Computer-Aided Design for Microelectromechanical Systems (MEMS)
Y. C. Lee1, B. McCarthy2, Jiankuai Diao2 and Zhongxia Zhang3, and K. F. Harsh4
1
Professor, Department of Mechanical Engineering
2
Graduate Research Assistant, Department of Mechanical Engineering
3
Graduate Research Assistant, Department of Chemical Engineering
4
Research Associate, Department of Mechanical Engineering
NSF Center for Advanced Manufacturing and Packaging of
Microwave, Optical, and Digital Electronics (CAMPmode)
University of Colorado, Boulder, CO 80309-0427, USA
leeyc@colorado.edu (e-mail address)
Abstracts
With the advancements of MEMS foundry
services and CAD tools, MEMS devices can be costeffectively designed and prototyped. Here, four designs
that utilize these tools are presented: 1) a flexure design
used to reduce the device warpage resulting from the
mismatch in thermal expansion coefficients between the
device and the substrate for flip-chip bonded MEMS, 2)
a digitally positioned micro-mirror using multiple
contacts, 3) a large MEMS flap that achieves uniform
movement for fluid mixing, and 4) a three-dimensional,
solder-assembled MEMS device that has been optimized
for minimal deviation from the desired assembly angle.
Keywords: Microelectromechanical systems,
computer-aided design, Coventor, optimization, optical
MEMS, flexure
I. INTRODUCTION
Microelectromechanical systems (MEMS)
technology enables the fabrication of both sensing and
actuating devices that can be integrated with
microelectronic, optoelectronic and microwave devices
to create advanced microsystems. These devices,
fabricated using semiconductor processing techniques,
are very low cost while still achieving complex and
precise nano-scale movements. Hundreds of MEMSbased sensors and actuators have been demonstrated, and
the number of applications for them is growing [1]. Also,
these micro-mechanical devices will be enabling
components for many novel microsystems to be
developed in the 21st century. The following quote from
a US National Research Council’s 1997 report describes
the potential impact of MEMS well. “Many people in the
field of microelectromechanical systems (MEMS) share
the belief that a revolution is under way. As MEMS begin
to permeate more and more industrial procedures, not
only engineering but society as a whole will be strongly
affected. MEMS provide a new design technology that
could rival, and perhaps even surpass, the societal
impact of integrated circuits (ICs).” [2].
In general, there are four major activities critical
to MEMS advancement. They are foundry fabrication,
computer aided design (CAD), packaging and reliability.
1
During the last several years, progress in the
establishment of foundry services and CAD tools has
been very impressive. Figure 1 shows the cross-section
of a well-known foundry process, MUMPs (Multi-User
MEMS Processes), with its polysilicon and silicon oxide
layers [3]. The oxide layers are sacrificial layers and are
removed with HF after fabrication. MUMPS is only one
example of the many surface micro-machining, bulk
micro-machining, and LIGA foundry processes available
today though.
Several foundry services, including MUMPs,
also offer a cost-sharing program that significantly
reduces low-volume prototyping costs. Rather than
committing to a whole wafer, a research group can
submit a design for a 1cmX1cm square. An example of
such a design square is shown in Figure 2. That design is
then integrated with others on the same wafer and the
cost is split accordingly. Our research center takes this
concept a step further. As can be seen in Figure 2, the
1cmX1cm square is divided up into 25 2mmX2mm
sections that are allocated to individual researchers
within the center. Thus, after the fabricated chips are
received from MUMPs, they are sub-diced again to
provide each researcher with his/her own design. In this
way, an individual at an academic institution can
fabricate a prototype for approximately $128 using the
MUMPs process.
With foundry services well established, MEMS
design, prototyping and manufacturing procedures are
very similar to those established for application-specific
integrated circuits (ASIC.) With this new manufacturing
capacity, there is a need for a large number of educated
MEMS designers who can design MEMS for integrated,
application-specific microsystems. Fortunately, the
newest versions of MEMS CAD tools are so userfriendly that any engineer can apply them to MEMS
design after studying about 10 tutorials. As proof of this,
we have developed a MEMS design course at the
University of Colorado at Boulder. In only one semester,
most of the students were able to learn and apply the
CAD tool to the synthesis and analysis of original and
useful MEMS designs.
