REVIEW OF SCIENTIFIC INSTRUMENTS
VOLUME 69, NUMBER 3
MARCH 1998
Measurement of forces and spring constants of microinstruments
M. T. A. Saifa) and N. C. MacDonald
School of Electrical Engineering and The Cornell Nanofabrication Facility, Cornell University,
Phillips Hall, Ithaca, New York 14853
~Received 7 July 1997; accepted for publication 8 December 1997!
Micromachines, including micro scanning tunneling microscope and high force ~'0.1 mN and
higher! actuators, are examples of a new class of silicon-based microinstruments for atomic scale
surface imaging and modification, as well as submicrometer scale material investigations. A large
class of such microinstruments and sensors consist of actuators, such as interdigitated comb drives,
that generate force, F, in the form F5 b V 2 , where b is a constant and V is the applied voltage. Such
actuators often move a large distance during actuation so that the restoring force, R, of the springs
varies nonlinearly with x, i.e., the springs behave as hard springs, and R5K 1 x1K 3 x 3 1...
~restoring force is similar for 1ve and 2ve x due to symmetry!. b and K i , i51,3,..., of the microactuators usually differ from their design values due to processing nonuniformities. Hence,
evaluation of these constants becomes necessary for each actuator, and is essential for the
instruments that are employed to study materials’ behavior on a small scale. In this article, a
methodology is developed to calibrate microinstruments, i.e., to evaluate the values of b and K i ,
i51,3,... . The method is based on buckling of a long slender beam of known dimensions, and made
of a material with known property ~elastic modulus!. Buckling is achieved by an axial compressive
force on the beam applied by the actuator of the microinstrument. b and K i are derived from the
relation between the applied voltage on the actuator, and the post buckling deformation of the beam.
The beam is designed and cofabricated with the actuator, and hence the calibration mechanism is
integrable with each microinstrument. An analysis is provided to estimate the possible errors in
calibration due to errors in the measured dimensions of the calibrating beam. It is shown that the
calibration error increases linearly with the error in the measured linear dimensions. The
applicability of the method is demonstrated by fabricating microinstruments, which, prior to their
calibration, are employed to apply torque on single crystal silicon bars ~in the form of pillars!, until
the bars fracture. The instruments are then calibrated, and the calibrated values of b and K i are used
to evaluate the torques applied on the bars at different voltages. Stresses to fracture of the bars are
also estimated. The torsion experiment is an example of the application of integrated
microinstruments for small scale material studies. © 1998 American Institute of Physics.
@S0034-6748~98!00703-5#
I. INTRODUCTION
tural beams!. During actuation, the movable component
moves by deforming the springs. For small motion, the
springs behave linearly. For larger motions, frequently encountered in MEMS, the nonlinear terms of the springs become dominant.5 Thus, calibration of a microinstrument involves evaluation of ~1! the force generated due to a known
applied input ~voltage, temperature, magnetic field!, and ~2!
the spring constants including the nonlinear terms. Note that
the microinstruments that are used as sensors may also need
calibration. Since calibration must be carried out for each
instrument independently, and since the small size scale limits the application of an external calibrating device, hence the
calibration mechanism must be designed and integrated with
each microinstrument.
In this article, a methodology is developed to calibrate
the force generated by microinstruments and their spring
constants including the nonlinear terms. The method is based
on buckling of a long slender beam of known dimensions,
and made of a material of a known elastic modulus. Actuators that generate force, F5 b V 2 , by actuating a set of comb
capacitors6 are considered. Here b is a constant, to be deter-
Micromachines, including microscanning tunneling
microscope1–3 and high force actuators,4 are examples of a
new class of silicon-based microinstruments for atomic scale
surface imaging and modification, and submicron scale material investigations. A well-known problem for any instrument, including microinstruments, is their calibration. For
many macroinstruments, the calibration scheme is well developed, such as, load cells, available commercially, that
calibrate the force generated by loading instruments for material studies. The calibration of newly developed microinstruments, based on microelectromechanical system
~MEMS! technology, is complicated due to their small size.
MEMS microinstruments usually consist of a set of actuators
that generate force by electrical, thermal, or magnetic actuation. The actuators have a stationary and a movable component, the latter being held by a set of springs ~usually struca!
Current address: Mechanical and Industrial Engineering, University of Illinois, Urbana, IL 61801.
0034-6748/98/69(3)/1410/13/$15.00
1410
© 1998 American Institute of Physics
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Rev. Sci. Instrum., Vol. 69, No. 3, March 1998
M. T. A. Saif and N. C. MacDonald
mined, and V is the applied voltage between the movable and
the fixed combs. In principle, b 5 e 0 Nh/d, where e 0 is the
permitivity constant, N is the number of comb capacitors, h
is the height of the comb capacitors, and d is the gap between them. h and d for the all the combs are usually designed and fabricated identically, but deviation from their
design values during fabrication renders b unknown. The
restoring force, R, of the actuator spring is given by R
5 ( i51,3,...K i x i , where K i , i51,3,... are the unknown spring
constants, to be determined, and x is the actuator displacement. In principle, K i depends on the material properties and
the dimensions of the springs. Thus, in principle, K i can be
designed by a proper choice of materials and dimensions.
However, the deviation of the actual dimensions from the
designed values render K i unknown.
The methodology of calibration is demonstrated by fabricating two pairs of microinstruments. Each pair has two
identical instruments. Each instrument consists of an actuator
with 2000 comb capacitors and a buckling beam for calibration. The pair is first employed to apply torque on a short
single-crystal silicon ~SCS! pillar until the pillar fractures.
Thus, two pairs fracture two pillars. The cross-sectional areas
of the pillars are approximately 131 m m2 and 1.5
31.6 m m2, measured by a calibrated scanning electron microscopy ~SEM!. The applied voltage on the actuators and
the angle of twist of the pillars are recorded during the experiment. After the experiment, the actuators are calibrated.
