JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2008
213
An Electrothermomechanical Lumped Element
Model of an Electrothermal Bimorph Actuator
Shane T. Todd, Student Member, IEEE, and Huikai Xie, Senior Member, IEEE
Abstract—This paper reports a simple electrothermomechanical lumped element model (ETM-LEM) that describes the behavior of an electrothermal bimorph actuator. The ETM-LEM
is developed by integrating an electrothermal LEM of a heater
with a thermomechanical LEM of a bimorph actuator. This new
LEM uses only one power source in both the electrical and thermal
domains. The LEM provides a simple and accurate way of relating
the output mechanical response of a bimorph actuator to the
electrical inputs. The model shows that the tip angular rotation of
the bimorph actuator is linearly proportional to its average temperature change. The LEM predicts a linear relationship between
both the average temperature change and bimorph tip angular
rotation versus voltage when operated above a certain voltage.
The LEM is used to predict the rotation angle of a fabricated
electrothermal bimorph micromirror in response to the electrical
inputs and produces results that agree with finite element model
simulations and experimental data within 15% for all measured
parameters.
[2006-0097]
Index Terms—Bimorph, electrothermal modeling, electrothermomechanical modeling, lumped element modeling, micromirror,
thermal actuator, thermomechanical modeling.
I. I NTRODUCTION
E
LECTROTHERMAL bimorph actuators are common in
MEMS devices, and these actuators have been used in
applications, including micromirrors [1]–[5], RF switches [6],
nanoprobes [7], IR detectors [8], and read–write cantilevers
for data storage [9]. Compared with other types of actuators
such as electrostatic actuators, electrothermal bimorphs can
achieve large mechanical displacements because of the large
strain difference that is created when using materials with different coefficients of thermal expansion (CTEs). Electrothermal
bimorphs can also be fabricated using standard IC processing
methods and materials and can easily be integrated into CMOS
compatible devices.
To understand the actuation behavior of an electrothermal
bimorph actuator, one must consider the actuator’s electrothermal response to an electrical input and thermomechanical response to a rise in temperature. Thermomechanical models
Manuscript received May 23, 2006; revised April 26, 2007. This work was
supported by the National Science Foundation under award BES-0423557.
Subject Editor N. Aluru.
S. T. Todd was with the Department of Electrical Engineering, University of
Florida, Gainesville, FL 32611-6200 USA. He is now with the Department of
Electrical and Computer Engineering, University of California, Santa Barbara,
CA 93106-9560 USA (e-mail: stodd@ece.ucsb.edu).
H. Xie is with the Department of Electrical and Computer Engineering,
University of Florida, Gainesville, FL 32611-6200 USA (e-mail: hkx@ufl.edu).
Digital Object Identifier 10.1109/JMEMS.2007.908754
of bimorph actuators are well established [10]–[12], and they
have been used for MEMS bimorph actuators [1]–[3]. Meanwhile, electrothermal models have been developed for various MEMS heaters, including microbridge heaters [13]–[15],
and free-end heaters [2], [7], [16]–[19]. Furthermore, circuitequivalent lumped element models (LEMs) have been developed for structures with resistive heating elements [14], [20],
[21]. For example, a simple electrothermal LEM (ET-LEM)
with a single thermal power source was developed for a heater,
assuming a uniform temperature distribution [20]. However,
most heaters exhibit a nonuniform temperature distribution
along with a nonzero thermal coefficient of electrical resistivity
(TCR), which results in a complex relationship between the
electrical inputs and the thermal outputs of a device. In this
case, the heater may be divided into an electrical and thermal coupled network containing a finite number of elements.
Each element consists of temperature and voltage nodes, a
thermal power source, and thermal and electrical resistors.
This approach has been demonstrated by Mastrangelo [14] and
Manginell et al. [21]. Generally, these models are not solvable
analytically and must be simulated in a circuit simulator such as
SPICE.
In this paper, we introduce an ET-LEM of a heater with
a nonzero TCR and a nonuniform temperature distribution
using a single equivalent thermal power source. Using this new
ET-LEM, a simple electrothermomechanical LEM (ETMLEM) has been developed to describe both the electrothermal
and thermomechanical behaviors of a bimorph actuator with
an embedded heater. Compared to previously reported models,
the major advantages of this model include the use of only one
power source in both the electrical and thermal domains of the
LEM, the reduced complexity of the derived equations, and the
ability to predict the bimorph mechanical response to electrical
inputs (including current, voltage, and power).
The modeling strategy employed here is to first develop a
thermomechanical LEM (TM-LEM) of the bimorph actuator
and an ET-LEM of the heater separately and then integrate them
into a single ETM-LEM. The development of the LEMs follows
the methods described by Senturia [20]. In the next section,
the electrothermal bimorph used in the model will be briefly
introduced. In Section III, the TM-LEM is developed using
the bimorph thermomechanical equations. In Section IV, the
ET-LEM is derived using a previously reported electrothermal
transducer model [16]. In Section V the ET-LEM and TM-LEM
are combined to form an integrated ETM-LEM that is used
to derive equations for the bimorph tip angular rotation in
response to power, current, and voltage. Section VI compares
the model results to finite element model (FEM) simulations
1057-7157/$25.00 © 2008 IEEE
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JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2008
III. B IMORPH A CTUATOR TM-LEM
The thermomechanical equations which describe a bimorph
deformation in response to a temperature change are well
known, and they were established a long time ago [10], [11].
