Analog Integrated Circuits and Signal Processing, 29, 151–158, 2001
C 2001 Kluwer Academic Publishers. Manufactured in The Netherlands.
High-Level Fault Modeling in Surface-Micromachined MEMS
N. DEB AND R. D. (SHAWN) BLANTON
Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213
E-mail: ndeb, blanton@ece.cmu.edu
Abstract. MEMS structures rendered defective by particles are modeled at the schematic-level using existing
models of fault-free MEMS primitives within the nodal simulator NODAS. We have compared the results of
schematic-level fault simulations with low-level finite element analysis (FEA) and demonstrated the efficacy of
such an approach. Analysis shows that NODAS achieves a 60X speedup over FEA with little accuracy loss in
modeling defects caused by particles.
Key Words: MEMS, NODAS, FEA
1.
Introduction
MicroElectroMechanical Systems (MEMS) has matured into a multi-disciplinary area with significant
applications in industry. One trend in MEMS is towards higher degrees of integration consisting of many
mixed-domain devices. As a result, new methods of
hierarchical design for microsystems consisting of
mixed-domain components are increasingly needed
[1]. In addition, robust fault models and test methods are required to ensure high yield and reliability
of MEMS products. Hence, analysis and design tools
capable of assessing and preventing faulty MEMS behavior are necessary to ensure end-quality, in particular
for the the many life-critical applications of MEMS.
Surface-micromachined MEMS are devices where
the micromechanical structure is fabricated using
layers of thin films deposited and selectively etched,
a process that is similar to the fabrication of electronic
circuits. Most commercial applications use surface
micromachining because of its well-developed infrastructure for depositing, patterning and etching thin
films for silicon integrated circuit technology. Early applications of this technology include the digital mirror
display [2] and the accelerometer [3]. These industrial
successes and the existence of developed design expertise, stable fabrication services, and electromechanical modeling tools have made surface-micromachining
a natural choice for developing a MEMS testing
methodology.
MEMS misbehavior can result from a variety of
sources including particulate contaminations, stiction,
layout curvature, layout etch variations, side-wall angle, and package tilt. In prior work, we have performed
three-dimensional finite element analysis (FEA) of particulate contaminations [4]. Other fault modeling and
fault classification techniques have also been investigated [5–9]. Although accurate, the amount of time
to perform FEA is significant. The aim of this work is
to develop particulate fault models using basic
MEMS primitives. MEMS misbehavior discovered
through low-level process simulation and FEA is represented at the schematic-level using existing twodimensional models of fault-free MEMS primitives.
Our approach is similar to that of [6,7] since NODAS
uses an analog hardware descriptive language. However, our approach is different from the schematiclevel simulation of faulty comb-drives discussed in [7],
where separate one-dimensional models of fault-free
and faulty comb-drives are utilized. We use existing
“good” models of MEMS primitives to model particulate faults by changing the topology of the nominal
design, an approach which is very similar to stuckat-fault modeling used in digital circuit testing [10].
Also, unlike the work in [6], we do not represent
non-electrical behavior with equivalent electrical
models. Instead, we use nodal models that permit the
simultaneous use of electrical and non-electrical models thereby eliminating the need for inter-domain
translation.
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Deb and Blanton
The rest of this paper is organized as follows. Section 2 gives a brief overview of the schematic-based
simulator NODAS. Section 3 presents our case study
of a MEMS accelerometer structure. In this section, we
describe the different types of defects caused by particulates and our approach to modeling these defects
using the “good” models utilized in NODAS. We also
compare NODAS simulation results with that of FEA.
Finally, Section 4 presents our conclusions and areas
of future work.
2.
NODAS for MEMS Simulation
NOdal design of Actuators and Sensors (NODAS) [11]
is a schematic-based simulator and a nodal matrixsolver similar to Spice except that it can simulate nonelectrical devices as well. It uses the concept of a physical system or nature consisting of through and across
variables described in Kirchhoffian network theory. For
translational mechanics, force is the through variable
and displacement is the across variable. For rotational
mechanics, moment is the through variable and angular displacement is the across variable. Although
these natures are analogous to the electrical current
(through variable) and electrical voltage (across variable), NODAS does not integrate mixed-domain behavior into Spice by translating a mechanical nature
into an electrical analogue. Instead, NODAS uses modern analog hardware descriptive languages like VHDLAMS and Verilog-A that allow the simultaneous use of
electrical and non-electrical natures. Behavioral models of mechanical elements (e.g., beam and plate) and
electromechanical elements (e.g., electrostatic gap)
have been developed with separate mechanical and
electrical terminals. The models are parameterized
functions of the geometric and physical properties of
the MEMS elements.