Calibration results are then used to evaluate the torque applied on the samples as well as their failure stresses. The
torsion experiment is useful for studying the effect of large
strain gradients in small samples on material strength.7
II. METHOD OF CALIBRATION: MEASUREMENT OF
FORCES AND SPRING CONSTANTS
A. Buckling of a long slender beam
A long slender beam subjected to an axial compressive
force buckles when the force exceeds a critical value of8,9
P cr5H p k /L ,
2
2
~1!
where the constant H depends on the boundary conditions,
e.g., H54 for beams with both ends clamped. k is the minimum flexural rigidity, and L is the length of the beam. k
5EI when the beam is made of a single material with modulus of elasticity, E, and with minimum area moment of inertia, I. I depends on the cross sectional geometry, such as
I5bh 3 /12 for a rectangular cross section with width b and
height h. In this article, the fabricated beams have a trapezoidal cross section, and the corresponding I is evaluated ~see
Sec. III!. If the beam has an initial imperfection, such as
being slightly bent, then it does not buckle suddenly under
compression, but bends gradually along the transverse direction with increasing load. The transverse deformation increases rapidly with a small increase of load as the load
approaches P cr . Buckling of a beam with both ends clamped
has been analyzed in Ref. 4. Here, the analysis is extended.
For convenience, the notation adopted in Ref. 4 is used.
Consider the beam in Fig. 1. Its length, measured along
the axis, is L, and its initial imperfection is modeled by its
1411
FIG. 1. Schematic of an actuator and its calibration beam with initial imperfection j. The total length of the beam along its axis is L. C 1 and C 2 are
the points of inflection.
cosine shape with an amplitude j /2. L does not change due
to buckling. Such assumption is reasonable since the effect
of axial shortening on buckling is small.9 The total deformation of the buckled beam—initial deformation, j, plus the
deformation due to applied load P—is approximated by a
cosine curve with an amplitude D/2. The validity of the cosine shape is verified experimentally later ~see Sec. VI!. Let
v 52 p /L, s be the coordinate along the beam axis from the
point of inflexion C 1 , y 0 (s) and y(s) be the initial and total
deflections at s. Then
y 0 5 ~ j /2! sin~ v s !
and y5 ~ D/2! sin~ v s ! .
~2!
At any section of C 1 C 2 , the internal moment M is related to
curvature8 by
M du du0
,
5 2
k ds ds
~3!
where u and u 0 are defined in Fig. 1. Using
M 52 Py, sin u 5
dy
5y 8 ,
ds
y9
du
5
ds ~ 12y 8 2 ! 1/2
in Eq. ~3!,
2
y 90
y9
Py
5
2 1/22
k ~ 12y 8 !
~ 12y 80 2 ! 1/2
S
'y 9 11
S
~4!
1 2 3 4 5
y8 1 y8 1
y 8 6 1...
2
8
16
2y 09 11
D
D
1 2 3 4 5
y 8 1 y 08 1
y 8 6 ... .
2 0
8
16 0
~5!
The second equality holds when y 8 2 !1. Using Eq. ~2! in Eq.
~5!
P
1 v4
D sin~ v s ! 5 v 2 ~ D2 j ! sin~ v s ! 1
~ D 32 j 3 !
k
2 22
3sin~ v s ! cos2 ~ v s ! 1
3 v6
~ D 52 j 5 !
8 24
3cos4 ~ v s ! sin~ v s ! 1
5 v8
~ D 72 j 7 !
16 2 6
3sin~ v s ! cos6 ~ v s ! ... .
~6!
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1412
Rev. Sci. Instrum., Vol. 69, No. 3, March 1998
M. T. A. Saif and N. C. MacDonald
Multiplying both sides of Eq. ~6! by sin(vs) and integrating10
with respect to s from 0 to L/2, and using P cr54 p 2 k /L 2 , the
load displacement relation is obtained:
S D
P
v 2D 2
3 vD
5 ~ 12 j /D ! 1
~ 12 ~ j /D ! 3 ! 1
P cr
32
64 2
S D
25 v D
3 ~ 12 ~ j /D ! 5 ! 1
1024 2
4
d 5 v 2 D 2 L/165 p 2 D 2 /4L,
6
~ 12 ~ j /D ! 7 ! 1... .
~7!
The relative displacement, d, along the axial direction
between the end points of the beam after buckling is given
by4
d5
v 2D 2L
@ 12 ~ j /D ! 2 # .
16
in the rest of the paper, it is assumed that j 50, and that the
beam is initially free from any net axial force. Then, Eqs. ~7!
and ~8! reduce to
~8!
The accuracy of Eq. ~7! @with terms up to ( j /D) 3 # and Eq.
~8! have been verified by nonlinear 2D finite element ~FE!
analysis in Ref. 4. It is shown that, for a beam, 500 mm long,
1 mm wide, with j 51.5 m m, Eqs. ~7! and ~8! show excellent
correspondence with FE results as long as D,50 m m ~10%
of L! and corresponding d ,10 m m.
B. Initial imperfection j
So far, imperfection of a long slender beam is modeled
by its initial bent shape that coincides with the first mode of
buckling with an amplitude j. A ‘‘perfect’’ beam must be
straight, uniform in cross section along the length, have zero
net stress on any cross section, and satisfy the required
boundary conditions, such as clamped at both ends with zero
slope. An imperfect beam buckles under a load less than P cr .
For example, an initially bent beam, deforms transversely
under compressive load less than P cr , but the rate of deformation with load increases rapidly as the load approaches
P cr . Hence the buckling is gradual. Whereas for an initially
straight beam, it is sudden, i.e., d P/dD50 at P5 P cr .
Due to optical lithography based fabrication, micromechanical beams are straight with high accuracy. Any appreciable deviation ~e.g., j .0.5 m m of a 500 mm long and 1
mm wide beam! can be detected by SEM. The boundary
conditions of micromechanical beams can be ensured by appropriate design of supports.
