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

Choo, H.

Demmel, J.

Garmire, D.

Govindjee, S.

Kant, R.

Muller, R. S.

Séquin, C. H.

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Diamagnetically Levitated MEMS Accelerometers

Caltech Authors - Main

1203
1-4244-0842-3/07/$20.00©2007 IEEE
TRANSDUCERS & EUROSENSORS ’07
The 14th International Conference on Solid-State Sensors, Actuators and M
icrosystems, Lyon, France, June 10-14, 2007
DIAMAGNETICALLY LEVITATED MEMS ACCELEROMETERS 
D. Garmire1, H. Choo1, R. Kant2, S. Govindjee3, C. H. Séquin1, R. S. Muller1, J. Demmel1
1Berkeley Sensor & Actuator Center, Berkeley, CA, USA 
2Stanford University, Palo Alto, CA, USA 
3ETH 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 )(tx??  is: 
( ) )()()(1)( tztKztzD
M
tx ????? −+−= , (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
110
-8
10
-7
10
-6
10
-5
Frequency of Vibration
St
an
da
rd
 D
ev
ia
tio
n 
of
 E
rr
or
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 trend
1/Hz trend
2nd derivative error
Typical Suspension
Levitated Suspensions
2EM
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1-4244-0842-3/07/$20.00©2007 IEEE
TRANSDUCERS & EUROSENSORS ’07
The 14th International Conference on Solid-State Sensors, Actuators and M
icrosystems, 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): 
( ) Azhz
wh
BFgAh
graphite
graphite
loadtotal ???
?
???
?
+≈+
2
0
5
16235.0 µπ
χρ , (2) 
where htotal is the total plate thickness, ? is the 
plate density (2300 kg/m3), g is 9.8 m/s2, A is the 
plate area (8mm × 8mm), Fload is the vertical load 
applied to the plate, ? is graphite’s magnetic 
susceptibility (450×10-6), µ0 is 4?×10?7 N·A?2, B
is the magnetic flux density (1 Tesla for NdFeB), 
w × w is the magnet size (100 µm × 100 µm), and 
hgraphite 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)
L
ev
ita
tio
n 
H
ei
gh
t (
m
)
Modeled Data
Measured Data
4 6 8 10 12 14
x 10-4
0
2
4
6
8
x 10-4
Thickness (m)
Le
vi
ta
tio
n 
H
ei
gh
t (
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
N
S
Graphite Square
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1-4244-0842-3/07/$20.00©2007 IEEE
TRANSDUCERS & EUROSENSORS ’07
The 14th International Conference on Solid-State Sensors, Actuators and M
icrosystems, 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)
a
cc
el
e
ra
tio
n
 
(g)
levitated accelerometer
ADXL
0.2 0.4 0.6 0.8 1-1
-0.5
0
0.5
1x 10
-3
time (seconds)
a
cc
e
le
ra
tio
n
 
(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 10.05
0.052
0.054
0.056
0.058
time (seconds)
an
gl
e 
(ra
di
an
s)
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 Hz2) 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 Hz2 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) 
M
ea
su
re
d 
A
cc
el
er
at
io
n 
(g
) 
A
cc
el
er
at
io
n 
(g
) 
A
cc
el
er
at
io
n 
(g
) 
A
ng
le
 (r
ad
ia
ns
) 
Time (seconds) 
Time (sec nds) 
Ti e (seconds) 
Noise has 6.0µg standard deviation 
Time (seconds) (b) 
Applied impulse to the table 
Applied impulse to the table
(a) 
2EM
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1-4244-0842-3/07/$20.00©2007 IEEE
TRANSDUCERS & EUROSENSORS ’07
The 14th International Conference on Solid-State Sensors, Actuators and M
icrosystems, 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 B2/? haluminum/htotal = 
100 Hz, where c is aluminum’s conductivity 
(37.7×106 /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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Diamagnetically Levitated MEMS Accelerometers

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