Phd Thesis from Dr. Frabrizio Cacchione

1. Mechanical characterization and simulation of
fracture processes in polysilicon Micro Electro
Mechanical Systems (MEMS)
Tesi da presentare per il
conseguimento del titolo di Dottore di Ricerca
Politecnico di Milano
Dipartimento di Ingegneria Strutturale
Dottorato in Ingegneria Strutturale, Sismica e Geotecnica - XIX Ciclo
di
Fabrizio Cacchione
Aprile 2007

2. Mechanical characterization and simulation of fracture processes in
polysilicon Micro Electro Mechanical Systems (MEMS)
Tesi di Dottorato dell’ Ing. Fabrizio Cacchione
Relatore:
Prof. Ing. Alberto Corigliano
Aprile 2007
Dottorato in Ingegneria Strutturale, Sismica e Geotecnica del Politecnico
di Milano
Collegio dei Docenti:
Pietro Gambarova (Coordinatore)
Annamaria Cividini
Claudia Comi
Roberto Contro
Alberto Corigliano
Claudio di Prisco
Marco di Prisco
Alberto Franchi
Carmelo Gentile
Cristina Jommi
Giulio Maier
Paolo Negro
Roberto Nova
Anna Pandolfi
Roberto Paolucci
Maria Adelaide Parisi
Umberto Perego
Federico Perotti
Alberto Taliercio

3. Acknowledgments
The first acknowledgment goes to Dr. Benedetto Vigna and to all the persons in
the Mems Product Division of STMicroelectronics who believed in me and gave
me the opportunity to continue my research in the very interesting and
stimulating field of microsystems, by sponsoring the PhD grant.
I desire to express my gratitude to my supervisor Prof. Alberto Corigliano for his
scientific guide and for his aid along these three years. During this thesis I had
the opportunity to interact with many researchers of the scientific board at the
Department of Structural Engineering of the Milan Polytechnic: I would like
thank to them all for the suggestions and the kind support. I would like to
express my gratefulness to: Prof. Attilio Frangi to have introduced me to Fortran
programming, for the very helpful suggestions he gave me in many occasions
and for transmitting me every time enthusiasm and a 'positive-thinking' way to
face everyday problems; Dr. Aldo Ghisi, for the support given with free fall
simulations, but mostly for his capacity to listen to my problems and to give me
right indications and suggestions; Dr. Stefano Mariani, a very precious person,
with lot of new and interesting ideas and with a personality I really appreciate;
Prof. Anna Pandolfi, who helped me in the solution of some problems when I did
not know how to get out.
Many thanks to all the guys in the Department I met by the way, I spent some
nice time with in different occasions, and shared a piece of my life with.
Finally, I wish to thank for the constant love and comprehension the persons who
always make my dreams possible, to whom this work is dedicated.
To Elisa and to my family.
Milan, April 2007
Fabrizio Cacchione

4. Contents
1 Introduction.................................................................................................... 7
1.1.Engineering motivations........................................................................ 7
1.2.Objectives, methodology and outline of the thesis............................ 8
1.2.1.Methodology.................................................................................... 8
1.2.2.Thesis layout.................................................................................... 9
2 Micro Electro Mechanical Systems (MEMS)........................................... 11
2.1.General description............................................................................... 11
2.2.Design of a suspended structure with a simple micromachining
process.......................................................................................................... 12
2.3.Applications........................................................................................... 13
2.3.1.Pressure sensors............................................................................ 13
2.3.2.Accelerometers.............................................................................. 15
2.3.3.Gyroscopes..................................................................................... 17
2.3.4.RF switches.................................................................................... 19
3 Mechanical characterization of polysilicon as a structural material for
MEMS................................................................................................................ 21
3.1.Polysilicon as a structural material in MEMS.................................... 21
3.2.Testing methodologies.......................................................................... 23
3.3.Quasi-static testing................................................................................ 25
3.3.1.Off chip tension test...................................................................... 26
3.3.2.Off chip and on-chip bending test............................................... 27
3.3.3.Test on membranes (Bulge test)................................................... 29
3.3.4.Nanoindenter-driven test............................................................. 30
3.4.High frequency testing......................................................................... 32
3.4.1.Fatigue testing and on-chip structures....................................... 33
3.4.2.Fatigue mechanisms...................................................................... 33
4 Weibull theory applied to the study of polysilicon strength................36
4.1.General concepts................................................................................... 36
4.2.Basic theory............................................................................................ 37
4.3.Statistical size effect and stress gradient effect.................................. 39
4.3.1.Effect of the modulus.................................................................... 40
4.3.2.Effect of the volume...................................................................... 41
4.3.3.Effect of the stress distribution.................................................... 42
4.4.Application of Weibull approach to 3D structures........................... 46
4.4.1.Failure probability computation.................................................. 46
I

5. 4.4.2.Weibull parameters identification............................................... 47
5 On-chip testing of ThELMATM
polysilicons............................................ 49
5.1.The ThELMATM
process........................................................................ 49
5.2.Study of the role of grain structure and surface defects in thin
polysilicon.................................................................................................... 52
5.2.1.Out of plane structure description.............................................. 52
5.2.2.Numerical models for the computation of electrostatic fields.54
5.2.3.Data reduction procedure............................................................ 57
5.2.4.Results and discussion.................................................................. 61
5.3.Design and testing of a new test structure for the mechanical
characterization of thick polysilicon......................................................... 63
5.3.1.Structure description..................................................................... 63
5.3.2.Data reduction procedure............................................................ 68
5.3.3.Result and discussion................................................................... 69
5.3.4.Estimation of fracture parameters............................................... 73
6 Numerical representation of polycrystals................................................ 86
6.1.Materials and microstructures............................................................. 86
6.2.Voronoi tessellation as a tool for the creation of numeric
polycrystals.................................................................................................. 88
6.2.1.Definition and properties............................................................. 89
6.2.2.Centroidal Voronoi tessellation................................................... 91
6.2.3.Creation of interphases................................................................. 93
6.2.4.MEMS structures tessellation....................................................... 94
6.3.Constitutive models for polysilicon.................................................... 96
6.3.1.Elastic tensor for cubic crystals.................................................... 96
6.3.2.Non linear cohesive crack model.............................................. 101
7 Non linear mechanical simulation algorithms for MEMS.................. 107
7.1.Overview on MEMS mechanical simulations.................................. 107
7.2.Direct step by step dynamic analysis................................................ 108
7.2.1.Explicit Methods.......................................................................... 109
7.2.2.Implicit Methods......................................................................... 111
7.3.Dynamic relaxation algorithm for quasi-static simulations........... 112
7.4.Cohesive fracture algorithm.............................................................. 116
7.5.Pure explicit algorithms for micro-scale simulations: drawbacks.119
7.6.Implicit–explicit integration scheme................................................. 121
7.7.FETI algorithms for multi-domain problems................................... 123
7.7.1.Comberscure-Gravouil algorithm............................................. 123
II

6. 7.7.2.Application of the domain decomposition algorithm to multi-
connected domains with crack propagation..................................... 128
7.8.Mass scaling and final remarks......................................................... 136
8 Parametric study of polycrystalline silicon............................................ 139
8.1.Methodology adopted........................................................................ 139
8.2.Young's modulus evaluation and prediction.................................. 139
8.2.1.Grain size effect........................................................................... 140
8.1.1.Texture effect............................................................................... 142
8.3.Fracture properties.............................................................................. 144
8.3.1.Grain orientation effect............................................................... 146
8.3.2.Specimen size effect.................................................................... 147
8.1.2.Grain size effect........................................................................... 149
8.3.3.Introduction of defects................................................................ 152
9 Fracture and shock assessment of MEMS accelerometers...................158
9.1.Methodology adopted........................................................................ 158
9.2.Shock assessment of an uniaxial accelerometer.............................. 159
9.2.1.Global level.................................................................................. 159
9.2.2.Device level.................................................................................. 161
9.2.3.Local level..................................................................................... 163
10 Conclusions............................................................................................... 166
10.1.Achieved results................................................................................ 166
10.1.1. Experimental results................................................................ 166
10.1.2. Numerical procedures and algorithms.................................. 167
10.1.3. Parametric simulations............................................................ 168
10.1.4. Simplified approach for the shock assessment..................... 168
10.2.Future prospects................................................................................ 168
10.2.1. Fatigue and fracture testing.................................................... 168
10.2.2. Numerical models.................................................................... 169
References....................................................................................................... 170
III

7. Introduction
1.1. Engineering motivations
The diffusion of micro-electro-mechanical-systems (MEMS) devices in a
wide range of applications is rapidly increasing over the years. This
depends on the relatively low cost of the applications produced with
micromachining technologies with respect to equivalent products
fabricated using standard technologies.
One of the key factors on the final cost of the products is the size of the
device: scaling down the size of the products makes possible an increase
of the number of devices that can be produced with the same amount of
material and with the same fabrication time and therefore it lowers the
price per produced unit.
In the same time, from a structural engineering point of view, the design
of a micro-structure made of very tiny beams and plates arises a number
of questions. It is not completely clear if at this scale it is still possible to
use the same methodologies, models and tools commonly used at the
macro-scale. In the case of a positive answer, it should be interesting to
know the limit, intended as a characteristic structural dimension, for the
application of classical structural engineering tools (like structural beam
and plate theories). Among many, some open questions at the micro-scale
are: if the material can be modeled as homogeneous and isotropic; how
the microstructural morphology plays its role in the determination of
mechanical properties of the material; if the material mechanical behavior
would be the same at the micro-scale and at the macro-scale and how
would it be possible a direct measure of the mechanical properties of the
material.
From an academical point of view, these questions opened the way to
new research fields that can be seen as a meeting point for the classical
structural engineering, material science and engineering, physics and
chemistry.
1