However, to the best of the authors’ knowledge, an analytical
model for the mechanical response of a bimorph to a temperature distribution has not been accurately established. In
this section, it will be shown that the tip angular rotation of
a bimorph is linearly proportional to its average temperature
change. A TM-LEM will be developed where this relationship
will be implemented.
A. Development of the TM-LEM
Fig. 1. Schematic of a cantilevered bimorph.
and experimental measurements of an electrothermal bimorph
micromirror.
II. E LECTROTHERMAL B IMORPH A CTUATOR
A thermal bimorph is composed of two material layers with
different CTEs, as shown in Fig. 1. Typically, the layers consist
of one material with a high CTE, such as a metal like Al,
and another material with a low CTE, such as a dielectric like
SiO2 . When the bimorph temperature decreases or increases,
the high-CTE material will contract or expand, respectively,
more than the low-CTE material, yielding a bending moment
to change the bimorph curvature. This curvature change results
in an angular rotation and a vertical displacement at the end of
the bimorph, which can be used for actuation in many types of
devices.
In electrothermal bimorph actuators, a resistive heater is used
to generate a temperature change. The heater can be externally
attached to the bimorph [7] or embedded in the bimorph [4].
Since the embedded heater layer is typically much thinner than
the two layers of the bimorph, its mechanical contributions are
ignored in the following analyses. Fig. 1 shows a schematic
of a cantilevered bimorph actuator with a top layer material
of thickness t1 and width w1 and a bottom layer material of
thickness t2 and width w2 . All parameters of the top and bottom
layers will be denoted by subscripts “1” and “2,” respectively.
For simplicity, the widths of the material layers are assumed
to be equal (i.e., wA = w1 = w2 ). The length and the total
thickness of the bimorph actuator are, respectively, LA and
tA = t1 + t2 . The radius of curvature of the bimorph ρA is
defined as positive when the bimorph curls in the positive
z-direction, as shown in Fig. 1. In the next section, we will
develop the TM-LEM of the bimorph actuator.
Consider a bimorph actuator at an ambient temperature of
T0 that is uniform about the beam. An applied moment due
to an internal strain in the bimorph can result from residual
stress present in the material layers at T0 or if the temperature
changes. In this analysis, the residual stress is assumed to be
uniform about the material layers. The curvature of a bimorph
beam can be represented in terms of both the applied moment
caused by the internal strain and the composite stiffness of the
bimorph shown as [11], [22]
M0 + MT
1
=
ρA
EI
(1)
where ρA is the radius of curvature, M0 and MT are the
moments caused by the residual stress and temperature change,
respectively, and EI is the composite stiffness of the bimorph.
The composite stiffness of the bimorph can be found using the
transformed-section method [11], [12], [22] and is expressed as
wA t41 E12 + t42 E22 + t1 t2 E1 E2 4t21 + 6t1 t2 + 4t22
EI =
12
t1 E1 + t2 E2
(2)
where E is the biaxial elastic modulus of either layer, and
all other parameters are defined in Fig. 1. The biaxial elastic
modulus is given by
E =
E
1−ν
(3)
where E is the elastic modulus of either layer, and ν is the
Poisson ratio of either layer. The applied moment caused by
the residual stress in the bimorph can be derived by integrating
the stress across the thickness of the bimorph [22] and is
given by
M 0 = mA
σ1
σ2
− E1
E2
= mA ∆ε0
(4)
where σ is the residual stress in either layer, mA is a parameter
that we call the moment coefficient, and ∆ε0 is the difference in
strain in the material layers caused by the residual stress. In this
convention, tensile stress is positive, and compressive stress is
TODD AND XIE: ELECTROTHERMOMECHANICAL LUMPED ELEMENT MODEL OF ELECTROTHERMAL BIMORPH ACTUATOR
215
negative. The moment coefficient is defined to simplify notation
in the model and is given by
mA =
wA tA t1 t2 E1 E2
.
2 (t1 E1 + t2 E2 )
(5)
For a bimorph with an embedded heater, the temperature
change resulting from actuation will be nonuniform about the
bimorph length and is given by the temperature distribution
T (s). This means that the strain due to the temperature distribution is nonuniform about the bimorph and will depend on
the position along the length. The bimorph moment caused by
the temperature distribution will also be distributed along the
length and is given by
MT (s) = mA ∆εT (s)
(6)
or
MT (s) = −mA (α1 − α2 ) (T (s) − T0 ) = −mA ∆αT ∆T (s)
(7)
where ∆εT (s) is the position-dependent strain difference
caused by the temperature distribution, s is used to represent
the position along the length of the bimorph, where s follows
the contour of the bimorph deformation (as shown in Fig. 1),
and α is the CTE of either layer.
Since the total bimorph moment is distributed about the
length, the curvature will also be position-dependent. This is
why it is difficult to analytically derive the tip displacement of
the bimorph. However, it is possible to analyze the tangential
angle at the tip of the bimorph (tip angle) by integrating the
moment distribution about the length. The tip angle of the bimorph can be found by integrating the arc angles of differential
sections across the length of the bimorph, which yields [23]
LA 1
·
θ=
EI LA
LA
(M0 + MT (s)) ds
0
=
LA mA
EI
σ1
σ2
− E1
E2
− ∆αT ∆T = θ0 − θT
(8)
where ∆T is the average temperature change of the bimorph,
θ0 is the initial tip angle at T0 due to the residual stress, and θT
is the tip angular rotation due to the temperature distribution.