Four such MEMS design cases are presented in
the following sections. The first three designs are
representatives of term projects carried out by the
students taking the MEMS design course. The first case
considers the flexure design that is the top concern in
almost every MEMS device design. In particular, a
flexure is optimized to reduce the effect of the thermal
expansion coefficient mismatch between a silicon device
and a ceramic substrate used for packaging. The second
case illustrates the use of multiple contacts to achieve
digital positioning for a novel micro-mirror. The third
design focuses on the thermo-mechanical behavior
through a design for MEMS flaps for enhanced fluid
mixing. All the design work is carried out using
Coventor’s CoventoreWare [4].
In addition, a fourth design case is presented
with an automated design optimization procedure for a
three-dimensional, solder-assembled MEMS device.
Such an automated procedure is not available in the CAD
tools today, but we would like to present this case since
the procedure is so important that it should be part of
most MEMS design activities. We expect MEMS CAD
tools to include such a design optimization option in the
near future.
II. FLEXURE DESIGN
This first case details the optimization of a
flexure design that is used in many MEMS devices.
Appropriate flexure designs are critical for MEMS
because they are currently the only way to reliably allow
for device movement and actuator control. Other options,
such as rotating hinges and sliding contacts, are not yet
reliable enough for the majority of MEMS applications
[5]. However, due to processing limitations, it is often
difficult to design flexures that are compliant enough in
the desired direction while still providing enough
resistance in other directions to prevent undesirable
movement and stiction.
Figure 3 illustrates a typical case where a good
flexure is needed [6]. Due to a difference in thermal
expansion coefficients between the flip-chip bonded
MEMS device and the substrate, cooling after bonding
can cause the MEMS to buckle. As shown in Figure 3,
flexures that could keep the device planar and intact
would be extremely useful. To keep the device planar
and intact, the flexure must be compliant in the in-plane
axial direction while still being stiff enough in the out-ofplane z-direction to prevent stiction. The following
optimization provides general trends in the flexure
stiffness ratio that can be used in flexure design.
II.1 Method
Figure 4 shows the flexure design chosen for
optimization. The optimization is done with AutoSpring,
a CoventorWare module, by individually varying the
fold length, fold spacing, and number of folds. These
parameters are shown in Figure 4. AutoSpring provides
an accurate representation of a typical flexure application
because it moves the free end of a flexure through a
range of specified displacements with a zero-slope
2
constraint at the free end of the flexure. The constraint is
typical of MEMS flexures because they are normally
attached to a larger moving plate (see Figure 3). For each
displacement in AutoSpring, a reaction force and a
spring constant are calculated. The spring constant
values are then fitted to a second-degree polynomial so
that nonlinear and linear springs can be analyzed.
For the optimization, each variation of the
flexure design is submitted to AutoSpring where it is
subjected to displacements of -10µm, -5µm, 5µm, and
10µm in both the vertical and axial directions. From the
resulting spring constant polynomial, only the linear k1
value is used for comparison because the k0 coefficient is
zero without the presence of residual stress and the k2
coefficient is generally several orders of magnitude less
than k1, indicating that the nonlinear effect is fairly small.
This agrees with the expectation that the flexures behave
elastically and linearly.
The goal of the optimization is to maximize zstiffness or bending stiffness while minimizing x or axial
stiffness. The stiffness ratio of z-stiffness to x-stiffness is
calculated and plotted for each case to better observe
trends in the flexure behavior. In this case a higher
stiffness ratio is desirable because maximum xdisplacement for a given z-displacement means the
flexure can absorb a significant mismatch in thermal
expansion coefficients without succumbing to stiction.
II.2 Results
Before proceeding with the optimization, a
partial validation of the model is done by comparing the
CoventorWare and analytical results for a basic
cantilever beam. This validation confirms that the 2 x 2
x 1.5µm 27-node, brick element size is acceptable for
predicting displacements. Following the model
validation, a total of 36 separate analyses are carried out
for the optimization. A total of four fold lengths, 10, 30,
50, and 70µm are used. For each fold length, three fold
spacings are used, and for each fold length and fold
spacing combination, three configurations using one
fold, two folds, and three folds respectively are analyzed.
These parameters are displayed in Figure 4 for reference.
Finally the distance from the fixed end to the free end of
all flexures is kept to 50µm for the sake of comparison
and because many flexures are designed to fit in a predefined space. The only exception is the case with three
folds and a 10µm fold spacing, which takes up more
room by definition. Surface plots of the z-stiffness to xstiffness ratio for all cases are shown in Figure 5, where
a cubic interpolation formula is used to plot the surface
between data points.