8. 1.2. Objectives, methodology and outline of
the thesis
The aim of this thesis is to investigate and to give an answer to some of
the questions formulated above, combining both experimental and
numerical techniques to study the mechanical behavior of the most
common structural material used for microsystem applications,
polysilicon. The main objectives are:
• the investigation of elastic and fracture properties of the material;
• the critical evaluation of the fracture assessment techniques used
by the scientific and industrial community;
• the possibility to accurately simulate the mechanical behavior of
the material in order to use fast an reliable numerical tools instead
of very time and money consuming experimental campaigns to
understand the role of the microstructure on mechanical
properties.
• Since impacts and shock waves are one of the principal causes of
failure for MEMS devices, algorithms and models were studied in
order to simulate the mechanical response of the structure and a
simplified approach for the shock assessment was proposed.
1.2.1. Methodology
The experimental work done in this thesis is accomplished using diverse
techniques: optical and electron scanning microscopy were used to
measure and control the layout of the structure designed, the fracture
paths and the operating conditions of the devices; electrical
measurements were performed using high resolution electronic
instrumentation and electrical schemes quite common for mechanical
characterization at the microscale, such as voltage generators to apply the
structural load and capacimeters to measure the displacement of the
specimens.
Numerical analyses were performed using both three-dimensional and
bi-dimensional models. The Ansys Inc. software was used in the
microstructure design phase and for linear elastic static simulations.
2

9. Abaqus program was instead used for explicit dynamics simulations
performed for the shock assessment and discussed in Chapter 9. A non
-linear static and implicit-explicit dynamic code was instead developed
and used to carry out parametric fracture simulations and dynamic
fracture simulations. This code includes the traditional displacement
finite element formulation enriched by other advanced tools, such as: the
microstructural description of the material; the run-time insertion of
interface elements in order to simulate the fracture propagation and a
multi-domain integration algorithm used to bypass some computational
difficulties connected with explicit integration algorithms.
1.2.2. Thesis layout
After this introduction, in Chapter 2 a short introduction to Micro Electro
Mechanical Systems (MEMS) is presented. Some commercially available
devices are described, together with an example of a simple deposition
process invented to give an idea on how the fabrication of a suspended
microstructure is possible.
In Chapter 3 the state of the art on the mechanical characterization of
polysilicon is given and the most important experimental techniques used
are described. The major advantages of each technique are underlined as
well as the results achieved and some critical consideration about the
progresses that should be done to have a more accurate mechanical
material characterization.
The Weibull approach, commonly used to characterize fracture properties
of brittle materials is presented in Chapter 4. The capability of this
method in foreseeing the fracture loads of micromachined components in
complex stress conditions is carefully analyzed and a numerical
procedure for the reduction of Weibull parameters from an experimental
data set is described.
All the experimental work done is treated in Chapter 5. Two
microstructures, designed and subsequently tested are described and the
results obtained by the use of the on-chip approach are discussed. In the
final part of the Chapter, some hypotheses for a fracture mechanics
material characterization are considered.
3

10. Chapter 6 is focused on the numerical representation of polycrystals by
the use of routines that are capable to divide a domain into tassells, whose
shape looks like a typical granular structure of a polycrystal. Besides, the
constitutive models adopted to describe material behavior are presented.
Numerical algorithms used in the microstructural analyses are addressed
in Chapter 7. The main problems and peculiarities regarding the solution
of the dynamic problem at the micro-scale are examined and the solutions
adopted for the reduction of the computational cost of the simulations are
tackled.
The influence of the grain size on the elastic properties of the material, as
well as the role of a privileged grain orientation is investigated in the first
part of Chapter 8. In the second part of the chapter the role of the
microstructural morphology, the effect of the size of the structure and the
one of a possible defect distribution on the overall fracture properties are
analyzed.
A possible simplified approach for the mechanical shock assessment of a
MEMS accelerometer is given in Chapter 9. The problem is divided into
three simpler sub-problems with a computational cost by far smaller than
the one of the initial problem solved monolithically.
Work for the future and research perspectives are confined to the final
Chapter 10, where also the conclusions of this work are summarized.
4

11. Chapter 2
Micro Electro Mechanical
Systems (MEMS)
2.1. General description
Over the years conventional microelectronics has continued to make
spectacular advances, following Moore’s law to deliver undreamed
processor speeds, integration density and memory capacity. However,
this phenomenon has been accompanied by an increase in specialization
where most engineers focus on some specific aspect of a circuit design
and refine it to a very fine art.
More recently, however, a new application of integrated circuit
fabrication technology has emerged that has brought a significantly new
approach to design and development: Micro-electro-mechanical systems,
or MEMS. These devices are constructed using fabrication techniques
familiar to the semiconductor industry, manufacturing components in
large batches on silicon wafers. The difference is that they combine with
electrical and electronic components also mechanical, optical or fluidic
elements.
MEMS extend the functionality of silicon components into many new
applications such as accelerometers or laboratory-on-chip products. A
key benefit of this technology is that it builds on well known
manufacturing techniques, allowing also the use of older equipment since
the lithography is not deep submicron. Moreover, since the components
are made side-by-side on wafers and with an extremely well controlled
process they can be much more precise and repeatable than similar
products manufactured in other ways.
5

12. Microelectronic integrated circuits (ICs) can be thought of as the "brains"
of systems and MEMS augments this decision-making capability with
"eyes" and "arms", to allow microsystems to sense and control the
environment. In its most basic form, the sensors gather information from
the environment through measuring mechanical, thermal, biological,
chemical, optical, and magnetic phenomena; the electronics process the
information derived from the sensors and through some decision making
capability direct the actuators to respond by moving, positioning,
regulating, pumping, and filtering, thereby controlling the environment
for some desired outcome or purpose.
Today the most common MEMS devices are mechanical sensors that
measure acceleration, rotation and pressure. On the other side, the most
common MEMS used not for sensing some external mechanical stimulus,
but to act on a system is the inkjet thermal printer head chip, that
represents the most known product in a very fast expanding technology:
microfluidic systems. Hundreds of microscopic channels are created
inside this chip and are normally filled with ink. Small resistive heating
elements in each channel can be independently driven so that the
temperature in a limited area rises to around 800°C in a few
microseconds, vaporizing part of the ink to propel the ink volume out of
the channel towards the paper, painting a small dot.
2.2. Design of a suspended structure with a
simple micromachining process
In this section an example will be given on how a micromachining
process can be used for the design of a suspended structure. For this
purpose, a surface micromachining process is invented. This process was
thought to be as simple as possible, in order to give to the reader an idea
of a MEMS fabrication process. The result of the process is a cantilever
beam clamped to the substrate.
It is worth noting that all of the steps described are just the sum of many
technological processes, that have the aim to create an eigen-stress free
material, with a very precise thickness and dimensional control, without
6

13. defects and with desired physical-chemical properties.
With the aid of Figure 2.1 it is possible to describe the process. Starting
from the monosilicon substrate (Figure 2.1(1)) it is possible to grow a film
of silicon oxide (Figure 2.1(2)). This layer is needed to create the gap
between the movable structure and the substrate, as will be clarified later,
and it is not strictly necessary for the construction of structural parts of
the MEMS. The subsequent step consists in creating a square hole into the
silicon oxide layer (Figure 2.1(3)). The polysilicon film is then deposited
on the oxide (Figure 2.1(4)). The cavity made into the oxide is thus filled
with polysilicon. In the place of the hole is therefore created the
mechanical connection between the polysilicon and the substrate. An
etching process is then needed to give the desired shape to the
mechanical structure (Figure 2.1(5)). At this point the structure is not free
to move, because it is glued to the substrate by the oxide layer. The last
step consists into the removal (usually with a dry acid attach) of the oxide
(Figure 2.1(6)).
2.3. Applications
This section gives an introduction to some MEMS sensors currently in use
and explains their principles of operation. The section covers pressure
sensors, accelerometers, gyroscopes and RF switches, that (excluding
microfluidics MEMS), are the most common micromachined sensors.
2.3.1. Pressure sensors
Pressure sensors are one of the most commonly used forms of MEMS
sensors. They are found in a wide and expanding area of applications
ranging from blood pressure monitoring, washing machines, car tires and
exhausts to hydraulic systems and aeronautics.
Pressure sensors can be classified into two main families, distinguishing
the way they sense pressure.
7

14. Figure 2.1. Simple surface micromachining process. (1) Silicon substrate. (2) Oxide
growth. (3) Oxide etching. (4) Deposition of the structural layer. (5) Structural
layer etching. (6) Oxide removal.
The most simple and common ones are the piezoresistive devices. They
are principally thin plates with the external boundary clamped to the
substrate, as shown in Figure 2.2(a). When the pressure acts on the top
surface of the device, causes the plate to deform. Nearby the edges of the
membrane, where the strains are maximum, are usually placed four
piezo-resistors. Thus, when the plate undergoes bending, the resistance of
the piezo-resistors changes. By measuring the resistance changes of the
system, it is possible to know the value of the pressure acting on the
diaphragm.
Another common layout of pressure sensors relies on a different working
principle. As it is possible to see in Figure 2.2(b) , the entire system acts as
a capacitor. In fact, when the pressure deforms the plate, it moves toward
an electrode placed on the bottom. Diminishing the distance between the
plate and the electrode causes the total capacitance to increase.
8

15. Figure 2.2. (a) Side and top views of a piezoresistive pressure sensor (from Johnson,
1992); (b) side view of a capacitive pressure sensor (from Lee and Wise,1982)
Therefore, knowing the relationship between the deformed configuration
of the plate versus the capacitance variation and the one between the
applied pressure versus the deformed shape of the plate, it is possible to
measure the pressure starting from the capacitance changes.
2.3.2. Accelerometers
Accelerometers or inertial sensors are widely used within the aerospace,
defence, automotive, marine and consumer industries. In the aerospace
industry they are used for flight stabilization of aircraft and rockets and
navigation. Automotive applications include vehicle stability systems,
rollover prevention systems and aids to navigation as well as impact
sensors in airbags. Naval and marine applications include ship
stabilization and navigation. The most common MEMS accelerometer
designs fall into two categories: capacitive and piezoresistive.
Figure 2.3 illustrates the working principle of a very simple capacitive
accelerometer. The device can be schematized as a second order
mechanical system, composed by a mass, a spring and a damper.
9

