MEMS Amplification of Piezoelectric Strain for In-Plane
Actuation
by
Nicholas J. Conway
B.S., University of California at Berkeley (2001)
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
at the
LB
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
8 2003
-LIBRARIES
June 2003
@ 2003 Massachusetts Institute of Technology
The author hereby grants to Massachusetts Institute of Technology permission to
reproduce and
to distribute copies of this thesis document in whole or in part.
Signature of Author ...........................
Department cdflechanic
ngineering
i?/ 9 May 2003
Certified by .............................
- Sang-Gook Kim
Professor of Mechnical Engineering
hesis Supervisor
A ccepted by ...................................
Ain A. Sonin
Engineering
of
Mechanical
Professor
Chairman, Committee on Graduate Students
BARKER
I
g7
li
I,
MEMS Amplification of Piezoelectric Strain for In-Plane Actuation
by
Nicholas J. Conway
Submitted to the Department of Mechanical Engineering
on 9 May 2003, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
To provide a competitive actuating solution, microelectromechanical systems (MEMS) based
actuators need low operating power and form factors. Piezoelectrics provide substantially higher
work output/volume for a given voltage, when compared to other actuating solutions. In addition, macro-scale piezoelectric based actuators provide the basis for many precision actuating
solutions. To this date, however, there are few examples of precision piezoelectric devices fabricated in-situ at the MEMS level. There are two main factors to attribute this to: limited
strain of the piezoelectric thin films, and fabrication difficulties. This thesis provides the solution to these two issues, as well as introduces a scalable design for developing compact in-plane
actuating solutions for MEMS devices. Applications of low voltage, high displacement in-plane
actuators include MEMS RF switches for mobile electronics applications as well for micro- and
nano-transport.
The actuator was designed using axiomatic design to address four functional requirements.
The first requirement of high amplification is addressed using the angular advantage of bending
beams. The second requirement of high force output is addressed by the high mechanical
The third requirement that the
efficiency of the amplifier through use of flexural pivots.
amplifier end-effector move in a straight line is obtained through use of a four-bar guiding
linkage design. The arrayable aspect of the design is addressed by an electrode bus which
allows the actuators to additively move in series, and in parallel.
The device is fabricated using the piezoelectric thin film of lead zirconate titanate (PZT),
deposited in a sol-gel form and the resist SU-8. Fabrication was done at the Microsystems
Technology Laboratory (MTL) at MIT. Preliminary results indicate the design achieved strain
amplification in excess of 10:1, and can be arrayed successfully.
Thesis Supervisor: Sang-Gook Kim
Title: Professor of Mechnical Engineering
3
4
Acknowledgements
I'd like to thank Professor Sang-Gook Kim for believing in my idea.
My office-mates
Stan Jurga, Yong Shi, Raj Sood, Chee Wei Wong and Dr. Yong-bae Jeon for helping me
out whenever I needed it. Professor Triantaphyllos Akylas and Professor George Barbastathis
provided support and gave me academic enthusiasm.
The extra support of the staff at MTL,
including Paul Tierney, Dave Terry, Kurt Broderick, Vicky Diadiuk, Dennis Ward, and Bob
Bicchieri. My family for giving me support throughout my life. My friends here at MIT, Chris
Dames, Ivan Skopovi, Ali Tabaei, Jeff Hiller, Denise Poy, and Dave Willis for helping me get
started here at MIT. My friends in Westside for helping me stay grounded and my lady, Alicia,
for the smiles at the beginning and end of the day.
5
6
Contents
1
Motivation
11
2
Piezoelectric actuation
14
3
4
2.1
The piezoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2
Polarization and hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3
Sol-gel thin film PZT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4
Strain induced by an electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
21
Design
3.1
Amplification (FR 1 with DP1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2
Constraining the end effector with a parallel guiding linkage (FR 2 ,DP 2 ) . .
3.3
Efficiency (FR 3 with DP 3 ) . . .
3.4
Powering an array of actuators with an electrode power bus (FR 4 with DP 4 )
.. . . . .
. . . . . . . . . ..
26
. .
. . . . . . . . . . 29
.
30
33
Modeling the device
4.1
Static Analysis: Amplification ratio expectations. . . . . . . . . . . . . . . . . . . 33
4.2
Dynamic Analysis
4.3
The coupled device dynamics: expectations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
. . . . . . . . . . . . . . . . . . . . . 44
5
Fabrication
48
6
Results and Discussion
54
6.1
Q uality of PZT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.2
A ctuation Test
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7
7
Conclusion and Future Work
7.1
Conclusion
7.2
Future Work
.........
. . . . . . . ...
58
.......................................
. ..
. . . . ..
. . . . . . . . . . . . ..
58
. . . . . 58
A Process Details
64
B MATLAB Scripts
68
C Miscellaneous
73
8
Nomenclature
T
Stress, internal
D
Electric displacement
P
Polarization
Y
Elastic modulus
S
Strain
E
Electric Field
6T
Free dielectric constant
ES
Clamped dielectric constant
k 33
Longitudinal coupling constant
k 31
Transverse coupling constant
k1 5
Shear coupling constant
dij
Piezoelectric constant (Strain/field)
eij
Piezoelectric constant (Stress/field)
SE
Compliance at constant field (short circuit)
sD
Compliance at constant electrical displacement (open circuit)
cE
Stiffness at constant field
k
Torsional stiffness linear constant
kL
Axial stiffness linear constant
K
Material stiffness
9
10
Chapter 1
Motivation
To provide a competitive actuating solution in the embedded electronics world, microelectromechanical systems (MEMS) based actuators need low operating power and voltages, and a
small form factor. Traditionally, a majority of MEMS devices use electrostatic actuation. The
positives of electrostatic actuation include high micro-scale displacement, ease of fabrication,
and linearity of actuation, balanced by the negative of a large form factor required to obtain a
large micro-scale forces, at high operating voltages.
Thin film piezoelectrics for MEMS, con-
versely, have the exact opposite qualities: small displacement, difficult fabrication, non-linear,
high power output (forces) at a small form factor and operating voltages.
Robust fabrication
and displacement or strain amplification of micro-scale piezoelectrics would enable an actuator
that has good performance in all five qualities.
Figure 1-11 shows a comparison of different types of actuators.
As is evident,
the piezo-
electric PZT has not only a substantially higher work per volume ratio than many other types
of actuators, but also operates at high cycling frequencies.
Note that this data is for bulk
PZT. Thin film PZT can withstand much higher electric fields. Assuming the elastic modulus
of thin film PZT changes very little from the bulk value, the equation given in Table C.1 for
piezoelectric work per volume 2 , gives 1.2x10 7 J/m 3 . This is two orders of magnitude larger
'From: P. Krulevitch, A.P. Lee, P.B. Ramsey, J.C. Trevino, J. Hamilton, and M. A. Northrup, "Thin Film
Shape Memory Alloy Microactuators," JMEMS Vol. 5, No. 4. 1996 pp 270-282. Added to the figure by the
author is the thin film PZT point.
2
Thin film PZT typically has strain coefficients about 50% of the bulk value, so let d 33 = 200xlV12 m/V.
Thin film PZT has demonstrated it can withstand electric fields as high on the order of 100V/ yim[20]. This comes
11
108
la SMA
i04 Zsolid-
thin film
+1bSMA
S PZT
3 thermo.
pmaic
106-
5ae=,
04 themal
expanion
6ae.s
e
1.05-
7&PZT
#
* Sbe..
+8 muscle
1S--
6b e-.
5Ce n..
103
9W9pbbble
7b Zat O
1 02
1
10
102
1.
)4
101
106
167
Cycling Frequency (Hz)
Figure 1-1: Work per volume versus cycling frequency for various microactuators. Numbers refer to
Table C.1, which gives details on the calculations. Ideal values, shown with unfilled diamonds, represent
the energy available for actuation. Other values (filled diamonds) are based on actual microactuator
data.
than the bulk value in the table, putting thin film PZT actuators in the same regime as shape
memory alloys, but with the added benefit of faster response times[261.
Piezoelectrics have a maximum extensional strain limit on the order of 0.1%. For MEMS
applications where a displacement of 10 pim might be desired, a piezoelectric would need to
be on the order of 1 cm long if it were directly coupled to the end effector 3 .
For a practical
micro-scale application, that form factor is unacceptable. Therefore, it is imperative that we
develop a means of amplification of piezoelectric strain[3].
In the last decade, several methods have been used to amplify piezoelectric strain in macrofrom the relation between electric displacement, D, and the electric field, E, which the authors approximate as
(Y and d 33 are the elastic modulus and strain coefficient respectively): work/volume= ID - E :: Y (d33E)2
3
An end effector is the part of an actuator that interfaces the non-fixed components in the mechanical system.
12
scale size actuators, these include multi-morphs (cantilevers), moonies[4], rainbow ceramics[3],
and hydraulic piston based actuators [2].
More recently, the moonie-type actuator was refined
by Jaenker to form a hybrid actuator using parallel guiding linkages and flexural pivots.
Of
theses solutions, only multi-morphs have been implemented in full at the MEMS level to this
author's knowledge [5].
A hydraulic piston actuator with a peak amplification of 50:1 has
been implemented at the MEMS level, but was assembled a microfabricated device with a
piezoelectric stack and had dimensions of 20x20x10mm[27].
MEMS piezoelectric multi-morphs fabricated thus far, operate normal to the wafer plane,
due to constraints of fabrication, and as such, there are limitations on the geometry and path of
the end effector. Residual stress levels of films in the multi-morph must be carefully monitored
in order to set and control the end effector position. The moonie-type design, in which the
displacement of the end effector is significantly uncoupled from stress in the actuating member
as neither end of the actuating member is fixed solves this problem4 .
In-plane actuation5 is important for the design of sophisticated compliant mechanisms, as
compliant mechanisms can most easily be micro-fabricated by 2-D photolithography.