The plots in Figure 5 indicate that fold length is
the most critical parameter in determining the stiffness
ratio. For example, increasing the fold length from 10µm
to 70µm for the 1 fold case with a 5 µm fold spacing
provides a 2000% increase in the stiffness ratio. The
dominance of the fold length parameter is partly because
the values used cover a much wider range than the other
parameters, but that also reflects typical MEMS design
constrains. It is normally easier to vary the fold length to
a greater extent than it is to vary the number of folds or
fold spacing given an axial length constraint. The fold
spacing is also an important factor as smaller spacing
provides significantly better performance. For instance
going from 10µm spacing to 5µm spacing for 2 folds and
a 70µm fold length yields a 21% increase in the stiffness
ratio. The one parameter that does not have a large effect
is the number of folds. Going from 1 fold to 3 folds with
a 70µm fold length and 5µm fold spacing yields a 4%
decrease in the stiffness ratio. In this case, there is only a
small change because the two stiffnesses are decreasing
at the same rate as the number of folds increases. In
summary, an optimal flexure design will include the
longest fold length and smallest fold spacing possible.
The number of folds can then be used to tailor the
stiffness to a given application.
III. DIGITALLY POSITIONED
MICROMIRRORS
This next case moves from the individual
component level to the device level as it details the
improvement of a digital micromirror design through
simulation. Digital micromirrors will be key devices for
many optical applications where precision position
control is required such as optoelectronic modules for
optical communication, free space optical interconnects,
optical switching and other systems [7-9]. In these
applications, digital devices are preferred over analog
ones because the analog devices require complicated
closed-loop control electronics, which result in high cost
and power consumption. A micro-mirror with precise,
digitally controlled positioning or tilting does not suffer
from the analog limitations, however. Thereby, it can use
open-loop control electronics, reducing the system
complexity and power consumption.
III.1 Method
A novel multilevel digital micromirror design has
been reported by Zhang et al. [10]. The primary design
of the digital micromirror and its principle of operation
are shown in Figure 6, where 6-(a) shows a SEM picture
of the mirror, which is fabricated using MUMPs. The
mirror is connected to its supporting beams by two
flexures. The two electrostatically driven legs along the
other two sides of the mirror are designed to tilt it in the
positive and negative directions. The poly2 layer is used
for each leg or top electrode, which is made of four
identical plates connected by thin, flexible beams. The
bottom electrodes, constructed of poly0, are four separate
plates located right under the top electrode, each of
which can be supplied with voltage independently and
sequentially to define the tilting angles of the mirror. As
illustrated in Figure 6-(b), the snap-down of each
successive electrode section corresponds to a separate
digital angle. Also, to achieve a larger rotation range prestressed beams are used for the support beams to increase
the gap between the top and bottom electrodes [11].
3
Although the digitally positioned micromirror has been
fabricated and has shown good performance in
preliminary experiments, the desired precise digital
control has not been completely obtained. The
mechanical structure needs to be modified and optimized
To optimize the mechanical structure, the CoventerWare
finite element code with 27-node brick elements is used.
Through a careful study of the effects of mesh size on the
convergence time and accuracy, the most efficient finite
element meshes for the top beam contain extrude solid
elements with the largest dimension specified as 50 µm.
Manhattan elements with a size of 10µm x 10 µm x 5 µm
are used for the bottom electrodes. As shown in Figures
7-a and -b, due to symmetry, only half the device is
modeled. Simulations predict the combined mechanical
and electrostatic behavior of the micromirror; bimorph
support beams and movable drive legs as a total system.
Some minor structures such as dimples, etch holes, etc.
are removed for simplification. First, the bimorph effects
used to generate the large initial gap are simulated and
compared to the deformation observed in experiments.
The model’s z-displacement is 14.0 µm, which is close
to the experimental result of about 13.2 µm. In addition,
due to warpage, there is an initial angle of about 0.7°
measured; this angle is simulated by applying a uniform
downward load (0.0005Mpa) on the top beam. Finally, in
the MUMPs process, the height of the dimples used
between the top electrodes and the substrate is 0.75 µm.
This distance is used as a surface constraint for the top
electrode in the model to terminate vertical surface
movement before contact between the electrodes occurs.
CoventorWare’s Cosolve module is used to simulate the
hysteresis behavior of the model. In order to precisely
simulate the digitally controlled tilting behavior of the
mirror, a two-volt increment is used.