16. Figure 2.3. Schematic of a capacitive accelerometer
When the external acceleration acts on the system, the mass moves in the
opposite direction of the acceleration. The displacement is then sensed by
a capacitive system. It is formed by a number of electrodes that are
connected with the mass and therefore are free to move and others that
are clamped to the substrate. The displacement of the mass causes a
change in the capacitance measured by the electrostatic system. Thus
from the direct measure of this change is possible to compute the value of
the external acceleration.
The mechanisms of a piezoresistive accelerometer is very similar to the
one of the piezoresistive pressure sensor. Some piezoresistors (in black in
Figure 2.4) are diffused on the top surface of the device and electrically
connected in order to form a Wheatstone bridge. When the acceleration
acts on the system, the proof mass reacts moving and deforming the thin
plate it is suspended above (as schematically shown in Figure 2.4). The
deformation of the plate causes the resistance change of the
piezoresistors. The electronic circuit senses the resistance change and
computes the magnitude and direction of the acceleration from the
electric signal. MEMS accelerometer performance relies on many of the
same parameters as MEMS pressure sensors. The primary metrology
issues are accurate thickness control and accurate dimensional
manufacturing of the proof mass.
10

17. Figure 2.4. Schematic of a piezoresistive accelerometer
Of relevant importance are the material properties, such as Young’s
modulus, which affect device sensitivity, particularly in suspension
beams. Sidewall orthogonality and parallelism is also an issue for
accelerometers as deviations from parallel sidewalls can lead to device
non-linearity and failure.
2.3.3. Gyroscopes
As with accelerometers, gyroscopes are increasingly being used in
consumer products. In parallel to this, applications in the traditional
gyroscope markets, such as aeronautics and defense, are expanding.
Macro-scale gyroscopes normally use a large mass flywheel rotating at
high speed, however the entity of frictional forces, that cause a premature
failure of the system and lubrication techniques, not sufficiently efficient
at this scale, prevent this technique being viable for microdevices. As a
result, most micromachined gyroscopes use a mechanical structure that is
11

18. driven into resonance and rotation excites a second resonance due to the
Coriolis force (see Figure 2.5).
Considering the mechanical scheme of the gyroscope, it is possible to
write the equations of motion in the local frame. They are:
{m ¨xx ˙xkx x=F xm x ˙2
m y ¨2m ˙y ˙
m ¨yy ˙yk y y=F ym y ˙2
−m x ¨−2m ˙x ˙
, (2.1,2.2)
where m is the mass of the system, x and y the local reference
coordinates, x and y the damping coefficients, k x and k y the elastic
stiffnesses, Fx and Fy the forces acting on the system and  is the rotation
of the local frame with respect to an inertial system.
Figure 2.5. Schematic of a MEMS gyroscope
In order to measure the angular rate with this gyroscope, the x mode is
driven sinusoidally using Fx at an amplitude of Fd
and a pulsation of d.
The pulsation d is usually significantly higher than the specified
bandwidth of the gyroscope. At this high frequency, the terms x ˙2
, y ˙2
,
y ¨ and x ¨ are small and can often be neglected. The equations of
motion become:
12

19. {m ¨xx ˙xkx x=F x2m ˙y ˙
m ¨yy ˙yk y y=F y−2m ˙x ˙
, (2.3,2.4)
When the gyroscope experiences an external angular velocity, ˙=, the
2 ˙x term in equation (2.4) causes the y mode to vibrate at the driven
frequency d with an amplitude that is proportional to the angular rate,
. Therefore from the y output signal, the value of the angular rate can
be obtained.
An obstacle to this technology is the small size of the Coriolis force
compared to the driving force that is applied in an orthogonal direction.
One way of countering this is using structures with a high output over
input ratio, as for instance structures vibrating at the resonance of the
sensing axis. Unsatisfactory alignment of the drive mechanism with the
axis of freedom can also result in cross-talk, which can overshadow the
relatively small Coriolis force.
MEMS gyroscope performance relies on many parameters including the
dimensional and material properties of the resonators and the suspension
beams. Resonant frequencies of the gyroscopes are often measured with
vibrometers, as extra modes can cross-talk with Coriolis induced modes.
Suspension beams require dimensional measurements including sidewall
verticality and thickness measurement.
2.3.4. RF switches
High frequency switches are used in wireless communications and radar
systems for switching between the transmit and receive paths, for routing
signals to the different blocks in multi-band/standard telephones, for RF
signal routing in phase shifters used in phased-array antennas, and
numerous other applications. RF-MEMS switches offer significant
benefits over semiconductor switches in terms of high isolation (in
particular over 30 GHz), low loss over a wide frequency range, extremely
low standby power consumption and excellent linearity characteristics.
However, the main drawbacks remain the relatively high drive voltages
and slow response.
13

20. Figure 2.6 shows a very common RF switch layout. When the line that
brings the signal is at the same potential of the bridge, no forces act on the
system and therefore the switch is in the up state. If a constant voltage is
applied on the signal line, an electrostatic force arises that pushes the
bridge toward the line. If the voltage is bigger than a threshold value, the
bridge collapses on the line causing a change in the state of the switch,
that goes in the down position.
Figure 2.6. Side and top views of a RF switch (from Tilman, 2003)
14

21. Chapter 3
Mechanical characterization of
polysilicon as a structural
material for MEMS
3.1. Polysilicon as a structural material in
MEMS
Polysilicon is by far the most common structural material in MEMS
applications. It is used for a huge variety of applications, mainly because
of two factors:
• the existence of well-established deposition technologies, in which
polycrystalline silicon had a very important role since the
beginning of the microelectronics era;
• the excellent physical properties of this material. Its Young’s
modulus is higher than that of titanium and comparable with that
of steel. The rupture resistance at the micro-scale stands in the
range of the one of the best construction steels, while its density is
less than aluminum's one. Thermal properties, too, make it a very
good material for high temperature applications. Thermal
conductivity is very high, while the thermal expansion coefficient
is very small and the melting point is only one hundred degrees
less than iron’s one.
Since polysilicon is an aggregate of mono-crystalline silicon grains, its
properties depend on the properties of the crystallites composing it, on
their shapes, on their orientation and on the physical characteristics of the
15

22. grain boundaries between different grains. This means that the overall
properties of polysilicon will be strongly influenced by the process used
for the deposition; process data like the deposition temperature and the
deposition pressure will all influence the final properties of the material.
Besides the dispersion of the physical properties due to different
processes, it has to be added that at the micro-scale the measure of these
properties is a very difficult and challenging task. Earlier work conducted
by several researchers revealed significant differences in the measured
values of the elastic properties and of the nominal strength of polysilicon
without providing in-depth explanations for such a variety of
observations. A main question arose, therefore, as to whether the newly
evolving test methodologies were adequately precise. In pursuing that
question, a round robin study [Sharpe et al., (1998)] demonstrated the
inconsistency of measured modulus and strength values, even when
specimens from the same source were examined. The material in that
work was fabricated in close physical proximity from the same wafer of
the same run and in the same deposition reactor at the Microelectronics
Center of North Carolina (MCNC, now Cronos-JDS Uniphase). In that
round robin effort, the elastic modulus values differed considerably,
namely from 132 to 174 GPa, and the strength also demonstrated a rather
wide dispersion, ranging from 1.0 GPa, for specimens tested in tension, to
2.7 GPa for specimens tested in bending. A second round robin
examination, conducted on material fabricated at the Sandia National
Laboratories [La Van et al., (2001)] also showed signs of inconsistent
rupture strengths demonstrating a dependence on specimen size and
measurement technique. A relatively new round robin test was carried
out by [Tsuchiya et al., (2005)]. The specimens, produced with the same
process, in the same wafer, were distributed to five different research
groups. In this case the obtained modulus varied from 134 GPa to 173
GPa, while fracture strength varied from 1.44 GPa to 2.51 GPa. As a result
of these findings, it appears advisable for any micro-fabrication facility
not to use the properties cited in the literature for final design and
verifications but to identify the most feasible measurement technique and
to conduct measurements for every fabrication run. However, while
16

23. individual measurements provide values of some effective modulus and
strength for a particular material, such measurements can not be identified
unequivocally as mechanical properties of polysilicon. Variation of these
properties in micro-machined components is often on the order of 50% or
larger, frequently with no particularly strong correlation to the silicon
deposition process or crystal structure.
3.2. Testing methodologies
Resorting to some definitions provided by ASTM (American Society for
Testing and Materials) standards for testing at macro-scale, among
material properties of interest in the context of the present discussion are:
the Young’s modulus, defined as the slope of the linear part of the stress-
strain curve; the Poisson’s ratio, which measures the lateral expansion or
contraction when the material is subjected to uniaxial stress in the linear
range; the fracture strength, i.e. the normal stress at the beginning of
fracture; the response to cyclic loading in terms of the S-N curve, which is
a plot of the applied stress versus number of cycles at rupture. In order to
measure material properties one should be able to construct a specimen
according to a given design, apply an external input in terms of forces or
displacements and measure the specimen response using direct
procedures, in the sense that the variable of interest should be (almost)
directly measured. All these steps are fully standardised at the macro-
scale and are currently applied for testing construction materials like steel
and concrete. Unfortunately, these practices cannot be easily applied at
the scale of MEMS. In particular one has to resort to fully or partially
indirect approaches. E.g., in order to measure the Young’s modulus,
cantilever beams in bending are often utilised; deflection is measured and
the property of interest is computed on the basis of an analytical or
numerical model of the beam. Even during on-chip tension tests some
sort of inverse analysis has to be performed since, in general, only
capacitance variations are measured directly while deformations are
obtained on the basis of a numerical model. Many testing methodologies
have been proposed in the scientific literature for the extraction of static
17

