A new class of thermal microactuators, Z-shaped thermal
actuator, is introduced in comparison with the wellestablished
V-shaped thermal actuator. Though they share
many features in common, Z-shaped thermal actuator offers
several advantages: compatibility with anisotropic etching,
smaller feature size, larger displacement, and larger variety
of stiffness and output force. While the Z-shaped thermal
actuator was modeled analytically and verified by
multiphysics finite element analysis (FEA), the beam width
and length of the central beam were identified as the major
design parameters in tuning the device displacement,
stiffness, stability and output force. Experimental
measurements were taken on three arrays of Z-shaped
thermal actuator with variable parameters. Results agreed
well with the finite element analysis. The development of Zshaped
thermal actuator is applicable in simultaneous
sensing and actuating applications. During the quasi-static
test of individual Z-shaped thermal actuator, the average
temperature in the device structure was estimated based on
electric resistivity at each actuation voltage.
Nomenclature
fij Compliance coefficients
Fx Internal (reaction) axial force
P Virtual unit force
M Internal (reaction) moment
α Thermal expansion coefficient
U Deflection at the tip
L Length of the long arm beam
l Length of the central beam
w Beam width
h Beam thickness
E Young’s modulus of silicon
I Beam moment of inertia (=w3h/12)
k Stiffness of a Z-shaped beam
kp Thermal conductivity of silicon
V Applied voltage across a Z-shaped beam
ρ Resistivity of silicon
Introduction
Electrostatic actuator [1] and electrothermal actuator [2-4]
are the two major in-plane actuators in microelectromechnical
systems (MEMS). Electrostatic actuators,
also known as comb drive actuators, have an output force
typically on the order of 1 μN when actuation voltage is more
than 30 V [5], while thermal actuators can easily generate a
force of 1 mN at an actuation voltage around 5-10 V [6].
Since comb drive actuators require high actuation voltage (>
30 V), which is not compatible with microelectronic power,
and large area of comb structures, thermal actuators have
attracted significant attentions in recent years, as they are
demonstrated as a compact, stable and high-force actuation
apparatus [4].
Thermal expansion is the operating principle of all kinds of
thermal actuators. As for in-plane thermal actuator, Ushaped
and V-shaped thermal actuators have been explored
and implemented for a few years. Former, also known as
thermal actuator [8-10], employs asymmetrical thermal
beams with different cross-sectional areas (cold arm and hot
arm). The locus of motion of single U-shaped thermal
actuator is an arc, which a pair of U-shaped thermal actuator
could translate linear motion. The latter, also known as bentbeam
thermal actuator [6,7,11-13], utilizes thermal
expansion of symmetric, slanted beams to generate
rectilinear displacement of the central shuttle. The V-shaped
thermal actuators, especially, have been implemented in
many applications including linear and rotary microengines
[6], nanoscale material testing systems [11,12], and
nanopositioners [14].
The advantages of V-shaped thermal actuators are large
force (on the order of mN), huge stiffness, low actuation
voltage and small features size. However, the slanted beams
in V-shaped actuator usually are not oriented along a
crystalline orientation, so that anisotropic etching cannot be
used for fabricating these structures, which largely limits the
available fabrication methods as well as materials. Also, the
slanted beams pose challenges for fabricating small
features, which deteriorates as the beam width gets close to
the resolution of photolithography (typically ~ 2 μm). Though
large stiffness makes the V-shaped thermal actuator a
perfect displacement controlled actuator, it cannot be used
as a simultaneous sensor and actuator as the electrostatic
devices. Thus an additional load sensor is always required
for such applications as nanomechanical testing [12] and
nanomanufacturing [15].
The Z-shaped thermal actuator introduced in this paper is a
new class of thermal actuators with Z-shaped beams for inplane
motion. It offers a large range of stiffness and output
force that is complementary to the comb drives and VProceedings
of the SEM Annual Conference
June 7-10, 2010 Indianapolis, Indiana USA
©2010 Society for Experimental Mechanics Inc.
T. Proulx (ed.), MEMS and Nanotechnology, Volume 2, Conference Proceedings of the Society for Experimental Mechanics Series 2,
shaped actuators. Similar to V-shaped actuators, the
structure and operating principle of the Z-shaped actuators is
first described and modeled analytically. Finite element
multiphysics simulations were then performed to verify the
experimental results that were measured in vacuum. The
devices were fabricated using the SOI-MUMPs (silicon-oninsulator
multi-user MEMS process) (MEMSCAP, Durham,
NC) with 10 μm thick silicon as the structural layer.
Concept and Modeling
A. Structure
Schematic of the Z-shaped and V-shaped thermal actuator is
shown in Fig. 1 for comparison. It is seen that the basic unit
of a Z-shaped actuator is a pair of Z- shaped beams and a
shuttle in the middle. Despite of principle similarity, Z-shaped
actuators rely on bending of the symmetric Z-shaped beams
induced by thermal expansion to achieve rectilinear
displacement of the central shuttle. Due to Joule heating, the
device is heated up when a current is passed through. The
temperature rise leads to thermal expansion of all the beams
especially the long beams; the long beams cannot expand
straight due to symmetry constraint of the structure, rather
they bend to accommodate the length expansion. As a result
the shuttle is pushed forward. Figure 1(c) shows a scanning
electron microscopy (SEM) image of a Z-shaped thermal
actuator with two pairs of Z-shaped beams.