To this
date, however, there are limited examples of piezoelectric devices fabricated in-situ6 at the
micro-scale, most of which are cantilever-type (not in-plane). There are two main factors to
attribute this to: limited strain and fabrication difficulties associated with piezoelectric thin
films.
This thesis provides the solution to these two issues, as well as introduces a scalable
compact design for developing in-plane actuating solutions for many MEMS devices.
4
Chapter 3: Design will make this clear. Refer to Figure 3-6 for an example of a macroscale-precision
piezoelectric actuator.
5Actuation in a plane parallel to the substrate surface on which a microdevice is fabricated. Typical substrates
are silicon, quartz, and pyrex wafers.
6
Many MEMS devices have been made by assembling a piezoelectric element with a separately fabricated
microfabricated structure.
13
Chapter 2
Piezoelectric actuation
In choosing a piezoelectric to work with, the author chose sol-gel lead zirconate titanate (PZT)
for three main reasons: 1) of piezoelectric materials available for micro-fabication, lead based
ceramics like PZT have relatively high piezoelectric strain coefficients compared to other piezoelectrics, 2) the author's research group has experience with sol-gel PZT, and 3) time constraints
on development prevented the search for other materials and methods of deposition (including
sputtering PZT). Compared to other piezoelectrics for MEMS such as ZnO and Polyvinyledenefluoride (PVDF), PZT has a significantly better piezoelectric strain coefficients[11].
2.1
The piezoelectric effect
The "piezoelectric" effect occurs due to spontaneous, or self, polarization of the crystal structure. Piezoelectric materials tend to have high dielectric constants and function well in capacitors. The ions in the crystal can be stable at non-polar equilibrium positions, and do not have a
center of charge symmetry in the crystal structure. When a stress is applied to a piezoelectric,
positive and negative charges builds up on surfaces, in a phenomena called the piezoelectric
effect.
The inverse or anti-piezoelectric effect manifests when an electric field applied across a
piezoelectric shifts the crystal ions, inducing a strain. For the sake of this paper the anti- prefix
will be dropped for brevity as it is the anti-piezoelectric effect that is used in actuators[14].
Piezoelectrics function well as quasi static-micropositioning actuators using feedback control to
compensate for any non-linearities, hysteresis, and drifts[24].
14
a)
P
b)
s
E
E
Figure 2-1: Typical hysteresis of a piezoelectric in response to an applied electric field. a) polarization,
and b) strain.
2.2
Polarization and hysteresis
Piezoelectric ceramics are randomly polarized prior to exposure to an external electric field.
Before using, the dipoles of each grain need to be alligned in order to get a uniform piezoelectric
effect, in a process called poling.
Thicker film dielectrics typically need to be poled at an
elevated temperature in a high electric field.
For the case of thin films (< 1 pm), the field is
usually high enough at lower voltages, that the film polarizes during operation. Increasing the
electric field for thin film PZT beyond a critical field (between 40 and 100 V/ Am depending on
film quality), results in dielectric breakdown.
Excessive current draw can also cause problems.
After polarization, when the applied field is small, the strain is proportional to the applied
electric field.
If the electric field becomes too large, the strain curve deviates from linear
behavior and hysteresis occurs due to polarization reorientation and polarization magnitude
changes'.
This generates a butterfly strain hysteresis curve[15].
Figure 2-la) shows a plot of a
typical polarization hysteresis, and Figure 2-1b) shows a typical strain hysteresis that results in
response to the application of an electric field. In each case, an initial polarizing field applied
leaves a remnant polarization, and a remnant strain.
'Every time an electric field is applied, the piezoelectric is re-poled to a certain extent.
15
Figure 2-2: Perovskite crystal structure of ABO 3 -[17
2.3
Sol-gel thin film PZT
The advantages of using PZT sol-gel are two fold: (1) the film composition is more uniform
than sputter-deposited PZT, and (2) the equipment costs are low, as only a spin coater, hot
plates and a furnace/RTA 2 machine are required to create the PZT. Spin coating the PZT
necessitates a flat substrate surface. Typically, a polished silicon wafer is used.
Since each
coating is less than 0.25 pm thick, the surface roughness and feature heights need to be on
that order 3 . Also, the thin coating means many coating steps are required to obtain ~1 pm
films. PZT is suspended in a solvent (propylene-glycol based), that needs to be removed in
order to allow the PZT to be annealed to the proper crystal structure. Pyrolysis, or drying,
on a hotplate after each coating removes the solvent. This process is critical as it defines the
adhesion (interface stress) of the film to the substrate beneath it. Lead is a highly volatile metal,
so it is important that diffusion of the lead to the substrate not occur during the annealing
process (650+ 'C). A diffusion barrier of Si0
2
(insulator) and Pt (electrode) is used to prevent
interdiffusion of Si and Pb.
In order to anneal to the proper piezolelectric crystal structure, the perovskite phase, the
PZT film needs nucleations sites to stimulate crystal formation during annealing.
2
3
To create
RTA: rapid thermal anneal
The coating thickness is particular to the PZT sol-gel used in this work, see appendix for details.
16
these sites, a bottom electrode layer of platinum/titanium on top of the silicon-dioxide is used.
TiO. forms perovskite nucleation sites by titanium diffusing through platinum to the PZT.
Without nucleation sites, PZT can form the wrong crystal structure (pyrochlore phase, poor
or no piezoelectric properties).
Figure 2-2 shows a general perovskite crystal structure.
PZT
is a solid solution consisting of PbZrO 3 and PbTiO 3 (A = Pb, B = Zr,Ti) . Below a certain
temperature (Curie temperature), the positively charged Pb, Zr, and Ti displace from the
positions shown in Figure 2-2 with respect to the negatively charge oxygen atoms and generates
The crystal lattice is elongated by the polarization to
a self-polarizing dipole in the crystal.
form a tetragonal, rather than the depicted cubic structure. X-ray diffraction crystallography
can confirm the proper crystal structure.
The necessity of the metal layer (as a diffusion barrier and crystal nucleation enhancer)
beneath the PZT limits the geometry of the substrate prior to PZT deposition.
Processing
is complicated further because the etch selectivity of annealed PZT is similar to SiO 2 .
The
solution to these problems proposed and used during this research is to pattern and etch the
sol-gel coated PZT film prior to annealing, when it has a much higher etch rate than SiO 2 (not
densified) [21].
2.4
Strain induced by an electric field
Lead-zirconate-titanate, ( Pb(Zr,Ti)0
3
or PZT) has a tetragonal crystal structure, and is a
member of the class 4mm point group4 . PZT behaves electromechanically as follows in equations
(2.3) and (2.4), relating the tensor-based constituitive properties of piezoelectric strain, the
strain coefficients, dmi, the compliance, s9, and the free dielectric constanct, &6k, to the strain,
Si, the stress, T, the electric field, Em, and the electric displacement, Dm[22]. For simplicity,
these equations ignore thermal effects, which can be important in determining the dielectric
constant and strain.
Si
= sET
+ dmiEm
(2.1)
4Crystal types are classes. Out of the 32 crystal classes, 20 are piezoelectric. Although piezoelectrics lack
full crystal symmetry, the symmetry that they do have allows for the simplification of the property tensors.
17
Dm
(2.2)
dmiTi + ernkEk
s2 s3
0
0
0
T1
0
Gd
SE
0
0
0
T2
0
0
d
0
T3
0
0
d33
S
s
S2
s2
E8
S3
SE sf
SE
3
0
0
13
E
13
3
+E2
31
S4
0
0
0
s4
0
0
T4
0
S5
0
0
0
0
sE
0
T5
d15
0
0
0
0
0
0
sE0
T6
0
0
0
T
1
0
0
(S6
-0
Di
D2
=
D3
0
6 0T1 0
-0
0
0
0
0
di
0
0
0
d15
0
0
d 31
d 33
0
0
0J
[d 31
(2.3)
d15 0
T
150
T3
+
<
24
(2.4)
0E
0
-
- T5
Ei
6T
3
E3
kT6)
For simplicity of design, it is convenient to utilize the transverse strain mode (contraction/expansion) of PZT. Figure 2-3 shows this transverse mode. An electric field is applied in
the 3-direction(determined by polarization) which is in the thickness direction (dimensioned by
t) of the film. When the field is applied, strain is induced in the 1,2,and 3 directions of the film.
In the 1- and 2- directions, strain is proportional to the d31 strain coefficient. In the 3-direction
it is proportional to the d 33 coefficient.
Equations (2.3) and (2.4) reduce to equation (2.5) in
the transverse mode.
S2 S3
D3
s11
1a2
S8E
E
S8E
E
13s3
1f3
d31
d31
S13
d31
T1
Esf3
d
Ef?
12d331
T2
833
d33
T3
d33
63_
11
13(2.5)
{E3
Strain is equal in the 1- and 2- directions, but the displacements, AL and AW are not
necessarily.
They are linearly dependent on L and W.
18
This generalization ignores the effect
S
t E
I
A2
|
9t
W
Figure 2-3: PZT actuated in the tranverse mode. Field in the E 3 direction.
of the stiffness of the structure, and mechanical coupling in general (Poisson effects, etc.).
To simplify the system further, we can ignore internal bias, or residual stresses in the PZT
generated during fabrication (T = 0).
To estimate the strain in the L direction only, we use
strain we use:
SL =
AL
=d
L
31
(2.6)
E 3.
For a capacitor E 3 can be approximated by,
_V
(2.7)
E3 = -,
t
Where V is the applied voltage across the capacitor, generating the electric field. Stress, -,
is, approximating PZT as isotropic with an elastic modulus YPZT,
-=
YPZTSL
=
V
t
YPZTd31--
(2.8)
The axial force, FPZT, is stress times cross sectional area. This is given by[18],
FPZT =
V
= YPzTd31WV.
o-A = O-Wt = YPZTSLWt = YPzTd31-Wt
t
(2.9)
Taking into account the structural stiffness, kL, the actual displacement, AL, becomes,
19
AL - FPZT
kL
where kL is the stiffness of the structure.