III. 2 Results
Using the simulation results, the tilting angles
of the mirror are calculated and plotted as a function of
the applied voltages. These model results are shown in
Figure 8 where they are compared to experimental
results as well. During the experiment, the mirror’s
tilting angle jumps from 0.36 to 2.8 degrees when the
voltage reaches 19.4 Volts. The angle then increases to
another level at 3.5 degree with 22 Volts applied. At this
voltage all the electrodes collapse onto the substrate and
there are no further angle changes even if higher voltages
are applied. When the voltage is reduced, the angle
remains constant because the release voltage
corresponding to the two closely connected surfaces is
usually much lower than the pull-down voltage. At 6.2
Volts, the angle is shifted from 3.5 to 1.9 degrees. At 4
Volts, the angle is changed again from 1.9 to 0.9 degrees.
As mentioned before, these digital angles with levels of
0, 2.8, 3.5, 1.9 and 0.9 degrees can be used to simplify
control electronics for optical switches. Of course, the
flatter and the longer these stages, the better for the
digital control. The MEMS device’s performance
measured demonstrate the feasibility of the digital
positioning concept, but it needs to be optimized for
better digital performance.
individual electrodes as well as widening the flexures.
As shown in the figure, the agreement between
the experiment and the simulation with 0.75 µm dimple
is not very good. The voltages required for the pull
down process are too high, and the number of digital
levels is different from the measured one. During the
experiment, the first two electrodes collapse at the same
time and result in a single digital angle of 2.8 degrees.
With 0.75 µm, however, these two electrodes collapse
sequentially and result in two different digital angles. It
is clear that the 0.75 µm assumed needs to be modified
for this particular device measured.
IV. MEMS FLAPS FOR FLUIDS MIXING
The third case turns from the demonstration
of a MEMS device for a generic application to the
modification of a generic MEMS device for a specific
application. In this case, MEMS microactuators for the
purpose of enhancing fluid mixing are improved through
the use of simulation [12]. In the field of fluid
mechanics, precise control over a given flow is often
necessary. Here, MEMS are used as the controlling
mechanism. The flow geometry to be controlled is
presented in Figure 11. The actuators are mounted on
either side of the jet, providing a disturbance that is
naturally amplified as it travels downstream. The driving
mechanism for the first set of flaps is electro-thermal,
which is capable of providing both the deflection and
force necessary to disturb the flow [12]. The actuators
are mounted on a ceramic substrate using flip-chip
bonding with solder. This approach allows for several
MEMS actuators to be accurately positioned on a
ceramic substrate, which is then placed next to the jet.
The effect of the actuators is significant, as shown by the
images in Figure 11. In the absence of forcing, flow is
laminar and featureless; however, when excited, largescale vortices arise that enhance the mixing of the jet
with the surrounding air. This phenomenon is known to
have a profound effect on combustion processes.
If the top electrode-to-solid surface gap is
changed to 0.1 µm, results are close to the measured
angle-vs-voltage relationship. The magnitudes are close
and the numbers of digital levels are matched. But, it
should be noted that the simulated results are not mesh
independent. Finer meshes cannot be used due to the
limitations of the computer hardware. As a result, 0.1 µm
should be used as a reference only. It is difficult to
determine the correct gap resulting from the use of
dimples. More studies are needed to characterize this
gap. Nevertheless, the simulations provide us with a
more insight into the digital positioning. Large gap is
desirable for the future device design because it achieves
a higher number of digital angles and the voltage region
for each digital angle is flatter than those cases with only
0.1 µm gap.
After a series of trial-and-error simulations with
different flexures, support beams and plates, a better
performing design is developed and is shown in Figure 9.
The dimensions of the three electrostatically driven
plates are increased sequentially from the center region
to the end away from the mirror. The positions of the
three top plates are also moved towards the mirror for
better tilting. This design is the “Modified Design –1.” It
is further improved to be “Modified Design –2” by
increasing the width of the flexures to adjust tilting
angles. The simulation results for the three designs are
shown in Figure 10. Because 0.75 µm is the goal for the
future device designs, this is the gap used for the study.
Compared with the original design, the Modified Design
–1 is an improvement because it adds a new, third digital
level. The Modified Design –2, with the wider flexures,
adjusts the tilting angles further so they approach a more
uniform distribution with angles close to 1, 2, 3 and 4
degrees, respectively.
The comparisons among different designs and
gaps demonstrate that an already working device can be
improved through the use of simulation without the need
for prototyping. The model of the initial micromirror
design provides the nature of the relationship between
the numerical and experimental results. The subsequent
revisions to the model show that device performance can
be improved by modifying the shape and placement of
4
IV.1 Method
Although the electro-thermal flaps have been
shown to work, a new design driven by electrostatic
forces is being developed to simplify the fabrication
procedure. A model of the new design is shown in
Figure 12. The flap is composed of poly2 and gold strips.
