SDPS-2014
Printed in the United States of America, June, 2014
2014 Society for Design and Process Science
PROBABILISTIC MODELING OF MICRO-ELECTRO-MECHANICAL
SYSTEMS (MEMS)
Doug Harris, Diane McNulty, Rajiv Shah
Department of Transdisciplinary Science
University of Texas at Dallas
Dallas, Texas 79409-1021, USA
ABSTRACT
INTRODUCTION
Micro-Electro-Mechanical Systems (MEMS) are a
fast-developing technology that has a potential to
permeate most engineering and medical applications. For
this technology to continue expanding, issues regarding
the cost of manufacturing and reliability of the devices
have to be addressed. To improve the reliability,
probabilistic design methodologies are potent in both the
modeling and testing of high-performance MEMS. The
benefit of probabilistic design approaches is a more
rational basis for making design decisions that balance
component or system efficiency with reliability or safety.
Clearly there are numerous constituents (e.g., electrical,
fluidics, and chemical) of MEMS, this paper will focus on
the modeling of the mechanical properties of MEMS. The
objective of this paper is to use probabilistic techniques on
a capacitive accelerometer. The probabilistic software
NESSUS is used for the analysis. The parameters of the
accelerometer that impact the reliability most are
identified.
Micro-Electro-Mechanical Systems (MEMS)
Three general categories of MEMS include actuators,
sensors, and passive structures (Maluf, 2000). Sensors are
transducers that convert mechanical, thermal, or other
forms of energy into electrical energy; actuators do
exactly the opposite. Passive structures are devices in
which no transducing occurs. There exist various
techniques in manufacturing MEMS. Epitaxy, sputtering,
chemical vapor deposition, evaporation, and spin-on
methods are common techniques used to deposition
uniform layers of silicon, metals, insulators, or polymers
(Maluf, 2000). Lithography is a photographic process for
printing images onto a layer of photosensitive polymer
that is subsequently used as a protective mask against
etching. Several different materials are used in MEMS,
but polysilicon is currently the most widely used material
(Sharpe, et al., 1999).
The behavior of MEMS devices is limited by the
strength of critical features such as oxide cuts joining
layers, thin ligaments, pin joints and hinges (LaVan and
Buchheit, 2000). Wereszczak et al. (Wereszczak, et al.,
2000) examined the effects of specimen size,
crystallographic orientation, loading rate, and surface
condition on the tensile surface strength. In his paper, the
researcher provided results that could be used by package
designers and end-users to optimize or predict the preservice and service probabilistic mechanical reliability of
silicon devices. Mechanical testing of polysilicon is
relatively new endeavor that has developed within the past
10 years (Sharpe, et al., 1999). Sharpe et al. (Sharpe, et
al., 1999) showed that the stress-strain curve of thin-film
polysilicon could be measured. Chen et al. (Chen, et al.,
2001) used biaxial flexure specimens to characterize the
material. Li performed microscale tension test of single
crystalline silicon (Li, 2001).
NOMENCLATURE
a
b
E
g( )
h
KI
KIC
L
p
pf
u
x
Y
Z(X)





=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
Half crack length, (m)
Cross-sectional width, (m)
Modulus of elasticity (Pa)
g-function
Cross-sectional height, (m)
Stress intensity factor (Pa m0.5)
Fracture Toughness (Pa m0.5)
Beam length, (m)
Probability
Probability of failure
transformed space
Relative displacement (m)
Shape factor
Response function
Percentage error
Standard deviation
design stress
mean
Poisson's ratio
Probabilistic Design Methodologies
Analysts and designers are confronted with numerous
uncertainties and product variability. They must consider
manufacturing tolerances, material properties, loads, and
1
boundary conditions, and must design with these
uncertainties in mind to ensure that products are reliable
and safe (Cesare and Sues, 1999). The probability-based
load and resistance factor designs, and conventional safety
factor-based deterministic designs, in terms of capacity
reduction factor and load factors are essentially parallel to
each other. However, the distinction of probabilistic
design is that it addresses the design conservatism through
treatment of the uncertainty in the random variables, the
conservatism used in selecting the design values, and the
desired underlying reliability (Haldar and Mahadevan,
2000a). Reliability is a quality characteristic which deals
with sustained, failure free performance that meets or
exceeds our customer's needs and expectations. Reliability
is defined as the probability that a system, subsystem, or
component will function without failure, when operated
under specified conditions, for the duration of a specified
mission or time/usage period. Hence, reliability is a
measurable characteristic while reliability engineering is a
means to impact the reliability measure.