YPZTd31WV
kL
_
(2.10)
For a PZT member, considering the stiffnesses
of surrounding electrodes and dielectrics of each layer, j, kL is,
Y2Aj .Z
k_ =
L
Strain gradients across the thickness of a thin film result in bending.
(2.11)
Symmetry of stress in
the actuating member about its neutral axis is important to prevent bending and to maintain a
transverse mode. Piezoelectric actuators operating in an transverse mode described, can have
frequency responses as high as 100kHz.
Operating in a bending mode (like a multi-morph),
the response is in the range of 100Hz to 10kHz[16]. The micro-scale offers an opportunity to
increase natural frequencies further because mass sizes are so small.
20
Chapter 3
Design
The axiomatic design process[6] was chosen such that the goals and design space of the project
could be clearly defined.
In short, axiomatic design consists of relating the functional require-
ments (FRs) ,or goals of a project, to the design parameters (DPs)of a particular design upon
Two key guiding axioms in the design process are: (1) attempt
which the goals will depend.
to maintain the independence of each functional requirement and (2) minimize the information
content of the system, that is, to keep the design simple.
the constraints of the design and manfacturing space.
All this must be done subject to
An uncoupled design is a design such
that a particular functional requirement can be adjusted independent of other functional requirements of the system.
A decoupled design is a design such that functional requirements
may be achieved by the selected design parameters, but the functional requirements can only
be satisfied by adjusting design parameters in the proper order.
A coupled design is one such
that the functional requirements are not inpendent of each other. The following equations provide a simple matrix representation of each design type, to better understand these concepts.
Equations (3.1), (3.2), and (3.3) depict uncoupled, decoupled and coupled designs respectively.
Notice that the uncoupled design is diagonal, the decoupled design is triangular (lower), and
the coupled design is neither1 .
FR1
X
0
FR2
L0
X
DP1
J
DP2
'For a thorough discussion of axiomatic design, refer to Nam P. Suh's "Axiomatic Design" [6].
21
(3.1)
FR1
X
0
DP1}
FR2
X
X
DP2
FR1
X
X
DP1
FR2
X
X
DP2
(3.2)
(3)
To embark upon this design, a design paradigm was needed for several reasons.
because this is a proof of concept project, it is important to keep the goals in mind.
First,
MEMS
devices tend to have highly coupled designs, decoupled at best, because of the serial nature of
fabrication, material constraints, and fabrication constraints 2 . Therefore in MEMS design it
is challenging to keep functional requirements independent [25].
This design is very uncoupled
from a purely design standpoint, but the micro-fabrication process leaves the design decoupled
due to the forward and backward coupling nature of thin film fabrication.
Functional Requirements (FRs):
FR 1 : Amplify piezoelectric strain by a factor of 10 or more
FR2 : End effector motion must remain in a straight line.
FR3 : High efficiency of energy transfer
FR 4 : Device must be arrayable
Chosen Design Parameters (DPs):
DP 1 : Angle of amplifying structure
DP 2 : Four-bar linkage controls the rotation of the end effector direction
DP 3 : Dimensions of flexural pivots to minimize energy loss.
DP 4 : Electrode bus must conduct potential in such a way as not to impede amplifying
dynamics or efficiency.
Constraints:
C 1 : Form factor of 500 pm X 500 ptm
C 2 : Photolithography precision (2
1 am
precision with facilities used)
C 3 : Etching fabrication precision and adhesion
2
MEMS devices are usually fabricated in place. That is, parts are not assembled, but rather built into the
device. This makes fabrication a coupled process. Parts fabricated earlier are affected by later parts (backward
coupling), and later parts depend on the geometry of prior fabricated parts (forward coupling).
22
C
y
or
jowl
SLFP's
Figure 3-1: Compliant parallel guiding linkage amplifying structure. Small length flexural pivots (SLFP)
act as pivot points.
C 4 : PZT strain limit of ~0.1%
This thesis is to prove the concept of a device that the FR's comprise, and as such, there
will be no more functional requirements.
Process variables and flows have been determined
Figure 3-1 depicts the 2-D top view of the
during production and are given in the appendix.
design, which satisfies the functional requirements.
3.1
Amplification (FR 1 with DP 1)
Figure 3-2 represents the design's amplification principle.
Two ends of a link are constrained
to move along two perpendicular walls. The angle, 0, that the beam makes with the horizontal
wall, fully defines the link's position in space.
Let the left end of the link be the input and
the right end be the output, defined by x and y respectively. If 0 changes, then the change in
y versus the change in x is clearly going to be,
23
y
x
Figure 3-2: Amplification principle
Ay
Ax
sinG - sin00
cos 0 - cos 0 0
2 cos - (0
2 sin . (0
+ Oo) sin 21 (0 - 00)
(0+Oo) sin (0 -0o)
Cos 1 (0
-sin'
+0o) sin - (0 - 00)
+ 0o) sin - (00 - 0)
1
= - cot -(0+00),
where the displacement amplification, Ay/Ax, is a cot 0-like relationship.
Ay/Ax, from 0 to 45 degrees, with
0
(3.4)
2
If we plot
0 = 0,we get the plot shown in Figure 3-3.
For small angles, the displacement amplification is huge, with a negative change in x causing
a positive change in y. For the small strain of piezoelectrics, this technique would be ideal for
amplifying the strain.
Since the strain limit is small, operating an amplifier around a small
initial angle would be ideal.
Also notice that amplification is only dependant upon the initial
angle, 00, and the actual angle, 0. When 0 is less than 450, the mechanism is a displacement
amplifier and a Fi,, is reduced. When 0 is more than 45 , the structure becomes a force amplifier
and a displacement reducer. For maximum displacement amplification, 0 should be as close to
0
as possible. For maximum force amplification, 0 should be as close to 900 as possible.
We now can decompose the DP for the amplification ratio into two parameters due to the
PZT strain limit constraint.
DP
1
: The initial angle,
0
0, of the actuating arm.
DP1 2 : The angle 0.
Since we are not decomposing the associated functional requirement, it is evident that this
is a redundant design. As will be evident in the analysis section, the output force does not follow
24
0
-20
-40
Ay/Ax
-60
-80
-100
-120
5
10
15
20
25
0(deg)
30
35
Figure 3-3: Plot of the amplification ratio versus theta.
25
40
0
= 0.
45
a reciprocal relationship, but neglecting to losses due structural stiffness (the hinges/pivots),
the output force F0 .t is,
|Fout| = |FinIt an 0.
Eq. (3.5) satisfies conservation of energy (integrating from
3.2
(3.5)
f()
F tdy =
x(
Findx).
Constraining the end effector with a parallel guiding linkage
(FR2 ,DP 2 )
The amplifying mechanism in Figure 3-1 is created in a high-aspect-ratio MEMS material
such as by etching the structure in a silicon wafer, or the spin-on SU-8 resist. An actuating
member connects locations A and A' on one face of the structure (parallel to the xy plane).
This member is made up of the active material, PZT. Let A and A' be the input surfaces. Let
location B be fixed and C is free to move. As locations A and A' are drawn together a certain
distance, that displacement gets amplified non-linearly by the four four-bar linkages that connect
B to A,A'to C, the end effector. The guiding linkage ensures that the end effector surface,
C, will stay aligned to the fixed surface at B (the end effectors will not rotate) and external
disturbances will preferentially result in end effector motion in the y direction, and input surface
motion (A and A') in the x direction, thus satisfying FR 3 . The actuating member's symmetric
boundary conditions (it is suspended between points A and A'), significantly uncouples effects
of bending moments induced by stress from the amplifying mechanism 3 . Because the amplifying
mechanism is designed to be least stiff in the x and y directions, the majority of the motion
will occur in those directions.
The parallel guiding linkage effect is accomplished using small length flexural pivots (SLFP's),
which are compliant pivots that closely approximate a pure rotation as all of the bending stress
is concentrated at the SLFP rather than the surrounding material (see Howell[7] for a full discussion of SLFP's). Bending occurs at the location and directions of least resistance, in the xy
plane, because the high-aspect-ratio structure is very thick at all points other than the middle
3This is important in microactuator designs because residual stress in thin films can often undesirably deform
actuators when they are released. This is a particular problem in cantilever-type designs.
26
F_
in
Figure 3-4: Idealized model of the mechanism.
of flexure hinge. This is similar to a big resistor and a small resistor in parallel; most of the
current goes through the small resistor as the small resistor can be likened to a wire when
compared to the big resistor. Figure 3-4 shows an equivalent model of the device. The SLFP's
behave as torsional springs and the device itself is made up of four parallel guiding linkages.
Lines represent links and spirals represent hinges. The input force is Fi, and the output force
is Fost.
Using symmetry we can look at one of the four linkages at a time. Figure 3-5 shows
that the dynamics of the system can be described using one generalized coordinate, 0.
The amplifying mechanism design is similar to the macro-scale design by Jaenker[23] (see
Figure 3-6), but with the difference that the pivots are 1) notch-type hinges and 2) each hinge's
longitudinal axis is aligned with the normal to the direction of the hinge motion, ensure that
each hinge is bending in its most compliant mode, leaving the least resistive directions, the
x and y.
The asymmetric notch hinge design limits stress concentration since each hinge
is operating in a non-fully reversed stress regime, and also concentrates bending in a smaller
27
y
k
F
k
out
2
F.
in
Figure 3-5: Symmetry model
Figure 3-6: A hybrid design by Daimler-Benz AG Research. ~10cm wide. Peter Janker, Frank Hermle,
Thomas Lorkowski, Stefan Storm, and Marc Wettemann, "Development of High Performing Piezoelectric
Actuators for Transport Systems". Actuator 98: 6th International Conference on New Actuators, 17-19
June 1998, Bremen, Germany, pp 181-185.
28
Figure 3-7: Asymmetric notch hinge
region, more closely behaving like an ideal pivot (see Figure 3-7). The PZT film thickness is
small, less than 1 Iam, so the voltage required to actuate is small. The amplifying mechanism's
structural stiffness should compensate for bending moments generated by the off-neutral axis
PZT member (high-aspect-ratio).