The gold-on-polysilicon bimorph effect lifts up the plate,
and the applied electrostatic voltage moves the plate
back toward the substrate to create the air puffs for fluid
mixing. This design is a scaled up version of the bimorph
design reported in [11]. The size of the flap is about
300µm × 900 µm.
For the first part of the simulation, the creation
of the initial gap by the bimorph effect is modeled,
wherethe stress-free temperature is assumed to be 75 oC.
Figure 13 shows the displacement of the flap when the
temperature drops from 75 to 20 oC . Because of the
symmetry of the device, only half of the flap is
simulated. The maximum displacement of about 25um in
the z-direction occurs at the edge of flap. This is an
increase from the initial 2.75 µm gap between the nitride
and poly2 layers to what is now a 27.75 um gap. The
large displacement achieved with such a large gap is
needed for the flap to purturb the fluid. However, there
are two major problems in this design:
• The z-displacement along the ydirection (Figure 13) is not uniform
and the maximum z-displacement
occures at the end of the device. This is
not desirable for 2-D flow control. The
z-displacement should be more
uniform in the y-direction.
• The pull-down voltage calculated is
about 103 V, which is too high for this
particular application.
These two problems are solved by subsequent design
improvements. The first improvement is the optimization
of the gold strip widths on each flap to achieve a uniform
displacement. Figure 13 shows that the gap is larger at
the end than that at the middle. As a result, to get a
uniform gap the gold strips are made narrower at the end.
Assume the gap d is a functional of the strip width:
d = f ( w( y ))
(1)
The design objective is to find a width
distribution w(y) such that d becomes a constant.
Assume w(y) is continuous and differentiable up to as
many terms as needed.
We can expand the w(y) as Taiylor’s series:
y
y
w( y ) = c0 + c1 + c2 ( ) 2 + L
( 2)
l
l
where l is the length from the center of the gold
strip in the middle to the center of the gold strip at the
end. The origin of y is at the center of the middle gold
strip. The direction of y is opposite to the y-direction
shown in Figure 13 though. For simplicity, using the
first-order guess of w(y), we can get
y
c y
(3)
w( y ) = c0 + c1 = c0 (1 + 1 )
l
c0 l
In simulation c0 is set to be the width of the
middle gold strip, so there is only one variable, c1 , to be
determined. The y can only take discrete values such
that:
y m
=
l
n
( 4)
where n is the total number of gold strips in the
half flap and m means mth gold strip from the center and
takes values: m = 0, 1, 2, L , n − 1 . The width of
the gold strips varies according to this linear distribution.
An appropriate c1 is calculated for a uniform gap along
the y-direction. The bifurcation method used is described
as follows:
•
Assuming
c1
= −1 , the vertical
c0
displacement is larger at middle.
•
Assuming
c1
= 0 , the vertical
c0
displacement is larger at end. There
exists a value of
c1
∈ (−1,0) such
c0
that the displacements at end and at
middle can be the same.
Continuing the iteration several times, the
value of 0.8 for the ration is found to meet the
uniformity requirement.
5
IV.2 Results
Figure 14 shows the width distribution of the
gold strips and the z-displacements at the edge. The
difference in z displacement at edge along the y direction
is reduced from about 4µm (Figure 13) to about 1µm
(Figure 14). In addition, the design is further improved
to reduce the pull-down voltage by using flexures to
connect the flap plate to the anchor. Figure 15 shows the
improved model and the corresponding simulaton results.
Here
c1
= −0.6 is used, which is obtained by the
c0
bifurcation method introduced above. Such flexures
reduce the pull-down voltage from 103 to 70 Volts. In
this case, the stress-free temperature is assumed to be
170 oC due to a thermal annealing following fabrication.
In summary, electrostatic microactuators to control a
planar jet have been designed using simulation. Design
iterations in Coventorware resulted in a more uniform
actuator tip displacement and a reduction in actuation
voltage from 103V to 70V.
V. OPTIMIZING MEMS FOR SOLDER
SELF-ASSEMBLY
The final case does not involve the use of CoventorWare
for design iteration or optimization, but rather introduces
a more advanced type of automated optimization. As
MEMS devices progress and become more elaborate,
understanding the complex interactions between the
many possible design variables becomes difficult to
accomplish by manually running all the design iterations
in a MEMS CAD software package available today. One
solution to this type of problem is the use of automated
design optimization algorithms to find appropriate design
variations. This methodology is common in engineering
design but is vastly underutilized for MEMS.