The ability to quantify the uncertainty of complex
engineering structures subject to inherent randomness in
loading, material properties, and geometric parameters is
becoming increasingly important in the design and
analysis of structures (Riha, et al., 2000). Some common
probabilistic approaches include Monte Carlo simulation,
response surface methodology, and the most probable
point methods.
The Monte Carlo simulation is a statistical sampling
experiment. It involves a repeated generation of random
variates and a subsequent determination of the system
response for each realization of the variates by using
deterministic methods (Spanos and Zeldin, 1998). Proper
statistical processing provides estimates of useful response
statistics.
The most probable point methods belong to the limit
state approximation methods. The main methods are the
MPP methods are the First-order Reliability Method
(FORM) and Second-Order Reliability Method (SORM).
According to Thacker et al. (Thacker, et al., 2001)
advance mean value group of the can handle complicated
but well-behaved response functions. These group of
methods includes the mean value method (MV), advanced
mean value method (AMV), and improved advanced
mean value method (AMV+) In the mean value method,
the value of the Z-function is defined as (Thacker, et al.,
2001)
Z

i 1 X i
n
Z MV  Z (  )  
( Xi  i )  H( X )
Haldar and Mahadevan (Haldar and Mahadevan, 2000b)
describe these methods in detail in their book.
Probabilistic Techniques for MEMS
Chen et al (Chen, et al., 2001) presented a
probabilistic structural analysis and design of a silicon
micro-turbo-generator rotor. These researchers showed a
possibility to execute design trades between
turbomachinery performance, fabrication efforts, and
structural reliability. The techniques presented can also be
applied to others highly stressed MEMS devices.
Objective
The objective of this paper is to use probabilistic
techniques on a capacitive accelerometer. The accuracy of
the probabilistic techniques will be assessed for the
accelerometer. The parameters of the accelerometer that
impact the reliability most will be identified.
PROBLEM FORMULATION
Model
A typical capacitive accelerometer is shown in Figure
2. In this accelerometer the beam-mass structure is
sandwiched between two Pyrex glass plates by
electrostatic bonding. The movement of the seismic mass
changes the capacitances between the mass and the two
fixed electrodes attached to the glass. The differential
capacitance between the two capacitors measures the
acceleration. The input variables used the probabilistic
analysis are shown in Table 1. The random variable
considered are Half crack length (a), Cross-sectional
width (b), Modulus of elasticity (E), Cross-sectional
height (h), Fracture Toughness (KIC), Beam length (L),
Relative displacement (x), and Poisson's ratio ().
Probabilistic Analysis
The procedure followed in MEMS material
probabilistic analysis procedure is shown in Figure 1. This
procedure is an extension of that proposed by Thacker and
Millwater (Thacker and Millwater, 1991). As shown in
this figure, the procedure contains two main components:
material characterization to extract statistical material
parameters and probabilistic analysis (includes finite
element analysis) to determine the structural response.
For the model analyzed in this paper, the stress
intensity factor KI can be defined as
K I   Y a
(2)
(1)
i
where Y is the shape factor, a is the half crack length, and
 is the design stress. In this equation the design stress
(3)
   f (a, b, E , h, L, x, )
where E is the modulus of elasticity,  is the Poisson’s
ratio, a is the half crack length, b is the width of the
specimen, h is the thickness of the specimen, and x is the
where H(X) are the higher order terms while ZMV is a
random variable representing the sum of the first order
terms. AMV improves MV compensating for the errors
incurred by the Taylor’s series truncation. AMV+
improves AMV by using an updated expansion point,
which often obtained by MV (Thacker, et al., 2001).
2
relative displacement. The parameters a, b, E, h, L, x, and
 are random variables. Therefore KI will have a joint
probability of the random variables in Equation 3. Since
both KI and KIC are random variables, then another
random variable Z can be introduced as
Z  K IC  K I
(4)
or generally
Z ( X )  g ( X 1 , X 2 , , X n )
(5)
where the g is a function of the random variables that
constitute KI. Mathematically, the g-function in Equation
4 is referred to as a performance function or limit state
function (Haldar and Mahadevan, 2000a). Thus if KI is
larger than KIC, the fracture toughness of the material, the
crack will propagate and the structure will fail. The failure
surface or the limit state of interest can then be defined as
Z = 0. In this case the probability of failure can be defined
as
p f  P( Z  0)
(6)
or
z zf ( x , x , x )dx dx dx
pf  
g () 0
1
2
n
1
2
n
The first natural frequency of the beam-mass element
of the capacitive accelerometer can be formulated as (Lust
and Wu, 1998)
22.37
Ebh 3
.