The average moment due to the temperature distribution can
be represented in terms of the average temperature change, i.e.,
M T = −mA ∆αT ∆T . Although the temperature distribution
along the bimorph beam is not uniform, it is the average
temperature that determines the tip angular rotation. This is
a convenient result because only a single parameter, which is
the average temperature change of the bimorph, is needed from
thermal analysis to predict the tip angular rotation. It should
be noted that (8) is valid for large deflections because it is
based on the moment–curvature relationship that is valid for any
curvature that causes stresses within the linear-elastic range of
the beam materials [22]. The integration over ds in (8) follows
the contour of the deformed bimorph, and it does not involve an
approximation of the curvature.
Fig. 2. TM-LEM of the bimorph actuator.
The resultant TM-LEM is shown in Fig. 2, where the bimorph moment is the effort variable and the tip angular rotation
rate θ̇ is the flow variable. Thus, the tip angle is simply the
integral of the flow variable. Therefore, the bimorph can be
modeled as an equivalent capacitor, and the charge of the capacitor represents the output tip angle. In the TM-LEM, the effort
and flow variables are conjugate power variables. The applied
moment is divided into two voltage sources: a dc source, which
represents the initial moment applied by the residual stress,
and a voltage-controlled voltage source, which represents the
additional moment applied by a change in temperature. The
compliance of the bimorph is represented by a capacitor with
a capacitance that is equal to the bimorph length-to-stiffness
ratio. The analysis of the TM-LEM is trivial because there is
only one passive element in the circuit. It is possible to include
the transient behavior in the TM-LEM by inserting inductors
to represent the equivalent masses of the structures and the
dissipative behavior by inserting resistors to represent damping.
Representing the curvature and tip angle in terms of the total
moment and the stiffness is clumsy because these terms involve
a complicated combination of variables. It is more convenient
to rearrange the curvature in (1) and tip angle in (8) in terms of
the total bimorph thickness and a parameter called the curvature
coefficient. The curvature coefficient is given by [11], [24]
βρ = 6
(1 + m)2
1/mn + m3 n + 4m2 + 6m + 4
(9)
where m = t1 /t2 is the thickness ratio, and n = E1 /E2 is the
biaxial elastic modulus ratio. The curvature coefficient is a
unitless parameter that varies from 0 to 1.5. The ratio between
the moment coefficient and the stiffness yields the curvature coefficient to the total thickness ratio (mA /EI = βρ /tA ). Thus,
the position-dependent curvature and the tip angle can be,
respectively, expressed as
σ1 σ2
−
− ∆αT ∆T (s)
E1 E2
σ1 σ2
LA
θ = βρ
−
∆T
= θ0 − θT .
−
∆α
T
tA
E1 E2
βρ
1
=
ρA (s) tA
(10)
(11)
These are much more convenient forms because they represent the curvature and tip angle in terms of the bimorph dimensions and a unitless variable that can only be equal to a value
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Fig. 3. Schematic of the electrothermal bimorph with embedded heater used in the electrothermal model.
within the range of 0–1.5. Now that we have established that
the average temperature change is the variable which controls
the tip angular rotation of the bimorph, we must determine the
average temperature change due to the actuation of the heater.
IV. H EATER ET-LEM
In this section, we will develop an ET-LEM that can be used
to predict the average temperature change of an electrothermal
heater. The ET-LEM can then be integrated with the TM-LEM
to form the ETM-LEM. We are applying the ET-LEM to a bimorph actuator with an embedded heater, although the ET-LEM
could be applied to many different types of electrothermal
transducers. The ET-LEM uses temperature change and power
as the effort and flow variables in the thermal domain, and
voltage and current as the effort and flow variables in the
electrical domain. Resistors are used to represent the thermal
resistance in the thermal domain and the electrical resistance
in the electrical domain. We will first develop the thermal
domain LEM and then later add the electrical domain to form
the ET-LEM.
temperature distribution is assumed to vary only along the
length of the actuator and is given by T (s). The change in
temperature at a position along the length of the actuator is
∆T (s) = T (s) − T0 . By ignoring convection and radiation on
the actuator surface, the temperature distribution of the actuator
is given by [16]
s
s2
+ f RTL
(12)
∆T (s) = P RTA − 2 + f
2LA
LA
where P is the total power dissipated by the electrical resistor,
RTA is the conduction thermal resistance of the actuator, f is a
parameter called the balancing factor, and RTL is the equivalent
external thermal resistance that the actuator sees at its leftside boundary. The actuator conduction thermal resistance is
given by
RTA =
LA
κA wA tA
(13)
where κA is the composite combination of the thermal conductivities of the layers in the actuator. The balancing factor is a
very important unitless parameter that varies from zero to one
and is expressed as [16]
A. Development of the ET-LEM
A schematic showing the structure of a bimorph actuator
with an embedded heater used to generate the electrothermal
model is shown in Fig. 3. The heater is actuated by applying a
voltage or current to the embedded electrical resistor. Before
the heater is actuated, the actuator temperature is equal to
the substrate and ambient temperature T0 . Upon actuation, the
f=
RTA /2 + RTR
RTL + RTA + RTR
(14)
where RTR is the equivalent external thermal resistance that the
actuator sees at its right-side boundary. The balancing factor
measures the relative importance of the actuator conduction
thermal resistance and external thermal resistances.