24. mechanical properties of polysilicon [e.g. Sharpe, (2002)]. Limiting the
attention to silicon MEMS, a first general classification of test procedures
can be made between off-chip and on-chip devices. In both cases the micro-
device is generally produced by deposition and etching procedures. On-
chip test devices [see e.g. Corigliano et al., (2004)] are real MEMS in
which actuation and sensing is performed with the same working
principles of MEMS. On-chip devices rely on the fact that all the
mechanical parts needed to load the specimen and the majority of the
ones (electrical or optical) needed for the measure of displacements and
strains are built together with the specimen during the micromachining
fabrication process. Usually, in these structures two main parts (the
actuator and the sensing devices) can be found. In many cases these
consist in a large number of capacitors that can be used for the creation of
an electric field, this in turn causes a force to act onto the specimen or for
the measure of a capacitance which is directly related to the displacement
of the specimen itself. The advantage of on-chip testing methods is linked
to the ease of fabrication and of use (usually without costly equipments)
and to the fact that complicate handling and alignment of the specimen
are avoided. The major drawback is that the force developed by on-chip
actuators can be insufficient to break specimens in quasi-static conditions
and that the maximum displacement is also limited, in the order of the
micrometer. On-chip testing of MEMS devices is especially advocated
since the thin-film microstructure and state of residual stress is a strong
function of micro fabrication process steps. Nevertheless it requires
accurate modelling and numerical/analytical analyses of the whole
device. An off-chip test [see e.g. Sharpe, (2002)] generally resorts to some
sort of external gripping mechanism actuating the force and an external
sensor measuring the response of the specimen. All the experimental set-
ups that use an external apparatus (load cells, micro-regulation screws,
etc..) in order to create a stress state into the specimen are usually
included in the category of off-chip testing. In this case a lot of attention
has to be paid during the handling of tiny MEMS specimens, during the
system-specimen alignment and to the specimen gripping systems. The
challenge of picking a specimen only a few micron thick, place it into a
18

25. test machine and perform the test is a formidable task. The main
advantage of-off chip methodologies is that the forces and the
displacements can be relatively high to break even several micrometers
thick specimens in pure tension and that many different configurations
can be set up to create an a priori desired multi-axial stress state into the
specimen. In principle, material parameters for MEMS, and in primis the
Young’s modulus E, can be determined exploiting several test devices.
Among others: tension tests, bending of cantilever beams, resonant
devices, bulge tests, buckling tests. Clearly, the most direct approach is
the tension test, but unfortunately this is not always applicable since it
requires the deployment of considerable forces at the micro-scale in order
to produce sensible deformation in the specimen. Hence a wealth of
alternative solutions have appeared in the literature. In the following two
Sections the experimental mechanical characterization of polysilicon as a
structural material for MEMS is discussed. It was decided to group the
most important experiments in two principal families:
• quasi-static testing, used for the characterization of Young’s
modulus, Poisson’s ratio and fracture properties [Section 3.3];
• high-frequency testing, used for the characterization of fatigue
properties [Section 3.4].
3.3. Quasi-static testing
The very first tests carried out for the quasi-static mechanical
characterization of the material can be traced back in the 80’s [Chen and
Leipold, (1980)]. The volume of the specimens used were approximately 1
cm3
, huge if compared with typical MEMS dimensions. From the first half
of 90’s an increasing interest for this problem arose and the consequence
was that a large number of test typology was designed. In the following
of this section a selection of the test devices and set ups considered as the
most common and interesting is presented. The classification is based
upon the actuation mechanism and on the way the system response is
read.
19

26. 3.3.1. Off chip tension test
This is the most common and important technique [Bagdahn et al., (2003);
Chasiotis and Knauss (2002); Chasiotis and Knauss (2003); Chasiotis,
(2006); Chung-Seog Oh et al., (2005); Knauss et al., (2003); Sharpe et al.
(1997); Sharpe et al. (1999); Sharpe, (2002); Tai and Muller, (1990);
Tsuchiya et al., (2005); Tsuchiya et al., (2002)] for the measurement of
mechanical properties for MEMS applications. A MEMS specimen is
produced and then placed on a testing system, Figure 3.1. Usually the
specimen is gripped to the system with the aid of UV curing adhesives or
via electrostatic gripping. This is the way to re-create common macro-
scale testing techniques at the micro-scale. The displacement is imposed
on the specimen by the use of piezo-transducers, with a resolution in the
order of the nanometer. The load cells read the applied load with an
accuracy of some µN. Specimen displacements can be measured either
with the use of optical systems [Chung-Seog Oh et al., (2005)] or using
laser interferometry [Sharpe et al., (1997)] or via digital image correlation
[Chasiotis, (2006)]. The results obtained using this technique cover the
most important quantities for mechanical design with polysilicon. The
measure of Young’s modulus and in some cases of Poisson’s ratio [Sharpe
et al. (1997)], together with the rupture strength are the most common for
all the research groups that worked with off-chip tension testing. Besides,
it is important to underline that with this kind of setup it was possible to
study the scale effects due to specimen’s dimension and the stress
gradient acting in the specimen. Very recently [Chasiotis, (2006)] it was
also possible to determine the fracture toughness KIC of polysilicon.
This result was achieved by means of a nano-indentation nearby the
specimen that caused a crack to propagate through the substrate and
partially involve the specimen. At the end of the process it was thus
possible to have a pre-cracked specimen, necessary for a complete
fracture mechanics characterization.
Figure 3.2 shows other possible configurations for off-chip tension tests
[Sharpe, (2002)]. In Figure 3.2(a) a tensile specimen is first patterned onto
the surface of a wafer and then the gauge section is exposed by etching of
20

27. the wafer. The larger ends are gripped in a test machine. In Figure 3.2(b) a
specimen is fixed to a die at one end and actuated by means of an
electrostatic probe at the other end.
Figure 3.1. Typical scheme of a tension test. (From Tsuchiya et alii, 2002)
Figure 3.2. Off-chip tension test specimens: (a) supported in a frame; (b) off-chip tension
test specimen fixed at one end (From Stanley et alii, 2002)
3.3.2. Off chip and on-chip bending test
Out of plane bending of test specimens is generally performed via an off-
chip apparatus as in Figure 3.3 where a cantilever beam is deflected by a
diamond stylus. The deflection of the free end is measured and the
Young’s modulus is obtained through inverse modeling of the cantilever
beam. However, if the beam is long, forces are small and difficult to
calibrate; if the beam is short, forces are higher but inverse analysis of the
21

28. Figure 3.3. Off-chip out-of-plane bending of a cantilever beam (From Hollman et alii,
1995)
beam is more involved.
Doubly supported silicon-micromachined beams can also be used to
study the out of plane bending of materials (see Figure 3.4). In this case a
voltage is applied between the conductive polysilicon or micromachined
beam and the substrate to pull the beam down. The voltage that causes
the beam to make contact is a measure of the beam stiffness. Residual
stresses in the beams and support compliance cause significant vertical
deflections, which affect the performance of these micro-machined
devices. Tests need to be supported by models of the devices that takes
into account the compliance of the supports and the geometrical
nonlinear dependence of the vertical deflections on the stress in the beam.
Figure 3.4. On-chip out-of-plane bending (From Kobrinsky et alii, 1999)
In-plane bending is a classical test for MEMS since several structural parts
of accelerometers are subjected to this kind of deformation. A typical on-
chip layout is presented in Figure 3.5, where a cantilever polysilicon beam
attached to a moving mass (on the left) is subjected to bending induced
by the fixed rectangular block (on the right). Actuation is performed by
22

29. means of interdigited comb-finger capacitors. This test is often conducted
to establish the flexural strength of the cantilever beam as a structural
component, but requires considerable care when employed for the
evaluation of the Young’s modulus due to the uncertainties in
geometrical parameters and model of the beam. Additional informations
and a wide discussion on the on-chip approach can be found in Chapter 5.
Figure 3.5. On-chip in-plane bending (Courtesy of MEMS Production Divisionof
STMicroelectronics)
3.3.3. Test on membranes (Bulge test)
The bulge test is one of the earliest techniques used to measure the
Young’s modulus, Poisson’s ratio and/or residual stress of non-
integrated, free-standing thin structures. This testing method relies on the
use of thin polysilicon membranes (circular, square, or rectangular in
shape, see e.g. Figure 3.6), relatively easy to design and realize, bonded
along their periphery to a supporting frame. [Jayaraman et al., (1998);
Tabata et al., (1989); Yang and Paul, (2002); Ziebat, (1999)].
Microfabrication techniques are particularly well suited for the creation of
such test structures with reproducible and well-defined boundary
conditions. During the test the membrane is loaded with a pressure
difference acting on the top and bottom surfaces. The membrane deforms
23

30. and its profile is measured with a profilometer. Usually the deflection at
the centre is recorded as a function of the applied pressure. Several
analytical or semi-empirical formulas exist correlating deflection to elastic
properties. From this test it is possible to measure:
• the biaxial elasticity modulus;
• the Poisson’s ratio;
• the nominal rupture strength;
• the internal stresses (measuring the buckled configuration of the
membrane);
Figure 3.6. Typical scheme of a bulge test. (From Ziebat, 1999)
One of the shortcomings of this methodology is that sometimes the
membranes separates from the substrate before the end of the test.
Moreover, since the mechanical response of the membrane varies with the
third power of its thickness and the fourth power of its lateral dimension,
it is necessary to have a good fabrication technology and an accurate
measure of the thickness of the layer.
3.3.4. Nanoindenter-driven test
Nanoindenters are often used for the mechanical characterization of thin
films. Basically there are two ways of using nanoindenters: film
nanoindentation and the use of a nanoindenter as an actuator to load
MEMS structures. Hardness (indentation) tests are routinely used to
characterize large-scale structures. In direct analogy, considerable efforts
have been made to develop nanoindentation techniques to characterize
microscale structures, and commercial instruments have been developed
[e.g. Li and Bhushan, (1999)]. Nanoindentation experiments [Xiadong Li
and Bhushan, (1999); Ding et al., (2001); Kim et al., (2002); Chung-Seog Oh
24

31. et al., (2005)] are performed allowing the tip to penetrate the film under
study. During the penetration in the layer, an elastic plastic stress state is
generated. This is the main reason why the elastic characterization is
done during the unloading phase, when the tip starts going backward to
reach the rest position. The results of this experimental test is a force vs.
penetration depth plot (Figure 3.1) and the projected area of contact
under the indenter. Young’s modulus and fracture properties are
computed using some semi-empirical formulas. The main advantage of
this method is that there is no need for an ad hoc designed specimen; it is
sufficient to have a portion of material large enough to apply the
nanoindenter. However, the application of this technique to thin films is
complicated by several factors including substrate effects and pile-up of
material around the indenter. Another possibility is to use the
nanoindenter tip as an actuator to perform a sort of off-chip test (Figure
3.7). In these tests the tip moves an extremity of the specimen, causing a
stress state in it. The applied load is measured with a piezo scanner, while
there are different ways to measure the displacement of the specimen,
like interferometry [Espinosa et al., (2003)] or the measure of the
displacement of the tip with aid of a laser beam and photodiodes
[Sundarajan and Bhushan, (2002)]. This methodology could be very
accurate, but it needs a very expensive instrumentation.
Figure 3.7.Force vs displacement plot of a nanoindentation. (From Xiadong Li and
Bharat Bhushan, 1999).
25

