mems-based force sensor: design and applications
TRANSCRIPT
MEMS-Based Force Sensor: Design and ApplicationsDaniel López, Ricardo S. Decca, Ephraim Fischbach, and Dennis E. Krause
Micromachined force detectors are extremely sensitive instruments capableof measuring forces as small as 1015 N. We describe one such instrumentthat combines a novel micromachined torsional oscillator with aninterferometric position-sensing mechanism that allows fine control ofvertical scans. The design, fabrication, and operation of MEMS-based forcesensors are described. The sensitivity and unique features of micromachinedtorsional oscillators allow us to undertake experiments that set newconstraints on corrections to Newtonian gravitational forces at separationsbelow 1 mm. Moreover, the measurements provide the most precisecharacterization of quantum vacuum fluctuation forces to date.© 2005 Lucent Technologies Inc.
mechanical systems (MEMS) is used to describe
miniature devices with characteristic dimensions on
the order of 1 mm that are fabricated using integrated-
circuit-like techniques. Industries as different as
automaking, aeronautics, cellular communications,
chemistry, acoustics, and lightwave systems are
exploring the potential capabilities of these micron-
sized machines.
Similarly, high-sensitivity MEMS-based sensors
are being used and developed in all areas of science
[4, 11, 13]. Perhaps the most ubiquitous example
of their use is as micromachined tips of atomic
force microscopes (AFMs) [2, 24, 44]. These micro-
machined cantilevers are very popular and are being
used extensively as physical, biological, and chemical
transducers [35]. The attractiveness of this structure
resides in its simple design, its reliability and low-cost
fabrication process, and its flexibility, which allows it
to be used in a variety of conditions (e.g., in liquid or
gaseous environments, in low or high temperatures,
IntroductionThe development of high-sensitivity force sensors
has always been associated with progress in science
and technology. In science, the ability to sense small
forces is of fundamental importance for the discov-
ery and complete characterization of new phenom-
ena. In technology, sensitive force sensors are
continually used in mature and novel applications
that have a profound impact on economies and on
societies in general.
In the late 1960s, engineers and scientists began
to realize that silicon and other semiconductors could
be used to fabricate not only discrete and integrated
electronic circuits, but also sensors with increased per-
formance and sensitivity. More recently, thanks to the
development of widely available lithographic and
etching technologies, the miniaturization of force sen-
sors has become increasingly popular. In particular,
the use of silicon micromachines as miniaturized sen-
sors is just beginning to impact the most diverse areas
in science and technology [3]. The term microelectro-
Bell Labs Technical Journal 10(3), 61–80 (2005) © 2005 Lucent Technologies Inc. Published by Wiley Periodicals, Inc.Published online in Wiley InterScience (www.interscience.wiley.com). • DOI: 10.1002/bltj.20104
62 Bell Labs Technical Journal
from the axis of rotation (see Figure 1). When
torsional motion is induced in the suspended plate by
the action of an external force, the oscillator plate will
rotate an angle u that can be quantified by using one
or both of these electrodes.
The restoring torque associated with uniform and
small rotations of the plane around the axis defined
by the springs is given by
(1)
Assuming the structure has uniform thickness,
(2)
Here u is the angle of rotation in radians, G is the
modulus of rigidity, l is the length of the torsional
element (see Figure 1) and K is a geometrical factor
that takes into account the geometry of the torsional
elements.
If the torsional springs have a rectangular cross
section with then K is given by [40]
(3)
If the torsional springs have a rectangular cross section
with w t, then K is given by [40]
(4)
For the particular case of torsional springs with square
cross section (i.e., w t),
The spring constant of a straight rod under bend-
ing motion (i.e., a cantilever) is governed by [40]
(5)
where I is the moment of inertia, which in our case
is proportional to wt3. Notice that for devices with
the beam bending is in the “hard” direction.w 6 t
kbending 3EI
l3,
K Ksquare t4.
K t3w
3c1 0.63
t
wa1
t4
12w4b d .
K w3t
3c1 0.63
w
ta1
w4
12t4b d .
w t,
ktorsion aKG
lb.
T ktorsion u.
Panel 1. Abbreviations, Acronyms, and Terms
AFM—Atomic force microscopeFET—Field effect transistorLPCVD—Low-pressure chemical vapor
depositionMEMS—Microelectromechanical systemsRIE—Reactive ion etchSEM—Scanning electron microscope
in the presence of high magnetic fields, and in high-
radiation settings).
In this paper, we describe the fundamentals and
applications of alternative micromachined force sen-
sors that have advantages over cantilever transduc-
ers. The MEMS-based force sensors described in this
paper are micromachined torsional oscillators. These
sensors present a low coupling with the environment
while maintaining a large quality factor Q. The effect
of external vibrations is considerably reduced, when
compared to cantilever oscillators, due to the intrin-
sically lower coupling of a torsional oscillator to modes
that involve a displacement of the center of mass.
Moreover, these sensors can be designed to have
extremely high force sensitivity. Indeed, they are so
precise that their sensitivity is limited by fundamental
limits like the thermomechanical noise limit.
The sensitivity and unique features of microma-
chined torsional oscillators allow us to undertake ex-
periments that set new constraints on corrections to
Newtonian gravitational forces at separations below
1 mm. In order to investigate the nature and charac-
teristics of these forces, we need a force sensor that
allows measuring the interaction between two bodies
when the distance between them is reduced below
1 mm. In this paper, we describe a torsional MEMS
sensor that achieves a force resolution of 1015 N at
room temperature with a quality factor Q of several
thousands.
We show in this paper that MEMS-based force
sensors are ideal force detectors and that the force
sensitivity they provide is orders of magnitude better
than that previously achieved.
MEMS DesignIn this section, we discuss the design of a simple
micromachined torsional structure that can be used as
the sensor element in a MEMS-based force sensor.
A typical micromachined oscillator (see Figure 1)
consists of a micromachined plate suspended by tor-
sional elements above a substrate. These torsional
elements are springs in the form of either straight rods
of uniform cross section or serpentines. Serpentines
are advantageous because, for the same sensitivity,
they can be arranged in a more compact structure. In
addition, a couple of electrodes used for sensing and
actuation are located underneath the plate and away
Oscillatorplate
Oscillatorplate
Electrode
Torsion rod
l
L
w
The SEM image is from a micromachined oscillator with dimensions 500 m 500 m 3.5 m.