TODD AND XIE: ELECTROTHERMOMECHANICAL LUMPED ELEMENT MODEL OF ELECTROTHERMAL BIMORPH ACTUATOR
The thermal LEM will be developed by using the temperature
distribution equation shown in (12) in solving for the maximum, average, and boundary temperature changes. Although
the model ignores the surface convection on the actuator, it
can include the surface convection that is present in the regions
adjacent to the actuator, represented by the equivalent thermal
resistances at the boundaries (RTL and RTR ). The equivalent
thermal resistance between two temperature nodes is found
by dividing the temperature difference between the nodes by
the total power dissipated between the nodes. The maximum
temperature is found by determining the position of the maximum temperature ŝ and then evaluating ∆T̂ = ∆T (s = ŝ).
The position of the maximum temperature can easily be found
by taking the derivative of (12) and setting it to zero, which
yields ŝ = f LA . Thus, the maximum temperature change is
given by
∆T̂ = P (f 2 RTA /2 + f RTL ).
(15)
At the position of maximum temperature, the slope of the
temperature distribution will be zero, and the temperature will
decrease directly on each side. This means that all of the power
dissipated by the heater to the left of the maximum temperature
node ∆T̂ will flow to the left toward RTL . Similarly, all of the
power dissipated by the heater to the right of ∆T̂ will flow to
the right toward RTR . The total power dissipated to the left of
∆T̂ is equal to the left-side actuator volume times the power
density which yields f P , and the total power dissipated to the
right of ∆T̂ is equal to the right-side actuator volume times
the power density which yields (1 − f )P . The current source
that represents the total power dissipated is placed at ∆T̂ . The
temperatures at the left and right boundaries of the actuator can
be found by, respectively, evaluating ∆TL = ∆T (s = 0) and
∆TR = ∆T (s = LA ) using (12), which yield
∆TL = f P RTL
(16)
∆TR = (1 − f )P RTR .
(17)
Now, the question becomes how to split the actuator conduction thermal resistance on the left and right sides of ∆T̂ . The
actuator conduction thermal resistances on each side of ∆T̂ are
denoted as RAL and RAR . It was shown earlier that the total
power flow at the left and right boundaries of the actuator are
f P and (1 − f )P , respectively. Thus, we have
RAL = (∆T̂ − ∆TL )/(f P ).
(18)
Substituting (15) and (16) into (18) yields
RAL = f RTA /2.
(19)
Similarly, we have
RAR = (∆T̂ − ∆TR )/ [(1 − f )P ] = (1 − f )RTA /2. (20)
Note that (14) is needed to obtain (20). We now have enough
information to construct a basic thermal LEM using a single
power source at the maximum temperature node and equivalent
thermal resistances between the maximum temperature and
boundary nodes. The basic thermal LEM is shown in Fig. 4,
217
Fig. 4. Basic thermal LEM.
where the balancing factor plays a crucial role in the thermal
behavior of the heater. The balancing factor f is given its name
because it determines where the maximum temperature is located and then balances the power flow and actuator conduction
thermal resistance on each side of the maximum temperature
node. Notice that half of the total actuator conduction thermal
resistance (RTA /2) is split on each side of the maximum
temperature node by f . The quadratic temperature distribution
across the actuator causes the actuator conduction thermal
resistance to be divided by two.
The thermal LEM in Fig. 4 does not include a node for the
average temperature. As shown in (8) and (11), an average
temperature node is needed to determine the tip angle of the
bimorph actuator. The average temperature change is found by
L
evaluating ∆T = (1/LA ) 0 A ∆T (s)ds of (12), which yields
∆T = P [(f − 1/3)RTA /2 + f RTL ] .
(21)
The location of the average temperature node ∆T in the
LEM depends on the relative values of the actuator conduction
thermal resistance and external thermal resistances, as represented by the balancing factor. If f ≥ 2/3, ∆T will exist to
the left of ∆T̂ . If f ≤ 1/3, ∆T will exist to the right of ∆T̂ . If
1/3 < f < 2/3, ∆T can exist on either side of ∆T̂ . In the LEM
considered for a cantilevered bimorph actuator, it is assumed
that the left-side external thermal resistance RTL is less than
the right-side external thermal resistance RTR because RTL
connects directly to the substrate, whereas RTR depends on
convection to dissipate heat (see Fig. 3). This places ∆T to the
left of ∆T̂ .
To insert ∆T into the thermal LEM, we must find the
equivalent thermal resistance between ∆T̂ and ∆T as well as
the equivalent thermal resistance between ∆T and ∆TL . The
equivalent thermal resistance between ∆T̂ and ∆T is given by
(∆T̂ − ∆T )/(f P ) = (f − 1 + 1/3f )RTA /2
(22)
which is obtained by using (15) and (21). Similarly, by using
(16) and (21), the equivalent thermal resistance between ∆T
and ∆TL is obtained as
(∆T − ∆TL )/(f P ) = (1 − 1/3f )RTA /2.