32. Figure 3.8. Typical AFM driven test. (From Sundarajan and Bhushan, 2002).
3.4. High frequency testing
Polysilicon in MEMS technology is used for the fabrication of resonators,
gyroscopes and other devices that oscillate at high frequencies during
their whole life. One of the most important failure mechanisms for such
systems is fatigue. Fatigue is usually interpreted as a phenomenon caused
by the motion of the dislocations present in the material that can coalesce
during the stress cycles to form micro-cracks. Micro-cracks then join
together to form one or more macro-cracks that cause the failure of the
structure. Polysilicon is a brittle material and there is no dislocation
motion under temperatures of about 900° C, therefore it is not expected to
be prone to fatigue in usual operating conditions. Nevertheless, in the
second half of ‘90s some groups in the US started working on this subject
and found that also polysilicon can undergo fatigue after a large number
of cycles, typically more than 109
, combined with high stress levels. In
26

33. order to reproduce experimentally fatigue failures, it is very important to
work with experimental set ups that can allow a relatively high frequency
testing (at least 1kHz) which in turn allow to reach a large number of
cycles in reasonable time. On-chip tests are usually the best choice for this
kind of study because they can work at high frequencies and due to the
fact that with some electrical control system it is quite easy to perform
multiple fatigue tests at the same time.
3.4.1. Fatigue testing and on-chip structures
As pointed out in the previous sections, on-chip test systems make in
general use of electrostatic actuation between a fixed (stator) and a
movable part (rotor) to load the specimen with a desired level of stress.
The force developed by the actuator is proportional to the actuation area
and inversely proportional to the gap between the rotor and the stator. It
turns out that to have a force sufficiently large to induce fatigue into the
specimens one should have an highly scaled lithography, a thick
polysilicon layer and a design area big enough for the thousands of
capacitors needed for the actuation. These requirements are the main
reason why on-chip testing is not common for quasi-static
characterization. Since the MEMS is a dynamic system, the force needed
to move the seismic mass decreases if one loads the structure with a time-
varying force at a frequency close to the resonant frequency of the system.
This is exactly what is usually done in fatigue tests for MEMS, in these
cases a reasonable low voltage is used to bring the specimen to fatigue
rupture. Very interesting results were obtained e.g. in [Bagdahn and
Sharpe, (2003)].
3.4.2. Fatigue mechanisms
As discussed in the previous section, it has been experimentally shown
that fatigue in polysilicon is a possible failure mechanism. Nevertheless,
the reasons why fatigue rupture occurs are not yet completely clear and
understood. Among the most active groups in the study of fatigue in
polysilicon, are those of Pennsylvania State University and of Case
Western Reserve University. These two groups proposed the most known
27

34. and accepted interpretations for fatigue mechanisms in polysilicon. The
first one after an experimental campaign conducted with specimens like
that shown in Figure 3.9, [Muhlstein et al., (2001); Muhlstein et al., (2002);
Muhlstein et al., (2004)] proposed a mechanism called reaction layer fatigue
(Figure 3.10) which can be summarized as follows: the native oxide is
formed, when the polysilicon is first exposed to air, as one of the final
steps of the process; the oxide thickens in the highly stressed regions and
becomes the site for environmentally assisted cracks which grow in a
stable way in the layer; when the critical size is reached, the silicon itself
fracture catastrophically by trans-granular cleavage.
The second group [Kahn et al., (2000); Kahn et al., (2002); Kahn et al.,
(2004)] after experiments conducted on specimens like the one shown in
Figure 3.11, showed that even an high stress state cannot cause an
appreciable growth of the native oxide, thus excluding the pure
environmentally assisted fatigue. On the other side, the Authors noticed
that increasing the level of humidity, fatigue life decreases. They did not
propose a specific fatigue mechanism for polysilicon.
Fatigue testing remains one of the most open research area in the field of
mechanical characterization of polysilicon. The mentioned researches are
only examples of a wider discussion now active in the scientific literature
[see e.g. also Ando et al., (2001)].
Figure 3.9. Test structure adopted in (From Muhlstein et alii, 2001)
28

35. Figure 3.10. Reaction layer fatigue mechanism. (From Muhlstein et alii, 2002)
Figure 3.11. Test structure adopted in (Kahn et alii, 2000)
29

36. Chapter 4
Weibull theory applied to the
study of polysilicon strength
4.1. General concepts
Brittle materials, such as materials used in MEMS applications, together
with common engineering ceramics, rocks and concrete have been widely
used for structural components for their excellent resistance to heat,
corrosion, and wear. But brittle materials also break easily and their
strength, i.e., the maximum stress they can withstand, varies
unpredictably from component to component even when nominally
identical specimens are tested under the same conditions. This variability
is not the result of a wrong preparation of specimens, but it is an intrinsic
characteristic of the material that cannot deform plastically under the
action of growing loads. Therefore, the strength of a brittle material is not
a well defined quantity and has to be described with respect to fracture
statistics. Since the 70's Weibull distribution [Weibull, (1951)] was widely
and successfully used by a number of engineers and scientists for the
study and characterization of the strength of brittle materials [Stanley and
Inanc (1984), Bažant (1991)]. This is the reason why it has been recently
applied also to the study of rupture phenomena in polysilicon MEMS
[Chasiotis and Knauss (2003), Corigliano et al. (2004), Corigliano et al.
(2005), Sharpe et al. (1997)]. Weibull theory essentially gives a way to
estimate the failure probability of a mechanical system, starting from the
computation of the probability of failure of its weakest part, the theory is
therefore also known as the weakest link approach. By means of the Weibull
approach it is possible to take into account the experimental scatter of
30

37. strength values typical of brittle materials, the statistical size effect and
the dependence of the probability of failure on the stress distribution.
4.2. Basic theory
The application of Weibull approach to a uniformly stressed uniaxial
specimen gives the following equation for the probability of failure Pf:
P f =1−exp
−
V
V r
〈−u
0
〉+
m
, (4.1)
where: V is the volume of the bar, u, 0, m are material parameters and
the 〈 x〉+ symbol denotes the positive part of x. The parameter u, often
called threshold stress, represents the stress below which the probability of
failure is null. The parameter 0
is the value of the stress that added to u
gives the failure probability of the 63.2% to break a specimen with a
volume V r. m, known as Weibull modulus, is inversely proportional to the
spread of the rupture strength of tested specimens. In the weakest link
framework, the failure probability of the entire structure is defined as the
probability that just one element (the weakest) of the structure fails.
If a body is loaded with a uniaxial but non homogeneous state of stress,
the expression of the probability of failure takes into account the
contribution to the total failure probability of every infinitesimal part.
Thus, equation (4.1) can be written as:
P f =1−exp
−
1
V r
∫
V
〈 x−u
0
〉+
m
dv '
. (4.2)
In the case of a multi-axial, non uniform stress state it is usually assumed
that cracks form in the planes normal to the principal stresses
1 x, 2 x ,3 x. Thus the probability of failure is given by:
P f =1−exp
−
1
V r
∫
V
∑
i=0
3
〈i  x−u
0
〉+
m
dv '
. (4.3)
31

38. The above equation is of general applicability and has been obtained by
iterating the hypothesis of statistical equivalence of all elementary parts
which constitute the volume V. It is also important to notice that the
fracture criterion based on principal stresses in a 3D situation is another
strong assumption which could be substituted by another one, giving rise
to a different stress function in the integral of equation (4.3). The general
expression is here applied under the assumption that parameter u=0,
this means that all level of stresses have an influence on the probability of
failure. Equation (4.3) is then re-written in a more compact way:
P f =1−exp
−
1
V r
∫
V
 x
0

m
dv'
, (4.4)
having defined the equivalent stress   x as:
 x≡∑
i=0
3
〈i x〉+
m

1
m
. (4.5)
The above relations can be used in order to estimate the probability of
failure Pf of a given structure or solid once the Weibull parameters m and
0 are known and the elastic distribution of stresses has been computed
via analytical formulae or numerical solutions, e.g. by means of the FE
method. Parameters m and 0 are usually experimentally determined
starting from a series of uniaxially tensile tests on cylindrical specimens
of volume V; in this simple case equation (4.1) reduces to:
P f =1−exp
−
V
V r

0

m
. (4.6)
Weibull parameters can be identified also from a specimen or structure
loaded in a multiaxial situation with a non-uniform stress distribution.
Let us re-write equation (4.4) in a form similar to (4.6):
P f =1−exp
−
1
Vr
∫
V
 x
0

m
dv'
≡1−exp
[−
V
V r
nom
0 
m

]. (4.7)
32

39. In (4.7) the nominal stress nom and the function  have been defined as:
nom≡max
x
[  x ] (4.8)
  x ,V , m≡
1
V
∫
V
 x
nom

m
dv' (4.9)
nom represents a nominal stress in the non uniformly stressed specimen
or structure, which acts as a scaling parameter for the elastic response.
The function  depends only on the normalized stress distribution in the
linear elastic response and is therefore independent from the load level.
4.3. Statistical size effect and stress gradient
effect
One of the main features of the Weibull approach relies on the fact that it
is capable to predict many typical aspects of the mechanical behavior of
brittle materials. In the MEMS scientific community, many researches
were performed in order to understand the importance of the size on the
value of the maximum stress sustainable by the structure [Ding et al.
(2001)] as well as the importance of the stress distribution inside the
mechanical component [Jadaan et al. (2003)], that can have a big impact
on the value of the peak load that causes the collapse. In order to compare
the behaviour of different structures, it is possible to define a critical
stress level 0
*
that gives the 63.2% of failure probability for a body under
the action of assigned loads. From equation (4.7) it follows:
V
V r
0
*
0

m
=1 . (4.10)
Therefore 0
*
can be written as:
0
*
=0V r
V 
1
m
, (4.11)
33

40. that expresses the dependence of 0
*
on the parameters , V and m.
4.3.1. Effect of the modulus
Weibull modulus m can be considered as a measure of the spread of the
failure stress around its mean value. As shown in Figure 4.1, this value
governs the slope of the failure probability plot in its linear part. If the
value of the modulus is small, the failure distribution presents a big
spread. As the modulus increases, the probability distribution becomes
narrower, until, when m∞, the cumulative probability function
becomes a step function. It means that a stress level lower than 0 cannot
cause a failure into the structure, while a stress level bigger that 0 for
sure causes the mechanical collapse. Weibull modulus affects the
importance of the size effect, too. If one takes equation (4.11), once fixed
the value of V, V r and , and plots the value of the parameter 0
*
against
the value of the modulus, as done in Figure 4.2, he notes that for large
values of m, 0
*
approaches to 0. This means that for large values of the
modulus the size effect vanishes.
Figure 4.1. Failure probability with different values of the modulus
34