W
Figure 1.Torsional structure with rectangular plate and electrodes below.
Bell Labs Technical Journal 63
In order to optimize the mechanical behavior of
the oscillator, the torsional springs should have the
right compliance with respect to the rotation of the
plate about the appropriate axis and they must also
exhibit sufficient stiffness to limit the excitations of
other modes of motion. Both longitudinal and trans-
versal serpentine springs (see Figure 2) have sections
that are mainly in torsion and sections that are mainly
in bending. In the transversal serpentine spring bend-
ing dominates, while in the longitudinal configura-
tion most of the structure is in torsion.
The factor G is related to the physical properties of
the materials used in the torsional elements. G is
connected to the Young’s modulus E and Poisson’s
ratio n by
For silicon, E 130 GPa and n 0.22; consequently
MPa.
The torsional resonant frequency of the oscilla-
tor is given by
(6)
In equation 6, I is the moment of inertia of the oscil-
lator. For a thin rectangular plate [40]
I mW2
12
LW3tr
12,
frestorsion fo
1
2p Bktorsion
I.
G 50
G E
2(1 n).
Torsion
L
L
L
Torsion
Bending Bending
Figure 2.Basic spring types suitable for use in MEMS oscillators.
64 Bell Labs Technical Journal
where m is the oscillator’s mass, r is its density and L,
W and t are, respectively, the length, width, and thick-
ness of the oscillator plate.
MEMS FabricationThe most standard processes for MEMS fabrica-
tion are bulk and surface micromachining. A detailed
description of these processes can be found in [32]
and [45]. In order to make clear the basic structure of
our oscillators, we present a short description of the
process used to fabricate them. The MEMS oscillators
used in our experiments were fabricated using a
three-layer monolithic surface micromachining
process. A sketch of the microfabrication process flow
is shown in Figure 3.
The process begins with 100 mm n-type (100) sil-
icon wafers of 1–2 cm resistivity. The surface of the
wafers is first heavily doped with phosphorus in a
standard diffusion furnace using POCl as the dopant
source. This reduces or prevents charge feed through
to the substrate from electrostatic devices on the sur-
face. Next, a 0.6 mm low-stress low-pressure chemical
vapor deposition (LPCVD) silicon nitride layer is de-
posited on the wafers as an electrical isolation layer.
This is followed directly by the deposition of a 0.5 mm
LPCVD polysilicon film, Poly 0. Poly 0 is then pat-
terned by photolithography, a process that includes
coating the wafers with photoresist, exposing the pho-
toresist with the appropriate mask, and developing
the exposed photoresist to create the desired etch
mask for subsequent pattern transfer into the under-
lying layer. After the photoresist has been patterned,
the Poly 0 layer is etched in a reactive ion etch (RIE)
system. This polysilicon level is used to define the
electrodes, wiring, and ground contacts.
A 2 mm silicon oxide sacrificial layer is then de-
posited by LPCVD and annealed at 1050 °C for 1 hour
in argon. This layer of SiO2, known as first oxide, is re-
moved at the end of the process to free the first me-
chanical layer of polysilicon. The sacrificial layer is
lithographically patterned with the “dimples” mask
and the dimples are transferred into the sacrificial SiO2
layer by RIE. The nominal depth of the dimples is
750 nm.
The wafers are then patterned with the third
mask layer, ANCHOR1, and reactive ion etched. This
step provides anchor holes that will be filled by the
Poly 1 layer. After ANCHOR1 has been etched, the
first structural layer of polysilicon (Poly 1) is deposited
at a thickness of 2.0 mm.
Wafer/SiN/Poly0 Etching of Poly11. 6.
2. 7.
3. 8.
4. 9.
5. 10.
Etching of Poly0 to defineelectrodes and ground
Deposition and etchingof second oxide
Etching of oxide to definedimples and anchors
Deposition of metal
Deposition of Poly1 HF release
Deposition of first oxide Deposition and etchingof Poly2
Figure 3.Sketch of a typical three-layer surface micromachining process.
Bell Labs Technical Journal 65
Subsequently, this layer of polysilicon is litho-
graphically patterned and etched using RIE tech-
niques. Once again, another film of SiO2 is deposited,
patterned, and etched, and this is followed by the
deposition, patterning, and etching of the second
structural polysilicon film.
To reduce stress in the structural films, the wafer
is annealed at high temperatures. Then, a 0.5 mm
Figure 4.Schematic of the experimental setup showing its main components: micromachined oscillator and metallic sphere.
66 Bell Labs Technical Journal
thick metal layer (Cr/Au) is deposited to create bond
pads and electrical connections.
Finally, the devices are immersed in HF acid to
selectively dissolve the SiO2 (i.e., the sacrificial lay-
ers) without etching the polysilicon (i.e., the struc-
tural layer).
A CO2 critical-point chamber is used to release
the devices without stiction problems. In a critical-
point dryer chamber, the temperature and pressure
of the liquid containing the MEMS devices is in-
creased beyond the CO2 critical point and, as a con-
sequence, the liquid transforms into a gas in a
continuous way, avoiding the presence of the liquid-
gas interfaces characteristic of a phase transition [32].
Experimental SetupWhen using mechanical oscillators to measure
forces, one has to confront the coupling of the oscil-
lator with environmental vibrations. Compared with
cantilever oscillators, torsional oscillators are less
sensitive to mechanical vibrations that induce a
motion of the center of mass. Moreover, the minia-
turization of the oscillators yields an improvement in
their quality factor and sensitivity [41]. Consequently,
it is advantageous to use microelectromechanical tor-
sional oscillators to measure small forces. The exper-
imental arrangement used in our experiments is
shown schematically in Figure 4. This arrangement is
general enough to be of use in any situation in which
the forces acting between two different materials must
be measured as a function of the distance separating
them. The dimensions and spring of the MEMS os-
cillator is designed to take into account the magni-
tude of the forces to be measured. The remainder of
the assembly consists of a sphere that can be brought
in close proximity to the oscillator plate (see Figure 4).