(23)
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JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2008
Fig. 5. ET-LEM, including the average temperature node.
Notice that the sum of these equivalent thermal resistances
is equal to the total equivalent thermal resistance that exists
between ∆T̂ and ∆TL which is f RTA /2.
The ET-LEM, which includes the average temperature
node and the power supplied to the thermal domain by the
electrical domain, is shown in Fig. 5. The power source in
the thermal domain is replaced by a voltage- or currentcontrolled current source that represents the heat supplied to
the thermal domain from the power dissipated by the electrical resistor. Notice that a temperature-dependent electrical
resistor exists in the electrical domain. The total electrical
resistance depends on the average temperature change and is
given by
ρE0 LA
(1 + ξ∆T ) = RE0 (1 + ξ∆T )
RE =
wE tE
(24)
where wE is the width of the electrical resistor, tE is the
thickness of the electrical resistor, ρE0 and RE0 are, respectively, the initial electrical resistivity and resistance at T0 , and
ξ is the TCR. This temperature-dependent resistor represents
the thermal feedback to the electrical resistance that exists
when the heater temperature rises. It is possible to expand the
ET-LEM to include transient behavior by inserting capacitors
which represent the capacitance in the electrical domain and
heat capacity in the thermal domain.
B. Electrothermal Response
where RT1 is defined as the equivalent average thermal resistance when convection is ignored on the surface of the actuator
and is given by
RT1 = (f − 1/3)RTA /2 + f RTL .
(26)
Solving for ∆T in terms of current and voltage, respectively,
yields
I 2 RE0 RT1
1 −I 2 RE0 ξRT1

1  4ξRT1 2
∆T (V ) =
V + 1 − 1
2ξ
RE0
∆T (I) =
=
2V 2 RT1 /RE0
.
4ξRT1 /RE0 V 2 + 1
1+
(27)
(28)
The positive TCR of the electrical resistor causes the electrical resistance to increase when the temperature increases. The
temperature-dependent electrical resistance causes the current
to provide positive feedback and the voltage to provide negative
feedback to the power and temperature as shown in (27) and
(28). At small current and voltage, the power and temperature
are quadratically related to the current and voltage as shown in
the numerators of (27) and (28). At larger current and voltage,
the temperature increases enough to appreciably change the
electrical resistance, causing the power and temperature to be
nonquadratically related to the current and voltage as shown
by the feedback terms in the denominators of (27) and (28).
The positive feedback from the TCR to the current causes
the average temperature change to approach infinity when the
current approaches the critical current, which is given by IC =
Now that we have a complete ET-LEM, the relationships
between temperature changes and electrical inputs can be obtained using circuit analysis. We will limit this analysis to solve
for the average temperature change since it is the parameter
that determines the bimorph tip angular rotation, as shown in
Section III. Solving for the average temperature change in terms
of power yields
1/RE0 ξRT1 . The negative feedback from the TCR to the
applied voltage causes the average temperature change to be
approximately linear with the voltage when half of the critical
voltage
is passed, where the critical voltage is given by VC =
∆T = P RT1
RE0 /ξRT1 . From the way we have defined the critical current and critical voltage, the initial electrical resistance is equal
(25)
TODD AND XIE: ELECTROTHERMOMECHANICAL LUMPED ELEMENT MODEL OF ELECTROTHERMAL BIMORPH ACTUATOR
Fig. 6.
219
ETM-LEM of the electrothermal bimorph actuator.
to the ratio between them, which is given by RE0 = VC /IC .
The linear relationship between the average temperature change
and applied voltage is expressed as
∆T (V VC /2) ≈
RT1 /ξRE0 · V.
(29)
The linear actuation range of the heater can be very beneficial
to many types of devices where linear actuation is important. By
equating (27) and (28), we obtain the I–V characteristic of the
heater, which is given by
I=
1+
2V /RE0
The average convection coefficient could represent nonuniform
convective heat transfer on a cross section of the actuator by
including a shape factor, as discussed by Lin and Chiao [13]
and Huang and Lee [18]. The convection thermal resistance
could include a different-shaped cross section by modifying the
area dimensions used in (31). It could also include conductive
heat transfer normal to the actuator surface by modifying (31)
to include a conduction term. If convection is considered on
the surface of the actuator, the equivalent average thermal
resistance is given by [16]
RT2
.
(30)
4ξRT1 /RE0 V 2 + 1
= RCA
This equation is a much simpler form of the heater I–V
characteristic compared with other models reported in the literature [13]–[15], [18]. The I–V characteristic demonstrates
that voltage provides negative feedback to current. At small
voltage, the current is linearly related to voltage by Ohm’s
Law as shown in the numerator of (30). At larger voltage,
the increased electrical resistance causes the current to become
nonlinear with voltage as shown by the feedback term in the
denominator of (30).
2 [1−cosh(a)]−a(rL +rR ) sinh(a)
+1
a2 (rL +rR ) cosh(a)+a (1+a2 rL rR ) sinh(a)
(32)
where a = RTA /RCA , rL = RTL /RTA , and rR =
RTR /RTA . The relationships derived in (25)–(30) can
include actuator convection by replacing RT1 with RT2 .
Equation (32) was derived from an electrothermal transducer
model that considered convection on the surface of the
actuator [16].