41. Figure 4.2. 0
*
/0 ratio as a function of the Weibull modulus
4.3.2. Effect of the volume
Weibull approach relies on the probability of finding a critical defect in
the structure. Therefore it appears natural that a structure with a larger
volume has a worse mechanical behaviour than a smaller one and
mathematically this can be seen considering equation (4.11), plotted in
Figure 4.3.
Figure 4.3. 0
*
/0 ratio as a function of the volume
35

42. Figure 4.4: Schematics of a pure tension on specimens with different volume
To make an example it is possible to consider two structures, as
schematized in Figure 4.4, with respectively a volume equal to V and 3V,
loaded with the same uniaxial tensile stress . Since the stress is constant
in every point of the volume, the value of the parameter , computed
applying equation (4.9), is one. Considering then the same value of the
representative statistical volume V r for both structures, it is possible,
making use of equation (4.6), to compute the failure probability in the two
cases, as shown in Figure 4.5. The plots indicates that, fixing the level of
the applied stress, the size effect associated to the volume is interpreted as
a bigger failure probability of the larger structure with respect of the
smaller one.
Figure 4.5.Failure probability of the structures of Figure 4.4
4.3.3. Effect of the stress distribution
The stress distribution in a brittle structure plays a very important role in
determining the maximum load applicable. Let us for instance consider
36

43. two different load conditions schematically drawn in Figure 4.6. The left
one represents a specimen with an homogeneous state of stress (pure
tension) and the right one represents a specimen with a non uniform
stress field (bending cantilever) with the same maximum value of the
former case.
Figure 4.6: Schematics of a pure tension and a pure bending specimens
In a pure tensile loading condition, every single material point
contributes to the total failure probability, while in the bent cantilever the
contribution is given only by a small portion of the body, near to the
clamped edge, that is highly stressed. It turns out that the failure
probability of the single tension specimen is higher than that of the
bending specimen. In the Weibull approach, the parameter that takes into
account the stress distribution is . In the two scenarios considered, using
the definition given in equation (4.9) and fixing the values of the modulus
and of the volume, the value of  in the pure tension case is:
tension=1 . (4.12)
In the case of the bending cantilever, using the Euler-Bernoulli beam
model, disregarding the shear stresses and considering the reference
frame of Figure 4.7, the positive tensile stress in a generic point P≡x , y
of the body is:
 x , y =F
 L−x
I
〈 y 〉+ , (4.13)
where F is the applied force, L the is length of the beam and I is the
moment of inertia of the section.
37

44. Figure 4.7: Reference frame for the cantilever beam
The maximum value of the stress is therefore:
nom=
F L
I
h
2
, (4.14)
where h is the height of the section.
Applying the definition of  given in equation (4.9), it yields:
=
1
V
∫
V
 x , y
nom

m
dv '=
1
Lt h
∫
0
L
∫
0
h
2
[tL−x
L
2y
h 
m
]dx dy , (4.15)
having defined t as the thickness of the beam. The final value of ,
computed solving equation (4.15) is:
=
1
m12
2m1
. (4.16)
Equation (4.12) can be used to compare the failure probability of the pure
tension structure against that of the bending cantilever. The value of the
stress that causes a failure probability equal to 63.2%, upon the
substitution of the value of  in equation (4.7) in the pure tension case is:
0
*tension
=0
V
Vr

−
1
m
, (4.17)
while for the bent beam is:
0
*bending
=0
V
V r

−
1
m
[m12
2m1
]
1
m
. (4.18)
38

45. Combining equations (4.17) and (4.18), one can express the ratio between
the critical stress 0
*bending
and 0
*tension
:
0
*bending
0
*tension
=[m12
2m1
]
1
m
. (4.19)
This function, plotted in Figure 4.8, shows that the stress necessary to
have a fixed value of failure probability for the bent beam is always
bigger than the one for the purely tensioned structure.
Figure 4.9 is a plot of three failure probability curves. The continuous line
refers to the pure tension test and the dashed curves refer to a bending
cantilever with two values of Weibull modulus.
Figure 4.8: Values of
0
*bending
0
*tension as a function of the Weibull modulus
39

46. Figure 4.9: Influence of  on the failure cumulative plot
4.4. Application of Weibull approach to 3D
structures
The computation of the failure probability of a structure or the
determination of Weibull parameters on a pure tension specimen or on a
simple cantilever beam are relatively easy, a closed form solution in fact
exists for expression of the stress field. When the shape of the structure
and the loading conditions do not allow for any simplifying hypotheses
in order to achieve a closed form solution, these operations may present
some practical difficulties. For this reason two different routines were
implemented. The former one makes use of the results of a linear elastic
FE solution to compute numerically the effect of the volume and that of
the stress distribution in the structure. The latter needs both the FE
solution to compute the value of the parameter  and the experimental
results for the minimization of the objective function that allows for the
determination of Weibull parameters.
4.4.1. Failure probability computation
Most of the effort necessary for the computation of the failure probability
is due to the computation of the coefficient  (Equation ), that is a
40

47. measure of how in-homogeneous is the stress inside the body. Once
discretized the body into Finite elements, from equation (), making use of
the linearity property for the integral operator, the equation that gives the
value of coefficient  can be rewritten as:
≡
1
V
∫
V
 x
nom

m
dv'=
1
V
∑
i=0
n.el
∫
V el

i
el
 x
nom

m
dv' , (4.20)
where n.el is the total number of elements of the discretization, i
el
 x is
the stress norm into the i-th element and V el is the volume of the i-th
element. Using Finite element solution in terms of stresses, equation
(4.20) can be approximated as:
≈
1
V
∑
i=0
n.el
∫
V el
i
FE
 x
nom

m
dv' , (4.21)
where i
FE
 x is the approximated stress field into the i-th element.
Then, solving numerically the integral in (4.21) by the means of the
Gaussian integration, the final expression of  becomes:
≈
1
V
∑
i=0
n.el
∣J i∣∑
j=1
n.gp
w jel
FE
x j 
nom

m
, (4.22)
Where ∣Ji∣ is the determinant of the Jacobian of the i-th element, n.gp is
the number of Gauss points and w j is the j-th Gauss weight.
Since the Weibull modulus m can be even in the order of fifty, the number
of Gauss points needs to be as large as possible. As an example in this
work the number of Gauss points for the integration of quadratic
tethraedra was chosen to be 11.
Once computed , the expression of the failure probability as a function
of the maximum value of the stress norm nom is given by equation (4.7).
4.4.2. Weibull parameters identification
The identification procedure is composed of two separate steps. The first
one consists in the identification of the modulus m and of the structural
parameter 0
*
. In order to accomplish this, a least square minimization is
41

48. performed on an objective function. Being the experimental data the
couples of values P f i
exp
,i
exp
, the objective function to minimize is:
m ,0
*
= ∑
i=1
numtest
[Pf i
exp
−Pf i
exp
]
2
. (4.23)
After the minimization of , once obtained m and 0
*
, it is possible to
compute the value of the parameter  as shown in the previous
paragraph. Then, manipulating the equation (4.11), the expression of the
material parameter 0 becomes:
0=0
*

V
V r

1
m
(4.24)
Since the right hand side of the (4.24) is known, it is finally possible to
compute 0.
42

49. Chapter 5
On-chip testing of ThELMATM
polysilicons
5.1. The ThELMATM
process
The test devices have been produced following the surface
micromachining process ThELMA (Thick Epipoly Layer for
Microactuators and Accelerometers) which has been developed by
STMicroelectronics to realize in-silicon inertial sensors and actuators.
The Thelma process permits the realization of suspended structures
anchored to the substrate through very compliant parts (springs) and
thus capable of moving in a direction orthogonal to the plane of the
wafer, such as the structure described in paragraph 5.2 or in a plane
parallel to the underlying silicon substrate, such as the one described in
paragraph 5.3. The process flow exploits several state-of-the-art
integrated circuit technology steps, together with dedicated MEMS
operations, like high aspect ratio (trench), dry etch and sacrificial layer
removal for structure release.
This technology is more complex than the Surface Micromachining but
allows to obtain silicon structures with a relatively large thickness; this, in
turn, increases the vertical surfaces and the global capacitance in
electrostatic actuators which move parallel to the substrate.
The process consists of the phases concisely enumerated hereafter and
schematically illustrated in Figure 5.1.
43

50. Figure 5.1. Schematic illustration of Thelma surface micromachining process.
(1) Substrate thermal oxidation. (2) Deposition and patterning of horizontal
interconnections. (3) Deposition and patterning of a sacrificial layer. (4) Epitaxial
growth of the structural layer (thick polysilicon). (5) Structural layer patterning by
trench etch. (6) Sacrificial oxide removal and contact metalization deposition.
44