In the most general case, the sphere is evaporated
with one material and the oscillator is coated with
another. As the sphere approaches the oscillator plate,
the acting forces between the different materials
Bell Labs Technical Journal 67
induces a torque in the plate, forcing it to rotate
around the torsional axis. The induced angular dis-
placement of the plate can be measured using the
electrodes located below it.
The sphere used in the experiments is glued with
conductive epoxy to the side of an Au-coated optical
fiber, establishing an electrical connection between
them. The entire setup (i.e., the MEMS oscillator and
the fiber-sphere) is rigidly mounted into a can, where
low pressure (e.g., 105 Torr) can be maintained. The
can has built-in magnetic damping vibration isolation
and is mounted onto an air table. This combination of
vibration isolation systems yielded base vibrations
with zrms 0.05 nm for frequencies above 100 Hz.
The fiber-sphere assembly can be moved verti-
cally by the combination of a micrometer-driven and
a piezo-driven stage. The micromachined oscillator is
mounted on a piezoelectric-driven xyz stage which,
in turn, is mounted on a micrometer-controlled xy
stage. This combination allows positioning the coated
sphere over specific positions on the coated plate. The
separation zi between the sphere and the Si-plate is
controlled by the z axis of the xyz stage. A two-color
fiber interferometer-based closed-loop system is used
to keep zi constant. The error in the interferometric
measurements is found to be zirms 0.25 nm; it is
dominated by the overall stability of the closed-loop
feedback system. Because this error is much greater
than the actual mechanical vibrations of the system,
the closed loop is turned off while data acquisition is
in progress.
A force F(z) acting between the sphere and the
plate produces a torque
(7)
where b is the lever arm between the sphere and the
torsional axis of the micromachined oscillator (see
Figure 4). In all the cases reported in this paper,
u 105 rad 1. Under these circumstances,
(8)
where Cright (Cleft) is the capacitance between the right
(left) electrode and the plate (see Figure 4).
Consequently, the force between the two metallic sur-
faces separated by a distance z is F(z) kC, where k
is the proportionality constant.
u r ¢C Cright Cleft,
V
T bF(z) ktorsionu,
To provide a calibration of the proportionality
constant k between F and C, it is very useful to apply
a known potential difference between the coated
sphere and the MEMS oscillator. Of course, this cali-
bration needs to be done at a sphere-plate distance
that is large enough to ensure that the dominant
interaction is electrostatic in nature. In this case, the
net force can be approximated by the electrostatic
force Fe between a sphere and an infinite plane [47],
(9)
Here eo is the permittivity of free space, Vsphere is the
voltage applied to the sphere, Vo is the residual po-
tential difference between the materials when they
are both grounded, and cosh u (1 zR), where
z zmetal 2do, R is the radius of the sphere, and do is
the average separation between the layers when the
test bodies come in contact (so that zmetal 0) and is
primarily determined by the roughness of the films.
By measuring, at constant z, the angular deflec-
tion u (and hence the electrostatic force) as a func-
tion of the applied voltage difference, the minimum
in Fe can be experimentally determined. Once
the value Vo is found, the equation (9) can be used to
determine three different parameters:
• The proportionality constant k between the
sphere-plate force and the measured difference
in capacitances,
• The radius R of the coated sphere, and
• The parameter do.
By performing a dynamic measurement, we di-
rectly use the high-quality factor of the micromachined
oscillator to increase the force sensitivity [11, 19]. In
this approach, the separation between the sphere and
the oscillator plate is varied as z A cos(wres t), where
wres is the resonant angular frequency of the oscillator
and A its amplitude. The solution for the oscillatory
motion yields [11, 19],
, (10)
where for for and Fx is the
force acting between the sphere and the oscillator
plate.
QW 1vo 2ktorsionI
v2res v2
o c1 b2
Iv20
0Fx
0zd
a
n1
coth(u) n coth(nu)
sinh(nu).Fe 2peo(Vsphere Vo)
2
+
–
+–
––
––10kHz 1K 1K
560k+15V 1k
Resistors values are in Ω, capacitors are in F, and inductors are in H.
MEMST
470n
470n
470u470100
18p2V
5u
5u
10M
10M
Cright
Cleft
V1
V2
1u
20k
–15V
To lock-inamplifier
14u
14u
__
++OP2710n
AD8292SK152
2SK152
1u 2k
10k100k
100k
10n
10G 10G10
1.5k 100u 100n
10n
10n20M–
–
++
Bridge Amplifier
Figure 5.Schematic of the electronic circuit used to measure the difference in capacitance between the two sides of theMEMS oscillator plate.
68 Bell Labs Technical Journal
Unlike the static regime where forces are mea-
sured, in the dynamic regime the force gradient
is measured by observing the change in the resonant
frequency as the sphere-plate separation changes.
The circuit used to measure the capacitance is
schematically shown in Figure 5 [1].
The field effect transistor (FET) of the amplifier
stage is placed as close as possible to the oscillator to
minimize the effect of parasitic capacitances. The
smallest angular deviation that can be detected in this
configuration is given by
(11)
where dC is the minimum detectable change in C,
Co ≈ (eoww )dg is the capacitance between the oscil-
lator and each electrode when no forces are present
(w and w are the dimensions of the electrodes), CT
20 pF Co is the parasitic capacitance of the mea-
surement circuit (determined by the capacitance of
the FET transistor plus parasitic capacitances to
ground), dV 10 nV/Hz12 is the input noise of the
amplifier, and Vi (i 1, 2) 1 V is the DC potential
used to correct for initial asymmetries in the circuit
and linearize its response. In obtaining equation (11),
we have assumed that the plates are infinite and we
W
du dC
Co
2dg
w dV
Vi
CT
Co
2dg
w ,
0Fx0zhave used the fact that the maximum angular devia-
tion umax 1.
The effect of du on the force dFelectronic is given by
(12)
where A(v) is the frequency response of the micro-
machined oscillator for a harmonic torque with con-
stant amplitude. When taking into account the white
thermal noise dFthermal, the total noise in the force is
.
