í

7

N·A

í

2

,

B

is the magnetic flux density (1 Tesla for NdFeB),

w

×

w

is the magnet size (100 μm × 100 μm), and

h

graphite

is the graphite-layer thickness.

We verify Eq. 2 by placing weights onto a

levitated plate of pyrolytic graphite (7.7×10

-5

N)

while monitoring the vertical displacement

through a side-mounted microscope (Fig. 3a). We

reduce tilting of the proof mass by positioning the

weight while monitoring the angular change of a

reflected laser beam. We also compare adding

additional pieces of pyrolytic graphite (Fig. 3b).

Figure 2: Two equipotential surfaces of the

diamagnetic field above alternating north- and

south-pole magnets (potential increases closer to

the magnets).

0.7

0.8

0.9

1

1.1

1.2

1.3

x 10

-3

0

2

4

6

8

x 10

-4

Force (N)

Levitation Height (m)

Modeled Data

Measured Data

4

6

8

10

12

14

x 10

-4

0

2

4

6

8

x 10

-4

T

h

i

c

k

n

e

s

(

m

)

Levitation Height (m)

Modeled Data

Measured Data

Figure 3: Theoretical (Eq. 2) and experimental

measurements of the levitation height of the proof

mass (a) under varying load and (b) with varying

thicknesses of graphite.

4. PROOF-OF-CONCEPT RESULTS

To test the MEMS levitated accelerometer, we

mount a fabricated SOI proof mass to several

layers of pyrolytic graphite. We levitate it above

NdFeB magnets and align the electrostatic-sensing

combs to it using a 3-axis stage (see Fig. 4a). We

measure and compare the responses of our

levitated accelerometer to those of the ADXL 203

accelerometer when both are subjected to tapping

impulses applied to the support table (see Fig. 4).

We reflect a laser beam off of the proof mass and

record its position on a CCD image sensor to

measure out-of-plane tilting motions of the plate

(Fig. 6). By processing the results using off-chip

circuitry, we show that the noise level of the

levitated accelerometer is at least a factor of 8.4

smaller than the noise level of the ADXL 203

using only 0.1V for the differential sense (specific

results are at 5 Hz). Upon excitation of the air

table, a higher-frequency (40 Hz) tilting-mode

coupling is measured. We can reduce the

amplitude of this mode by packing more magnets

under the levitated proof mass.

(a)

(b)

N

S

Graphite Square

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TRANSDUCERS & EUROSENSORS ’07

The 14th International Conference on Solid-State Sensors, Actuators and Microsystems, Lyon, France, June 10-14, 2007

At rest, the standard deviations of the ADXL

and our levitated accelerometer

measure,

respectively, 285 μg and 34 μg when we use a 0.1

V sense signal and average signals over 0.1

seconds. A larger sense voltage increases the

sensitivity of the detector (0.34 μg possible at a

10V applied signal). Note that added aluminum,

produces the desired damping of in-plane motion.

Figure 4: (a) Schematic of experimental setup;

(b) microscope photo of levitated proof mass;

(c) the magnets under the levitated proof mass.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-0.01

0

0.01

time (seconds)

acceleration (g)

levitated accelerometer

ADXL

0.2

0.4

0.6

0.8

1

-1

-0.5

0

0.5

1

x 10

-3

time (seconds)

acceleration (g)

levitated accelerometer

ADXL

Figure 5: Comparison of an ADXL 203 and the

levitated accelerometer excited (a) and at rest (b).

0

0.2

0.4

0.6

0.8

1

0.05

0.052

0.054

0.056

0.058

time (seconds)

angle (radians)

Figure 6: Tiling of the levitated proof mass occurs

in the first 0.4 seconds and then stabilizes.

Next, we examine the motion of a diamagnetic

proof-mass (D/M = 73 Hz, K/M = 4 Hz

2

) with a

fast

phase-shifting

interferometer.

This

interferometer uses a vibrating MEMS plate to

achieve rapid phase-shifting and therefore a high

sampling rate (50 Hz) [5] (Fig. 7). We use a Fisher

Scientific U56001 vibration generator to supply a

1.6 N impulse to the 300 Kg Newport air table.

From the response measured by the levitated

accelerometer (Fig. 7), we are able to determine

the table’s suspension values, K/M

§

11.1 Hz

2

and

D/M

§

2.3 Hz. The noise in the system has a

standard deviation of 6.0μg, not the limit of the

detector but the limit of isolating the table from

surrounding vibrations.