51. • Substrate thermal oxidation. The silicon substrate is covered with a
2.5 µm-thick layer of permanent oxide obtained with a thermal
treatment at the temperature of 1100 °C.
• Deposition and patterning of horizontal interconnections. The first 0.7
µm-thick polysilicon layer (poly1) is deposited above the thermal
oxide; this layer is used to define the buried runners which are used
to bring potential and capacitance signals outside the device, or
eventually as a thin structural layer.
• Deposition and patterning of a sacrificial layer. A 1.6 thick oxide layer
is deposited by means of a Plasma Enhanced Chemical Vapor
Deposition (PECVD) process. This layer, together with the thermal
oxide layer, forms a 4.1 µm-thick layer which separates the
moving part from the substrate and which can be considered
analogous with the sacrificial layer in a Surface Micromachining
process.
• Epitaxial growth of the structural layer (thick polysilicon). The
polysilicon is grown in the reactors, thus reaching a thickness of
15 µm.
• Structural layer patterning by trench etch. The parts of the mobile
structure are obtained by deep trench etch which reaches the
oxide layer.
• Sacrificial oxide removal and contact metalization deposition. The
sacrificial oxide layer is removed with a chemical reaction; in
order to avoid stiction due to attractive capillary reactions, this is
done in rigorously dry conditions. The contact metalization is
deposited; this will be used to make the wire-bonding between the
device and the metallic frame.
45

52. Figure 5.2. Section view of epitaxial polysilicon
The final structure of silicon produced by the Thelma process is given by
a series of single crystals in the shape of vertical columns (see Figure 5.2),
each of them grown in an epitaxial reactor.
5.2. Study of the role of grain structure and
surface defects in thin polysilicon
5.2.1. Out of plane structure description
The first on-chip device discussed in the present work is shown in Figure
5.3(a) and (b) [see also Cacchione et al., (2005)]. Figure 5.3(c) is a zoom of
the central part where the 0.7 µm thick beam specimens are placed and
shows the deformed shape of the specimens caused by the application of
the force. The actuator is an holed plate of 15 µm thick polysilicon and it
is suspended on the substrate by means of four elastic springs placed at
the four corners (see Figure 5.4(b)).
46

53. Figure 5.3. (a) Sem image of the structure; (b) top view of the structure; (c) specimens
working principle
Figure 5.4. (a) Cross section next to the specimens; (b) sensor and suspension spring
detail
The holed plate is also connected to the thin polysilicon film specimens
placed at the centre, as shown in Figure 5.4(a). The two symmetric
specimens are in turn connected on one side to the holed plate, while on
the other are rigidly connected to the substrate, therefore behaving as a
couple of doubly clamped beams. The holes in the plate are due to the
47

54. etching process for the elimination of the sacrificial layer, thus allowing
for the movement of the plate with respect to the substrate. The
movement in the direction orthogonal to the substrate is obtained by
electrostatic attraction. The application of a voltage on the plate produces
an electrostatic force that pushes the plate toward the substrate, so that
the whole plate and the substrate work as a big parallel plate electrostatic
actuator. When the plate moves towards the substrate, the couple of
specimens undergoes bending, as shown in Figure 5.3(c). It is important
to remark that only the squared part of the holed plate acts as an actuator,
while the holed rectangular parts added to each side of the plate act as
sensors (see Figure 5.4(b)). These sensors are designed in order to
measure the capacitance variation with respect to rectangular electrodes
placed under the thick polysilicon mass. From the capacitance variation is
then possible to compute the displacement of the actuator in the direction
orthogonal to the substrate. The length of each specimen is 7 µm; in order
to force the rupture in a priori chosen section, as shown in Figure 5.3(c),
their cross section changes with a linearly varying width which decreases
from 3 µm to 1 µm.
5.2.2. Numerical models for the computation of
electrostatic fields
Given the complex geometry of the actuator and of the sensing system, it
was not possible to use simple analytical formulae to compute the force
developed by the plate and the displacement measured by the sensing
system in order to achieve the final force vs. displacement experimental
plot. For this reason a series of numerical analysis was performed.
Actuator
FE simulations were done to understand the electrostatic behavior of the
plate, the influence of the holes and of the fringing perimetrical fields on
the force developed by the actuator.
The analyses were carried out varying the gap between the plate and the
substrate, in order to get the force vs. gap relationship. The direct result of
the analysis was the electrostatic energy of the system. The numerical
48

55. results were then fitted by means of a 6th
order polynomial. The force
developed by the actuator and the energy contained in the electric field
are linked by the equation:
Fact=−∂ E
∂x , (5.1)
where x is the displacement variable and E is the electrostatic energy.
Therefore computing the derivative of the polynomial expression of
energy with respect to the displacement, it is possible to compute the
force.
These results were compared with the analytical formula for the
electrostatic force developed by an infinitely thin square plate, without
holes, with the same lateral dimensions, taking into account perimetrical
fringing effects. As it is possible to see in Figure 5.5, in the gap range
available for the actuation (4.2÷1.8 µm), the two curves are almost
superposed, thus allowing to use a very simple analytical formula to
compute the force Fact developed by the plate:
Fact=
1
2
0
[ l2
g0−x
2 
P
2
1
g0−x]V
2
, (5.2)
where 0 is the permittivity of vacuum, l the side of the plate, V the
applied voltage, P the perimeter of the plate, g0 the gap at rest and x the
displacement. The results of the simulations show that the holes present
in the plate reduce the amount of the net surface of the capacitor but, due
to the non-negligible lateral dimensions of the sidewalls, the electrostatic
field inside the holes is not null and consequently compensates the lack of
net surface of the holed plate.
49

56. Figure 5.5. Actuation force vs. gap plots
Sensing
The distance between the rotor and the stator part of the sensing
electrodes is 1.6 µm when zero voltage is imposed between rotor and
stator; it decreases untill 0.2 µm when the maximum voltage is applied.
The surface of the sensing electrodes is 36 x 157 µm2
. This means that if
one would like to use FE to simulate the system when voltage is high, he
would be forced to discretize the system with a very fine mesh in order to
avoid high aspect ratios in the elements.
Figure 5.6. Bem mesh of the capacitive sensor
50

57. This is the main reason why the Boundary Elements Method
implemented in the commercial software CoventorwareTM
was used. This
method needs only the discretization of the surfaces of the electrodes
(Figure 5.6), to compute the capacitance of the sensing system and its
variation with the gap.
The analysis were carried out decreasing the gap with a step equal to 0.1
µm; the results were subsequently fitted (Figure 5.7) to get the expression
below:
x=
0.2598
C 2
1.3112C−0.0315
, (5.3)
where x, in µm, is the gap and C, in pF, the measured capacitance.
Figure 5.7. Gap versus capacitance variation plot
5.2.3. Data reduction procedure
Tests were carried out at room temperature and at atmospheric humidity,
with a probe station mounted on an optical microscope. The experimental
setup is shown in Figure 5.8.
51

58. Figure 5.8. Scheme of the experimental setup
The input voltage given to the structure and the variation of the
capacitance induced by the displacement of the rotor is measured
connecting an Agilent Precision LCR Meter between two pads. The LCR
resolution, in the range of measures of interest in this work, is ±1 fF. A
slowly increasing voltage is applied in order to induce quasi-static
loading conditions in the specimen. The experimentally determined
capacitance vs. voltage plots are transformed in force vs. displacement
plots shown in Figure 5.9, by making use of the relationships between
capacitance and displacement and between voltage and electrostatic force
described in the previous section.
Figure 5.9. Displacement vs. Force plot
Starting from the force-displacement plot, the force acting on the
specimens was obtained by subtracting the part equilibrated by the elastic
52

59. suspension springs in the four corners of the holed plate (see Figures 5.3
and 5.4). Then from the net force vs. displacement curve, it was possible
to obtain the Young’s modulus of the material and the nominal rupture
strength with the aid of a linear elastic Finite Element (FE) analysis
performed on the 3D FE mesh of Figure 5.10, as described later on.
One half of the specimen is modelled and discretized and a fixed uz
displacement is imposed on the anchor point of the specimen to the plate.
On the anchor points between the substrate and the oxide underneath the
specimen a null displacement is imposed .
Figure 5.10. Finite Element mesh and mesh detail used for mechanical simulations
Making the hypothesis of linear elastic material and small displacements,
the global mechanical behavior results to be linear and the specimen’s
stiffness is proportional to the Young’s modulus of the thin layer:
F=K sp uz , K sp=E k sp , (5.4, 5.5)
where F is the applied force, Ksp is the elastic stiffness and ksp is the
stiffness divided by the Young's modulus E. By making the additional
hypothesis that relation (5.5) holds also for the real structure, with the
same coefficient ksp, it is possible to find the value of Young’s modulus
from a linear fit of the experimental force vs. displacement plot, which
allows to compute the experimental stiffness Ksp. Due to the linear
53

60. proportionality stated by (5.5) it is in fact possible to write:
ksp=
K sp
FE
E
FE =
K sp
Ex
E
Ex , E
Ex
=E
FE Ksp
Ex
K sp
FE  . (5.6, 5.7)
where the superscript (FE) refers to the simulation and the superscript
(Ex) to the measured quantities. The determination of the rupture
strength was based on the same hypotheses made to find relations (5.4
and 5.5). The same linear proportionality between the applied force and
the maximum stress in the specimen can be stated both for experimental
and for simulated results:
Fz
 FE
max
 FE =
Fz
Ex
max
Ex , max
Ex 
=max
FE Fz
Ex
F z
 FE . (5.8, 5.9)
Relation (5.9) was applied in order to find the maximum stress at rupture,
substituting to Fz
Ex
the experimental value of the force at rupture, to Fz
FE
the value of the force equivalent to the applied load in the FE analysis and
to max
FE 
the maximum value of tensile stress found in the FE simulation, in
the section where rupture occurred experimentally (Figure 5.11). It is
important to remark that, besides the hypotheses of homogeneity and
linearity, relations (5.6, 5.7) and (5.8, 5.9) hold only if the geometry of the
specimen is carefully reproduced in the FE model. The final FE mesh was
therefore obtained after SEM images in order to reproduce the real
geometry.
54