We note that, by performing a measurement at reso-
nance, dFelectronic is reduced by a factor Q. Because the
Q of micromachined oscillators in vacuum can be
quite large (i.e., several thousands), the total noise in
the force can be approximated by
(13)
where kB is Boltzmann’s constant and T is the tem-
perature. It is evident from this equation that, work-
ing at resonance, an improvement of the sensitivity
on the order of Q12 can be achieved. Moreover, the
noise in the force can be reduced by designing a
MEMS sensor with optimum ktorsion and vo.
dF(v) dFthermal 1
bB4ktorsionkBT
voQ,
dF(v) 2(dFthermal)2 (dFelectronic(v))2
dFelectronic(v) ktorsiondu
bA(v),
V
Spring detail
Figure 6.Scanning electron micrograph showing the MEMS oscillators and a detail of the springs used.
Bell Labs Technical Journal 69
In the particular case of the experiments described
in the next section, we have used a micromachined
oscillator 3.5 mm thick with a 500 500 mm2 heavily
doped polysilicon plate suspended at two opposite
points by serpentine springs, as shown in Figure 6.
The springs are anchored to a silicon nitride (SiNx)-
covered Si platform. When no torques are applied,
the plate is separated from the platform by a gap
mm. Two independently contacted polysilicon
electrodes located under the plate are used to measure
the capacitance between the electrodes and the plate.
For the oscillator used in these experiments, we cal-
culated the torsion spring ktorsion 9.5 1010 Nm/rad.
In our case, w 2 mm (the width of the spring), t
2 mm (the thickness of the spring), Lspring 500 mm
(the length of the spring), and ESi 130 GPa (Young’s
modulus for Si). This value is in good agreement with
2
the measured value ktorsion 8.6 1010 Nm/rad.
We used Al2O3 spheres, with nominal diameters rang-
ing from 100 mm to 600 mm. The spheres were coated
with a 1 nm layer of Cr, followed by a 200 nm
thick layer of gold. The sphericity of the Al2O3 spheres,
as measured on a scanning electron microscope
(SEM), was found to be within the specifications of
the manufacturer. As an example, a 600 mm diameter
ball was found to have an ellipsoidal shape with
major and minor semi-axes of (298 2) mm and
(294 0.5) mm, respectively. Deposition-induced
asymmetries were found to be smaller than 10 nm,
the resolution of the SEM.
The separation zmetal between the two metallic sur-
faces (see Figure 4) is given by
(14)zmetal zi zo zgap bu,
70 Bell Labs Technical Journal
where b is the lever arm between the sphere and the
axis of the oscillator, zo is the distance the bottom of
the sphere protrudes from the end of the cleaved fiber,
and zgap includes the gap between the platform and
the plate, the thickness of the plate, and the thick-
ness of the metal layer. An initial characterization of
zo, obtained by alternately gently touching the
platform with the sphere and the bare fiber, yielded
zo (55.07 0.07) mm. Using an AFM, we deter-
mined that zgap (5.73 0.08) mm.
ApplicationsIn this section, we describe two experiments that
show the advantages of using MEMS-based force sen-
sors. In both cases, the measurements obtained with
the MEMS sensors were superior in sensitivity to pre-
vious measurements obtained with state-of-the-art
sensors; thus, MEMS sensors appear to be a promising
technology with which to detect unexplored proper-
ties of new forces.
Considerable attention has been devoted recently
to experimental searches for new forces over ultra-
short distances [18, 22, 26, 27, 28, 31, 37]. These
experimental searches have been prompted by theo-
retical propositions suggesting that deviations from
the inverse-square law, a characteristic of Newtonian
gravity, should be evident in measurements of
gravitational forces in the sub-mm range.
Gravity was the first of the fundamental forces
(i.e., gravity, electromagnetism, the strong force, and
the weak force) to be described mathematically, but it
is still the worst characterized. The main reason
for this is that gravity is the weakest of all the funda-
mental forces. Only recently has the validity of
Newtonian gravity for objects separated by distances
500 mm been experimentally confirmed [29].
These data are extremely important in constraining
any deviation of Newtonian forces in the particular
range of distances studied. A variety of experimental
studies have been trying to measure gravitational
forces at distances smaller than 100 mm [35, 36, 46],
but they all have been limited by experimental
resolution and by background forces of a quantum
nature.
The deviations from Newtonian gravity are
characterized [23, 30] by a Yukawa-modified potential
V(r),
(15)
where VN (r) is the Newtonian gravitational potential
for two point masses m1 and m2 separated by a dis-
tance r, G is Newton’s constant, and a and l are con-
stants. The strength (relative to Newtonian) of any
new interaction is characterized by the constant a and
the range of the interaction is characterized by l. The
non-Newtonian effects are generally constrained in
the phase space a a (l).
At short distances (i.e., small l), the forces arising
from a Yukawa-modified potential are relatively
weak, because the effective interacting masses are
themselves small. Moreover, background disturbances
play a more important role than a larger distance,
making it more difficult to search for new forces over
short separations. For experiments with typical
dimensions 1 mm, the dominant background arises
from the Casimir force [8, 10], which is dominant at
short distances and has not been completely charac-
terized with the required level of precision [15]. At
separations of 100 nm, the Casimir force between
an infinite 5 mm thick Au slab and a 300 mm radius
gold sphere is about 7 orders of magnitude larger than
the Newtonian gravitational force.
In the first experiment, we measured and charac-
terized the Casimir force between two different metals
with a precision 100 times better than previous meas-
urements [11]. These experiments are useful to un-
derstand the origin and behavior of the Casimir force.
By comparing the measured Casimir force to that pre-
dicted by theory, it is possible to impose new limits on
hypothetical corrections to Newtonian gravity [15].
Since the Casimir force is much stronger than gravity
at short distances, a very precise comparison between
theory and experiments is required. Unfortunately,
the uncertainty in the dielectric properties of the sam-
ples used prevents the calculation of the Casimir force
between real bodies to an accuracy of better than a
V(r) VN(r)[1 aerl] Gm1m2
r[1 aerl],
0.000
The data were obtained at two different separations zbetween the metallic layers.