The design to be presented is for solderassembled, 3-D MEMS devices [13]. One of the most
common methods for manufacturing MEMS devices is
by using surface micro-machining. Due to the nature of
thin film deposition technology, a fundamental problem
with surface micro-machining is its inability to produce
highly three-dimensional structures. A common solution
is to fabricate flat, 2D hinged components that can be
lifted or rotated into assembled structures. Such
structures are very common in many MEMS and
microelectronics fields, namely micro-optics [14]. The
draw back of hinged designs is that they need to be
assembled after fabrication. The traditional way to
perform this assembly is to do it manually or use
additional MEMS mechanisms to assemble devices
automatically [15,16]. Manual assembly usually consists
of rotating the plates by hand using high precision micromanipulators. This form of assembly is not practical for
mass assembly and manufacturing though, and is rarely
effective. Mechanism driven assembly is also insufficient
because these MEMS mechanisms are often large and
complex, and thus negate many advantages inherent in
MEMS devices.
A new method of assembling MEMS has been
developed that has made cost-effective mass
manufacturing of assembled 3D MEMS structures both
practical and feasible. This method uses the surface
tension properties of molten solder or glass as the
assembly mechanism [17-19]. The solder method
involves using a standard hinged plate with a specific
area metalized as solder wettable pads. Once the solder
is in place, it is heated to its melting point, and the force
produced by the natural tendency of liquids to minimize
their surface energy pulls the free plate away from the
silicon substrate (Figure 17). Solder is a predominant
technology for electronics assembly and packaging. It is
not only used for electrical connections, but also for submicron accuracy alignment in many packaging
applications such as optoelectronic passive alignment
[20]. Using solder, hundreds or thousands of precision
alignments can be accomplished with a single batch
reflow process, and the cost/alignment can be reduced by
orders of magnitude. In addition, solder provides high
quality mechanical, thermal and electrical connections.
Figure 18 is an actual solder self-assembled
plate that is 400 microns square and was assembled with
an approximately 200 micron diameter solder sphere.
With the development of this technology, the focus of
work on solder self-assembly has changed from
demonstrating that hinged MEMS can be assembled, to
refining the precision to which they can be assembled.
The predominant source of error is the deformation that
results from process-induced stresses within the
structure. The deformation is reduced by using finite
element modeling and optimization algorithms to
optimize the solder self-assembly structure for maximum
assembly angle precision.
V.1 Method
The basic building block of a solder selfassembly structure consists of a single solder sphere with
a hinge and mechanical lock on either side (Figure 19). A
typical plate is between 100-500 microns on either side.
A 600 micron wide by 200 micron tall structure is
chosen for optimization because ample experimental
samples are available to confirm the predictions.
For this case, the parameters to be varied are: the
contact position of the mechanical lock and plate, the
width and height of the solder pad, and the position of
the hinge. The only constraint is that the solder pad
should remain large enough to be practical for solder
deposition processing. The finite element software
ABAQUS is used to model the structure and extract
relevant data, and the optimization algorithm NLPQL
[21] is used to optimize the variables. The optimization
process is as follows: starting with an initial guess for the
variables, the model results and the derivative of those
results with respect to each variable is evaluated. These
guesses are used to create an ABAQUS input file, which
is then processed, and the results extracted. Since this is
not an analytical function to be optimized, the derivatives
are found using a simple finite difference method. These
values are then used by the NLPQL program to generate
6
a new prediction. This process is repeated until the
solution converges to some desired tolerance. Since this
process could run for many iterations, it is important that
an efficient model is used. For this reason Kirchoff
composite shell elements are used over standard 20 node
3D solid elements. The resulting computation time per
model evaluation is reduced by approximately 95%. The
average time to converge to an optimum solution, when
run on a 500 MHz DEC Alpha (DEC personal
workstation model 500au) with 768 MB of RAM
running DEC Unix V4.0 is about 30 minutes and
involves ten step iterations and 40 model evaluations.
The plate is modeled using composite shell
elements, but the solder is simulated with standard threedimensional solid elements. The interaction of the
kickstand and hinges is modeled using contact surface
approximations rather than by including the actual hinge
and kickstand structure into the model. This greatly
increases the computational efficiency of the model
without affecting accuracy. The accuracy of the model is
gauged by comparing the predictions to experimental
data. Figure 19 shows one such comparison in which a
200 by 1800 µm solder self-assembly plate is modeled,
fabricated, and measured interferrometrically. It is found
for all cases that the model prediction fell within the data
variation.
