JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 6, NO. 3, SEPTEMBER 1997 257
Electrostatic Curved Electrode Actuators
Rob Legtenberg, John Gilbert, Stephen D. Senturia, Fellow, IEEE, and Miko Elwenspoek, Associate Member, IEEE
Abstract— This paper presents the design and performance
of an electrostatic actuator consisting of a laterally compliant
cantilever beam and a fixed curved electrode, both suspended
above a ground plane. A theoretical description of the static
behavior of the cantilever as it is pulled into contact with the
rigid fixed-electrode structure is given. Two models are presented:
a simplified semi-analytical model based on energy methods,
and fully three-dimensional (3-D) coupled electromechanical nu-
merical simulations using CoSolve-EM. The two models are
in qualitative agreement with each other, and predict stable
actuator behavior when the beam deflection becomes constrained
by the curved electrode geometry before electrostatic pull-in can
occur. The pull-in behavior depends on the shape of the curved
electrode. Test devices have been fabricated by polysilicon surface
micromachining techniques. Experimental results confirm the
basic theoretical results. Stable behavior with relatively large
displacements and forces can be generated by these curved
electrode actuators. Depending on the design, or as a result of
geometrical imperfections, regions of unstable (pull-in) deflection
behavior are also observed. [212]
Index Terms—Actuator, electrode shape, electrostatic, theoret-
ical model(ling).
I. INTRODUCTION
E
LECTROSTATIC actuation is very attractive for mi-
croelectromechanical systems because of good scaling
properties to small dimensions, high-energy densities, and
relative ease of fabrication. However, electrostatic actuators
which are able to generate relatively large displacements and
large forces are difficult to design as a result of a geometric
discrepancy. Large-displacement actuators (e.g., comb drive
structures) require displacements perpendicular to the major
field lines, leading to small forces. Large-force actuators
(e.g., parallel-plate structures) require small gaps and a dis-
placement in the direction of the major field lines, thus
implying small displacements. Several actuator designs have
been reported employing curved structures in order to generate
large displacement and large forces. A curved electrode has
been applied in a microactuator for aligning optical fibers
[1]. Actuators have been presented where a large vertical
displacement is obtained by an S-shaped film sandwiched
between planar electrodes [2]. Another design employs a
deformed membrane which is pulled against a glass plate by
electrostatic forces [3]. Also, active joints have been proposed
Manuscript received May 17, 1996; revised December 4, 1996. Subject
Editor, N. de Rooij. This work was supported in part by the Dutch Technology
Foundation (STW) and by the Advanced Research Projects Agency under
Contract J-FBI-92-196.
R. Legtenberg and M. Elwenspoek are with MESA Research Institute,
University of Twente, 7500 AE Enschede, The Netherlands.
J. Gilbert and S. D. Senturia are with the Microsystems Technology
Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139
USA.
Publisher Item Identifier S 1057-7157(97)05809-5.
that employ a bent beam electrode that is pulled against a rigid
counter-electrode [4]. Furthermore, a distributed electrostatic
microactuator using wave-like electrodes [5] and an electro-
static moving-wedge actuator for use in a microrelay [6] have
been presented. All these actuators use curved structures with a
specific shape that are deflected by electrostatic forces toward a
counter-electrode, and generate displacements that are normal
to the wafer surface.
In order to investigate the basic phenomena of these ac-
tuators, the static behavior of cantilever beam structures that
are deformed by electrostatic forces along curved electrodes
has been studied. Special attention has been given to the
effect of the electrode curvature on the static behavior of the
actuators. The dynamic properties of comparable structures
have been presented elsewhere [7]. Test devices, consisting
of a laterally compliant cantilever beam and a fixed curved
electrode, both suspended above a ground plane, have been
fabricated by polysilicon surface micromachining techniques
[8]. Experimental data from these structures are compared with
theoretical results.
II. D
ESIGN
Fig. 1 shows the basic design of the cantilever beam and
the curved electrode. The gap distance between electrodes is
small near the clamped edge of the beam and increases with
the position along the length of the beam. When a voltage is
applied across the gap, an electrostatic force is created that
deforms the beam along the outline of the curved electrode.