0

2

4

6

8

-1

-0.5

0

0.5

1

x 10

-4

Figure 7: The fast phase-shifted interferometer (a)

measures the acceleration (b) of the 300 kg air

table after a 1.6 N impulse is applied to it.

(a)

(b)

(c)

(a)

(b)

Measured Acceleration (g)

Acceleration (g)

Angle (radians)

Time (seconds)

N

oise has 6.0

μg

standard deviation

Time (seconds)

(b)

A

pp

lied im

p

ulse to the table

A

pp

lied im

p

ulse to the table

(a)

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TRANSDUCERS & EUROSENSORS ’07

The 14th International Conference on Solid-State Sensors, Actuators and Microsystems, Lyon, France, June 10-14, 2007

5. IDEAL FABRICATION AND DESIGN

We show the feasibility of a MEMS process to

fabricate a levitated accelerometer with the

necessary

levitation

height

and

dynamic

properties. Evaluating Eq. 2, we calculate a

levitation height of 26 μm for a 2 μm-thick

pyrolytic graphite layer [4] on top of a 20 μm-

thick silicon plate with square magnets (100 μm

on a side) fabricated on the substrate. The magnet

arrays can be made separately and bonded to the

device to reduce complexity. Thus, the plate is

separated by a 6 μm-gap from the substrate and

magnets. Depositing a 0.122 μm layer of

aluminum on top of the plate results in eddy

current damping of D/M =

c

B

2

/

ȡ

h

aluminum

/

h

total

=

100 Hz, where

c

is aluminum’s conductivity

(37.7×10

6

/ohm m).

Fig. 8 shows a conceptual view of a processed

device. The magnets can be made separately and

placed in a backside trench or opening. Additional

magnets should be placed above to keep the proof

mass in place during the release of the proof mass

from its surrounding, and to lessen out-of-plane

tilting effects during operation.

Figure 8: The levitated proof mass construction.

Differential

comb-drives

are

used

in

electrostatic sensing for accurately determining

position. Differential vertical-offset combs can be

fabricated directly in SOI to electrostatically sense

out-of-plane motion [3] (Fig. 9).

Figure 9: Vertical sensing using offset combs.

The levitated accelerometer can be designed to

use either electrostatic sensing or interferometric

sensing. As demonstrated, using fast phase-

shifting interferometer techniques, we can

inexpensively and rapidly monitor lateral motions

at frequencies above 100 Hz with a precision

better than 2 nm per frame.

6. CONCLUSIONS

We have shown that diamagnetically levitated

accelerometers perform well at measuring low-

frequency vibrations below 5 Hz. Using

electrostatic measurements, we achieve 34 μg

sensitivity at a 0.1 V sense signal. Using

interferometer measurements, we achieve 6 μg

sensitivity. Furthermore, we have shown that such

accelerometers are possible in MEMS.

REFERENCES

[1]

B. Damrongsak and M. Kraft.

Electrostatic

suspension control for micromachined inertial

sensors employing a levitated-disk proof mass

.

Proc. MME 2005 Conference, Sept. 2005, pp.

240-243.

[2]

M. V. Berry et al.

Of flying frogs and

levitrons

. Eur. J. Phys. Vol. 18, 1997, pp. 307-313.

[3]

H. Choo et al.

A simple process to

fabricate self - aligned, high - performance

torsional microscanners: demonstrated use in a

two-dimensional scanner

. 2005 IEEE/LEOS

Optical MEMS, August 1-4, 2005, pp. 21-22.

[4]

Kostecki et al.

Surface studies of carbon

films from pyrolized photoresist

. Thin Solid Films,

396, (2001).

[5]

H. Choo et al.

Fast, MEMS-Based, Phase-

Shifting Interferometer

. Solid-State Sensor and

Actuator Workshop, June 4-8, 2006, pp.94-95,

Hilton Head, SC USA.

(attached on backside)

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