Lastly, the design takes advantage of batch micro-fabrication
such that no assembly is required. Jaenker claims an amplification ratio of 10.
3.3
Efficiency (FR 3 with DP 3 )
We want the output force of this amplifying transmission to follow the reciprocal tan
To do this, we look to the efficiency of
0 (cotangent) relationship as closely as possible.
a piezoelectric member and amplifier system, with associated stiffnesses
respectively.
KPZT
and Kamp,
The amplier and piezoelectric member are acting as springs in series. If we look
to the mechanical efficiency of the amplifying structure, it would be the ratio of work of the
piezoelectric member and amplifier to the work of just the piezoelectric member.
Since the
work done is proportional to the stiffness, we can say,
K1
7
lmechanical = Wamp+PZT
WPZT
Kamp
+
KPZT)
KPZT
1+
KpzT
(3.6)
Kamp
Equation (3.6) says that the stiffer the amplifying structure is, the higher the efficiency.
Therefore, if we concentrate deformation to specific points with low stiffness about the axis
of desired deformation, we can make the design highly efficient, allowing for maximum force
output 4 . The stiffness of the stucture is a function of how many flexural pivots there are. The
4
Jaenker et al. derive the same result for mechanical efficiency[23].
29
Figure 3-8: Electrode potential bus powering three actuators in series. Loops are compliant to prevent
increased device stiffness.
addition of a four-bar linkage increases the stiffness, so a mildly coupling effect is that not only
is it dependent on the hinge dimensions, but also the number of hinge.
The efficiency of the
structure will effect the force transmission ratio.
3.4
Powering an array of actuators with an electrode power
bus (FR 4 with DP 4 )
In order to actuate devices in series, some sort of electrode, or power bus is necessary to transmit
charge to the piezoelectric.
Figure 3-8 shows one possible design in place.
The bus connects
to the top and bottom electrodes of the actuator. Here it is connecting three actuators in series
and is symmetric about each actuator.
Any electrode bus designed, cannot be stiffer than the
SLFP's, to maintain fuctional indepedence. Therefore, a compliant electrode bus was designed,
taking into account photolithography constraints C 2 .
The electrode bus is made out of 2000
on top of 200
A
A of platinum (bulk elastic modulus ~171 GPa)
of titanium (bulk elastic modulus ~116 GPa), deposited by electron-beam
evaporation. Because of Ti/Pt is required for making sol-gel PZT, using it as the electrode
bus does not add additional fabrication steps. The electrodes have a "robust" line width of
11 /m, and have a compliant "loop" hinge structure to make the bus stiffness less than that
SLFP stiffness, while maintaining conductivity integrity. While the complex structure of the
30
loop hinge limits strain concentration, it makes an exact stiffness calculation difficult, so the
following very rough, conservative calculation will be made.
The equivalent length of this
hinge structure in the active range is 120 Itm. This hinge structure has a torsional stiffness of
roughly 5,
kr
For this case 1 = 120[pm, YbulkPt
=
YI
(3.7)
7r-y2--.
171 GPa, -y = 0.852, and is the characteristic radius,
and I is the cross sectional area moment of inertia of equation 4.16, where b = 2000
A,
and
t = 11 pm. This yields a ke of 7.2 x 10-8 N-m/rad. Compare this to the stiffness of the hinges
mentioned above, for SU-8 with a 2 /um thick hinge that is 10 ,am long, and we get 1 = 10 /1 m,
Y = 4 GPa[13, and where t = 2 pim, and b = 30 [im. This yields a kh of 8x10- 9 N-m/rad,
which is an order of magnitude smaller. Admittedly, the ratio ke/kh should be much less than
1 and not greater than 1, as was the case with the original design based around silicon rather
than SU-8 (silicon is 40 times stiffer than SU-8). A couple of factors should help of the hinges
should be less stiff than calculation makes them out to be. Residual stress in platinum makes
the film bend out of plane. This changes the area moment of inertia significantly, since the
electrode can bend through the thickness of the film as opposed to the line width, lowering
the stiffness by orders of magnitude, as in Figure 3-9. For a proof-of-concept design, this is
acceptable, even though stress levels in the platinum/titanium on SiO 2 were not determined.
The experimental "floating" nature of the loops provides a significant fabrication challenge
that, once achieved, will enable more robust future designs. Functional requirement coupling is
intended to be ignorable because the ke/kh ratio. Figure 3-10 shows the effect in reality, from
an image captured by an optical microscope.
In the end, the design matrix is made as shown in equation (3.8):
FR1
X
0
0
0
DP1
FR2
0
X
0
0
DP2
FR3
0
X X
0
DP
0
0
0
X
FR4
FR4
ausing
the pseudo rigid body model of Howell (eg 5.73)[8].
31
DP4
De4
(3.8)
no stress
no curl
stressed
(curled)
Figure 3-9: Curling of platinum electrodes due to residual stress
Figure 3-10: Image of a loop on a real released device, bending out of plane due to residual stress
32
Chapter 4
Modeling the device
4.1
Static Analysis: Amplification ratio expectations.
Let us perform a static analysis of the symmetry modell:
Proceeding with Lagrangian analysis:
F = Fni + Fo(ti1
z = -ri cos
(-
-
rl sin 0) 3
(4.1)
(4.2)
There are four springs, but only two different spring constants, giving the factor of 2 in
front.
The Langrangian, X,is given by,
X=T*-V.
(4.3)
The potential energy, V,of the system is given by
V = 2 [ki ( -_ oi)2 +
k2 (0 - 002)2
(4.4)
All calculations assume that the amplifier is so stiff in the out of plane direction that bending out of plane
can be ignored. Note the coordinate system.
33
and for the static case the kinetic coenergy, T*, is,
T* = 0.
(4.5)
The equations of motion can be determined through,
d
Of-\
(4.6)
7-'K~J)
'9 qj
where q is the generalized coordinate, and the generalized force, -j is,
N
Ef - O
(4.7)
i=1
For this model, q = 0 and R = Z. Computing the derivatives we get,
(4.8)
= r1 sin Oi - r cos Oj,
so,
Fir1 sin 0 - Foutri cos 0,
-
809-2-
[ki (0-
00 1) + k 2 (0-0
02 )],
(4.9)
(4.10)
Putting it all together we get,
Firl sin 0 - Fostri cos 0 = 2 [ki (9- Go1) + k2 (0 - 002)] ,
->' Fot = Fi, tan 0 -
2
rl cos 0
[ki (0 - 001) + k2 (0-002)],
(4.11)
(4.12)
Ignoring the structure stiffness, the output force is proportional to the tan 9 of the input
force. For small 9 angles, order 1 ', this is about 2%.
So Foot is about 2% of Fin after the
stiffness of the amplifying mechanism is overcome. Assuming small stiffness,and a PZT input
34
force is order 5 mN , then the output force is then the output force should be about 100 pN
which should be satisfactory for a MEMS applications, satisfying FR 2 . This is reasonable for
thin film PZT at 10 V, 100 pm wide, and an elastic modulus of 60 GPa, similar to SiO 2 , as in
equation (4.13). The piezoelectric force expected from the member is, ignoring the electrode
stiffness[18],
FPZT = Fin = YPZTd31WV,
(4.13)
where YPZT is the elastic modulus of PZT, d 31 is the piezoelectric strain coefficient of
PZT, W is the width of the member, transverse to the membranes length, and V is the applied
voltage. Increasing the width of the PZT member will increase the input force. The arrayability
would allow n-actuators to be operate in parallel giving n-times the force output, at a cost of
size.
If F,,
is zero, we can get an equation that relates the stiffness of the structure,
Fin
2
=
ri sin G
[(ki + k2 ) 0 - k 190 1 - k 2 00 2],
(4.14)
For small length flexural pivots (see Figure 4-1), the torsional spring constant, k, can be
approximated as (Howell),
YI
k= - 1
1
(4.15)
and the virtual pivot is at the the halfway point, 1,
2' and Y is the elastic modulus of the
pivot material, and I is the area moment of inertia of the cross section,
bt 3
I= .
12
(4.16)
It is important to note that the use of equation (4.15) is an approximation, of Howell's
approximation. Since the design in Figure 3-1 depicts asymmetric notch or circular flexure
hinges, the true stiffness of the hinge will be stiffer than equation (4.15) predicts, but the pivot
point will be more localized at 1.
this proof of concept design.
The author uses this relatation with that knowledge for
See Lobontiu[12] for a more complete treatment of asymmetric
35
x
Z
t
Figure 4-1: Small length flexural pivot bends primarily in the y-direction when loads are applied with a
component in the y-direction
circular flexure hinges.
The following are the device dimensions:
r, = 165 ptm
E = 4 GPa (Elastic modulus of SU-8)
t
b
=
2 pm
30 prm
L =10 pam
A MATLAB script was written to calculate the static deflection values. Using symmetry,
the input force equation was multiplied by four, and the displacement magnitudes in the x- and
y-directions were multiplied by two to get the full device static displacement.
Figures 4-2 and 4-3 show the static mechanics of a design with a 5
respectively.
and a 10 initial angle
The angular scales on the plots go until the strain limit of PZT is geometrically
reached at 0.1% This occurs at a change in x of 0.19 pm since the half length of the active
PZT region is 190 pam. Notice that in the d) plots, there is a trend of decreasing amplification
ratio with increasing 0, therefore, to maximize amplification, 0 should be as small as possible.
Conversely, the input force required decreases with increasing 0.
the the 50 plots are much more linear than the 1' plots.
with a 5
The devices tested were designed
for a safety factor, since residual stresses in the piezoelectric member can affect the
initial angle dramatically.
for the 5
The last observation is that
Minimum amplifcation occurs at the PZT strain limit and is 10.7
design and around 30 for the 1
design, satisfying FR1 .