0.575 0.600 0.625 0.650 0.675 0.700 0.725
0.001
0.002
0.003
0.004
0.005
0.006
0.007
(m
rad
)
VAu (V)
z 3 mz 5 m
Figure 7.Dependence of the angular variation u as a function ofthe applied voltage to the sphere.
Bell Labs Technical Journal 71
few percent. In the second experiment, we used the
isoelectronic technique [33] to set limits on new forces
over the 0.1 1 mm distance scale. In this experi-
ment, we used a configuration in which the Casimir
background is minimized, allowing us to compare the-
ory with experiments more readily.
Casimir ForceThe Casimir effect is a true quantum effect. In its
simpler form, it represents the attraction between two
parallel, neutral metallic layers in a vacuum [38]. The
interaction is the direct manifestation of the zero-
point energy fluctuations modified by the presence
of boundary layers. No such interaction exists within
the field of classical electrodynamics. H.B.G. Casimir,
who published his results in 1948 (see [10]), was the
first one to deal with the infinite energy of the
vacuum fluctuations. He found that for two perfect
metallic layers separated by a distance z the Casimir
force is attractive and is given by [10, 38]
(16)
where c is the speed of light, is the Planck constant
divided by , and S is the area of the layers.
Due to the experimental difficulties of maintaining
the alignment of two parallel planes, the setup used to
measure this force is similar to the one defined in the
section “Experimental Setup,” in which one of the in-
teracting surfaces is chosen to be spherical. In this case,
using the proximity force approximation, the attrac-
tive Casimir force is given by [11, 19]
(17)
with R being the radius of the sphere.
Using the experimental configuration shown in
Figure 6, we measured, for the first time, the Casimir
force between two different metals: copper and gold.
The sphere was coated with Au, as described in the
section “Experimental Setup,” and the MEMS oscilla-
tor with 200 nm of Cu. The force between the
two metals was determined by measuring the angle u
FCsphere
p3c R
360 z3 ,
2p
FC p2Uc S240 z
4,
(proportional to C) as a function of the separation
between the layers.
The proportionality constant between F and C
was found by applying a known potential difference
between the layers, which were separated by 3 mm
and 5 mm, as described in “Experimental Setup.” The
measurements are shown in Figure 7, which illus-
trates the dependence of u and, hence, of the electro-
static force, on the applied voltage VAu.
The force between the two metals is F(z) = kC,
where C is measured to one part in 5 105 using the
bridge circuit shown in Figure 5 [1]. The constant k is
determined by equation (9) to be k 50280 6 N/F.
The electronic noise of the amplifier stage used (i.e.,
10 nV/Hz12) is equivalent to an angular deviation
109 rad/Hz12. This noise level is much smaller
than the thermodynamic noise at the measuring fre-
quency kHz fo 687.23 Hz [41]:
(18)
In equation (18), we used T 300K and the quality
factor of our oscillator Q 8000. Using these values,
a force sensitivity of dF (dtorsionSu1/2)b
1.4pN/(Hz)12 is obtained.
Su12 c 2KBTQ
p foktorsion
a fo
fb2 d 12
3 107 radHz12
Wf 10
du
80
60
40
20
01000800600400200
F C (
pN
)
z (nm)
(a)
z (nm)
(b)
2
1
0
1
21000800600400200
F Cth
F Cex
p (
pN
)
R 100 mR 50 m
R 150 mR 300 m
(a) Casimir force between an Au-coated sphere and a Cu-coated plate as a function of the separation z. Data fordifferent nominal radii are shown.(b) Difference between theory and experimental data for the 300 m radius sphere.
Figure 8.Casimir force: theory and experiment.
72 Bell Labs Technical Journal
Figure 7 shows that the minimum in the electro-
static force occurs for Vo 0.6325V, which reflects the
difference in the work function of the Au and Cu lay-
ers. This value was observed to be constant for z in the
range 0.2 mm to 5 mm, and it did not vary when
measured over different locations in the Cu layer.
Once the value Vo was determined, we set Vsphere
VAu Vo 0.6325V to ensure that the electrostatic
force is identically zero and that the Casimir force be-
tween the Au-coated sphere and the Cu-coated mi-
cromachined oscillator can be measured. The results
obtained are shown in Figure 8.Figure 8a shows the Casimir force for spheres of
different radii. As expected, the Casimir force decays
very rapidly with z and is strongly dependent on the
radius of the sphere. In order to compare the data
with the theoretical predictions, the expression of the
force has to be modified to include the finite conduc-
tivity of the metals and the specific roughness of the
layers [15]. A detailed description of the Casimir force
is presented in [15]. The difference between theory
and data (including roughness and finite conductivity
corrections) is shown in Figure 8b for the case of
R 300 mm. The agreement between theory and data
is extremely good over the whole range of distance.
The experimental relative uncertainty at 188 nm is
0.27%, several times lower than the most precise
experiments reported [25, 34, 39, 42, 43].
To improve the sensitivity of our experiments, we
implemented the dynamic approach described in
“Experimental Setup.” In the dynamic approach, the
vertical separation between the sphere and the MEMS
is periodically changed as z zmetal A cos(wres t).
In the current experiment, the force acting be-
tween the sphere and the oscillator plate is the
Casimir force, so, in equation (10), Fx FC. Our
MEMS oscillator has a moment of inertia I ≈ 4.6
1017 kgm2, and A was adjusted between 3 nm and
35 nm for values of zmetal between 0.2 and 1.2 mm,
respectively. We foundvo 2p (687.23 Hz) and b22I
6.489 108 kg1. With an integration time of 10s,
using a phase-lock loop circuit [14], changes in the
resonant frequency of 10 mHz were detectable [15].
This allows the force to be detected with a sensitivity
of dF 6 fN/Hz12, a factor of 3 larger than the ther-
modynamic noise. However, unlike the static regime
where forces are measured, in the dynamic regime
the force gradient is measured using the relation
shown in equation (10). The gradient is
quantified by detecting the change in the resonant
0FC0z
200
250
150
100
50
01000800600400
P C (
mPa
)
z (nm)
(a)
z (nm)
(b)
1.0
0.5
0.0
0.5
1.0
1000800600400
P Cth
P C
exp (
mPa
)
R 150 mR 300 m
(a) Casimir pressure between an Au-coated sphere and a Cu-coated plate as a function of the separation z. Data from twodifferent spheres are shown.(b) Difference between theory and experiment for the data obtained with the 300 m radius sphere.