(10)
2
12 AL4
The natural frequency consists of five random variables,
namely, A, b, E, h, and L. Using probabilistic analysis the
estimated mean frequency for a given probability level
will established.
f1 
RESULTS AND DISCUSSIONS
The cumulative probability distribution (cdf) of the
response function (Equation 8) is generated using Monte
Carlo simulation (see Figure 3). This cdf can be used to
determine the probability that the response value will have
a selected value. Using the same plot, it is observed that
the probability of failure, pf, is 0.03.
The cdfs were also generated using the SORM,
FORM, AMV and AMV+. Figure 4 depicts the cdfs
generated by the five methods. In this figure the
probabilities are plotted in the standard normal scale.
From this figure it can be observed that for the response
function (Equation 8), SORM, FORM, AMV and AMV+
are in agreement with the Monte Carlo. This is the case
because the response function is mildly nonlinear.
To better compare the various methods to the Monte
Carlo simulation, the percent errors are presented in the
Figure 5. From this figure it is noted that though the
performance of all the various method are within the
acceptable range, AMV diverts more at the tail ends from
the Monte Carlo simulation.
For the probability of failure level of pf = 0.001,
Figure 6 show the probabilistic sensitivity factor if each of
the random variables based on the FORM, AMV, AMV+
formulations. From this diagram it is clear that the crosssection width, b, of the capacitive accelerometer
contributes least to the reliability of the design. On the
other hand, the stress intensity factor, KIC, and the beam
length, L, contribute most to the reliability. For the failure
level used, the random variable b could be input as a
deterministic value for an analysis that would involve
coupling with a finite element analysis package. It is also
observed that all the three methods predict the
contribution trends.
Figure 7 shows normalized sensitivities for the
probability failure level of 0.001. These importance
factors take into account the sensitivity of pf with respect
to  and  of each random variable (Thacker, et al., 2001).
Again in Figure 7, KIC and L have the most contribution to
the reliability. According to Thacker et al. (Thacker, et al.,
2001) negative sensitivities indicate that a positive
increase in the  will result in a decrease in the computed
probability, while the opposite would be true for positive
sensitivities. Figure 8 depicts the change to the importance
(7)
To solve Equation 7, analytical approximation approaches
will be used. The probabilistic software that will be used
is NESSUS. NESSUS is a software for performing
probabilistic analysis of structural components (SwRI,
2002).
Since the input variables have differing influence on
the statistics of the output value, a measure called the
sensitivity index is used to quantify the influence of the
basic random variables (Haldar and Mahadevan, 2000a).
This parameter indicates the random variable that
contributes most to the reliability of the mechanical
component under consideration. Additionally, the
extracted information can be used to treat the variable
with a very low sensitivity as a deterministic value, thus
reducing the computation time for complex problems
(Haldar and Mahadevan, 2000b; Thacker, et al., 2001).
Model Parameters
For the capacitive accelerometer studied in this paper,
the g-function will be formulated as
15
. Exh
(8)
g  K IC 
Y a
(1   2 ) L2
where
15
. Exh
(9)
KI 
Y a
(1   2 ) L2
and the values for the parameters are listed in Table 1.
These values were extracted from data in published
literature (Kraft, et al., 1998; Li, 2001; Nemeth, et al.,
2001; Sharpe, et al., 1999; Sharpe, et al., 1997; Yi and
Kim, 1999). And where data was either missing or
incomplete, assumptions were made. The random
variables used all had normal distributions.
3
factor (with respect to ) as a function of the probability
of failure level. Figure 9 depicts the change to the
importance factor (with respect to ) as a function of the
probability of failure level. In this figure it is observed
that above a critical pf, the sense of the importance factors
changes.
The cdf of the stress intensity factor, KI, (Equation 9)
is shown in Figure 10. Again here the FORM, AMV, and
AMV+ yield acceptable results. Using this diagram for
given probability levels the value of KI can be established.
For example for a probability level of 0.5, KI is 1.2106
Pa.
Figure 11 depicts the cdf of the first natural
frequency, f1, (Equation 10) obtained from a Monte Carlo
simulation. The natural frequency ranges from
approximately 5106 Hz to approximately 18106 Hz. At
a probability level of 0.5, the estimated natural frequency
is approximately 18106 Hz .