The investigation of faulty MEMS behavior must
ensure that “good” models preserve their accuracy when
combined with fault models. In addition, it is desirable that the fault models do not degrade the performance of schematic-level simulation. Our approach so
far avoids making new models for representing faulty
behavior by using existing beam, electrostatic gap, anchor, and plate models to represent particulate defects
discovered through Monte Carlo process simulations
[9]. This approach, by definition, avoids the pitfalls described above.
3.
Accelerometer Analysis
The device used in our investigation is a single-axis
accelerometer. The mechanical structures used in this
accelerometer are constructed using a MEMS synthesis
tool similar to the one described in [12]. It is synthesized for minimum noise and minimum area with the
topology shown in Fig. 1(a) using the openly available
MUMPs fabrication process [13]. A key parameter of
the accelerometer is its resonant frequency f x in the
sensitive direction x. Resonant frequency is a function of the physical and topological properties of the
accelerometer and is therefore a good indicator for detecting the presence of defects. We use the resonant frequency deviation and the x-direction displacement to
gauge the amount of misbehavior caused by particulate
defects. However, other operational parameters such
as sensitivity, both same-axis and cross-axis, can and
should be used as well. The nominal value of the resonant frequency f x for our example design is 5.23 kHz.
3.1. Anchor Defects
Particles that create unwanted anchors between suspended parts of the MEMS device and the substrate are
called anchor defects. Conductive particles anchored
under the beams/shuttle/fingers will short the sense
signal to ground, which will obviously cause zero sensitivity. However, insulating particles will be less detrimental (and therefore less predictable) since their electrostatic effect will be confined to adding capacitance
to the substrate. Our investigation is therefore focussed
on the misbehavior resulting from insulating particles.
Since the accelerometer topology possesses fourfold symmetry, a single quadrant of the design suffices
for defect analysis. The displacement variation along
the sensitive axis x is monitored using the three nodes
N0, N1, and N2 shown in Fig. 1. Node N0 is located at
the center of the shuttle mass, while nodes N1 and N2
are located at corners of two adjacent U-spring beams.
Flexure Beam (FB1)
This type of defect is modeled as two beams separated by an anchor as shown in Fig. 2. A particle that
is spherical or ellipsoidal in shape has a small aspect
ratio, and is therefore sufficiently stiff to be modeled
as an anchor. Fig. 3(a) compares the variation of the
resonant frequency f x with anchor defects located at
various positions along flexure beam FB1 (Fig. 1). The
High-Level Fault Modeling
anchors
fixed fingers
153
movable finger
N2
shuttle
mass
N0
anchor
FB2
TB
Y
FB1
N1
U spring
Feedback
fingers
Sense fingers
Feedback
fingers
X
(a)
(b)
Fig. 1. Topology of a surface-micromachined accelerometer showing: (a) the layout with U-spring beams and electrostatic comb-drives for
sensing and feedback, and (b) the corresponding NODAS schematic model.
location of the particle is expressed as a percentage of
beam length with the terminal anchor serving as the
0% point and the FB1-truss intersection corresponding to the 100% mark. The deviation of f x reported
by NODAS matches closely to FEA predictions with
the maximum deviation being only 0.5%. Fig. 3(b)
illustrates the low-frequency displacement of the three
observation nodes N0, N1 and N2 for a constant external acceleration of ax = 1 m/s2 . For all three nodes,
the agreement between NODAS and FEA is again very
good. It must be noted from Fig. 3 that the displacement
of N0 agrees well with the typical behavior of lin-
ear second-order systems such as the accelerometer.
The relation x/ax = 1/(2π f x )2 holds true for the displacement x of N 0, external acceleration ax , and the
resonant frequency f x of the structure. This assumes
that for small, realistic displacements the good/faulty
structure has negligible non-linearity.
Flexure Beam (FB2)
Fig. 4(a) compares the variation of f x due to anchor
defects located at various positions along flexure beam
FB2. The location of the particle is again expressed as
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Deb and Blanton
Fig. 2. NODAS schematic model of accelerometer anchor defects and broken-beam defects: Flexure Beam Anchor—A particle that causes an
unwanted anchor under a flexure beam (lower right of the figure) is modeled using two smaller beam elements and an anchor element. Shuttle
Anchor—A particle that causes an unwanted anchor under the shuttle mass (middle left of the figure) is modeled by simply adding an extra
anchor element. Broken Beam—A beam broken into two unconnected parts (lower left of the figure) is modeled simply by two unconnected
beams having total length equal to that of the unbroken beam.