The major concern for micromechanical beams is the
internal stress.11 These stresses arise from multiple layers of
thin films that constitute the beam. Internal stresses may also
be imposed on the beam by the actuator through their interconnection, because the actuator itself might be subjected to
residual stresses. Hence, depending on the fabrication process, one needs to take adequate measures ~such as
annealing12,13 and choosing appropriate location for the attachment of the beam with the actuator! to free the beam
from internal stresses.
It has been shown in Ref. 4 that for a slightly bent beam,
initially free from any net axial force, the post buckling loaddeformation relation is close to an initially straight beam.
The post buckling deformation of the beam is not very sensitive to initial imperfections. The reason for such weak sensitivity to initial imperfection is described in Ref. 14. Thus,
F
P5 P cr 11
d
2L
13 ~ d /2L ! 2 1
~9!
G
25
~ d /2L ! 3 1... .
2
~10!
C. Force equilibrium
Consider again the actuator and the buckling beam in
Fig. 1 with j 50. P cr of the beam is known from its geometry
and material property @Eq. ~1!#. Here the constants K i , i
51,3,... and b of F5 b V 2 are unknowns. The displacement
of the actuator is denoted by d. Note that, for comb capacitors, the force does not depend on the displacement of the
moving comb, as long as the distance between the ends of
the movable and the fixed combs is large compared to the
gap between them. With the increase of voltage, the actuator
applies a compressive force on the buckling beam. When the
force approaches P cr the beam buckles and the actuator
moves towards right by an amount d by stretching the spring.
The maximum transverse deformation, D, of the beam at the
middle is related to the axial deformation d by Eq. ~9!. Force
equilibrium requires
F5 b V 2 5 P1R,
~11!
where P is the force shared by the buckled beam @Eq. ~10!#
and R is the spring’s restoring force
R5
(
i51,3,...
K id i,
~12!
i.e., the springs restoring force is similar for 1ve and
2ve d due to symmetry. From Eqs. ~10!, ~11!, and ~12!
V 25
S
d
P cr
25
13 ~ d /2L ! 2 1
11
~ d /2L ! 3 1•••
b
2L
2
1
Ki i
d
b
(
i51,3,...
D
~13!
which implies that in a buckling experiment the square of the
actuator voltage, V 2 , will functionally depend on d according
to Eq. ~13!. From such an experiment, one measures V and
the corresponding transverse deformation, D, of the buckled
beam. The axial deformation, d, can be calculated from Eq.
~9!. V 2 can then be plotted as a function of d. The data can be
best fitted to
S
V 2 5a 0 11
1
(
d
2L
i51,3,...
13 ~ d /2L ! 2 1
25
~ d /2L ! 3 1•••
2
a id i,
D
~14!
where ~a 0 , a 1 , a 3 ,...! are the least-squares parameters.
Comparing Eqs. ~13! and ~14!, the calibration parameters of
the actuator are obtained:
b 5 P cr /a 0
and
K i5 b a i ,
i51,3,...
~15!
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Rev. Sci. Instrum., Vol. 69, No. 3, March 1998
M. T. A. Saif and N. C. MacDonald
S
V 2 5a 0 111
1
(
i51,3,...
S D
x2 d 1
~ x2 d 1 !
13
2L
2L
2
1
1413
S D D
25 x2 d 1
2
2L
3
1•••
~18!
a ix i.
Comparing Eqs. ~17! and ~18!
b 5 P cr /a 0
FIG. 2. Schematic of an actuator and its calibration cantilever beam. The
total length of the beam along its axis is L.
K i5 b a i ,
and
~19!
i51,3,...
For most practical purposes, (x2 d 1 )/2L!1 ~e.g., x2 d 1
510 m m, 2L51000 m m!, and only K 1 and K 3 will be of
interest. Then the first two terms in the expansion of P @Eq.
~16!# will suffice. Equation ~17! gives
V 25
S
D S
D
d1
P cr
1
P cr
K3 3
12
1
K 11
x1
x .
b
2L
b
2L
b
~20!
For most practical purposes, only the linear and the cubic
spring constants, K 1 and K 3 , are of interest in which case the
first two terms in the parenthesis of Eqs. ~13! and ~14! would
suffice, since d is expected to be several orders of magnitude
less than 2L.
From a buckling experiment, the measured values of V are
related to the corresponding measured values of x5 d 1
1 d (D) according to
D. Cantilevered buckling beam
where a 0 , a 1 , and a 3 are the least-squares parameters of the
best-fit curve through the data. Comparing Eqs. ~20! and ~21!
In the discussion on the initial imperfection of the buckling beam, it was realized that internal stresses may be imposed on the beam by the actuator through their interconnection. In order to ensure that the buckling beam is free from
such stresses, the beam can be designed as a cantilever with
a small gap of d 1 between the actuator and the beam ~Fig. 2!.
The actuator in Fig. 2, when actuated, first stretches the
spring ~i.e., in reality, it bends the supporting structural
beams! and closes the gap, d 1 , at a voltage of V 1 . Since d 1
is small, the spring restoring force is linear. Hence
b V 21 5K 1 d 1 .
With further increase of voltage, the additional force is carried by the buckling beam until the force on the beam approaches P cr . After buckling, the actuator moves to the right,
and the actuator force is equilibrated by the force, P, on the
buckling beam and the restoring force, R, of the spring,
b V 2 5 P1R,
where, from Eqs. ~10! and ~12!
F
2L
(
K ix i,
P5 P cr 11
R5
i51,3,...
13 ~ d /2L ! 2 1
G
25
~ d /2L ! 3 1••• ,
2
x5 d 1 d 1 .
b5
P cr
~ 12 d 1 /2L ! ,
a0
~21!
K 1 5a 1 b 2 P cr/2L,
K 3 5a 3 b .
~22!
E. Displacement measurement using buckling beam
Displacement can be measured with nanometer resolution using the buckling mechanism. A new novel method is
proposed here as follows. The relative displacement of the
ends of the buckled beam, d, is related to the transverse
deformation, D, by Eq. ~9!. Thus, if L5500 m m, d
50.1 m m, D54.5 m m, i.e., the displacement is amplified by
45 times.15 A change of D, dD, is related to a change of d,
d d , by
d d 5 ~ p 2 D/2L ! dD.