V. I NTEGRATED ETM-LEM
C. Electrothermal Model With Actuator Convection
Although the ET-LEM ignored the convection on the surface
of the actuator, the aforementioned equations can represent
actuator convection by using a more general expression for the
equivalent average thermal resistance. The actuator convection
thermal resistance is defined as
RCA =
1
2hLA (wA + tA )
(31)
where h is the average convection coefficient on the surface
of the actuator. The factor of two exists because convection
dissipates heat on all four sides of the actuator. In general, we
assume that the convection coefficient is constant and uniform
on the actuator and on the regions adjacent to the actuator.
The TM-LEM and ET-LEM given in Sections III and IV can
be integrated to form the complete ETM-LEM shown in Fig. 6.
The ETM-LEM demonstrates that, in the electrical domain, an
input voltage or current causes the electrical resistor to dissipate
power and raise the actuator temperature in the thermal domain,
delivering an applied moment and angular rotation to the actuator in the mechanical domain. The link between the electrical
and thermal domains is a voltage- or current-controlled current
source, which represents the dissipated power delivered to the
thermal domain from the electrical domain. The feedback from
the thermal domain to the electrical domain is represented
by the temperature-dependent electrical resistor. The link between the thermal and mechanical domains is a voltagecontrolled voltage source that represents the applied moment
due to the expansion of the materials in response to a rise in
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Fig. 7. (a) Top-view schematic of the micromirror device. (b) Side view of a two-beam section of the micromirror device.
temperature. By performing a standard circuit analysis on the
LEM, one obtains the tip angular rotation in terms of power,
current, and voltage, which are given, respectively, by
θT (P ) = βρ
LA
∆αT RT1 P
tA
LA
I 2 RE0 RT1
∆αT
tA
1 − I 2 RE0 ξRT1


LA ∆αT  4ξRT1 2
θT (V ) = βρ
V + 1 − 1 .
2tA ξ
RE0
θT (I) = βρ
(33)
(34)
(35)
These equations can also include convection on the surface
of the actuator by replacing RT1 with RT2 , as discussed in
Section IV. It should also be noted that we only chose to
derive the tip angular rotation due to the temperature change
and not the total tip angle, which includes the initial angle
due to the residual stress. This was done because we will only
report experimental results which measure the angular rotation
in the next section. However, as shown in (11) and in the LEM
in Fig. 6, the model can measure the total angle due to both
the residual stress and the temperature change represented by
θ = θ0 − θT .
Fig. 8.
SEM photograph of the micromirror device.
VI. E LECTROTHERMAL B IMORPH M ICROMIRROR
The ETM-LEM described earlier is used to model the behavior of a previously reported electrothermal bimorph micromirror [4]. The micromirror design consists of four regions—a
bimorph-actuator region, a mirror plate region, a substrate
thermal isolation region, and a mirror thermal isolation region,
as shown in Fig. 7. An SEM photograph of the micromirror
TODD AND XIE: ELECTROTHERMOMECHANICAL LUMPED ELEMENT MODEL OF ELECTROTHERMAL BIMORPH ACTUATOR
221
TABLE I
GEOMETRIC AND MATERIAL PARAMETERS OF ONE TWO-BEAM SECTION OF THE MICROMIRROR DEVICE
device is shown in Fig. 8. The geometry and parameters of the
design are listed in Table I.
A. Device Design
The bimorph-actuator region is composed of an array of
72 bimorph beams that attach the mirror plate to the substrate.
Each bimorph beam comprises an SiO2 bottom layer, an Al top
layer, and a polysilicon layer embedded in the SiO2 . The embedded polysilicon layer serves as an electrical resistor for heat
dissipation. The mirror plate is composed of an Al reflective
surface and a thick single-crystal-silicon bottom layer to ensure
mirror flatness. The thermal isolation regions are composite
structures of Al and SiO2 .
The mirror plate tilts at an angle equal to the bimorph tip
angle. Thus, the ETM-LEM can be used to predict the mirror
rotation angle in response to an applied electrical input. Since
the device is symmetric about the bimorph-actuator array, one
two-beam section [shown in Fig. 7(a)] can be used in the models. A cross-sectional side view of a two-beam section is shown
in Fig. 7(b). The material properties used in the analytical
models were found from published data. The geometric and
material properties of one two-beam section of the micromirror
device are given in Table I.
The regions adjacent to the bimorph-actuator array dissipate
heat through conduction and convection. The external thermal
resistance to the left of the bimorph RTL is due to the conduction and convection in the substrate thermal isolation region and
is approximately equal to
RTL ≈
2RTis RCis
RTis + 2RCis
(36)
where RTis is the conduction thermal resistance of the substrate
thermal isolation region, and RCis is the convection thermal
resistance of the substrate thermal isolation region. The external
thermal resistance to the right of the bimorph RTR is due to the
conduction and convection in the mirror thermal isolation and
is approximately equal to
RTR ≈ RTim + RTm /2 + RCm
(37)
where RTim and RTm are, respectively, the conduction thermal
resistances of the mirror thermal isolation and mirror plate
regions, and RCm is the convection thermal resistance of the
mirror plate region. The more accurate forms of RTL and RTR
were given previously in [16]. The expressions for the bimorph
actuator and external thermal resistances complete the models
of the electrothermal micromirror.