Fig. 3. FEA and NODAS comparison of the variation of (a) the principal resonant frequency f x , and (b) the low-frequency displacements of
the nodes N0, N1, and N2 for an anchor defect located under flexure beam FB1 for an external acceleration of ax = 1 m/s2 .
a percentage of the beam length with the FB2-truss intersection serving as the 0% point and the FB2-shuttle
mass intersection corresponding to the 100% mark. The
agreement between NODAS and FEA is reasonable
and the maximum deviation is 1.7%. The location of
the particle has been restricted to less than 30% of the
length of FB2 because beyond this point the deviation
from the specified behavior of the structure is so great
that the fault is guaranteed to be catastrophic. Fig. 4(b)
illustrates the low-frequency displacement of the three
observation nodes N0, N1 and N2 for a constant external acceleration. For all three nodes, the agreement
between NODAS and FEA is good and the maximum
deviation is less than 4%. Note that the displacement
High-Level Fault Modeling
155
Fig. 4. FEA and NODAS comparison of the variation of (a) the principal resonant frequency f x , and (b) the low-frequency displacements of
the nodes N0, N1, and N2 for an anchor defect located under flexure beam FB2 for an external acceleration of ax = 1 m/s2 .
Fig. 5. NODAS schematic model of (a) a movable finger affected by an anchor defect, and (b) a pair of adjacent fingers that are welded together
due to an inter-finger particle defect.
values predicted for node N2 by NODAS and FEA are
almost zero for the range depicted.
placement to be less than 2 × 10−15 m, which for all
practical purposes is zero as well.
Shuttle
Movable Finger
Displacement of node N0 in the x direction for an anchor defect located under the shuttle mass is predicted
to be zero by NODAS. FEA results indicate the dis-
Here we consider anchor defects that affect movable
fingers. An anchor defect located under a movable finger
is assumed to have little impact on lateral inter-finger
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Deb and Blanton
Table 1. FEA and NODAS comparison for the accelerometer with a movable finger affected by an anchor defect for an external acceleration of
ax = 1 m/s2 .
Nodal Displacements
fx (kHz)
N0 (pm)
N1 (pm)
N2 (pm)
Location of
Anchor Defect
NODAS
FEA
NODAS
FEA
NODAS
FEA
NODAS
FEA
Under movable finger tip
9.05
8.89
311
326
160
168
183
180
Table 2. FEA and NODAS comparison for the accelerometer with a broken beam for an external acceleration of ax = 1 m/s2 .
Nodal Displacements
fx (kHz)
N0 (pm)
N1 (pm)
N2 (pm)
Location
of Break
NODAS
FEA
NODAS
FEA
NODAS
FEA
NODAS
FEA
FB1
FB2
4.48
4.50
4.50
4.53
1244
1235
1264
1247
701.1
695.9
676.5
667.4
1292
78.7
1343
1.5
and fringe capacitances. Hence, the electrostatic force
of attraction is assumed to be unaffected by this defect
type. To model this defect, we separate the
lumped comb-drive model into two smaller combdrives. Fig. 5(a) shows how a movable finger affected
by an anchor defect under its tip is modeled using good
element models. The anchored middle beam represents
the defective movable finger while the outer beams
represent the corresponding fixed fingers. Each gap is
modeled by an electrostatic gap element. Simulation
results listed in Table 1 show that the accelerometer
will suffer a catastrophic frequency deviation due to
the defect. Defect locations that extend from the finger
tip towards the finger base will cause spring stiffness
to increase even more, further aggravating the impact
of the fault. Hence, we have not tabulated the resonant
frequency and displacement values for these cases.
3.2. Broken Beam Defect
The NODAS schematic of Fig. 2 shows our model of
an accelerometer with a broken beam. Table 2 lists the
comparative simulation results obtained from NODAS
and FEA. Both simulations show that resonant frequency is reduced as expected since a broken beam
eliminates the stiffness of the U-spring structure. The
small difference in f x for breaks in FB1 and FB2 is due
to the difference in the effective masses rather than the
spring constant change. For the break on FB2, the displacements predicted by NODAS and FEA for N2 do
not agree well. Note that both values are quite low indi-
cating negligible displacement (<6% of nominal). For
all other nodes, NODAS agrees with FEA within 4%.
3.3. Inter-Finger Defect
A particle can weld a movable finger to an adjacent
fixed finger. If the connection is conductive, the
resulting fault will definitely be catastrophic due to
signal shorting. Our analysis is for defects made of
dielectric material. We decouple the electrostatic and
mechanical effects of the defect and model only the
mechanical effect. The mechanical behavior of the particle is modeled as a moderately stiff truss beam located
between the fixed and movable fingers which are themselves modeled as beams as shown in Fig. 5(b).