Chen, K.-S., Spearing, S. M. and Nemeth, N. N., 2001,
"Structural design of a silicon micro-turbo-generator,"
AIAA, Vol. 39, pp. 720-728.
Haldar, A. and Mahadevan, S., 2000a, "Probability,
reliability, and statistical methods in engineering design,"
John Wiley.
Haldar, A. and Mahadevan, S., 2000b, "Reliability
assessment using stochastic finite element analysis," John
Wiley & Sons.
Kraft, O., Schwaiger, R. and Nix, W. D., 1998,
"Measurement of mechanical properties in small
dimensions by microbeam deflection," Materials
Research Society Symposium - Proceedings, Vol. 518, pp.
39-44.
LaVan, D. A. and Buchheit, T. E., 2000, Testing of
critical features of polysilicon MEMS in de Boer, M. P.,
Heuer, A. H., Jacobs, S. J. and Peeters, E. (Eds), Materials
Science of Microelectromechanical System (MEMS)
Devices II, Materials Research Society, pp. 19-24.
Li, L., 2001, "Microscale tension test of single
crystalline silicon," Master thesis, University of
California, Los Angeles.
Lust, R. V. and Wu, Y. T. J., 1998, "Probabilistic
structural analysis - An introduction," Experimental
Techniques, Vol. 22, pp. 29-32.
Maluf,
N.,
2000,
"An
introduction
to
microelectromechanical systems engineering," Artech
House.
Nemeth, N. N., Palko, J. L., Zorman, C. A., Jadaan, O.
and Mitchell, J. S., 2001, "Structural modeling and
probabilistic characterization of MEMS pressure sensor
membranes," ANSYS, Inc.
Riha, D. S., Thacker, B. H., Millwater, H. R., Wu, Y.T. and Enright, M. P., 2000, "Probabilistic engineering
analysis
using the NESSUS software,"
41st
AIAA/ASME/ASCE/AHS/ASC
Structures,
Structural
Dynamics, and Materials Conference and Exhibit, pp. 112.
Senturia, S. D., 2001, "Microsystem design," Kluwer
Academic Publishers.
Sharpe, W. N., Turner, K. T. and Edwards, R. L., 1999,
"Tensile testing of polysilicon," Experimental Mechanics,
Vol. 39, pp. 210-216.
Sharpe, W. N. J., Yuan, B., Vaidyanathan, R. and
Edwards, R. L., 1997, "Measurements of Young's
modulus, Poisson's ratio, and tensile strength of
polysilicon," Proceedings of the IEEE Micro Electro
Mechanical Systems (MEMS), pp. 424-429.
Spanos, P. D. and Zeldin, B. A., 1998, "Monte Carlo
treatment of random fields: a broad perspective," Applied
Mechanics Review, Vol. 51, pp. 219-237.
Stark, B., 1999, "MEMS reliability assurance
guidelines for space application," JPL.
SwRI, 2002, NESSUS 7.0 Overview, Southwest
Research Institute.
CONCLUSION
The cdf of the response function is generated using
Monte Carlo simulation. Using this plot it is observed that
the probability of failure, pf, is 0.03.
It can be observed that for the response function,
SORM, FORM, AMV and AMV+ are in agreement with
the Monte Carlo. This could be attributes to the response
function being mildly nonlinear. The percentage error plot
shows AMV formulation deviates more at the tail ends
from the Monte Carlo simulation.
For the probability of failure level of pf = 0.001, it
was observed that the cross-section width, b, of the
capacitive accelerometer contributes least to the reliability
of the design. On the other hand, the stress intensity
factor, KIC, and the beam length, L, contribute most to the
reliability.
For a probability failure level of 0.001, the
normalized sensitivities (with respect to ) it was
observed that above a critical pf, the sense of the
importance factors changes. From the analysis it was also
noted that for a probability level of 0.5, KI is 1.2106 Pa.
At a probability level of 0.5, the estimated natural
frequency is approximately 18106 Hz .
REFERENCES
Bao, M.-H., 2000, "Micro mechanical transducers :
pressure sensors, accelerometers, and gyroscopes,"
Elsevier.
Cesare, M. A. and Sues, R. H., 1999, "Profes
probabilistic finite element system - bringing probabilistic
mechanics to the desktop," AIAA/ASME/ASCE/AHS/ASC
Structures, Structural Dynamics and Materials
Conference, Vol. 4, pp. 3040-3050.
4
Thacker, B. and Millwater, H., 1991, NESSUS: A New
Tool for Safer Structures, Southwest Research Institute.