36
a) Input force versus angle - 00 = 5 degrees
b) x and y displacement
versus angle - 00
=
5 degrees
4
100
80.
.3
60.
E2
40-
L.L
20
~1
08
0
0.085
--- ~
0.085
0.1
0.095
0.09
-
0.09
0 (rad)
d) Amplification Ratio versus angle -
c) x and y strain versus angle - 00 = 5 degrees
2.5.
C
0.095
0.1
0 (rad)
0=
5 degrees
11.6r
2
11 .4
1.5
11.2
1
E
0.5
IU.80r
/-
10.61
0.085
0.1
0.095
0.09
0.085
0.09
0 (rad)
0.095
0.1
0 (rad)
Figure 4-2: Different plots with an initial angle of 5 . In b) and c), dashed and solid lines are the yand x- direction plots respectively.
a) Input force versus angle - 00 = 1 degrees
6 00
b) x and y displacement
versus angle - 80 =
1 degrees
12
.
500-
10
1
400
-3
300-
E 6
200,
A
08
100
0
0.01
4
2
0.02
0.03
0.04
0.05
0z
0 (rad)
0.06
0.01
0.03 0.04
8 (rad)
c) x and y strain versus angle - 00 = 1 degrees
d) Amplification Ratio versus angle -80 = 1 degrees
8
0.02
0.05
0.06
60
55~
6
\1
.250
-ii
. 45
4
.2 40
E 35
2
30
00.01
0.02
0.03
0.04
0.05
25
0.01
0.06
0 (rad)
Figure 4-3: Different plots with an initial angle of
and x- direction plots respectively.
10.
37
0.02
0.03 0.04
8 (rad)
0.05
0.06
In b) and c), dashed and solid lines are the y-
-
4
3.5
3
E 2.5
E
2
CL
1.5
.
1
0.5
0
C
10
20
30
40
50
60
Force (gN)
70
80
90
Figure 4-4: Displacement versus input force for an initial angle
100
0
= 50.
12
10
8
E
6
E
a)
CO
O,
4
2
n
00
100
200
300
Force (siN)
400
500
Figure 4-5: Displacement versus input force for an initial angle
38
600
o
= 10
y
r2
r1
k
3
F
out
Figure 4-6: Dynamic model based on symmetry
Figures 4-4 and 4-5 plot the displacement of the end effector versus the required input force
under no external load. Notice the linearity of the output for 0o = 50.
In equation (4.13),
the force generated by the PZT is proportional the voltage applied, and so we can expect that
the output displacement should behave like Figures 4-4 and 4-5, with, especially in the case
where
0
0 = 50, displacement should be linearly related to applied voltage.
Since the exact
properties of the thin film PZT (EPZT and d 31 ) used in the design, have yet to be determined
experimentally, the conversion is omitted.
4.2
Dynamic Analysis
For the dynamic anaylsis, a more complicated model is needed as shown in Figure 4-6.
39
Z1
=
2
=
32
Z3
=
-ri cos 0i+(-
Zi+
Z2
dz~j
dt
=
dt
-
d7Z3
2
dti
(4.17)
r1 sin 0)J
22
2(- 2-
-
r2
(4.18)
rl sin O)j,
(4.19)
(cos Oi + sin Oj),
-(-cos Oi sin
(4.20)
),
rib sin 0i - rib cos O-j
-
(4.21)
2ribcosoj,
(4.22)
0 (sin Oi-
(4.23)
cos Oj),
+ -T0(sin
+ cos Oj),
(4.24)
j
(4.25)
=
r, sin Oi- r1 cos
=
-2r 1 cosoj,
(4.26)
Adding together the kinetic coenergies due to the linear motion of m 1,m 2 , and M 4 , and the
rotary motion of the beams M 3 and m 4 we get,
T*
=
=
(mi + M 4)
2
2
(ml + m 4 ) r 2 +
m
40
2
+2
(1jb2)
+2
(2ri0 Cos 0 2+
2
()1
j
J1
)
(4.27)
where J is the angular moment of inertia of the two rotating beams that comprise m
m4, about an end, with m3
3
and
.
= n4
J=2
m 3 (r2 + w2)
(4.28)
31
where rl is the length of the beam and w is the width.
The potential energy is twice the static condition (8 springs now),
V= 4
k1 (0 -- 001)2 +
k2 (0-
(4.29)
002)2 1
The generalized force,
-
(4.30)
= Finri sin 0 - 2F..tricos 0,
Computing the necessary derivatives for the equations of motion,
=
d af
d
(Mi
+ m4) r O + m 2 b (2r 1 cos 0)2±2
2b
=
=
(Ml
OS0 2
(JA)
(4.31)
*2
+ M 4 ) r1 0 + m 20 (2r 1 cos 0) + 2M 2 2 (2r 1 cos 0) (-2r1 sin 0) + 2JA,
~r~M1 + M4 +
2J
2
1
= -4m2r
2
+
4m2 cos 0 r1j -8m
2r
cos 0 sin9
sin OcosO - -4 [ki (0 - 0 0 1 ) + k2 (0 - 0
2
(4.32)
02 )],
(4.33)
we get,
ao
d (aX)
41
(4.34)
(2
J
M1 + M4 + 2J+ 4m2 cos 2 0
1
=
2-
r 2 - 4m2r cos 0 sin 02 + 4 [k1 (0 - 001) + k2 (0 - 002)]
Finr1 sin 0 - 2F 0etri cos 0.
(4.35)
When actuating individual devices, adding actuators in series, and in parallel, will only add
mass terms to the equation. When all actuators are actuated at the same time, this equation
of motion should hold.
If we linearize the equations of motion about an initial angle 0 o, and say 0o
plug in 0 = Oo
=
001
= 002,
we
+ e, and get,
M1
+ m 4 + 2 + 4m 2 cos 2 (Go + e)) r
1
-4m
=
2 r, cos (Oo + e) sin (Go + e) 2
4 [ki + k2] e
Finr1 sin (0o + e) - 2Ftricos (0o + e) .
Using the following identities, and knowing that e
(4.36)
< 1,
+ e)
=
cos Oo cose - sin 0o sin e = cos 0o - e sinGo,
(4.37)
sin(0+e)
=
sin00cose+cos0osine =sin00+ecos00
(4.38)
cos (Oo
Simplifying equation 4.36, term by term, and dropping terms that are second order or
greater,
m+
+
m4 +
+ M4 + 2+
m1
= M1
+ M4 +
2J
2
1
4m 2 (cos 0o- e sin 0o)
4m 2 (COS2
+1m
cs
-
0
42
r E
9)
2E cos 0 0 sin 0 0 + e 2 sin 20)
r
-4m
2 ri
(cos o - e sin Oo) (sin o + ecos o) 52
=
0
Fir1 sin (0o + e)
=
Finr1 (sin 0o
-2Fostri cos (Co
+ e) =
+ e cos Go)
-2Foutri (cosoo
-
esin 0).
Combining these simplifications, we get the following linearized equation of motion,
Mi1
m4 +
+ 4m 2 cos 2 o)
r 2+
4 [ki
+ k2 ] e
= Finr1 (sin 0 o + 6 cos 0o) - 2Foutri (cos Oo - e sin Go) .
(4.39)
Since piezoelectric strain is small, the travel of the actuator is limited, so equation (4.39)
should be satisfactory. If the input force and output force are set to zero, we can estimate the
amplifiers natural frequency from the homogenous differential equation of equation (4.40)
E = 0.
4 [ki + k 2]
+
M1
+ M4 + 2J+ 4m2 cOs 2 G
(4.40)
r
Equation (4.40) yields a natural frequency of:
Wn
-
2
k + k2
'r
Ml+M4
(4.41)
+ 2J+4M2COS2 00)
Since the mass of objects is proportional to the volume cubed, in the MEMS world, natural
frequencies of this design can be very high and easily picked by clever hinge design to determine
the stiffnesses k, and the link length ri. Given a design such that,
p = 2240 kg/ m3 is the desity of cured SU-8 (Microchem).
A 1 = (70 x 22 x 2 + 30 x 280 + 55 x 174) x (1 x 10-6)2 = 2.105 x 10-8 m 2 , is the surface
area of ml.
A 2 = (25 x 95) x
(I x 10-6)2 = 2.38 x 10--9 M2 , is the surface area of M 2.
A 3 = (141 x 20) x (1 x 10-6)2 = 2.82 x 10-9 M2 , is the surface area of M3-
43
b = 30 x 10-6 m, is the thickness of the SU-8.
0= 5 .
Given this data, the natural frequecy of the manufactured amplifier should be on the order
of 5 kHz.
Since the hinges should be stiffer in reality, as mentioned in the previous section,
the natural frequency should be slightly greater than this value. Provided 00 < 1, the natural
frequency of the device should be fairly constant over different designs.
4.3
The coupled device dynamics: expectations
In the real device, the actuating member is coupled to the real system, in parallel. Therefore,
the stiffness of the actuating member will effect the natural frequency.
Due to limited data
on the residual strain or elastic properties of the platinum, SiO 2 , and PZT itself we can not be
sure of a real value, but the derivation will be included here for a full treatment.
The potential energy of the system becomes, using equation (2.11):
V = 4 [k1 (0 - 001)2 +
k1 (0
)2
+ 2kr (cos 0 - cos 00)2
(4.42)
It is assumed that the member mass is small, so it's effects on the kinetic coenergy are small,
but it is possible to lump it together with ml.
= -4m
2
sin 9 cos 0 - 4 [k
2 rO
1
(0 - 00) + k 2 (0 - 00)]
+ 2kLr2(cos 0 - cos Oo) sin 0, (4.43)
For the whole equation of motion we get,
+ 4m2 cos2 ) rO - 4m2r cos 0 sin 0b2 +
m1 +m4 + 2
1
+4 [k 1 (0 - Oo1) + k2 (0 - 00)] - 2kLr2 (cos 0 - cos 0) sin 0
=
Fi.r 1 sin 0 - 2Foutricos 0.