After validation, the optimization program is
able to generate a prediction that significantly optimizes
the deformation in the plate. The values to be minimized
are the rms, average, and maximum deflections of the
plate. Figure 20 shows three sample cases for one design
optimization problem: a) a prediction in which there is
no lock or hinge contact, b) a prediction in which the
lock contact position and hinge have been placed, and c)
the algorithm prediction for lock, hinge position, and pad
dimensions that will result in minimum deformation.
Interestingly, the case in which the lock and hinge are
placed poorly resulted in a more severe deformation than
with no lock at all. The poor lock and hinge position
results in a maximum deflection of ~5.5 µm and a rms
deflection of ~3.4 µm, whereas, the prediction with no
lock or hinge results in a max deflection of ~4 µm and an
rms deflection of ~2.1 µm. Finally, the optimized
structure shows a significant improvement with a
maximum deflection of ~0.9 µm and rms deflection of
~0.6 µm.
The reason for the reduced deformation is likely
due to the lock and hinge constraints working against the
deformation resulting from solder shrinkage. The
shrinkage tends to cup the plate around it like a shroud.
By placing the hinge and lock near the edge of the plates,
they restrict the plate and force it back toward the desired
position. If the lock and hinge are placed too close to the
solder, they only amplify the deformation. If there are
too far out, the plate will bend significantly between
them and the solder joint.
The above work discusses the reduction of plate
deformation through design optimization. However, the
primary point of this optimization is angle precision. The
question is, with plate deformation, how precise an angle
can still be achieved ?
In one experiment, a design for a 400µm-by400µm-by-3.5µm basic solder assembly structure is
optimally design and fabricated. Six of these structures
are then assembled and measured. The average angle of
assembly is 89.78o. More importantly, the maximum
deviating sample was at 89.69o, which is a mere 0.09o
angle deviation from device to device. This demonstrates
assembly repeatability, and repeatability is really what is
important. The devices are designed to rotate to 90o , and
are off by approximately 0.2o. By changing a design
parameter (the length of the lock) this number can be
reduced and the variation in angle precision will be the
maximum angle deviation from device to device (in this
case 0.09o). Thus, the discussed automated optimization
procedure can in fact determine the relationship between
a variety of variables and use that relationship to
optimize a specific structure.
VI. CONCLUSIONS
Using the MUMPs foundry service and the
CoventorWare MEMS CAD tool, students at the
University of Colorado – Boulder have been able to carry
out numerous original and useful MEMS designs. Three
representative designs have been reviewed: 1) a flexure
design used to reduce the device warpage resulting from
the mismatch in thermal expansion coefficients between
the device and the substrate for flip-chip bonded MEMS,
2) a digitally positioned micro-mirror using multiple
contacts, and 3) a large MEMS flap that achieves
uniform movement for fluid mixing. In addition, an
automated design optimization procedure has been
developed for a three-dimensional, solder-assembled
MEMS device. Its optimization procedure for the
minimal deviation from the desired assembly angle was
presented and discussed.
VII. ACKNOWLEDGMENTS
The authors would like to express their
appreciation to Coventor, Inc. for their software support
for the establishment of the MEMS design course at the
University of Colorado – Boulder. CoventorWare has
been used as the MEMS CAD tool for the first three
design cases presented. The fourth design case was
supported by the Defense Advanced Research Projects
Agency (DARPA), the Air Force Research Laboratory,
Air Force Materiel Command, USAF, under agreement
number F30602-98-1-0219. In addition, we would like to
thank Mr. Jianglong Zhang and Mr. Zhichun Ma for their
collaborations during the designs of the digitally
positioned MEMS and the MEMS flaps.
VIII. REFERENCES
1.
7
Petersen, K., “Bringing MEMS to Market,”
Solid-State Sensor and Actuator Workshop
Hilton Head Island, South Carolina, June 4-8,
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R. S. Muller et al., Microelectromechanical
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3.
Koester, D., Majedevan, R., Shishkoff, A.,
Marcus, K., “Multi-User MEMS Processes
(MUMPs) Introduction and Design Rules”, rev.
4, MCNC MEMS Technology Applications
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15, 1996.
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CoventorWare, Coventor, Inc.,Cary, NC, USA,
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H. Fettig, J. Wylde, T. Hubbard, M. Kujath,
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Houston, TX, Nov. 1-6, 1998.