(a) (b)
(c)
a V-shaped thermal actuator before and after motion. Drawn
not to scale. (c) SEM image of the Z-shaped thermal
actuator. The black area is an etched hole underneath. I is
the current passing through thermal beams, while DC is the
power source.
B. Mathematical Modeling
Following assumptions have been made for analytical
derivation: central shuttle is rigid and its thermal expansion is
neglected; thermal expansion of short beam (with length l in
Fig.1) is neglected; small strains and displacements are
considered; average temperature rise in a Z-shaped beam is
given [11].
Mechanical response of the structure in Fig.1 can be
equivalently modeled by considering half of the structure
without the shuttle, as shown in Fig. 2. In energy method,
three reaction forces, axial force Fx, virtual force P and
moment M, can be obtained by solving the following set of
equations [8]
11 12 13
21 22 23
31 32 33
2
0
x f f f F TL
f f f P U
f f f M
⎡ ⎤ ⎡ ⎤ ⎡ αΔ ⎤
⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦ ⎢⎣ ⎥⎦
(1)
where
3 2
11
2
3
f L l Ll
EA EI EI
= + +
2 2
12
3
2 2
f L l Ll
EI EI
= +
2
13 2
f l Ll
EI EI
= − −
2 2
21
3
2 2
f Ll L l
EI EI
= +
2 3
22
8
3
f l L l L
EA EI EI
= + +
2
23
f 2L Ll
EI EI
= − −
2
31 2
f l Ll
EI EI
= − −
2
32
f 2L Ll
EI EI
= − − 33
f 2L l
EI EI
= +
Set virtual force P equal to zero, deflection in the y direction
is derived as
3
2
2
12
6
3
U TL
w l L l
l
αΔ
=
⎛ ⎞
+ ⎜ + ⎟
⎝ ⎠
(2)
The stiffness of a Z-shaped beam is given by
( )
( )
3 3 2 2
3 3 2 4 2 4 4 4 2 2 3
2 6
8 16 2 12 6
Ew h l Lw Ll
k
Ll wl wL wLl Ll wLl
+ +
=
+ + + + +
(3)
The internal force is given by
3 2 2
24
2 x
F TEIL
l Ll Lw
αΔ
=
+ +
(4)
The output force f is given by the product of displacement
and stiffness, f=kU. The stiffness of a Z-shaped actuator with
a pair of beams is 2k, thus the output force is 2f accordingly.
Fig.2. Free-body diagram of a single Z-shaped beam.
There are three possible modes of heat transfer: conduction,
convection, and radiation. Convection and radiation are
generally neglected in MEMS structures; conduction through
the air layer between the device and the substrate is a major
heat transfer mechanism for surface micromachined
devices, since the air layer is typically very thin (on the order
of a few μm) [8,10-12]. But in SOI devices, the only heat
transfer mechanism is heat conduction to the anchors across
the beams, since the underneath silicon substrate is totally
etched as shown in Fig.1(c).
The performance of Z-shape thermal actuator is geometry
dependent. The peak displacement of one single Z-shaped
beam at a given temperature can be increased by simply
increasing the length of the long arm (L). The device
thickness (h) is not related to the displacement, but affects
the stiffness and loading force in direct proportion. In our
design, all the long beams (L) length was 88 μm and
structural thickness is 10 μm as specified in the SOIMUMPs.
Widths of all the long beams and central beams are
the same for simplicity. The central beam length (l) and
beam width (w) are variables for parametric study.
Thermomechanical finite element analysis (FEA) was
performed to verify the mathematical modeling. A 2D multifield
plane element PLANE223 was used in the FEA
(ANSYS v11.0) simulation, which involves electric, thermal
and mechanical fields. A constant temperature increase of
400 K was applied to the entire Z-shaped beam as shown in
Fig. 2. Note that, for verification of the mathematical model,
all thermal properties of single crystalline silicon (SCS) are
constants at room temperature; for comparison with the
experimental results, however, all thermal properties of SCS
are temperature dependent as listed in Table 1 in the end of
the paper.
D. Comparison between Z-shape and V-shape
A systematic comparison between the Z-shaped thermal
actuators and the V-shaped thermal actuators was carried
out to further illustrate their characteristics, as shown in Fig.
3. Fig. 3(a)
between the FEA and analytical solution, which confirms the
validity of the analytical model. Figures 3(a) and (b) together
show that, for both actuators, the smaller the beam width,
the larger the displacement. However, fabricating a small
beam width (especially ≤2 mm) is more challenging for
inclined beams (V-shape). In this regard, the Z-shaped
actuators can achieve relatively larger displacement.