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TABLE II
TEMPERATURE-DEPENDENT MATERIAL PROPERTIES USED IN FEM SIMULATIONS
B. Models Compared to FEM Simulations and
Experimental Results
DC characterization of the micromirror was conducted in
an open-air environment by measuring the I–V characteristic
and mirror rotation angle over a range of applied voltages.
The mirror was positioned on a rotation stage. The electrical
resistance was determined at each actuation point by taking the
ratio of the applied voltage to the measured current. Although
the average temperature change could not be measured directly,
it was calculated from the measured electrical resistance by
using the polysilicon TCR with (24). The polysilicon TCR was
found to be 5.85 × 10−3 K−1 in a previous experiment [23].
The micromirror tilts at an angle of approximately 32◦ at room
temperature due to the residual stress present in the material
layers of the bimorph. We did not make precise measurements
of the residual stress present in the material layers; therefore,
we did not model the total angle that includes both the initial tip
angle due to the residual stress and the tip angular rotation due
to a temperature change. Thus, the following analysis focuses
only on the tip angular rotation resulting from actuation.
In the following analyses, the experimental data are compared to the models and FEM simulations. Both the LEM
and the model which includes actuator convection (using RT1
and RT2 , respectively) are considered. The model that considers the actuator convection is labeled as “actuator convection
model” in the following figures. FEM simulations of the electrothermomechanical behavior of the device were conducted in
CoventorWare. In the models, we assumed that most material properties were independent of temperature. This assumption is not accurate in reality because many of the thermal
and mechanical properties of Al, SiO2 , and polysilicon are
temperature-dependent. To simulate the effect of temperaturedependent material properties on the behavior of the device, we conducted one set of FEM simulations with mostly
temperature-independent material properties (all material properties being temperature-independent except for polysilicon
electrical resistivity) and another set which included first-order
temperature dependence of selected material properties. In
the following figures, we refer to the temperature-independent
FEM simulations as “FEM simulation 1” and the temperature-
Fig. 9. Plots of the mirror rotation angle (i.e., bimorph tip angular rotation)
versus the average temperature change.
dependent simulations as “FEM simulation 2.” The first-order
temperature-dependent terms used in the FEM simulations are
listed in Table II.
Fig. 9 shows the thermomechanical data where the mirror rotation angle (equivalently, the bimorph tip angular rotation) is plotted against the average temperature change for
model, FEM simulation, and experimental results. The LEM
and temperature-independent FEM simulation curves match
very closely, showing that the analytical model is very accurate when the material properties are temperature-independent.
The temperature-dependent FEM simulation curve lies slightly
above the other curves. This is largely due to the increase of the
Al CTE with temperature. The experimental curve lies between
the temperature-dependent FEM simulation curve and the LEM
curve, indicating that the temperature dependence of the Al
CTE may have been overestimated in the FEM simulation.
Fig. 10 shows the electrothermal data where the average
temperature change is plotted versus the following: 1) power;
2) voltage; and 3) current for model, FEM simulation, and
experimental results. Before the experiments were conducted,
the convection coefficient on the surface of the device was
TODD AND XIE: ELECTROTHERMOMECHANICAL LUMPED ELEMENT MODEL OF ELECTROTHERMAL BIMORPH ACTUATOR
Fig. 10. Plots of the average temperature change versus (a) power, (b) current,
and (c) voltage.
unknown. In the models, we assumed that the convection coefficient was the same in all regions of the device. To estimate
the convection coefficient, the experimental data of the average
223
temperature change versus the total power were fitted using
a quadratic equation, and the convection coefficient used in
the equivalent average thermal resistance RT2 in (32) was
adjusted until RT2 equaled the linear term of the quadratic
experimental fit. This approach yielded a convection coefficient
of approximately 133 W/(K · m2 ) on the surface of the device.
The LEM curve in Fig. 10(a) produces a slightly higher
average temperature change compared to the actuator convection model curve. This is due to the fact that the LEM
ignores the heat dissipation caused by convection on the surface
of the actuator. Notice that there is only a slight difference
between these curves in Fig. 10(a). This result suggests that the
ET-LEM can accurately predict the electrothermal behavior of
a bimorph actuator despite the fact that it ignores the convection
on the surface of the actuator. Remember that, although the
LEM ignores the convection on the actuator, it includes the
convection in the regions adjacent to the actuator. Therefore,
as long as much more heat is dissipated in the actuator by
conduction compared to convection, the LEM can provide a
reasonable approximation of the electrothermal behavior of
the device. In other words, the LEM is a good approximation
as long as the actuator convection thermal resistance is much
greater than the actuator conduction thermal resistance (i.e.,
RCA RTA ). The experimental data in Fig. 10(a) show that
the average temperature change was nonlinear versus the power.
At higher power, the slope of the average temperature change
versus the power decreased. The temperature-dependent FEM
simulation data also show this trend, but not to the same extent
as the experimental data. This could mean that the thermal conductivities in the real device were more temperature-dependent
than the thermal conductivities used in the simulation. It could
also mean that the tendency of the slope to decrease at higher
power was caused by a larger heat dissipation resulting from
the following: 1) a convection coefficient that increased with
temperature; 2) radiation; and/or 3) the Thompson effect [14].