The displacement is parallel to the wafer surface. In this way,
the shape of the curved electrode can be easily adjusted by
changing the mask design. To prevent a short circuit between
the beam and the curved electrodes, electrical insulation is
required, e.g., by applying a dielectric layer between the
structures or by using stand-off bumper structures that prevent
physical contact of the electrodes. While this work focuses on
cantilever beam structures, similar actuators can be fabricated
using microbridges or membranes.
Simple polynomials normalized to the maximum cantilever
tip deflection have been used for the shape of the curved
electrode
which can be described by the following
expression:
(1)
where
is the maximum gap distance of the curved
electrode,
is the position along the -axis, is the length of
the beam, and
is the polynomial order of the curve, .
For different values of
, the electrode shape is shown in
Fig. 2. As will be shown in the next section, the performance
1057–7157/97$10.00 1997 IEEE
258 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 6, NO. 3, SEPTEMBER 1997
Fig. 1. Schematic view of the curved electrode actuator.
Fig. 2. Normalized graph showing the shape of the curved electrode as a
function of
x
[see (1)].
of these actuators depends on the electrode curvature and can
become unstable beyond the so-called pull-in voltage.
III. A
CTUATOR MODELS
Two actuator models are presented. The first is a two-
dimensional (2-D) model based on analytical energy methods,
supported by numerical determination of required coefficients.
Of necessity, this approach requires several major approxima-
tions such as the neglect of fringing fields and omission of
the effect of the nearby ground plane. The second model is
a fully three-dimensional (3-D) self-consistent electromechan-
ical numerical simulation using the MIT MEMCAD System,
specifically, the CoSolve-EM module [9]. The energy approach
offers the benefit of analytical insight into structural aspects
of the design; the 3-D simulations provide the ability to check
the energy-model results, and to examine issues not included
in the energy model.
A. Energy Model
1) Unloaded and Unconstrained Static Behavior: Because
the gap spacing is small with respect to the electrode length it
is assumed that the electric field exhibits a one-dimensional (1-
D) field line distribution, i.e., a parallel-plate approximation.
As stated above, fringing fields are ignored.
When a dc polarization voltage is applied between the
capacitor electrodes, an electrostatic force is developed that
is inversely proportional to the square of the gap spacing.
This makes the force dependent on the deflection, leading to
nonlinear behavior. The static deflection
of a prismatic
beam with a transverse pressure
can be described by
the following nonlinear differential equation:
(2)
where
denotes the static electrostatic force per unit
beam length as a function of the position
and the drive
voltage
, is the bending stiffness, is the thickness of
the insulator,
the dielectric constant in air, the width of
the beam,
is the dielectric constant of the insulator, and
the shape of the electrode as a function of the position .
An analytical closed-form solution of the above equation
cannot be found. A simplified model is developed based
on the Rayleigh–Ritz method with small-deflection theory,
where an approximate solution to the differential equation is
constructed in the form of admissible trial functions containing
undetermined parameters, the parameter values being found
by a variational minimization of the total potential energy
[10], [11]. The total potential energy, denoted by
, can be
expressed as
(3)
where
and are, respectively, the strain energy of bending
and the electrostatic potential energy (the first integral of the
electrostatic force
, given by
(4)
and
(5)
The deflection profile of a uniformly loaded cantilever beam
has been used for the admissible trial function
(6)
where
is a constant to be determined. This function satisfies
the full set of boundary conditions of the beam and is therefore
LEGTENBERG et al.: ELECTROSTATIC CURVED ELECTRODE ACTUATORS 259
TABLE I
C
ALCULATED PULL-IN VOLTAGES FROM THE ENERGY MODEL,
FOR
POLYSILICON (
Ey
=150
GPa) CANTILEVER BEAMS
(
h
3
t
3
L
=5
3
2
3
500
m) WITH A MINIMUM GAP SPACING
d
of 2
m AND A
MAXIMUM DEFLECTION
max
of 30
m. A RELATIVE DIELECTRIC
CONSTANT OF THE INSULATOR
"
r
EQUAL TO
1
HAS BEEN USED
expected to give an excellent correlation between the results
from the exact solution of the differential equation and the
approximate solution [10].
The system is in equilibrium when the first variation of the
total potential energy with respect to the constant
equals zero.