~23!
The change of dD can be measured using a micromechanical
vernier scale @see Figs. 5~a! and 6~a!# and an optical microscope with a resolution of less than 0.5 mm. Thus at D
54 m m, and for L5500 m m, a change of d can be measured
within 20 nm. Clearly, the sensitivity of the measurement of
d d can be increased significantly if dD is measured in SEM,
or measured electronically using a capacitive sensor.
~16!
III. ERROR ANALYSIS
Thus
V 25
d
V 2 5a 0 1a 1 x1a 3 x 3 ,
S
S D
x2 d 1
P cr
~ x2 d 1 !
13
11
b
2L
2L
1
(
i51,3,...
Ki i
x.
b
2
1
S D D
25 x2 d 1
2
2L
3
1•••
~17!
From a buckling experiment one obtains the data for V and
d (D) @Eq. ~9!#, which, together with measured d 1 , gives a
best-fit curve
The possible errors in the calibration of parameters, b,
K 1 , and K 3 , obtained from the curve fitting algorithm, are
estimated due to variations in measurements. There are two
major sources of errors. These correspond to measurements
of: ~1! the cross-sectional dimensions of the calibrating
beam, and ~2! the initial gap, d 1 , between the actuator and
the calibrating beam. The modulus of elasticity, E, of the
calibrating beam may also introduce errors and should be
accounted for. In this article, however, SCS is used as the
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1414
Rev. Sci. Instrum., Vol. 69, No. 3, March 1998
M. T. A. Saif and N. C. MacDonald
Fig. 4 represents Eq. ~21! (V 2 5a 0 1a 1 x1a 3 x 3 ), the best-fit
curve through the experimental data of V 2 and x5 d 1 d 1 .
Due to error in d 1 , d d 1 , let the data be shifted to the right by
d d 1 . The dotted line is the best fit curve through the shifted
data, and is given by
V 2shift5b 0 1b 1 X1b 3 X 3 ,
X5x1d d 1 .
~27!
The dotted line can also be represented in terms of a 0 , a 1 ,
and a 3 by
V 2 5a 0 1a 1 ~ X2d d 1 ! 1a 3 ~ X2d d 1 ! 3 .
FIG. 3. Sectional parameters of the calibrating beam are defined.
Coefficients of Eq. ~27! can be estimated by minimizing
D1d d
* d 1d d1 (V shift2V) 2 dX with respect to b 0 , b 1 , and b 3 , i.e.,
1
beam material. Its modulus is well characterized in the literature which will be used, and its error will be considered
negligible.
1
hw 3 hc 3
1
1ch ~ w/21c/3! 2 ,
I5
12
18
~24!
where w, h, and c are defined in Fig. 3. These values are
measured by a calibrated SEM ~Stereoscan 440, Leica Cambridge Inc.! with 5 nm resolution. Error in I, dI, is due to the
equal absolute errors in w, h, and w 1 , namely, dw5dh
5dw 1 . Since w 1 5w12c, dc50. Thus
S
D
hw 2 w 3 c 3
1
1 1cB 2 1chB dw,
dI5
4
12 18
~25!
where B5w/21c/3.
Error in the evaluation of P cr is then given by
d P cr54 p 2 EdI/L 2 .
~26!
B. Error in the coefficients of Eq. „21…
Errors in the coefficients, a 0 , a 1 , and a 3 , are due to the
error in the linear dimension d 1 , d d 1 5dw. The solid line in
E
]
]b0
A. Error in moment of inertia, I
Figure 3 shows a typical cross section of a calibrating
beam ~see also an SEM micrograph in Fig. 9!. The minimum
moment of inertia, I, about the centroidal verticle axis for the
section is given by
~28!
D1d d 1
d 1 1d d 1
5
]
]b1
5
]
]b3
~ V shift2V ! 2 dX
E
E
D1d d 1
d 1 1d d 1
D1d d 1
d 1 1d d 1
~ V shift2V ! 2 dX
~ V shift2V ! 2 dX50,
where D is the maximum value of d 1 d 1 obtained experimentally.
Values of b 0 , b 1 , and b 3 can be obtained approximately
by noting that the initial slopes of the two curves of Fig. 4
will be similar, i.e., b 0 'a 0 2a 1 d d 1 , and b 1 'a 1 . Thus the
errors in a 0 and a 1 due to d d 1 are
da 0 5a 1 d d 1 ,
da 1 50
~29!
and V2V shift'a 3 (X2d d 1 ) 2b 3 X . Hence b 3 is obtained
from
3
E
]
]b3
D1d d 1
d 1 1d d 1
3
@ a 3 ~ X2d d 1 ! 3 2b 3 X 3 # 2 dX50
~30!
giving the error in a 3 ,
da 3 5b 3 2a 3
UE
5a 3
5
D1d d 1
d 1 1d d 1
~~ X2d d 1 ! 3 X 3 2X 6 ! dX
UY E
D1d d 1
d 1 1d d 1
D1d d 1
1 1d d 1
a 3 @ 2X 6 d d 1 /213X 5 d d 21 /523X 4 d d 31 /4# u d
D1d d 1
1 1d d 1
@ X 7 /7# u d
X 6 dX
.
~31!
C. Errors in b, K 1 , and K 3
The worst case errors are obtained from Eq. ~22!:
db5
U
UU
d P cr
P cr
~ 12 d 1 /2L ! 1 2 da 0 ~ 12 d 1 /2L !
a0
a0
1
U
U
P cr d d 1
,
a 0 2L
dK 1 5 u a 1 d b u 1 u d P cr/2L u ,
FIG. 4. Possible shift of x vs V 2 data due to error, d d 1 , in the measurement
of d 1 ~the gap between the actuator and the free end of the calibration
beam!.