The plots of the average temperature change versus the
current and voltage shown in Fig. 10(b) and (c), respectively,
demonstrate the effect of the polysilicon TCR on the electrothermal behavior of the device. Fig. 10(b) shows that the
average temperature change increases dramatically as the current approaches the critical current. This behavior demonstrates
the positive feedback of the increasing electrical resistance to
the applied current. Fig. 10(c) shows that, for voltages greater
than half the critical voltage, the average temperature change
becomes approximately linear with the voltage. This behavior
demonstrates the negative feedback of the increasing electrical
resistance to the applied voltage.
Model, FEM simulation, and experimental results of the I–V
characteristic of the device are shown in Fig. 11. The slope of
the initial electrical conductance is also included in Fig. 11.
As was mentioned previously, the initial electrical resistance is
equal to the ratio of the critical voltage to the critical current,
which is given by RE0 = VC /IC . This relationship is shown in
Fig. 11. Notice that, for small voltages, the slope of the I–V
characteristic is approximately linear and equal to the initial
electrical conductance. This occurs because, at small voltages,
the temperature change is too small to cause an appreciable
increase in the electrical resistance. At larger voltages, the
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to rotate about a fixed axis [34]. The experimental data match
the model and FEM simulation results within 15% for all of the
reported measurements on the micromirror device.
VII. C ONCLUSION
Fig. 11. Plots of the I–V characteristic of the electrothermal bimorph
actuator.
ET-LEM and TM-LEM of an electrothermal bimorph actuator have been developed. The models were integrated to form
an ETM-LEM which predicts the bimorph tip angular rotation
in response to an applied power, current, and voltage. The
thermal LEM demonstrated the importance of the balancing
factor in predicting the location of the maximum temperature
and the division of heat flow and conduction thermal resistance
on each side of the maximum temperature node. The ET-LEM
demonstrates how the TCR provides feedback from the thermal
domain to the electrical domain. The ET-LEM was applied
to a bimorph actuator and could also be applied to a wide
variety of electrothermal transducers. It was analytically and
experimentally verified that the bimorph tip angular rotation
is linearly proportional to the average temperature change.
It was also verified that the average temperature change and
tip angular rotation of an electrothermal bimorph actuator are
linear with respect to the voltage when actuated past half the
critical voltage. All of the experimental measurements agree
with the model and FEM simulation results within 15%.
ACKNOWLEDGMENT
The authors would like to thank A. Jain for helping in
collecting the dc characterization data.
R EFERENCES
Fig. 12. Plots of the mirror rotation angle (i.e., bimorph tip angular rotation)
versus voltage.
slope of the I–V curve decreases. The decrease in the slope
becomes more significant as the voltage becomes larger because
the temperature change becomes large enough to sufficiently
increase the electrical resistance. As the voltage becomes very
large, the slope of the I–V characteristic approaches zero,
demonstrating that the current of the device is always less than
the critical current. The critical current is labeled in Fig. 11 to
demonstrate this phenomenon.
Fig. 12 shows the electrothermomechanical data where the
mirror rotation angle is plotted against the voltage for model,
FEM simulation, and experimental results. The experimental
data in Fig. 12 demonstrate that the mirror rotation angle is
approximately linear with the voltage for voltages greater than
half the critical voltage. This experimental result verifies that an
electrothermal bimorph actuator can be linearly actuated with
the voltage when operated in a certain range. This property was
used in a multidegree-of-freedom micromirror design, where
the application of equal and opposite voltages to bimorph
actuators on either side of the mirror plate caused the mirror
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Shane T. Todd (S’03) was born in San Diego, CA,
in 1980. He received the B.S. (magna cum laude)
and M.S. degrees in electrical engineering from the
University of Florida, Gainesville, in 2003 and 2005,
respectively. He is currently working toward the
Ph.D. degree in electrical and computer engineering
at the University of California, Santa Barbara. His
graduate study is supported by a National Defense
Science and Engineering Graduate Fellowship.
He has coauthored three journal papers and five
conference papers and has one pending patent. His
current research interests include design, modeling, fabrication, and characterization of RF and optical MEMS devices.
Huikai Xie (S’00–M’02–SM’07) received the B.S.
and M.S. degrees in electronics engineering from
Beijing Institute of Technology, Beijing, China, the
M.S. degree in electrical engineering from Tufts
University, Medford, MA, in 1998, and the Ph.D.
degree in electrical and computer engineering from
Carnegie Mellon University, Pittsburgh, PA, in 2002.
From 1992 to 1996, he was a Research Faculty
Member and Lecturer with the Tsinghua University,
Beijing, working on various silicon-based chemical
and mechanical sensors. He spent summer 2001 at
Robert Bosch Corporation, Broadview, IL, designing 6-DOF inertial measurement units. He is currently an Associate Professor with the Department
of Electrical and Computer Engineering, University of Florida, Gainesville.
He has published over 90 technical papers and is the holder of four U.S.
patents, with eight pending patents. His present research interests include
micro/nanofabrication, integrated inertial sensors, microactuators, integrated
power converters, optical MEMS, optical imaging and fiber-optic sensors.
Dr. Xie received the 1996 Tsinghua University Motorola Education Award
and the 1996 Best Paper Award from Chinese Journal of Semiconductors. He
was named the 2006 Small Times magazine Best of Small Tech Researcher of
the Year Finalist.