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For Open-Loop Controlled Applications," The
Fithteenth IEEE International Conference on
Micro Electro Mechanical Systems (MEMS
2002), Las Vegas, Nevada, USA, January 2024, 2002.
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flip-chip
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(2001) pp. 76-87.
12. Zhichun Ma, T. Peacock, E. Bradley and Y.C.
Lee, “Solder-Assembled MEMS Flaps to
Enhance Fluid Mixing,” Proceedings of 2001
ASME International Mechanical Engineering
Congress and Exposition, November 11-16,
2001, New York, NY.
13. Kevin F. Harsh, Victor M. Bright, Yung-Cheng
Lee, "Design Optimization of Surface Micromachined Self-assembled MEMS Structures,”
The ASME International, Intersociety
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Exhibition (InterPACK'01), Kauai, Hawaii, July
8-13, 2001.
14. N.C. Tien, M. Kiang, M.J. Daneman, O.
Solgaard, K. Y. Lau, and R.S. Muller,
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2687, 27 January-2 February, 1996, San Jose,
CA.
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Drive Actuator with Mechanical Links for SelfAssembly of Three Dimensional MEMS”,
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Assembled Micro-actuated XYZ Stages for
Optical Scanning and Alignment”, Transducers
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8
20. Q. Tan, Y.C. Lee, “Soldering for
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Layers and Nominal Thickness in Microns
1.5
0.5
2.0
0.75
2.0
0.5
0.6
MCNC
Figure 1: Cross-sections of MUMPS (Multi-User MEMS Processes) Foundry Process
Figure 2: A typical 1cm X 1cm chip design using MUMPS
9
400 µm
Flexure
40 µm
110 to 25 oC
Figure 3: Warpage of a MEMS variable capacitor resulting from thermal mismatch
between silicon MEMS device and alumina substrate. Flexures can be used to reduce the
warpage.
Number of Folds
Fold Length
Fold Spacing
Figure 4: Parameters for the flexure design
Figure 5: Optimization plots for flexures with different stiffness ratios
10
Bottom
electrode
Top
electrod
Mirror surface
Pre-stressed
beam
Top electrode
Micromirror
Bottom
electrode
(a) Original position of
micromirror
(b) One electrode “snap down”
(c) Two electrodes “snap down”
(d) Three electrodes “snap down”
(e) Four electrodes “snap down”
Figure 6: (a) SEM photo and (b) pperating principle of the digitally positioned
micromirror
(a)
(b)
Figure 7: a) model for the simulation of the digitally positoned mirror and b) an example
of the tilting angles simulated
11
6
Tilting Angles of the Mirror (degrees)
5
4
Experimental Results
3
Simulation with 0.1 micron
2
Simulation with 0.75 micron
1
0
0
20
40
60
80
100
Voltage Applied (v)
Figure 8: Simulation results of the digital micromirror
Original design
Modified design
Figure 9: Original and modified designs
12
Tilting Angles of the Mirror (degrees)
5
4
3
2
Original Design
Improved Design -1
Improved Design -2
1
0
0
20
40
60
80
100
Voltage Applied (v)
Figure 10 Simulation results of the modified designs
Instabilities
are
born here
MEMS
Figure 11: The arrangement of MEMS flaps used to enhance fluid mixing. Flaps are
placed on either side.
Figure 12: Solid model with gold, polysilicon and nitride layers
13
ge
d
E
En
d
M
idd
le
y
Figure13: Contour of vertical z-displacement and the z-displacements at the edge of the
flap along the y-direction.
Figure 14: Optimized flap design and its z-displacements at the edge of the flap along the
y-direction
Figure 15: Optimized flap design with flexures and its z-displacements at the edge of the
flap along the y-direction
14
Hinged
Plate
(2)
(1)
Hinged Plate
Substrate
Substrate
Figure 16: Illustration of solder self-assembly of hinged MEMS plate. (1) During reflow. (2)
Equilibrium position.
Figure 17: Solder assembled three-dimensional MEMS device with kickstands
200 µm
1800 µm
Interferrometric
measurement of out of
plane deformation
Figure 18: Validation example in which the model prediction (bottom) matches the
interferometric measurement (top right)
15
Figure 19: Comparison of results: (top) prediction when no lock or hinges are used,
(middle) prediction when lock and hinge position are poorly chosen. (bottom) prediction
for optimum pad size and lock/hinge position.
16