Whether this equilibrium is stable or unstable is determined
by the second variation of the potential energy with respect to
. At the transition from a stable to an unstable equilibrium
both the first and the second derivatives of the potential
energy with respect to
are zero. Solving these two equations
simultaneously yields the pull-in voltage of the cantilever
and an implicit expression for the constant at pull-in
(7)
(8)
The calculated pull-in voltages for different polynomial orders
obtained by this energy method are listed in Table I.
Equations (7) and (8) give little insight in the effect of
different parameters. Some aspects will be discussed here.
Increasing the polynomial order
of the electrode curve
decreases the pull-in voltage while the maximum displacement
just before pull-in stays about the same. Thus by using curved
electrodes, the pull-in voltage can be lowered significantly,
resulting in large amplitude motion at lower driving voltages
as compared to a parallel-plate structure
. At voltages
above the pull-in voltage, the displacement cannot be con-
trolled because of the unstable pull-in event. The maximum tip
displacement at pull-in is independent of the beam properties
and only depends on the gap geometry. The tip displacement
at pull-in, calculated from the examples in Table I, is about one
third of the maximum gap spacing
which is comparable
to the behavior of a lumped parallel-plate spring model (see,
for example, [12]). The pull-in voltage strongly decreases
either with decreasing initial gap spacing at the clamped edge
of the beam or with increasing dielectric constant of the
insulating layer between the electrodes.
Fig. 3. Sketch of the constrained beam deflection model with external
loading force
P
.
2) Constrained Static Behavior: The pull-in voltage de-
creases with increasing polynomial order. However, for
polynomial orders above two it was found that the deflection
profile of the beam becomes constrained by the geometry of
the curved electrode before the pull-in voltage is reached. For
this situation, the model needs to be adjusted. In addition, a
force
, that is acting on the tip of the beam, has been added
in order to perform external work. This force will deform
the beam wherever it is not yet in contact with the curved
electrode.
It is assumed that the beam will be partly in contact with
the curved electrode and will partly be free standing and is
clamped at point
, as sketched in Fig. 3. The problem will
thus have a variable-boundary condition with respect to the
free-standing length of the beam. Therefore, the system must
be divided into two regions. From the clamped edge of the
beam to point
the beam is assumed to be in physical contact
with the curved electrode. Thus the deflection profile
will
be equal to the shape of the electrode
and the distance
between the electrodes is equal to
, the thickness of the
insulator. Beyond point
, the beam is free and is deflected
by electrostatic forces and by the external force
.
The expressions for the strain energy of bending and the
electrostatic potential energy become
(9)
and
(10)
An additional term, the work from the external force acting
on the tip of the beam, must be added to the total potential
energy [see (3)], and is given by
(11)
The admissible trial function of the deflection profile of the
cantilever beam now also depends on the contact distance
and applied force
for (12)
for (13)
260 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 6, NO. 3, SEPTEMBER 1997
Fig. 4. Modeled tip deflection as a function of the applied voltage for several
polynomial orders
n
of the curved electrode. When the polynomial order is
larger than
2
the behavior becomes stable up to the maximum tip deflection.
Otherwise, a pull-in voltage exists. Variable settings that have been used are:
E
y
=150
GPa,
h
=5
m,
t
=2
m,
L
= 500
m,
d
=2
m,
max
=30
m, and
"
r
=1
.
This system is in equilibrium when the first variation of
the potential energy with respect to the contact distance
equals zero and the first variation of the potential energy with
respect to the constant
equals zero. Solving both equations
simultaneously by numerical iteration gives the values of
and at any given applied voltage.
The tip deflection versus driving voltage for unloaded
designs with a polynomial order ranging from two to four is
shown in Fig. 4. For designs with a polynomial order above
,
stable behavior (no pull-in) up to the maximum tip deflection
is found. This is the result of the constrained beam deflection
which makes the beam zip along the curved electrode as the
voltage is increased.
The boundary between constrained and unconstrained be-
havior can be more easily understood when only the collapsed
part of the beam is considered (i.e.,
). In that case,
only the first terms on the right-hand side of (9) and (10) are
considered. The dependency of the system on the polynomial
order
can be found by looking at the first and second
derivatives of the total potential energy with respect to this
polynomial order
. A short mathematical exercise which,
for the sake of brevity, is not included here shows that; the
second derivative of the total potential energy with respect to
the polynomial order
for this simplified situation becomes
negative for
indicating unstable behavior; the second
derivative of the total potential energy with respect to
is
zero for
indicating an extremum; and that the second
derivative of the potential energy with respect to
becomes
positive
indicating stable behavior.