U
~32!
dK 3 5 u b da 3 u 1 u a 3 d b u .
It is noted in Eqs. ~25!–~29! that the errors in P cr and a 0 are
linearly dependent on dw5d d 1 . Equation ~31! shows that
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Rev. Sci. Instrum., Vol. 69, No. 3, March 1998
M. T. A. Saif and N. C. MacDonald
1415
FIG. 5. Microinstrument with torsion sample. ~a! Schematic shows the instrument with the actuator and a calibrating cantilever buckling beam. The region R 1
has a lower trench depth than the rest of the device. This reduces the height of the torsion pillars. Aluminum and silicon dioxide have been wet etched from
region R 2 . The figure is not drawn to scale. ~b! SEM micrograph.
da 3 , to first order, linearly varies with d d 1 . Since d d 1
!1 m m, the higher order terms of d d 1 in da 3 are small.
Thus the errors in the calibration parameters are also linearly
dependent on dw5d d 1 .
In the following, the applicability of the calibration
method is demonstrated by fabricating microinstruments
consisting of actuators and their corresponding calibrating
beams. The instruments are also employed to conduct a material test, namely, twisting of SCS bars until fracture. Each
bar is twisted by a pair of identical ~nominally! microinstruments. After the completion of the torsion tests, the instruments are calibrated, and their calibration coefficients are
used to back calculate the stresses to failure of the silicon
bars. The demonstration is an example of integrated microinstrumentation where the actuator, the material sample, and
the calibration mechanism are all designed, patterned, and
cofabricated.
IV. MICROINSTRUMENTS FOR TORSION TEST
Figure 5 shows the schematic of a pair of microinstruments ~L and R, not drawn to scale! for applying torque on a
SCS pillar. The pillar is attached to the substrate, while at the
top it is attached to a lever arm AB, released from the substrate. Each actuator is 1 mm apart from the lever arm. Two
of such pillars, designated by samples 1 and 2, are twisted
and fractured by two pairs of microinstruments.
Each microinstrument consists of an actuator with 2000
comb capacitors and a calibrating buckling beam. The actua-
tor has a rigid backbone, released from the substrate, and is
held by six beams ~springs!. It ~backbone! supports cantilever beams which carry the movable combs. A rigid frame, F,
is attached to the backbone of the actuator L. The calibrating
buckling beam is nominally 500 mm long. It is held by a
rigid support attached to the substrate. The free end of the
beam is nominally 1 mm away from the frame, F, of the
actuator. There is a vernier scale in the middle of the beam
which enables recording of the increments of the transverse
displacement, D, of the buckled beam with 0.8 mm resolution using an optical microscope. Thus, at a transverse deformation D54 m m a change of d can be measured with 20 nm
resolution.
Two microinstruments, L and R of Fig. 5, are identical
to each other. The instruments as well as the samples are
fabricated by the single-crystal reactive etching and metallization ~SCREAM! process.16,17 Torsion samples are formed
by partial release of silicon pillars. Region R 1 in Fig. 5 has a
lower trench depth than the rest of the device. This is
achieved by masking R 1 and by avoiding the second anisotropic silicon etch of the SCREAM process. Thus the height
of the sample pillars are kept short in order to ~1! reduce the
total angle of twist required for fracture, and more importantly, ~2! avoid metallization of the SCS torsion sample during sputtering, since a short sample pillar is veiled by the
overhang of the SCREAM sidewalls. SEM micrographs of
the fractured samples, shown later ~Fig. 7!, will reveal that it
is indeed the case. After fabrication of the instruments, alu-
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Rev. Sci. Instrum., Vol. 69, No. 3, March 1998
M. T. A. Saif and N. C. MacDonald
FIG. 7. ~a! A torsion sample and part of the actuators. ~b! Close up of the
torsion pillar. Also, the sample and the actuators are shown after stripping
off aluminum and silicon dioxide for clarity only. Testing is carried out with
aluminum and SiO2.
FIG. 6. ~a! A calibrating beam and a part of the actuator that it calibrates.
The cantilever beam is 500 mm long and 1 mm wide. Aluminum and silicon
dioxide have been stripped off from the beam and the surrounding frame.
Thus they are made of single-crystal silicon. The close up of the left free end
of the beam is shown in ~b!.
minum and silicon dioxide are wet etched from the region
R 2 . Thus the buckling beams are made of only SCS. Being
cantilevered, they are free from any net axial force.
The distance between the loading arm of the actuator
and the center of the torsion pillar is indicated by X in Fig. 5,
where X56 m m and 20 mm for the sample pillars 1 and 2.
Figure 5~b! shows the SEM micrograph of a microinstrument
with the calibration beam ~left end! and the torsion sample
~right end!. Figure 6 shows the details of a calibrating beam
and a close up of its left free end. Figure 7 shows the close
ups of a torsion sample and part of the actuators. Figure 7~a!
shows that the sample floor is at higher elevation @region R 1
of Fig. 5~a!# than the floor of the device. Figure 7~b! shows
the torsion sample before and after wet etching of aluminum
and silicon dioxide from the top and sides for clarity. Testing
is carried out with these films.
rapidly, while the actuators advance along the axial direction
slowly, thus twisting the torsional sample. The applied voltages on both the actuators, the transverse deformation, D, of
the buckling beams, and the angle of twist of the sample
@using the angular scale ~Fig. 5!# are recorded. Voltages are
increased until the pillars fracture. Figure 8 shows the twisting of a sample recorded under an optical microscope. Next,
the actuators are calibrated. It is noted that for the torsion
experiments the actuator displacement is small ~x2 d 1
,10 m m, 2L51000 m m!, and hence only K 1 and K 3 of the
actuator are of interest. Thus Eq. ~22! will be used for calibration. Calibration of one of the actuators, denoted by 1L,
is demonstrated in detail. The rest of the actuators ~1L, 2L,
and 2R! are calibrated identically.