Figure 9.Casimir pressure: theory and experiment.
Bell Labs Technical Journal 73
frequency as the sphere-MEMS separation changes.
Differentiating equation (17) with respect to z, one
obtains
, (19)
where is the force per unit area be-
tween two infinite metallic plates.
The experimental distance-dependence of PC (see
Figure 9) is shown in 9a for two different spheres.
Because, for all the spheres, the dynamic measurement
yields the pressure between parallel plates, all mea-
surements are expected to give the same results. The
small deviations observed arise from an incomplete
characterization of the roughness of each sphere, which
is equivalent to a shift in the separation sphere-MEMS.
This shift has been observed to be in the range of 5 nm
to 10 nm. In [15] and [19], a theoretical analysis of the
dynamic interaction, including roughness and finite
conductivity, is presented. In Figure 9b, we compared
our data with the theory for a 300 mm sphere. The ex-
perimental uncertainty at 260 nm is 0.26%, a marked
improvement on previous models [9], implying a 15%
agreement between data and theory.
Given the improved sensitivity of our measure-
ments of the Casimir interaction, we can set more
PC(z) (FCS)
0FC
0z 2p RPC(z)
strict limits on new forces acting over short separa-
tions. Using the difference between theory and ex-
periment shown in Figure 8b, we obtained [19] al
200 nm, 1013, which represents an improvement by
a factor of 4 over previously reported limits [21]. The
new results are summarized in Figure 10.In Figure 10, constraints from previous experi-
ments are also shown. Curve 1 was obtained by
Mohideen’s group [39], curve 2 was determined by
Lamoreaux [34], curve 3 was extracted from data
taken by Kapitulnik’s group [13], and curve 4 was ob-
tained by Price’s group [37]. In all cases, the region in
the (a, l) plane above the curve is excluded, and the
region below the curve is allowed by the experimental
results. As can be seen from Figure 10, the experiment
with a MEMS-based force sensor leads to the strongest
constraints in a wide interaction range (i.e., 56 nm
330 nm). The largest improvement, by a factor
of 11, is achieved at l 150 nm. We note that the
constraints obtained here almost completely fill in the
gap between those obtained by AFM measurements
and those obtained using a torsion pendulum. Within
this gap, the best previous restrictions were obtained
from old measurements of the Casimir force between
dielectrics, which were not as precise or reliable as
MEMS—Microelectromechanical systems
(meters)
Excludedby
experiments
MEMS force sensor
1
2
3
4
1020
1016
1012
108
104
100
107 106 105 104
Figure 10.Constraints on the Yukawa interaction constant aversus interaction range l obtained from Casimir forcemeasurements.
74 Bell Labs Technical Journal
those reported here for the Casimir pressure measure-
ments between metals using a micromachined oscilla-
tor [15].
Reduced Casimir BackgroundIn the experiments previously described, we
showed that, by using a very sensitive force sensor
and a detailed theoretical model of the Casimir force
[15], it is possible to infer new constraints on Yukawa
corrections to Newton’s law of gravity. Our experi-
ments produced the most precise comparison of the-
ory and experiment to date and yielded the limit
a 1013 for l 100 nm [15]. In order to improve
even further the limits on a a(l) without develop-
ing more precise theoretical calculations, it is essential
to perform experiments in which the enormous
Casimir background is suppressed.
Recently, an alternative approach that uses the
fact that Casimir forces depend mainly on the elec-
tronic properties of materials has been proposed [20,
33]. Since gravitational forces, and virtually all pro-
posed new forces, involve coupling to both electrons
and nucleons, experiments measuring force differ-
ences between materials with very similar electronic
properties (i.e., isotopes of the same element) should
be very sensitive to new interactions [33]. By measur-
ing force differences between iso-electronic materials,
the Casimir background is subtracted and detailed
modeling of the Casimir force is not required. Hence,
by comparing the force between a probe (the sphere)
and a substrate (the oscillator plate) containing two
materials with identical electronic properties, any dif-
ference in the force experienced by the two substrates
can be attributed to a force other than the Casimir.
An experiment like this can be performed with the
experimental setup described in “Experimental Setup”
by coating the plate of the micromachined oscillator
with the two different iso-electronic materials.
In the experiments reported below, rather than
using two isotopes of the same element, we compare
the force produced by two dissimilar materials like
Au and Ge covered by a thick layer of Au.
Figure 11 shows the details of the setup and the
sample used in the experiments. The left panel shows
a SEM picture of the micromachined oscillator with
the two metals evaporated symmetrically around the
rotation axis. A cross section of the metals used in
this “Casimir-less” experiment is also shown. By cov-
ering the bottom Au and Ge films with a thick layer of
Au (larger than its plasma wavelength, which is
135 nm), we can be confident that the Au-coated
sphere will “feel” the same Casimir force on both sides
of the sample. On the other hand, the gravitational
force acting on the Au sphere will change when the
probe is displaced from the Ge-covered side to the Au-
covered side. Thus, by directly comparing the forces
measured on the Au-and-Ge-coated substrate, a limit
on a a(l) can be obtained without using detailed
theories of the Casimir interaction.
Within the experimental configuration, and using
a Yukawa-like correction, as in equation 15, to the
Newtonian gravitational attraction, the difference in
force that the sphere will experience is given by [16, 17]
(20)
where G is the gravitational constant and the func-
tions KS and KP are associated with the layered struc-
ture of the sphere (S) and oscillator plate (P). These
functions are given by [16, 17]
KP [(rAu rGe)e(dPAudTi)l(1 edGel)].
KS [rAu (rAu rCr)ed SAul (rCr rS)e(dS
AudCr)l]
¢Fhyp(z) 4p2Gal3ezl RKS KP
Bell Labs Technical Journal 75
The densities of the materials used in our multilayer
are represented by ri and the thicknesses of the dif-
ferent materials are di.