3) Force Generation: The force generated by the actuator
is a function of the displacement of the tip. It can be found
numerically by setting the first variations of the potential
energy with respect to the contact distance
and the constant
to zero at an assumed tip displacement. The resulting external
force as a function of tip position is shown in Fig. 5. Actuator
dimensions are given in the figures; the effect of the dielectric
Fig. 5. Theoretical force generation of a quadratic order electrode curvature
as a function of driving voltage. The tip deflection is fixed at 0, 10, and 20
m. Also the effect of the relative dielectric constant of the insulating layer
is shown. Unless given in the graph, variable settings equal to Fig. 4 have
been used.
Fig. 6. Deflection profiles of the cantilever beam in Fig. 5 for several driving
voltages when the tip is constrained at zero deflection.
constant of the insulating layer between the electrodes is
also shown. Forces are typically a few micronewtons for
this example but increase with decreasing gap distance and
with increasing dielectric constant of the insulating layer. The
deformation of the beam is illustrated in Fig. 6, for the case
of the tip fixed at zero deflection. When the actuator is loaded
by displacement-dependent forces, for example, a spring, the
problem can be solved by substitution of the appropriate force-
displacement relation.
B. 3-D Coupled Electromechanical Simulations
In order to study 3-D effects like fringing fields and the
effect of the ground plane (see Fig. 7), simulations have
been performed using CoSolve-EM [9]. CoSolve-EM is a
software package that is capable of doing self-consistent
electromechanical analysis of complex 3-D structures. The
approach is based on a relaxation scheme combining a fast
multipole-accelerated scheme for the electrostatic analysis
(FASTCAP [13]) with a standard finite-element method for
the mechanical system analysis (ABAQUS [14]).
LEGTENBERG et al.: ELECTROSTATIC CURVED ELECTRODE ACTUATORS 261
Fig. 7. Example of a geometric model used in the CoSolve-EM simulations (order
=2
).
In the model of Fig. 7, is along the length of the beam,
is in a direction normal to the ground plane, and the
principal motion of the beam is in
-direction toward the
curved electrode. Because the cantilever beam contacts the
curved electrode, an interface had to be inserted between the
electrodes using CoSolve-EM. And because of convergence
problems in the presence of contact with the present version
of CoSolve-EM, the levitation in
direction of the beam,
which results from a lack of balance between fringing-field
forces on the top and bottom surfaces of the cantilever, had
to be suppressed [9]. The calculated tip deflection versus
driving voltage for quadratic, cubic, and fourth-order electrode
curvatures with an initial gap distance of 1
m are shown in
Fig. 8, together with results from the energy model.
It can be concluded that the results from the energy model
and the 3-D coupled electromechanics are qualitatively in good
agreement with each other. But there are two effects which
compete. The energy model neglects fringing fields. Therefore,
one would expect the 3-D simulations, which add the fringing
fields to the problem, to show larger displacements and lower
pull-in voltages. However, comparing the two models suggests
the opposite, reduced tip displacements and increased pull-in
voltage compared to the energy model. It is readily shown from
Fig. 8. Results of the unloaded tip deflection versus voltage behavior for the
energy model using a parallel-plate approximation and the 3-D finite-element
model in the presence of the ground plane. The dielectric constant of the
spacer is
1
, and a Youg’s modulus
E
y
of 150 GPa, a Poisson’s ratio
of
0
:
3
, beam dimensions
(
h
3
t
3
L
)
of
5
3
2
3
500
m, a minimal gap spacing
d
of 2
m, and maximum gap spacing
max
of 30
m have been used in
the calculations.
a comparison of 3-D simulations with and without the ground
plane that if fringing fields are included, then the effect of
![Fig. 2. Normalized graph showing the shape of the curved electrode as a function of x [see (1)].](/figures/fig-2-normalized-graph-showing-the-shape-of-the-curved-3cjc66px.webp)