44
(4.44)
Linearizing, the additional term becomes,
-2kLr2 (cos 0 - cos O0) sin 0
=
-2kir
[cos(Oo + e) sin(Oo + e) - cos Oo sin(Oo + e)]
=
-2ktr
[(cos 00 - e sin 00) (sin 00 +e cos 0o) - cos 0o (sin Oo + e cos Oo)]
=
-2kLr2 (sin
=
-2kLr| [-e sin 2 001
- e sin 2 0
0o cos 00
(4.45)
+ e Cos 2 00 - 62 sin 00 cos 00 - cos Oo sin 00 - 6 cos 2 00)
(4.46)
Combining these simplifications, we get the following linearized equation of motion,
(in1
=
Fi,
1
+ 4m 2 cos2
+ m4 +
(sin 0 0
r
+ (4 [k
1
+ k2 ] + 2kLrl sin 2
e cos 00) - 2Fouri (cos 0 0 - e sin 00) .
(4.47)
The stiffness of the actuating member adds an additional term to RHS acting as a moment
offset. In spite of this, looking at the homogeneous equation, the device natural frequency is
now,
on =
4 [k1 + k 2J + 2kLr2 sin 2 0
(m 1 + M4 + 2J + 4m2 cos 2 0(
1
r1
(4.48)
If kL >> k 1 or k2 , then the natural frequency becomes,
(4.49)
= Sin
sn 0
Mi1 + m
which is independant of r1, assuming r1
4
+
2
+ 4m 2 cos
(00
> w. Furthermore, if we take 0 < 1 (maximizing
amplification) we get,
Wn
= 00
2kL
(mi-n
+ 7 m4 +
4M2)
(4.50)
Essentially, the device has been reduced to a spring-mass system. The natural frequency
of the system is linearly related to the initial angle, 00. This seems to be indicative of the fact
45
that the larger the angle, the less force the PZT has to exert to move the device.
When Oo
approaches E, the PZT member is hardly taxed at all to move the amplifyer, if Oo is precisely
zero, the model system could not move, due to the increase in force. The actual device natural
frequency can be approximated using estimated material properties of PZT and Pt, ignoring
the thin layer of titanium.
ThePZT member is made up of a stack of SiO2, Ti, Pt, PZT, Ti,
and Pt, with the metals serving as electrodes.
YPZT ~~60 GPa
Ypt
95 GPa
Ysi0
tPZT
tpt
0.2 pm
tsio 2
0.4 pIm
WPZT = 130 /m
Wpt1 = 140 pam
~ 60 GPa
2
0.2,pm
Wsi0
2
= 140 pm
WPt2 = 100 Am
L = 380 pm
Solving for the natural frequency, we get w, = 60kHz, with a member stiffness kL
2.46 x 10 4 .
=
This is very large and should be noted that this is under ideal axial actuation of
the PZT member.
The maximum force that the PZT member can generate is limited by the width of the top
electrode.
Using approximate values for the PZT to get an estimate we get from equation
(2.9), and a d 31 = 100 x 101 2 C/N, and a potential of 15V, we get, FPZT = 9000jiN out of the
whole membrane Performing the static analysis we can get a relationship between input force
and displacement for the 5' design as shown in Figure 4-7. An input force of 9000pIN should
actuate the device to the strain limit of the PZT member.
46
3.53E 2.5E
2-
CL
.m 1.5-
0.50
0
500
1000
1500
2000 2500 3000
Force (MN)
3500
4000
4500
5000
Figure 4-7: Displacement versus input force for an initial angle 6 o
47
= 50.
Chapter 5
Fabrication
The fabrication was performed using UV lithographic techniques on 100 mm p-type silicon
substrates. Quality, size, and thickness of the silicon substrate is not important, other than
that the wafer be at least single-sided polished.
The silicon is used as a sacrificial layer only,
and plays a limited structural role.
Recent advancements in MEMS have demonstrated that SU-8 may be used for the development of in-plane compliant mechanisms[9]. Therefore, it is proposed to use SU-8, as opposed
to bulk silicon, to obtain a high-aspect-ratio compliant structure.
SU-8 is a photo-curable
epoxy-based negative resist. Since it requires no etching to pattern to obtain high-aspect-ratio
structures as well as demonstrating excellent mechanical properties[13], it is highly desirable to
use in a prototype design. Figures 5-1 through 5-5 show the process flow for device fabrication
depicting the top view of a device on the left and a cross section of a device on the right.
The fabrication process uses four different masks produced by Advanced Reproduction of
Andover, MA on chrome on soda lime glass using an e-beam writer. The first step is a deposition
of SiO 2 by thermal oxidation of a silicon substrate. In step two, lift-off is achieved in acetone, to
disolve the resist in the regions where no electrode is desired. The bottom electrode, electrode
bus, and alignment marks are formed in this step (mask #1).
coated, dried (pyrolysis), etched, and then annealed.
In step three, the PZT is spin
The proper perovskite crystal structure
is ensured by etching the PZT prior to annealing, leaving PZT only on top of Pt/Ti regions.
If PZT is annealed on top of the SiO 2 , regions of poor crystalization will form, causing stress,
cracks and other defects in the PZT (mask #2).
48
A 2 [Lm gap is left between one end of
Figure 5-1: Begin with growing 2000 A of thermal oxide on a silicon substrate
M
M:
Figure 5-2: Pattern and deposit bottom metal layer using liftoff of e-beam evaporated Ti (200 A) and
Pt (2000 A)
the bottom electrode and the bus, to allow the top electrode to connect to the bus and limit
PZT exposure to the SiO 2 . Step four is the same as step two, forming the over lapping (top)
electrode of the PZT member (mask #3).
The SU-8 deposition occurs in step 5.
The SU-
8 is spin coated onto the wafer, and carefully dried using slow ramping speeds to minimize
residual stress in the SU-8.
The device is released, first by etching away exposed thermal oxide
with CF 4 using a short RIE to expose the silicon substrate, followed by an isotropic xenon diflouride (XeF 2 ) etch to remove silicon underneath the device. The etch rate for our particular
XeF 2 etcher is roughly 10 pm/min. This all "dry" release is particularly desirable, because it
avoids problems associated with wet releases (stiction, etc.).
and specific process flows.
See the appendix A for detailed
The oxide left under the PZT member shifts the neutral axis and
will inevitably cause bending of the PZT member.
The oxide is left to maintain fabrication
simplicity.
Figure 5-6 is a model of the finished design as it should appear, minus the transperency of
49
no
Igo
Cl:40 BOE (buffered oxide etchant). Anneal at
Figure 5-3: Spin coat PZT. Dry. Etch in 400
650'C. See appendix
Figure 5-4: Pattern and deposit top metal layer using liftoff of e-beam evaporated Ti (200 A) and Pt
(2000 A).
Figure 5-5: Spin on SU-8, photo-pattern and bake. Release the device by RIE (CF 4 ) of oxide and XeF 2
etch of silicon using SU-8, Pt, and PZT as etch masks.
50
620 microns
I.5mm
670 microns
Figure 5-6: Solid model of finished actuator.
SU-8.
The exposed square areas on the left are bond pad areas such that the device can be
wire bonded or actuated by probe tips.
Prior to releasing, the device appears as in Figure 5-7. For this particular actuator, notice
the misalignment of the PZT from the top electrode.
This is caused by the difficulties in
aligning the chrome mask to the wafer alignment marks because the PZT dries dark.
Notice
the transparency of SU-8.
Figure 5-8 shows an SEM image of a released device. The out of plane bending of the device
and upwards curling the electrode bus can be attributed to the fact that SU-8 is non-conducting
and the device is not grounded, allowing charge to build up. Devices were observed to fail in
2
the SEM chamber due to this effect. Figure 5-9 shows two 1 cm dies connected with finished
actuators.
51
Figure 5-7: Unrealeased actuator. Notice misalignment of the PZT. This particular device was not
tested due to the misalignment because of potential leakage problems.
Figure 5-8: SEM image of device. Out of plane deflection due to the fact that the device is not grounded.
52
Figure 5-9: Two dies connect. Each die is 1 cm 2
53
Chapter 6
Results and Discussion
6.1
Quality of PZT
In order to ensure that the devices work, a verification of the PZT's polarization was performed
prior to actuation measurements.
The devices were tested using a probe station to determine
the unloaded displacement of the device, and the hysteresis curve of the piezoelectric. The
hysteresis curve was measured using an RT66A standardized ferroelectric test system from
Radiant Technologies, Inc. of Albuquerque, New Mexico, USA., connected to a PC running
Radiant Technologies CHARGE program. Force output was not tested due to time constraints.
The result in Figure 6-1 compares favorably with literature[20] with a maximum polarization
of ~50 pC/cm 2 , for 0.4 pm thick film at 15V with an area of 11.4x10- 4 cm 2 . The leakage current
is 2.58x10-
6
A at a resistivity of 1.65x10 8 Ohm-cm, both are within acceptable tolerances.
The
resistivity is two orders of magnitude lower than desired, but is still high such that little current
was drawn. More care during fabrication needs to be maintained to ensure lower leakage. Other
process runs demonstrated higher resistivity, so this is an achievable goal.
Figure 6-2 shows strong perovskite phases in the XRD. Only perovskite PZT phases peaks
are visible, other than a strong platinum peak. The pyrochlore phase, visible around 290 and
340 if present,is not detectable within the instrument resolution.
54
60Cm
40*,01iIOW4
E
0
0
N
-15
-20
-10
5
-0
**-20 -
155
10
2
20
.5
IL
-60 J
Potential (V)
Figure 6-1: PZT hysteresis of a 0.4
pm thick film at 15V
PZT/Pt/Ti/SiO,/Si annealed at 650 0 C for 30
m in.
500U -
4000
-
3000
-
N
CD
CD
C%4
U,
a)
2000
4
-A
-
aC
1000 -
020
30
40
60
2E
Figure 6-2: X-Ray diffraction crystallography of the PZT.