Fig. 3(c) and (d) show that the stiffness of the Z-shaped
actuators is about one order of magnitude smaller than that
of the V-shaped actuators, when the beam width ranges
from 2 to 8 μm. The stiffness of V-shaped actuators does not
change with the beam width; by contrast, that of Z-shaped
scales approximately with square of beam width, because Vshaped
actuators are mainly based on beam extension while
Z-shaped actuators are mainly on beam bending. Since the
stiffness of bending beam is proportional to the cube of the
beam width, Z-shaped actuators possess a large stiffness
range for different beam width For some applications that
requires simultaneous sensing and actuating functions, Zshaped
thermal actuator alone could also be consider as
senor at the same time, while V-shaped thermal actuator
should combine with a certain type of sensor.
Fig. 3(e) and (f) show that both actuators share the same
column effective length factor, which means they have the
same critical buckling force and possess similar level of
stability. The output force is in the range of 30 to 490 μN,
calculated by f=kU. Apparently, the output force of the Zshaped
actuators is smaller than that of the V-shaped
thermal actuators [6,11].
process in run 27. All the Z-shaped thermal devices
in our design have the same anchor-anchor distance
(412μm); central beam length and width are two design
parameters. Three arrays of Z-shaped thermal actuators with
different beam widths (2 μm, 4 μm and 8 μm) were
fabricated for parametric study and device optimization.
Within each array, the length of the central beam varies from
1 to 20 μm. One such array of the Z-shaped thermal
actuators is shown in Fig.4. The displacement was
measured using an optical microscope with the edge
detection method [16]. Resolution is calibrated as 81.5
nm/pixel. Fig.5 shows the displacement of the Z-shaped
thermal actuators of two arrays. The array with 2 μm and 8
μm beam width were actuated under 2 V and 3 V,
respectively. Multiphysics FEA results for both arrays with
the corresponding applied voltages are also plotted in the
Figure. The displacement results agreed quite well between
experiments and FEA, which are also in line with the
analytical modeling as shown in Figure 3(a).
Fig.4 An array of Z-shaped thermal actuator with the same
beam width of 4 mm.
00 2 4 6 8 10 12 14 16 18 20
100
200
300
400
500
600
700
800
Central beam length (μm)
Displacement (nm)
Width 2μm -- FEA
Width 2μm -- Experiment
Width 8μm -- FEA
Width 8μm -- Experiment
Fig.5 Measured and FEA simulated displacements of Zshaped
thermal actuator arrays with two different beam
widths (2 mm and 8 mm). The actuation voltage on the array
with 2 μm beam width is 2 V and that on the array with 8 mm
beam width is 3 V.
The following tests were carried out inside an SEM (JEOL
6400F) on a particular Z-shaped thermal actuator with 4μm
beam width and 20 μm central beam length. Displacement
was measured with the actuation voltage from 0 to 6 V. The
measured displacement is plotted with respect to the input
current, as shown in Fig. 6(a). The stiffness of the structure
was calculated to be 273.4 N/m based on the measured
dimensions. Figure 6(b) shows the dependence of electric
resistance of the structure on the input current. Assuming
the linear dependency of resistivity on temperature as listed
in Table 1, the average temperature in the device was
estimated [8,9], and also plotted in Fig.6(b).
0 2 4 6 8 10 12
0
200
400
600
800
1000
1200
Measured Displacement (nm)
Measured current (mA)
0
50
100
150
200
250
300
Estimated force (μN)
(a)
0 2 4 6 8 10 12
150
200
250
300
Measured resistance (Ω)
Measured crrent (mA)
400
600
Estimated temperature (K)
(b)
Fig.6 (a) Measured displacement and corresponding
(calculated) force as functions of input current. (b) Measured
resistance and corresponding (estimated) average
temperature change as functions of input current.
Conclusions
A new electrothermal actuator with symmetric Z-shaped
beams was developed in this paper. Compared to V-shaped
one, it offers some unique advantages such as compatibility
with anisotropic etching, larger displacement, and smaller
stiffness and output force. The variable stiffness and force of
Z-shaped actuators fill the gap between those of the comb
drives and V-shaped thermal actuators. Additionally a Zshaped
actuator with smaller stiffness could be used as a
simultaneous load sensor. Among all of the design
parameters, the length of the central beam and beam width
were identified as the major ones in tuning the device
features. The quasi-static experimental measurements of
three arrays of Z-shaped thermal actuators agreed well with
the FEA predictions.
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Table 1. Silicon properties used in simulations of Z-shaped thermal actuators
Material properties Unit Value Reference
Young’s modulus GPa 160 [17]
Poisson’s ratio - 0.28 [17]
Thermal conductivity (constant) Wm-1K-1 146 [4]
Thermal conductivity (temperature dependent) Wm-1K-1 kt(T)=210658T-1.2747 [17]
Resistivity (constant) Ωm 5.1×10-5 Measured
Resistivity (temperature dependent) Ωm ρ(T)=5.1×10-5[1+3×10-3(T-273)] [17]
Thermal expansion coefficient (constant) K-1 2.5×10-6 [4]
Thermal expansion coefficient (temperature dependent) K-1 α(T)=-4×10-12T2+8×10-9T+4×10-7 [4]
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