First, the buckling load, P cr , for the calibrating beam of
1L is evaluated. Cross-sectional dimensions of the buckling
beam are measured using a calibrated SEM ~Stereoscan 440,
V. TORSION EXPERIMENT AND CALIBRATION
Torsion experiment and calibration are carried out under
optical microscope using a probe station. Two microinstruments are employed to apply torque on a sample pillar ~Fig.
5!. With the increase of voltage on each instrument, its actuator approaches the calibrating beam. As the gap, d 1 ,
closes, the actuator also touches the lever arm of the torsion
sample. The voltage is increased continuously on the actuator until the calibrating beam just buckles. The voltage on
both the actuators are then increased equally. Consequently,
the transverse deformation of the calibrating beam increases
FIG. 8. Twisting of a torsion sample recorded under an optical microscope.
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Rev. Sci. Instrum., Vol. 69, No. 3, March 1998
M. T. A. Saif and N. C. MacDonald
1417
FIG. 9. The monitor beam for the calibration beam of actuator 1L.
Leica Cambridge Inc.!. Its resolution is less than 5 nm, verified by measuring a standard grid. The cross section ~Fig. 9,
trapezoidal due to processing! can be obtained by cleaving
the calibrating beam. Instead, the dimensions are obtained
from a test cantilever beam, identical in design and placed
adjacent to the calibrating beam. Figure 9 shows the test
cantilever beam. Note that the angle of tilt has been considered while measuring the section. From the cross-sectional
dimensions and the length of the calibrating beam, the
critical buckling load is calculated, P cr54 p 2 EI/L 2 532.55
mN, where18 E5E $ 110% 5168 900 mN/mm2, I51.19 m m4,
and L5494 mm, instead of 500 mm ~there is a small grid at
the middle of the beam spanning 6 mm to hold the vernier
scale!. Note that the cross sections of all the beams coincide
with the ~110! crystal plane of SCS. E 110 is the modulus on
the ~110! plane along the normal direction. The gap, d 1 ,
between the actuator and the free end of the calibrating beam
is measured using an SEM ~Stereoscan 440, Leica Cambridge Inc.!. Note that the actuator approaches the beam
@Fig. 6~b!# at two locations. The gap for each location is
measured. They are 0.81 and 0.83 mm. d 1 50.82 m m is their
average value. Next, the buckling experiment is conducted.
With actuation by voltage V, the actuator touches the
calibrating beam, and compresses it. The beam buckles as
the compressive force exceeds P cr . The transverse deformation, D, of the buckled beam ~using the vernier scale, Fig. 5!,
and the corresponding voltages are recorded. Figure 10~a!
shows such a buckled beam where the circles indicate the
deformation predicted by Eq. ~2! ~see also Sec. VI!. Figure
10~b! shows a close up of a buckled beam near the vernier
scales. The micrographs are recorded by an optical microscope. Figure 11 shows the values of V as a function of D.
The end displacements, d, of the beam, and hence the actuator displacements, x5 d 1 d 1 , are calculated from Eq. ~9!.
Figure 12~a! shows V 2 as a function of x. Note that buckling
begins abruptly at V 2 '560 V2. This implies that there was
very little initial imperfection in the buckling beam. For, if
there were, buckling would be gradual. The solid line in Fig.
12~a! ~x5 d 1 50.82 m m to x55 m m! is the best-fit curve,
V 2 5a 0 1a 1 x1a 3 x 3 , where a 0 5504 V2, a 1 578 V2/mm,
and a 3 50.94 V2/mm3. The vertical line, L 1 L 2 , in Fig. 12~a!
corresponds to the voltage required to generate the buckling
force, P cr , and L 0 L 1 represents the force-displacement relation of the actuator prior to touching the calibrating beam.
FIG. 10. ~a! Buckling of a calibrating beam. The circles denote the deformation predicted by Eqs. ~36! and ~37!. ~b! Close up of a beam near the
vernier scales. Both ~a! and ~b! are recorded by an optical microscope.
Using Eq. ~22! and the values of a 0 , a 1 , and a 3 , the calibration parameters are evaluated: b 50.065 m N/V2, K 1
55 mN/mm, K 3 50.061 mN/mm3. All the four actuators, 1L,
1R, 2L, 2R, are calibrated similarly, and their parameters
are shown in Figs. 12~a!–12~d!. The calibration parameters
are then used to evaluate the force applied by the actuators
on the torsion samples. The actuators 1L and 1R were employed to fracture sample 1, while 2L and 2R to fracture
sample 2.
A. Errors in the parameters of actuator 1 L
The cross-sectional dimensions of the calibrating beam
are w51.06, h57.15 m m, c50.19 m m ~Fig. 3!. Then, from
FIG. 11. Maximum transverse deformation, D, of the calibration beam of
actuator 1L at different applied voltages. The deformation occurs at midlength of the beam.
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Rev. Sci. Instrum., Vol. 69, No. 3, March 1998
M. T. A. Saif and N. C. MacDonald
FIG. 12. Actuator displacements obtained at different applied voltages ~shown squared! for four actuators, 1L,..,2R. The solid lines are best-fit curves. The
verticle lines represent the voltages required to initiate buckling. The part of the curves on the left side of the verticle lines corresponds to the actuators’
displacements prior to touching the calibrating beams.
Eq. ~25!, dI52.98dw. The coefficients of Eq. ~21! are a 0
5504 V2, a 1 578 V2/mm, and a 3 50.94 V2/mm3. The maximum actuator displacement during calibration experiment,
D55.11 d 1 m m55.92 m m. It is estimated that the SEM
measurement error dw550 nm. Then from Eqs. ~25!, ~26!,
~29!, ~31!, and ~32!, and b 50.065 m N/V2, K 1 55 mN/mm,
K 3 50.06 mN/mm3, I51.19 m m4, P cr532.55 m N, and d 1
50.82 m m,
dI50.149 m m4,
da 1 50,
d P cr54.07 m N,
da 0 53.9 V2,
da 3 50.027 V2/mm3,
d b 50.0086 mN/V2,
dK 1 50.67 mN/mm,
dK 3 50.0098 mN/mm3.