The right panel of Figure 11 shows a SEM picture
of the MEMS oscillator with the multilayer sample de-
posited on top. In our experiment, we used a bottom
layer of Au and Ge because it gives a large mass density
difference: rGe 5.32 103 kg/m3 and rAu 19.28
103 kg/m3. This bottom layer is capped by a homoge-
neous Au layer 150 nm thick. The fact that the thick-
ness of this layer is greater than the plasma length for
Au is sufficient to ensure that the underlying Au/Ge
composite has a negligible Casimir interaction with
the Au-coated spherical probe [15]. The MEMS oscil-
lator is similar to the one we used for the Casimir
measurements described in the previous section.
During the microfabrication process, we have si-
multaneously fabricated the MEMS oscillator and the
evaporation masks that allow us to deposit metals on
selected regions of the plate (see Figure 11). Once the
micromachined oscillator is released, we flip the cor-
responding mask over the oscillator plate and the
metals are evaporated without venting the chamber.
In the first evaporation, we deposited the following
multilayer: dTi 10 nm, dGe 200 nm, dPt 10 nm,
and 50 nm of Au. The Ti layer is used to increase the
adhesion of Ge to Si and the Pt layer is used to avoid
inter-diffusion of Au and Ge. The oscillator is then re-
moved from the evaporation chamber, the mask used
for the first evaporation is returned to its original po-
sition, and the second evaporation mask is flipped
over. The second evaporation is similar, the only dif-
ference being that an Au layer replaces the Ge layer:
dTi 10 nm, dAu 200 nm, dPt 10 nm, and 50 nm
of Au.
Once the second evaporation is done, a final
evaporation consisting only of Au is performed on top
of the previous multilayers. This last Au layer and the
final 50 nm of Au of the multilayers constitute .
In order to obtain maximum sensitivity to the
force, we used a dynamic technique in our measure-
ment of the force difference between the sphere and
the two sides of the oscillator. It would be desirable to
move the sphere parallel to the rotation axis of the os-
cillator plate, while keeping z constant, at a frequency
coincident with the resonant frequency of the oscilla-
tor. The problem with this approach is that the reso-
nant frequency of our MEMS device ( fo 700 Hz) is
dPAu
A cross section of the multilayer sample used is shown in the upper middle panel.
Pt
Ti
Figure 11.Scanning electron micrograph and schematic of the MEMS torsional oscillator used in the Casimir-less experiment.
76 Bell Labs Technical Journal
too large and moving the sphere over several tens of
microns at this frequency is experimentally compli-
cated. An analysis of equation (20) provides a solution
to this problem: the sphere is harmonically moved over
the interface at a frequency fx such that it will see an ef-
fective mass density described by a square-wave func-
tion with a characteristic angular frequency wx. In
practice, we used the electrodes under the oscillator
to induce an oscillation such that the separation be-
tween the sphere and the plate changes as zm zmo
dz cos(wz t) with zmo dz. Simultaneously, we moved
the micromachined oscillator’s stage along a direction
parallel to its axis (x-axis) such that the effective mass
density under the sphere is
(21)
where r (rAy rGe)2 and is the square-
wave function with characteristic angular frequency
wx. Introducing the above expressions in equation
(20), we find the time dependence of the force dif-
ference detected by the sphere
(22) a
k0
1
k!c (dz cos(wzt))
ld k,
¢Fhyp(zm, t) ¢Fhyp(zmo) ¢r ¢r(t)
2¢r
(t)
reff ¢r ¢r (t),
W
By selecting wz wx wr, the Fourier component at
wr of equation (22) is the only one providing a
significant signal-to-noise ratio. By using this hetero-
dyne technique, the signal to detect is at fr even
though no parts in the system are moving at this fre-
quency. For our experiment, we select fx (wx2p)
(wr140p) fr70 10 Hz. The signal at fz fr fx is
used to excite the oscillator plate, while the signal fx is
used to move it laterally along the x-axis.
In our experiments, the lateral displacement was
varied between 50 mm and 150 mm and the vertical
amplitude was 5 nm. The minimum detectable force
measured with this heterodyne technique (see
Figure 12) is shown in 12a.
We determined the root mean square value of the
force for integration times t ranging from 0.1 to
2000 sec. As is evident in Figure 12a, the results ob-
tained for are well described by the thermo-
dynamic limit in conjunction with the electronic
noise. For integration times larger than 3 seconds, the
root mean square value of the force starts to deviate
from this limit and for the measured force
becomes independent of the integration time t. The z
dependence of this integration-time-independent
force is shown in Figure 12b. Although the experi-
mental data show a finite force forF (t 200s, z)
t 7 200s
t 6 3s
F
(a) Minimum detectable force as a function of the integration time. Data are shown for a 50 m radiussphere at different separations of the sphere and the multilayered sample z.(b) Dependence of F
– measured at 100s as a function of the separation of z.
F– (N
)
1E-14
1E-15
1E-160.1 1 10
(s)
(a)
100 1000
F– (fN
)
6
1
2
3
4
5
100 200 300
z (nm)
(b)
400 500
z 300 nmz 500 nm
z 150 nm
Figure 12.Force as a function of integration time and sample separation.
Bell Labs Technical Journal 77
finite separations z, several experimental facts and un-
certainties prevent us from concluding that it origi-
nated from new physics. The large magnitude and
strong distance-dependence of this force indicate that
its origin is not Newtonian in nature. For the geome-
try used in our experiment, the Newtonian force
should have a magnitude of N and should be
independent of the separation sphere-plate. is not
associated with magnetic impurities in the materials,
because these forces are much smaller than the sen-
sitivity of our apparatus [13]. Moreover, it is not in-
duced by vibrations, because it takes a constant value
at a fixed separation, independent of t.
Some possible experimental errors that may have
an effect on the force detection are listed below:
• A geometrically induced background arises if the
motion of the probe is not parallel to the x-axis or
if the motion across the interface does not occur
at constant zmo. These effects can be sorted out
experimentally by performing the same mea-
surement, but only over one side of the sample.
The maximum force observed when doing the
experiment on one side of the sample showed a
residual force more than one order of magnitude
smaller than . Hence, we conclude that the re-
sults shown in Figure 12b are not caused by lack
of parallelism between the probe direction and
the torsional axis of the oscillator [16, 17].