55
60
mum
mrn~u
-
-
-
Figure 6-3: Side-by-side comparison of the end effector displacement at OV (left) and 14V (right). White
lines indicate location. 0 = 5 . Shows a displacement of 6 /Lm.
6.2
Actuation Test
The device tested had an initial design dimensions of an angle of 5
on a built-in scale with an
8 pm period, consisting of three actuators in series to make optical recoding of measurements
easier. The images were recorded and analyzed in imaging software to measure displacements.
The low accuracy of the measurements on the captured images occurs due to focusing limitations
of the optical microscope used, and the scale period.
Because of this, limited data points
(0,5,10, and 14 V) were collected, at an estimated accuracy of +1 pm. Extra data would not
provide a significant infromation suing this technique. Figure 6-3 shows the device displacement
comparison in a side-by-side manner at the field limit of actuation. While preliminary results
are encouraging, clearly, a more precise method of measurement is needed, such as a computermicro-vision system.
Figure 6-4 shows the actuator maximum displacement at ~2 ptm/actuator at 14 V, while
drawing no current. The analysis used to generate Figure 4-4b is based on geometric analysis
and is independent of device material properties. Therefore, if the dynamics of the actuator
56
8
E
2--
-
1-
0*
0
10
5
15
Potential (V)
Figure 6-4: Displacement versus voltage of the PZT end effector, for three actuators in series.
are behaving as expected, Figures 4-4b and d can be a benchmark. This corresponds to an
amplification ratio of ~11 per actuator, if the PZT member has contracted of ~0.18 pim, as
shown in Figure 4-4d. However, because the constriction of the PZT member was immeasurble
with resources available, it is unclear what the true amplification is. This compares unfavorably
(50%) with the estimate of Figure 4-7.
The critical factors that influence this discrepancy
are out of plane motion (caused by moments induced by the off-neutral axis PZT member),
electrodes being too stiff (system dynamics changed), poor PZT quality (d 31 not what was
expected), and leakage current between the top and bottom electrodes. Dielectric breakdown
occured at around 15V across the 0.4 pm thick PZT film, which corresponds to an electric field
of 37.5 V/ pm, similar to the expected value of 40 V/ pm.
57
Chapter 7
Conclusion and Future Work
7.1
Conclusion
An in-plane piezoelectric micro-actuator was designed and fabricated.
The hybrid actuator
demonstrated an amplification ratio estimated to be in excess of 10:1. Also demonstrated was
the ability of the actuator to be fabricated in arrays of microactuators, in series, or in parallel,
by means of an electrode, or power, bus. New fabrication techniques for PZT were developed,
including the patterning of the sol-gel film prior to annealing, and the release of a piezolectric
actuator using XeF 2 . A foundation has now been established for the development of devices
which use this actuator, and compliant devices using SU-8.
7.2
Future Work
To analyze the current design, foremost, more accurate measurements of device dynamics need
to be done in order to verify the actuation mode.
to be evaluated for a rigorous design treatment.
Next, material properties, (like d 31 ) need
Device resistivity, while acceptable, is two
orders of magnitude larger than desired currently, so increased care needs to be maintained when
fabricating the PZT future iterations to improve efficiency, allowing for fields on the order of
100V/ pm. The electrode bus, proven to be fabricable, should be redesigned less conservatively
in order to further approach ideal device mechanics.
To increase lateral displacement, and out of plane bending, the actuating member could be
58
b)
Figure 7-1: Membrane endpoint effects
further uncoupled from the amplifiying mechanism. Figure 7-1 shows the effects of different end
point conditions connecting the member, represented as a beam, to the amplifying mechanism.
Figure 7-1 a) shows the current design with the endpoints fixed to the amplifiyng mechanism.
Any out of plane bending of the member gets coupled into the the amplifying mechanism,
whereas in b), the member is connected by ideal pivots (in reality, they would be flexural
pivots).
The moments acting on the amplifying mechanism are decoupled, and only the forces
generated by the member would act on the amplifying mechanism.
Fully decoupling the
member is in the true spirit of the original design, however for the proof of concept stage, this
was not attempted.
In this way, the PZT member can be designed as a multi-morph, rather
that as a pure contractor, to increase device displacement.
Second, the development of a true electrode bus, such that individual actuators can be
addressed in an array of actuators would be desirable, for fully digital/analog control over a
membrane.
Serious wiring challenges exist, so perhaps a solution similar to the macro-scale
BRAID[19] could be accomplished, such that a "decoder" existed on each device to switch the
signal on and off, with the power bus doubling as a communication bus. Lastly, a single-crystal
silicon, rather than SU-8, based amplifiying mechanism was conceived and may be desirable for
silicon IC material properties.
Therefore, it would be important to show that this design can
be implemented with different materials.
Vast arrays of the actuators can be mechanical membranes, as a precursor to mechanical
materials.
The actuator can be applied to different applications, including, low power RF
switching (see Figure 7-2, early prototype), bio-manipulation, micro-manipulation, and possibly
59
"="g3ft
,
iamgfi
Figure 7-2: An actuated bi-stable beam.
Figure 7-3: A two-axis manipulator
for nano-metrology and manufacturing (see Figure 7-3, early prototype).
Virtually any low
and voltage power solution for MEMS can be accomplished, with great response characteristics.
Inverting the design, it can be used as a de-amplfying mechanism (gain less than 1) to enable
greater control for sub-nanometer alignment.
60
Bibliography
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Control, Vol.44, No.5, 1997, pp. 1140-1147.
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61
[9] Volker Seidemann, Sebastian Btitefisch, and Stephanus Btittgenbach. "Fabrication and
Investigation of In-Plane Compliant SU8 Structures for MEMS and Their Application to
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62
[20] Kazunari Maki, Nobuyuki Soyama, Satoru Mori, and Katsumi Ogi,
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3
"Evaluation of
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63
Hilton
Appendix A
Process Details
Table A.1: Metalization process
step
process
Resist
1
2
Prebake
description
HMDS first Image reverse resist AZ 5214E
Dispense speed to 500 rpm 6 sec
Spread speed to 750 rpm 6 sec
Spin speed 4500 rpm for 30 sec
30 -mm at 90'C in an oven
EV1 365-450 nm wavelength at 10 mW/cm2
Exposure total of 1.4 sec
3
4
5
6
Post exposure bake
30 min at 90'C in an oven. Be
Flood expose
2
EV1 365-450 nm wavelength at 10 mW/cm
Exposure total of 45 sec
Develop
Developer AZ 422 80 - 90 seconds
sure not to bake too long.
Electron-Beam Evaporator
a dTemescal Semiconductor Products
Metal deposition
200 titanium followed by 2000 of platinum
at a deposition rate of 1 A/sec.
Lift off
8
methanol --+ 2Achieved in acetone ->
propanol
Dry with nitrogen. Use ultrasound for small
features, and change acetone regularly to prevent metal particle adhesion. If particles accumulate, ultrasound in 2-propanol to get rid
of them.
64
Table A.2: Fabrication of thick PZT film. Courtesy of Dr. Yong-Bae Jeon and Nicholas Conway
For a 1 ,am thick maximum PZT film on platinum 2000 A on 2000 A silicon oxide.
[step
process
PT sol-gel
spin
[description
Mitsubishi Materials Co.
1
coating
PT (PbTiO3 ) Sol-gel, 3000 rpm
2
Pyrolysis
yStep
Heat to 380'C for 1 min on a hot plate
down 1 minute at 200'C, 1 minute 80'C
PZT
3
sol-gel
spin
coating
PZT sol-gel (F2), 2000 rpm.
Pyrolysis
Heat to 80'C for 1 min on a hot plate
Then 380*C for 5 min on a hot plate
Step down 1 minute at 200'C, 1 minute 80'C
Go to step 3 n-times
Get 0.2 pm per coat when annealed
Allow wafer to cool between coats.
4
5
Mitsubishi Materials Co.
HMDS first
Thick resist AZ 4620
6
Resist
7
Prebake
Static dispense for 9 sec -> 1500 rpm 9 sec
-- 3500 rpm 60 sec -> 5000 rpm 10 sec.
90"C in oven for 30 min.
Exposure
EV1 365-450 nm wavelength at 10 mW/cm2
Exposure total of 45 sec at intervals of 15 see
with 15 sec wait.
9
Develop
Developer AZ 440 for 4 minutes
10
Postbake
90"C in oven for 60 min
8
Etch in 100 HCl (17%):20 BOE (4%): 400 DI
Wet
11
etching
unannealed PZT
Strip
2
resist
12
Annealing
13
of
water (77%) for less than 10 sec.
Etching is visibly instantaneous.
Spray with DI water immediately after etch.
Achieved in acetone -+ methanol
- 2-propanol, Dry with nitrogen.
Ramp up wafer on custom stainless steel heat source
on a hot plate at 380'C. Move entire wafer on heat
sink into box furnace in air at 650'C for 20 min.
The heat source is a 1/16in thick and 5inX5in piece
of stock stainless steel with a 100mm wafer recess machined into it. The low thermal conductivity of steel
makes it act like a heat source, preventing excessive
thermal shock when moving wafer from the hot plate
to the box furnace.
65
-wafer seat
stainless steel plate
Figure A-1: Wafer sink schematic
Table A.3: SU-8 50 process
step
process
Spin coating
description
MicroChem Corp.SU-8 50 Negative Photoresist
Pour fast from bottle onto wafer.
500 rpm 5 sec --+ 5000 rpm 30 sec
1
Yields 30 pm thick resist.
2
Single hot plate:20 min at 65'C
Ramp to 95*C for 5 min
Allow to cool on hot plate slowly
Prebake
3
Exposure
Post exposure bake
4
EV1 365 -450 nm wavelength at 10 mW/cm'
sec
Exposure total of 25 sec at intervals of 5
wait.
with 15 sec
Hot plate: 65'C for 1 min
Ramp to 95'C for 4 min
Ramp to 105*C for 1 hour
Allow to cool on hot plate
slowly
MicroChem Corp SU-8 Developer PGMEA
5
5 min.