Fig. 6 gives the variation of f x and the nodal displacements for a particle attached between a pair of
movable and fixed fingers. The 0% point is the movable finger tip and the 100% point is the movable finger
base. These plots, unlike the previous ones, are nonmonotonic and both display an extremum. This phenomenon is due to the formation of a spring from the
welded fingers. As shown in Fig. 5(b), the spring consists of two flexure beams (FF1 and MF1) and one truss
beam. This spring model is valid if the connecting particle acts as a sufficiently stiff truss and is much smaller
compared to the lengths of FF1 and MF1. The total
length of the two flexure beams, L1 + L2, is constant.
If the flexure beam lengths are varied, the spring constant reaches its maximum value when the two flexure
beam lengths are equal, i.e., L1 = L2. Fig. 6 supports
High-Level Fault Modeling
157
Fig. 6. FEA and NODAS comparison of the variation of (a) the principal resonant frequency f x , and (b) the low-frequency displacements of
the nodes N0, N1, and N2 for different weld locations for a particle positioned between movable and fixed fingers for an external acceleration
of ax = 1 m/s2 .
our intuition as the location of the extrema in the plots is
approximately the 45% point. The extremum point p,
expressed as a percentage of the movable finger length,
is given by
1
L non-overlap
p=
1−
2
L movable
where L movable is the length of movable finger and
L non-overlap is the portion of the fixed finger that does
not overlap with the movable finger. Using the values
of L movable = 302.3 µm and L non-overlap = 23 µm from
our accelerometer design, we obtain p = 46.2%, which
agrees well with the 45% point indicated in Fig. 6.
4.
Conclusions and Future Work
Our approach for high-level fault modeling uses
NODAS for schematic-level simulation of MEMS affected by particulate contaminations. We have generated schematic-level fault models of particulate defects
by modifying the nominal topology with existing models of beam, electrostatic gap, anchor, and plate models. We have demonstrated that NODAS is capable of
simulating faulty MEMS with an accuracy comparable
to FEA for faulty behaviors resulting from single particulate contaminations, namely, anchor defects, interfinger defects and broken beams. NODAS has a significant speed advantage over FEA. For example, a linear frequency response simulation of an accelerometer
using FEA needs 230 s for 82 sample points. Under
similar conditions of system load, the same workstation running NODAS can simulate 1800 sample points
in 75 s. This is a speedup of more than 60X at the cost
of reduced but acceptable accuracy. Our results allow
us to claim that schematic-level MEMS fault modeling
using NODAS is feasible and even preferable to lowlevel modeling methods like FEA due to the enormous
reduction in simulation time achievable with minimal
loss of accuracy.
Schematic-level modeling of the electrostatic effects of particles located under and between fingers
is our next major area of focus. Specifically, particles
that fall in the gap without contacting both fingers can
have a significant effect on the sensitivity of the device.
Vertical stiction, a problem encountered frequently in
surface-micromachined MEMS, and other sources of
MEMS failure will also be investigated. The analysis
of multiple failure sources in the presence of manufacturing variations (such as layout under/over-etch,
layout curvature, side-wall angle, etc.) is also required
to develop a comprehensive understanding of MEMS
misbehavior.
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Nilmoni Deb received his B.Tech (Hons) degree
from Indian Institute of Technology, Kharagpur, India,
in 1997 and M.S. in Electrical and Computer Engineering from Carnegie Mellon University, Pittsburgh, USA,
in 1999. Currently, he is a Ph.D. student in the Department of ECE, Carnegie Mellon University, with broad
research interests in CAD for MicroElectroMechanical Systems with emphasis on Test for MicroElectroMechanical Systems.
Shawn Blanton is an associate professor in the
Department of Electrical and Computer Engineering
at Carnegie Mellon University where he is a member
of the Center for Silicon System Implementation. He
received the Bachelor’s degree in Engineering from
Calvin College in 1987, a Master’s degree in Electrical
Engineering in 1989 from the University of Arizona,
and a Ph.D. degree in Computer Science and Engineering from the University of Michigan, Ann Arbor in
1995. His research interests include the computer-aided
design of VLSI circuits and systems; fault-tolerant computing and diagnosis; verification and testing; and computer architecture He has worked on the design and
and test of integrated systems with General Motors
Research Laboratories, AT&T Bell Laboratories, Intel,
and Motorola. Dr. Blanton is the recipient of National
Science Foundation Career Award and is a member of
IEEE and ACM.