Thacker, B. H., Nicolella, D. P., Kumaresan, S.,
Yoganandan, N. and Pintar, F., 2001, "Probabilistic finite
element analysis of the cervical spine," Math. Modeling
and Sci. Computing, Vol. 13, pp. 12-21.
Wereszczak, A. A., Barnes, A. S. and Breder, K., 2000,
"Probabilistic strength of {111} n-type silicon," Journal of
Materials Science: Materials in Electronics, Vol. 11, pp.
291-303.
Yi, T. and Kim, C. J., 1999, "Measurement of
mechanical
properties
for
MEMS
materials,"
Measurement Science & Technology, Vol. 10, pp. 706716.
1.2
Probability of Failure, p f
1.0
0.8
0.6
0.4
0.2
0.0
-1.E+06 -5.E+05 0.E+00
5.E+05 1.E+06
2.E+06
2.E+06
Response Value, z
FIGURES AND TABLES
Fig. 3 Cumulative distribution function
MEMS Material
characterization
6
Primitive material
random variables
Standard Response Value, u
Primitive load
random variables
Primitive b.c.
random variables
Random variables
definition
Finite Element
analysis
Probabilistic
model
Structural
response
MEMS Failure
Probability
Numerical
probabilistic
approximation
Reliability model
response
MEMS Reliability
4
2
0
SORM
-2
Monte Carlo
FORM
-4
AMV
Fig. 1 MEMS material probabilistic analysis procedure
AMV+
-6
-1.E+06 -5.E+05 0.E+00 5.E+05 1.E+06 2.E+06 2.E+06
Response Value, z
Fig. 4 Cumulative distribution function in the
transformed space
20%
Fig. 2 Capacitive accelerometer(Bao, 2000)
Percentage Error,  (%)
Table 1 Input variables
Variable
Name
a
Crack length, (m)
Mean, 
Standard
Deviation, 
Distribution
3.45E-07
5.59E-08
Normal
b
Cross-sectional width, (m)
4.26E-05
4.15E-06
Normal
E
Modulus of elasticity (Pa)
1.59E+11
8.07E+09
Normal
h
Cross-sectional height, (m)
3.50E-05
2.77E-06
Normal
KIC
0.5
SORM
FORM
AMV
AMV+
15%
Fracture Toughness (Pa m )
1.52E+06
1.75E+05
Normal
L
Beam length, (m)
1.71E-04
1.09E-05
Normal
x
Relative displacement (m)
3.03E-06
2.76E-07
Normal
Y
Shape factor
1.10778
-
-

Poisson's ratio
0.2400
0.0220
Normal
10%
5%
0%
-5%
-10%
-15%
-20%
-4
-3
-2
-1
0
1
2
3
Standard Response Value, u
Fig. 5 Solution error
5
4
5
3
Probabilitic Sensitivity Factor
0.6
Probabilistic Sensitivity Factor
pf = 0.001
0.4
0.2
0.0
-0.2
-0.4
-0.6
FORM
AMV
-0.8
KIC
a
L
b
h
E
x
2
Probability, p
Probabilistic Sensitivity Factor
0.0
dp 
d p f
L
b
h
E
Random variables
x
5.E+05
1.E+06
2.E+06
2.E+06
3.E+06
Fig. 10 Cumulative distribution of the fracture toughness
Kic
A
L
b
h
E
x
v
1.2
1.0
dp 
d p f
1
0
-1
0.00
AMV
Stress Intensity Factor, Ki (Pa)
Probability, p
Probabilitic Sensitivity Factor
2
FORM
AMV+
0.0
0.E+00
4
3
0.4

7
5
0.6
0.2
Fig. 7 Sensitivity with respect to distribution parameters
6
0.8
Monte Carlo
-3.0
a
1.00
1.0
1.0
KIC
0.25
0.50
0.75
Probability of Failure, pf
pf = 0.001
2.0
-2.0
dp 
d p f
-2
1.2
dp 
d p f
-1.0
v
Fig. 9 Sensitivity with respect to distribution parameters
5.0
3.0
E
x
-1
Random variables
4.0
L
h
0
-3
0.00
Fig. 6 Probability sensitivity factors
A
b
1
AMV+

Kic
0.25
0.50
0.75
Probability of Failure, pf
0.8
0.6
0.4
0.2
1.00
0.0
0.E+00
Fig. 8 Sensitivity with respect to distribution parameters
5.E+06
1.E+07
2.E+07
2.E+07
Natural Frequency (Hz)
Fig. 11 Cumulative distribution for the first natural
frequency
6