Thus the percentage errors in b, K 1 , and K 3 are 13%, 13.5%,
and 16%, respectively.
B. Estimation of the linear spring constant, K 1
The linear spring constant, K 1 , of the actuator can be
estimated from the stiffness of the individual beams or
springs, and compared with the calibrated value. The stiffness, k 1 , of a beam, clamped at both ends, is given by8 k 1
512EI/L 3 , where L is the length of the beam, and E is the
modulus of elasticity of the material of the beam, I is the
moment of inertia of the cross section about an axis normal
to the direction of displacement. For a composite beam, EI
should be replaced by ( i E i I i , where E i and I i are the modulus of elasticity, and the moment of inertia of the ith material
section. Here, one of the beams of the actuator 1L is cleaved
to evaluate its spring constant and assume that the rest of the
five beams have similar constants. Figure 13 shows the micrograph of the cleaved section. The inner core is SCS,
coated on top and sides by SiO2 and sputtered aluminum.
The modulus of Al and SiO2 are close to 70 GPa4 and hence
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Rev. Sci. Instrum., Vol. 69, No. 3, March 1998
M. T. A. Saif and N. C. MacDonald
FIG. 13. Cleaved section of one of the six supporting beams ~springs! of
actuator 1L. Its cross-sectional dimensions are estimated to evaluate the
linear spring constant of the actuator, which is compared with the calibration
value.
they are taken as one material. The cross-sectional dimensions of the beam are estimated from the micrograph which,
together with L5500 m m, E SCS5168.9 GPa ~normal to 110
plane!, and the values of the elastic modulii of Al and SiO2,
give k 1 '0.65 mN/mm. Thus the spring constant of the actuator, K 1 56k 1 53.9 mN/mm, in contrast to the calibrated
value K 1 55 mN/mm. Note that the purpose of this comparison is simply to have an independent estimation of calibration, not validation.
C. Applied torque
Force, p, applied by the actuator at voltage, V, on the
lever arm of the torsion sample is
p5 b V 2 2 P2R
5 b V 2 2 P cr2 P cr
2L
2K 1 ~ d 1 d 1 ! 2K 3 ~ d 1 d 1 ! 3 . ~33!
In the torsion experiment, the axial motion of the actuator
was small prior to fracture, and hence the contribution of the
cubic term in Eq. ~33! is small. Torque, T, applied by the
two actuators ~e.g., 1L and 1R! on the torsion sample is
T5 ~ p L 1p R ! X,
FIG. 14. Angle of twist of two torsion samples at different applied torques
until fracture. The torques are computed from the applied voltages on the
actuators and their calibration parameters.
corresponding actuator displacements are 0.8 mm and 0.75
mm, respectively. Thus the applied torque is T 2
511848 mN mm.
An approximate shear stress to failure of the samples is
evaluated as follows. It is noted that the cross section ~parallel to the wafer surface! of the torsional sample pillars coincide with the ~100! direction of SCS. Also at mid-height of
the pillars, their cross-sectional area is minimum and is approximately a square. The minimum cross-sectional dimensions are estimated prior to fracture by a calibrated SEM
~Stereoscan 440, Leica Cambridge Inc.!. They are (1
31 m m2) and (1.531.6 m m2) for the samples 1 and 2, respectively.
The maximum shear stress on the cross section ~~100!
plane!, t max , along the edge of the rectangle is given by19
t max5
d
~34!
where p L and p R are the forces due to the left and right
actuators. Here X56 m m for sample 1, and 520 m m for
sample 2. Figure 14 shows the torque applied on the two
samples as a function of the angle of twist. The angle is
measured directly from the angular scale ~Fig. 5! as well as
from the value of d. The two angles match each other.
Voltages applied on the actuators 1L and 1R prior to
fracture of the sample 1 are 46 and 43.6 V, respectively. The
corresponding displacements, d, of the actuators are 0.5 and
0.425 mm. Equations ~33! and ~34!, together with the calibrated actuator parameters ~Fig. 12!, give the torque applied
on sample 1 prior to failure: T 1 51174 mN mm. Similarly,
prior to fracture of sample 2, the applied voltages on actuators 2L and 2R are 36 V ~same for both actuators!. The
1419
TG
,
b3
~35!
where G54.803 for a square section, and b is the width of
the section. Thus t max55653 MPa for sample 1 and t max
52630 MPa for sample 2 ~using b51.5 m m!. Figures 15 and
16 show the fractured surfaces of samples 1 and 2 from two
different angles. Note that there is no aluminum on the
samples except near the bottom. Aluminum did not reach the
samples during sputtering due to the overhang of the sidewalls ~Fig. 7!. From Figs. 15 and 16 it is clear that the failure
occurred along an inclined plane, most likely the ~111! plane
for which the surface energy is 1.23 J/m2 in contrast to ~100!
plane, where it is20 2.13 J/m2. Due to the inclined fracture
plane, two cusps are formed in case of the shorter sample 2.
These cusps are part of the lever arms of the torsion sample
which were released from the substrate prior to the torsion
experiment ~Fig. 17!. When the sample is tall and/or narrow,
as in case of sample 1, such cusp are small ~Fig. 15!, or they
may not form at all. The entire fracture surface is contained
within the length of the pillar.
Figure 14 shows that the torque-twist curve for the SCS
torsion samples are linear until failure, implying very little or
no nonlinear elastic or plastic deformation. This is attributed
to apparent low stress to failure @less than 10% of the shear
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1420
Rev. Sci. Instrum., Vol. 69, No. 3, March 1998
M. T. A. Saif and N. C. MacDonald
FIG. 17. The formation of two cusps on the fracture surface ~Fig. 16! of
sample 2 is explained. ~a! Micrograph of a thick and short torsion sample,
~b! schematic of the sample and the fracture surface. The point S on the
cross section has the highest shear stress.