Rinse with 2-propanol, and dry with nitrogen
6
Slowly ramp to 105+'C, bake for 30-60 min.
Cool slowly on hot plate.
Gets rid of excess solvent and hardens.
Develop
Hard bake
66
Other notes:
The oxide etch is accomplished with a PECVD/RIE machine: PlasmaQuest
The XeF 2 release etch is accomplished with MEMS Solution ES-2000XM manufactured by
SETech.
67
Appendix B
MATLAB Scripts
% Script for calculating static displacement of the actuator clear all; close all;
%Calculation of PZT Force Based on Device dimensions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
W = 100e-6; % Active region 100 microns wide
d31 = 100e-12;
E_PZT = 60e9;
V = 10;
% Piezoelectric strain coefficient (C/N) approx.
% Youngs modulus of PZT, between 10 and 90.
% 10 volts
- actuation volatage
Membranelength = (380e-6/2) % half length of piezoelectric membrane
F_PZT = V*EPZT*d31*W;
disp(['The PZT membrane maximum force is
'
num2str(F.PZT/1e-6)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
E = 4e9;
% 4 GPa stiffness of SU-8
I = (30e-6)*(2e-6)^3/12;
Li = 10e-6;
L2 = 10e-6;
k1 = E*I/L1;
k2 = E*I/L2;
ri = 165e-6;
r2 = 20e-6;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
68
'
micro-Newtons'])
%%%%%%%%%%%%%%%%%
For an initial angle of 5 degrees
Fin2 = 0;
theta02 = 5*pi/180;
% 5 degrees
theta = theta02:.001:pi/4;
for i = 1:length(theta);
Fin2(i) = 4*2/r1/sin(theta(i))*((kl+k2)*theta(i)-kl*theta02-k2*theta02);
% Factor of 4 because of four quadrants of symmetry
F_out2(i) = FPZT*tan(theta(i))-4*2/r1/cos(theta(i))*((kl+k2)*theta(i)
-kl*theta02-k2*thetaO2);
end
deltay = r1*(sin(theta)-sin(theta02));
deltax = abs(rl*(cos(theta)-cos(theta02)));
% Find the maximum strain of the PZT based on the geometry
xO = rl*cos(theta02);
yO = r2+rl*sin(thetaO2)+W/2;
xmax = 0.001*Membranelength; % 0.1 % strain
i = find(deltax < xmax);
i = max(i); % get the index of the max strain
/. Plot the input force required as a function of theta
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
subplot (221)
plot(theta(1:i),Fin2(1:i)/1e-6)
title('a) Input force versus angle - \theta_0 = 5 degrees')
xlabel('\theta (rad)')
ylabel('F-i-n (\muN)')
% Plot the the relative displacements as a function of theta
%figure
subplot(222)
7 Factor of two b/c of half symmetry (both sides move);
plot(theta(1:i),2*deltay(1:i)/e-6, '--',theta(1:i),2*deltax(1:i)/le-6);
69
title('b) x and y displacement versus angle - \theta_0 = 5 degrees')
xlabel('\theta (rad)')
ylabel('Displacement (\mum)')
% Plot the the relative strain as a function of theta
%figure
subplot(223)
plot(theta(1:i),100*deltay(1:i)/yO,'--',theta(1:i),100*deltax(1:i)/x);
title('c) x and y strain versus angle - \theta_0 = 5 degrees')
xlabel('\theta (rad)')
ylabel('% Strain')
% Plot the Amplification ratio as a function of theta
%figure
subplot(224)
plot(theta(1:i),
deltay(1:i)./abs(deltax(1:i)));
title('d) Amplification Ratio versus angle - \theta0
=
5 degrees')
xlabel('\theta (rad)')
ylabel('Amplification Ratio')
figure
0/V = FPZT/(EPZT*d31*W);
plot(Fin2(1:i)/1e-6, 2*deltay(1:i)/1e-6);
%title('Displacement versus Input Voltage - \theta_0 = 5 degrees')
xlabel('Force (\muN)')
ylabel('Displacement (\mum)')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%/%%%%%%
For an initial angle of 1 degree
Fin = 0;
theta01 = 1*pi/180;
% 1 degree
theta = theta0l:.001:pi/4;
for i = 1:length(theta);
Fin(i) = 4*2/r1/sin(theta(i))*((k+k2)*theta(i)-k1*thetaOl-k2*thetaO1);
70
% Factor
of 4 because of four quadrants of symmetry (4 times the stiffness)
F_out(i) = FPZT*tan(theta(i))-4*2/rl/cos(theta(i))*((kl+k2)*theta(i)
-k1*thetaO1-k2*theta01);
end
deltay = r1*(sin(theta)-sin(theta01));
deltax = abs(rl*(cos(theta)-cos(theta0l)));
% Find
the maximum strain of the PZT based on the geometry
xO = rl*cos(theta0l);
yO = r2+rl*sin(theta0l)+W/2;
xmax
=
0.001*Membranelength;
i = find(deltax < xmax);
i = max(i); % get the index of the max strain
% Plot the input force required as a function of theta
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure
subplot(221)
plot(theta(1:i),Fin(1:i)/1e-6)
title('a)
Input force versus angle - \theta_0 = 1 degrees')
xlabel('\theta (rad)')
ylabel('FJi-n (\muN)')
% Plot the the relative displacements as a function of theta
%figure
subplot(222)
% Factor of two b/c of half symmetry (both sides move);
plot(theta(1:i),2*deltay(1:i)/le-6,'--',theta(1:i),2*deltax(1:i)/le-6);
title('b)
x and y displacement versus angle - \theta_0 = 1 degrees')
xlabel('\theta (rad)')
ylabel('Displacement (\mum)')
% Plot the the relative strain as a function of theta
figure
71
subplot(223)
plot(theta(1:i),100*deltay(1:i)/yO,'--',theta(1:i),100*deltax(1:i)/x);
title('c) x and y strain versus angle - \theta_0 = 1 degrees')
xlabel('\theta (rad)')
ylabel('% Strain')
% Plot the Amplification ratio as a function of theta
%figure
subplot(224)
plot(theta(1:i), deltay(1:i)./abs(deltax(1:i)));
title('d) Amplification Ratio versus angle - \theta_0
=
1 degrees')
xlabel('\theta (rad)')
ylabel('Amplification Ratio')
figure
plot(theta(1:i),F-out(1:i)/1e-6)
title('Output force versus angle - \theta_0 = 1 degrees')
xlabel('\theta (rad)')
ylabel('F-o-u-t (\muN)')
figure
%V = FPZT/(EPZT*d31*W);
plot(Fin(1:i)/le-6, 2*deltay(1:i)/ie-6);
%title('Displacement versus Input Force - \theta_0
=
1 degrees')
xlabel('Force (\muN)')
ylabel('Displacement (\mum)')
rhoSU8 = 2239; %density of SU-8 kg/m^3
ml = rhoSU8*(70*22*2+30*280+55*174)*30*(le-6)^3;
m2 = rhoSU8*(25*95)*30*(le-6)~3;
m3 = rhoSU8*(141*20)*30*(1e-6)^3;
J = m3*((141e-6)^2+(20e-6)^2)/3
% natural frequency in Hertz
wn = 1/rl*sqrt(8*kl/(ml +m3+ J/r1^2+4*m2*(cos(theta02))^2))/(2*pi)
72
Appendix C
Miscellaneous
The following table shows many types of actuators and their assosiated work per volume
performance.4
73
Table C.1: Work Per Unit Volume for Various Microactuators. From P. Krulevitch, A.P. Lee, P.B. Ramsey, J.C. Trevino, J. Hamilton, and M. A. Northrup, "Thin Film Shape Memory Alloy Microactuators,"
JMEMS Vol. 5, No. 4. 1996 pp 270-282.
Actuator Type W/v (J/m 3 )
Equation
Comments
[
1. Ni-Ti SMA
2.5x10 7
0- -
6.0x10 6
o- - e
max,
one-time
_-=500MPa,
thousands
4.7x10
6
2k
e=5%
of
_a=300MPa,
2 Solid-Liquid
Phase Change
output:
cycles:
e=5%
k=bulk modulus = 2.2 GPa
(H 2 0) 8% volume change (acetamide)
3.
Thermopneumatic
1.2x10 6
4.
4.6x10 5
Thermal
measured values:
F=20 N,
6=50 pam, v =4mm x 4 mm x
50 pm 3
(E E+f (Ao) - AT) 2
Expansion
5.
Electromagnetic
ideal,
4.0x10
5
F
F
2.8x104P
1.6x10 3
1.8x10 5
V
F-6
F
8. Muscle
C
2
eV4A
ideal:
V
=
100
volts,
6=gap=0.5 /pLm
7.0x10 2
Piezoelec-
silicon,
measured
values,
variable
reluctance:
F= 0.28 mN,
3=250 ptm,v=100x100x250 pm3
measured values, external field:
Torque = 0.185 nN-m, v=
400x40x7 ptm 3
3.4x10 3
7.
tric
on
ideal,
variable
reluctance:
v=total gap volume, M,=1V-
=
sec/m
6. Electrostatic
nickel
s=substrate, f=filmAT=200
V
1.2x10 5
(d 3 3E): Ef
1.8x10 2
(d 33 E)2 Ef
1.8x104
(- - e)
measured
values,
comb
drive:
F=0.2mN (60 Volts),
v=2x20x3000 Lm 3 (total gap),
6=2 pm
measured values, itegrated force
array: v=device volume, 120
volts
calculated, PZT, Ef=60GPa
(bulk),
d3 3 =500
(bulk),
E=40kV/cm
calculated, ZnO, Ef=160GPa
(bulk),
d3 3 =12
(bulk),
E=40kV/cm
measured values: o-=350MPa,
6=10%
9 Microbubble
3.4x10
2
measured
values:
bubble
diam.=71 [Lm,F=0.9/pN,6=71 [Lm
74