61. Figure 5.11. Principal tensile stress contour plot
5.2.4. Results and discussion
The values of Young’s modulus obtained from 20 tests are 174 ± 9 GPa,
with a variation coefficient of 5%. The low dispersion around the mean
value confirms the quality and good reliability of the whole procedure.
The data concerning rupture of the specimens were interpreted in the
framework of Weibull approach, as discussed in Chapter 4. The Weibull
modulus obtained was m=5.13 and the Weibull stress 0
*
obtained
directly on the non uniformly loaded specimen is 2773 MPa with a
standard deviation equal to 648 MPa; the Weibull stress 0 obtained after
the reduction due to the stress gradient and size effect, i.e. the Weibull
stress of an equivalent, uniformly loaded, reference volume specimen is
2237 MPa. These results have been compared to the ones obtained with in
plane moving structures made up with the same LPCVD 0.7 µm thick
polysilicon [Cacchione et al., (2004)]. In this case the values obtained for
the Young’s modulus were 178 ± 2 GPa and the Weibull parameters
resulting from the data reduction were m=6.18 and 0=1840 MPa. The
value of the Young’s modulus remains substantially the same in the two
cases, meaning that the elastic properties of the material can be
considered constant along the thickness of the deposited layer and that in
this case there is no important influence of the crystalline granular
55

62. structure. Something different happens analysing the rupture properties
of the material. In fact the Weibull modulus remains almost the same,
while the value of 0 is about 20% bigger in the case of out of plane
loading condition. Figure 5.12 shows that trying to predict the failure
probability of the “out of plane structure” using the data obtained from
the in plane one, the result would be a little conservative.
Figure 5.12. Prediction of the failure probability of out of plane structures using in plane
failure data
The causes of this difference could be related to the fabrication process
and in a special way to the thin polysilicon etching step that defines the
geometry of the specimen itself. In fact the polysilicon sidewalls surfaces
are exposed to the gas during the etching, while the top surface does not,
being it covered with the resist mask. The etching could introduce on the
sidewalls surface micro flaws, micro defects and internal stresses, only
partially recovered during the subsequent process steps, which could
reduce the ultimate strength of the material.
56

63. Figure 5.13. A Sem image of the sidewalls
Figure 5.13 is a SEM image of the sidewalls of the polysilicon layer. It is
possible to see that the surfaces are characterized by the presence of
micro-grooves. These could be privileged sites for the fracture initiation
because of the stress concentration effect they can induce given their
relatively small fillet radius.
5.3. Design and testing of a new test structure
for the mechanical characterization of
thick polysilicon
5.3.1. Structure description
The second device designed and then experimentally tested during this
work is shown in Figure 5.14. Compared to the one presented in the
previous paragraph, this structure presents many differences. It was
designed to test the mechanical properties of the epitaxial polysilicon, the
'thick' layer deposited with ThELMATM
process. This film is almost
twenty times thicker than the poly1 one and therefore it is necessary a
bigger force to break the specimen. This explains the reason for the
relatively big dimensions of the structure, that takes up a 1600 by 2250
57

64. Figure 5.14. Top view of the structure and zoom of the specimen.
58

65. µm2
rectangular area. The actuation movement is in a plane parallel to
the one of the wafer and it is due to a number of comb finger capacitors, a
very common choice for MEMS structures. In the next sub-sections the
structure will be divided into four parts, separately discussed in detail.
Actuator
The electrostatic actuation is realized by more than four thousands comb
finger actuators. The comb fingers are grouped on specific structures
called arms (see Figure 5.15).
Figure 5.15. Actuation arm (the dashed area represents the fixed part).
Every arm contains 31 comb finger actuators and its capacitance, as a
function of the seismic mass displacement x is:
C arm=C031Ccomb=C031
20 t
g
x , (5.10)
where C 0 is the capacitance at rest of the system, C comb is the capacitance
of a single comb finger actuator. The symbol t represents the thickness of
the layer, equal to 15 µm, and g, equal to 2.2 µm , the gap between stator
finger and rotor finger. The force developed by every arm is:
Farm=
1
2
∂ Carm
∂ x
V
2
=31
0 t
g
V
2
, (5.11)
where V represents the applied voltage. The total number of arms na is
130. Hence the total force developed by the actuator can be expressed as:
Fact=na
1
2
∂C arm
∂ x
V
2
=4030
0 t
g
V
2
. (5.12)
59

66. Frame
The frame is a suspended structure that supports the actuator arms (see
Figure 5.16). In the central part of the frame six suspension springs
(colored in light gray in Figure 5.16) are placed. These springs avoid the
collapse of the structure onto the substrate during the actuation. They are
rectangular cross-sectioned l=291 µm slender beams. The in-plane
width is w=3.2 µm, while the out-of plane thickness is t=15 µm. By
assuming a Young's modulus E=145 GPa, as determined in Corigliano
et al. (2004), the linear stiffness of the six springs for a movement parallel
to the substrate as shown in Figure 5.16 can be easily computed:
kspring =6
12
EJ
l3 =6
12
Et w3
12l3 =17.35
 N
 m
, (5.13)
being J the moment of inertia of the section. With reference to Figure
5.14, the upper part of the frame is clamped to the specimen, that is thus
loaded with the force developed by the actuator that is not absorbed by
the spring system. It is worth noting that the suspension system was
designed in order to be as compliant as possible to leave almost all the
force ( the 90% of the force developed) to load the specimen.
Figure 5.16. Schematic of the frame (filled with dashed line). In black the anchor points,
in gray the six suspension springs
60

67. Sensing
In the upper part of the structure, as in Figure 5.14, there is the sensing
system. It is made up with six arms with 80 comb finger capacitors on
each (Figure 5.17).
Figure 5.17. Sensing arm (the dashed area represents the fixed part).
The total sensing capacitance, as a function of the displacement of the
specimen is:
C sens=C0 sens480
20 t
g
x . (5.14)
With the aid of FE electrostatic analysis a simulation of the sensing
system was performed. The results of the simulations show how the
analytical formula holds for a displacement up to 10 µm, that was never
reached during the experimental campaign.
Specimen
The specimen of the structure was designed in order to perform both
quasi-static and fatigue testing. It consists in a lever system that causes a
stress concentration in a very localized region (Figure 5.18). The specimen
can be divided into four parts :
• a beam that is the physical link between the frame and the
specimen;
• the lever, that transforms the axial action coming from the beam
into a bending moment acting in the notched zone;
• a notch, that is the most stressed part, where the crack nucleates;
• a part fixed to the substrate.
61

68. Figure 5.18. Deformed shape of the specimen and contour plot of principal tensile stress
of the notched zone
5.3.2. Data reduction procedure
The experimental setup and the instrumentation used is the same as that
of the out of plane thin polysilicon structure. In this case the electrical
scheme is slightly different because the actuator is used only to load the
specimen, while the displacement is measured with the sensing system
(see Figure 5.19). This configuration was chosen in order to avoid that any
possible deformation of the frame during the test could affect the
measure and to measure the displacement as close as possible to the
specimen’s area. The measured capacitance variation is used to compute
the displacement imposed on the extremity of the load beam by equation
(5.10).
Figure 5.19. Electrical scheme of the experimental setup
62

69. This causes the rotation of the lever arm and the creation of the desired
state of stress in the notch. The force produced by the actuator for every
imposed voltage is computed using equation (5.12). From this
information it is then possible to plot the force versus displacement
curves (Figure 5.21). Then, following the same procedure described for
the out-of-plane structure, it was possible to combine experimental tests
and Finite Element simulations to obtain the values of the Young's
modulus measured and the values of the maximum tensile stress
occurred in every test.
5.3.3. Result and discussion
A number of 31 structures, deposited on the same wafer, were tested. As
it is possible to see in Figure 5.20, the measures were very repeatable. The
force versus displacement plots shown in Figure 5.21 appear to be linear,
implying that the electrostatic behavior of the rotor and sensor parts are
correctly described by the analytical formulae used in the data reduction
procedure. From the slope of the force versus displacement plots it was
possible, as done with the out of plane structure, to compute Young's
modulus values. The values reduced are in agreement with the ones
obtained in [Corigliano et al., (2004)], confirming the overall quality of the
data reduction procedure.
Figure 5.20. Capacitance variation versus applied voltage plot
63

70. Figure 5.21. Force versus displacement plots
The mean value measured is 143 GPa, with a standard deviation of ±3
GPa. Even in this case, the data concerning rupture of the specimens were
interpreted in the framework of Weibull statistics, as discussed in
Chapter 4. As it is possible to notice in Figure 5.22, the experimental
results are clearly well interpolated by the Weibull cumulative
distribution function. The parameter identification allowed for the
computation of the Weibull modulus, that is m=25.76 and the Weibull
stress 0. This value, representing the level of stress that gives the 63.2%
of failure probability for a pure tension specimen with the same size as
the reference volume, is 3622 MPa.
Figure 5.22. Weibull plot of the experimental data (asterisks) and interpolating Weibull
cumulative distribution (dashed line)
64

71. After the testing, the specimens were investigated using an optical
microscope. The images show that the fracture starts from the notch as
predicted from the FE simulations carried during the design phase. From
the pictures shown in Figure 5.23, two main aspects can be caught:
• the crack path is often irregular and can be quite different from
one structure to another. This can be due to the crystalline
structure and grains orientation in the notched area. The grain
morphology and orientation is different from one structure to
another and has a very important impact on the crack propagation
direction;
• the crack starting point is not always the same. This is due to the
non uniform flaw distribution on the notch surface, caused by the
fabrication process. Flaw distribution is supposed to be
responsible of the scatter of the experimental fracture results.
65

72. Figure 5.23. Optical microscope images of broken specimens.
66

73. 5.3.4. Estimation of fracture parameters
Approximate evaluation of the energy release rate
From Finite Element elastic simulation it was possible to approximately
compute the critical value of stress intensity factor.
Figure 5.24. Finite Element mesh of the notch and J integral paths.
Considering the notch as it was a real crack, by the means of an internal
routine of the Ansys Inc. Finite Element code, a path for the J-integral
computation was defined (see Figure 5.24). The code then computes
numerically the value of J a for an assigned value of the external force
Fa=180N. Three different paths were used to compute J a, as shown in
Figure 5.24, with a negligible difference of the three computed values,
that resulted 1.282N/m. The value of the J-integral is proportional to
the square of the value of the applied force. If one assumes that the
proportionality coefficient is the same both for the Finite Element
analyzed test structure and for the experimentally tested ones (similarly
to what done for the Young's modulus and rupture stress determination),
he can obtain a formula that allows for the computation of J for every
assigned load F. That is:
67