FIG. 15. Fracture surface of sample 1, seen ~a! from 30° angle with verticle
and ~b! from 45° with verticle.
modulus ~80 GPa! for ~100! plane of SCS21,22#. The failure
might have initiated from a surface defect causing stress concentration.
Finally, note that the forces applied by the two actuators
on the lever arm of a given torsion sample are unequal, for
example, p L 598.2 m N and p R 597.9 m N for sample 1 prior
to fracture. The force differential, Dp5 u p L 2 p R u , induces a
small additional uniform shear stress on the cross section. It
also induces a small moment, M 5DpDz, on the pillar ~Fig.
18!. However, the moment arm, Dz, is small (,0.5 m m),
because the SCREAM beams are slightly tapered and the
point of contact between the lever arm of the torsion sample
and the loading arm of the actuator is towards the lower end
of the beam, as shown in Fig. 18 ~also see SEM micrograph
of Fig. 7!. Thus the effect of moment is also small compared
to the applied torque ( p L 1 p R )X, X56 or 20 mm, and the
above experiment generates almost pure torsion on the
sample.
VI. REMARKS
Here the major assumptions of the calibration method,
its limitations, and the design considerations of the calibrating beam are highlighted.
FIG. 16. Fracture surface of sample 2, seen ~a! from top, and ~b! from 45°
angle with verticle.
FIG. 18. The cross section of the torsion sample and the loading arm of one
of the two actuators are shown. Due to a small tapering of the SCREAM
beams, the point of contact between the loading arm and the lever arm of the
torsion sample is close to the neck region of the sample. Hence the flexural
moment arm, Dz, is small.
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Rev. Sci. Instrum., Vol. 69, No. 3, March 1998
M. T. A. Saif and N. C. MacDonald
~1! It is assumed that the first mode of buckling of the calibrating beam is a cosine curve, and an initially straight
beam ( j 50) deforms after buckling according to @Eq.
~2!#
D
y5 @cos~vs!21#,
~36!
2
where s is measured along the axis of the beam, with s
50 at the left support. The horizontal projection, x(s),
of the beam can be evaluated from
E EA
s
s
0
0
x~s!5 dx5
12 ~ dy/ds ! 2 ds.
~37!
Figure 10~a! shows the coordinates of a buckled beam by
circles, evaluated from Eqs. ~36! and ~37!, where L
5500 m m and D533.5 m m are used. D is obtained
from the deformation (AB) of the experimentally buckled beam shown in the micrograph. The actual buckled
beam is also shown in the figure. Clearly the cosine
buckled shape is a reasonable assumption even when the
transverse deformation, D, is large. It is, however, found
that x(s5500 m m)5494.41 m m, whereas the distance
between the ends of the buckled beam measured from
the micrograph is X5494.15 m m. The overestimation of
the predicted value is expected, since the axial compression of the beam due to the buckling load is neglected.
Close correspondence between the predicted and the experimental deformations of the buckled beam implies
that the beam deforms primarily at its first mode. Any
higher modes, if present due to imperfections, have negligible components in the deformation.
~2! In the model presented in this article, it is assumed that
the length of the calibrating beam does not change during buckling due to compression, i.e., the axial deformation of the beam is negligible compared to the axial displacement, d. For long slender beams this is a reasonable
assumption.9 However, if axial deformation becomes
important, then one needs to conduct a nonlinear finite element analysis of buckling to obtain the relation
P5P( d ) and d 5 d (D) numerically instead of Eqs. ~7!
and ~8!. Thus b V 2 5 P1R5 P( d )1 ( i51,3,..K i d i . The
experimentally obtained data of V and d (D) can be best
fitted to V 2 5a 0 P( d )1 ( i51,3,...a i d i giving b 51/a 0 , and
K i 5a i b .
~3! The accuracy of the calibration depends on the precision
of the calibrating beam. Thus, its prescribed boundary
conditions, and its uniformity along the length must be
ensured. The beam must be free from any axial compressive force other than the force applied by the actuator
during actuation. The elastic modulus of the material
must also be well defined.
~4! It is assumed that the compliance of the backbone of the
actuator of Fig. 5 is negligible, i.e., the springs do not
deform when the actuator applies compressive force P
, P cr on the calibrating beam. Such deformation is indeed negligible compared to the post buckling deformation, d, of the springs, provided the backbone and its
attached frame F ~Fig. 5! are rigid and P cr is small. One
can, however, reasonably account for such small displacement by using d 5 d e 1 p 2 D 2 /4L, where d e is the
1421
displacement of the actuator prior to buckling of the
beam. Note that if there is an initial gap, d 1 , between the
actuator and the buckling beam, then the actuator displacement is d 1 d 1 .
~5! While calibrating the actuator 1L for b, K 1 , and K 3 ,
only the first two terms in the parenthesis of Eqs. ~17!
and ~18! are retained, and the term G53 @ (x2 d 1 )/2L # 2
125/2@ (x2 d 1 )/2L # 3 as well as the higher order terms
are ignored. Such minor simplification is justified since
K 3 x 3 @G P cr within the range of interest of x(0 – 5 m m).
~6! The methodology developed here is readily applicable to
actuators that generate force, F5 b I, where I is a function of the input, e.g., I5V 2 for comb actuators, and
temperature for thermal actuators.
~7! The calibrating beam should be designed such that its
buckling load, P cr , is well within the force range of the
actuator.
~8! The fracture experiments reported in this paper are intended to demonstrate the application of microinstruments for material characterization. The data of the failure stresses should not be considered as conclusive.
~9! An optical microscope and a probe station were used to
carry out the calibration and torsion experiments to demonstrate the simplicity and flexibility of the experimental
setup. An SEM could be used to carry out the
measurements,23 or capacitine sensors can be designed to
measure the displacements.
~10! The method of calibration presented here needs verification by an independent check using a direct force
sensor. Work is currently underway, and will be reported shortly.
ACKNOWLEDGMENTS
The work was supported by ARPA and the National Science Foundation. The computing resources of the Materials
Science Center, Cornell University, were used to carry out
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