• Differences in local roughness. This will have a
huge impact when the separation between bodies
is comparable to the characteristic root mean
square roughness dr. The magnitude of this ef-
fects will decay as (drzmo)2 when zmo dr [23].
In our case, dr 4 nm, so, for zmo 200 nm, the
effect of roughness differences on the detected
force is 0.02%.
• The existence of patch potentials (i.e., local vari-
ations of the work function [12] of the metals)
can produce unwanted electrostatic forces. In our
samples [15, 17], the patch potential has a lateral
dimension of 0.3 mm and produces a maximum
force dFep(200 nm) 1 fN (ref).
In spite of all the experimental uncertainties listed
above, we can use the data of Figure 12b to impose
more strict limits on hypothetical forces. Using these
experimental values and assuming that any new force
W
F
F
1019
MEMS—Microelectromechanical systems
Curve 1 was obtained by Mohideen’s group [39], curve 2was determined by Lamoreaux [34], curve 3 was extractedfrom data taken by Kapitulnik’s group [13], and curve 4 wasobtained by Price’s group [37].
(meters)
Excludedby
experiments
MEMS force sensor
Casimir-lessconfiguration
1
2
3
4
1020
1016
1012
108
104
100
107 106 105 104
Figure 13.Values in the (a, l) phase space excluded by theexperiment.
should have an amplitude less than or equal to that of
the observed force, we obtained the a a(l) diagram
shown in Figure 13 [17].
Our experimental setup, in which the Casimir
background has been dramatically reduced, improves
by an order of magnitude the current limits for
Yukawa-like corrections to Newtonian gravity [19].
ConclusionsIn conclusion, we have described a MEMS-based
force sensor with ultra-high sensitivity for measuring
the interaction between different bodies. Currently,
resolution of fN is easily achievable at room tem-
perature. By changing the dimensions of the torsional
springs, improvement in the force resolution should
be possible. The sensitivity of our micromachined sen-
sor allows measurements that would be extremely
complicated using any other technique, such as the
observation of the Casimir effect and deviation of the
Newtonian gravity at distances below 1 mm.
Our primary objective with the reported mea-
surements has been to impose new constraints on
hypothetical corrections to Newtonian gravity. We
78 Bell Labs Technical Journal
have done so by using Casimir force measurements
between a sphere and a plate separated by 0.2 to
1.2 mm. These experimental results, along with a de-
tailed theoretical analysis, lead to new constraints on
Yukawa modifications of Newtonian gravity at short
distances. The MEMS-based force sensor we used al-
lowed us to measure the Casimir force between two
metallic surfaces with a resolution two orders of mag-
nitude better than previous experiments. Moreover,
we presented an experimental configuration in which
the Casimir background was suppressed, allowing us
to set new limits in the 10 to 100 nm distance scale.
By not having to resort to detailed theoretical de-
scriptions of the Casimir interaction and by using soft
micromachined force sensors, the experimental sen-
sitivity is greatly improved. The results reported here
are the first experiments done at these separations. We
found that for nm, which represents
an improvement of one order of magnitude over
previous limits.
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(Manuscript approved June 2005)
DANIEL LÓPEZ is a member of technical staff inthe Nanofabrication Research Group at BellLabs in Murray Hill, New Jersey; he is also amember of the New Jersey NanotechnologyConsortium. He received his B.Sc. degree(with honors) in physics from the University
of San Luis in Argentina and his Ph.D. degree, also inphysics, from the Institutio Balseiro in Bariloche,Argentina. At Bell Labs, he has worked onsuperconductors, MEMS-based devices for opticalcommunications, acoustics, and basic physics research.He was a member of the team that received the 2000Bell Labs President’s Gold Award for the outstandinglevel of innovation and technical excellencedemonstrated in the WaveStar® LambdaRouter project.Dr. López has co-authored over 100 papers in the fieldsof superconductivity, and he has been invited to givelectures at several international conferences anduniversities and research institutes worldwide. In thefield of micro- and nano-mechanical devices, he has co-authored several patents. He is a member of theAmerican Physical Society and has also been a panelistin technology-related conferences. His researchinterests include design and fabrication of MEMSdevices for communication networks, acousticalsystems, medical devices, and high-sensitivity forcedetectors.
RICARDO S. DECCA is an assistant professor of physicsat Indiana University-Purdue UniversityIndianapolis in Indiana. He received hisLicenciatura and Ph.D. degrees in physics atthe Instituto Balseiro in Bariloche,Argentina. His dissertation dealt with the
granular behavior of the recently discovered highcritical temperature LaSrCuO superconductor. For atime, he worked at Bell Labs, where he investigatedthe optical properties of highly correlated GaAs/AlGaAsdouble quantum wells. Dr. Decca has developed newways of using near-field scanning microscopy and iscurrently studying the interaction (i.e., resonantenergy transfer) between single quantum dots, theorganization of lipid bilayers, and the use of MEMS infundamental physics. He has authored and coauthoredmore than 60 journal and professional conference
papers and is a member of the APS, the AAAS, and theAAPT.
EPHRAIM FISCHBACH is a professor of physics at PurdueUniversity in West Lafayette, Indiana. Hehas a B.A. degree in physics from ColumbiaCollege in New York City and M.S. and Ph.D.degrees in physics from the University ofPennsylvania in Philadelphia. His research
interests include searching for new gravity-like forces.Dr. Fischbach is a fellow of the American PhysicalSociety.
DENNIS E. KRAUSE is an associate professor of physicsand Physics Department chair at WabashCollege in Crawfordsville, Indiana. Hereceived his B.A. degree in physics fromSaint Olaf College in Northfield, Minnesota,his M.S. degree in physics from the
University of Wisconsin-Milwaukee, and his Ph.D. inphysics from Purdue University in West Lafayette,Indiana. Before going to Wabash, he was a visitingassistant professor in physics at Williams College inWilliamstown, Massachusetts. Dr. Krause’s primaryresearch interests include the search for new Yukawaand inverse power law forces at sub-micron separationsand the theoretical investigation of the Casimir forceusing novel materials.