An Electrothermomechanical Lumped Element …JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 17, NO....

JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2008 213

An Electrothermomechanical Lumped ElementModel of an Electrothermal Bimorph Actuator

Shane T. Todd, Student Member, IEEE, and Huikai Xie, Senior Member, IEEE

Abstract—This paper reports a simple electrothermomechan-ical lumped element model (ETM-LEM) that describes the be-havior of an electrothermal bimorph actuator. The ETM-LEMis developed by integrating an electrothermal LEM of a heaterwith a thermomechanical LEM of a bimorph actuator. This newLEM uses only one power source in both the electrical and thermaldomains. The LEM provides a simple and accurate way of relatingthe output mechanical response of a bimorph actuator to theelectrical inputs. The model shows that the tip angular rotation ofthe bimorph actuator is linearly proportional to its average tem-perature change. The LEM predicts a linear relationship betweenboth the average temperature change and bimorph tip angularrotation versus voltage when operated above a certain voltage.The LEM is used to predict the rotation angle of a fabricatedelectrothermal bimorph micromirror in response to the electricalinputs and produces results that agree with finite element modelsimulations and experimental data within 15% for all measuredparameters. [2006-0097]

Index Terms—Bimorph, electrothermal modeling, electrother-momechanical modeling, lumped element modeling, micromirror,thermal actuator, thermomechanical modeling.

I. INTRODUCTION

E LECTROTHERMAL bimorph actuators are common inMEMS devices, and these actuators have been used in

applications, including micromirrors [1]–[5], RF switches [6],nanoprobes [7], IR detectors [8], and read–write cantileversfor data storage [9]. Compared with other types of actuatorssuch as electrostatic actuators, electrothermal bimorphs canachieve large mechanical displacements because of the largestrain difference that is created when using materials with dif-ferent coefficients of thermal expansion (CTEs). Electrothermalbimorphs can also be fabricated using standard IC processingmethods and materials and can easily be integrated into CMOScompatible devices.

To understand the actuation behavior of an electrothermalbimorph actuator, one must consider the actuator’s electrother-mal response to an electrical input and thermomechanical re-sponse to a rise in temperature. Thermomechanical models

Manuscript received May 23, 2006; revised April 26, 2007. This work wassupported by the National Science Foundation under award BES-0423557.Subject Editor N. Aluru.

S. T. Todd was with the Department of Electrical Engineering, University ofFlorida, Gainesville, FL 32611-6200 USA. He is now with the Department ofElectrical and Computer Engineering, University of California, Santa Barbara,CA 93106-9560 USA (e-mail: [email protected]).

H. Xie is with the Department of Electrical and Computer Engineering,University of Florida, Gainesville, FL 32611-6200 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/JMEMS.2007.908754

of bimorph actuators are well established [10]–[12], and theyhave been used for MEMS bimorph actuators [1]–[3]. Mean-while, electrothermal models have been developed for vari-ous MEMS heaters, including microbridge heaters [13]–[15],and free-end heaters [2], [7], [16]–[19]. Furthermore, circuit-equivalent lumped element models (LEMs) have been devel-oped 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 distributionalong with a nonzero thermal coefficient of electrical resistivity(TCR), which results in a complex relationship between theelectrical inputs and the thermal outputs of a device. In thiscase, the heater may be divided into an electrical and ther-mal coupled network containing a finite number of elements.Each element consists of temperature and voltage nodes, athermal power source, and thermal and electrical resistors.This approach has been demonstrated by Mastrangelo [14] andManginell et al. [21]. Generally, these models are not solvableanalytically and must be simulated in a circuit simulator such asSPICE.

In this paper, we introduce an ET-LEM of a heater witha nonzero TCR and a nonuniform temperature distributionusing a single equivalent thermal power source. Using this newET-LEM, a simple electrothermomechanical LEM (ETM-LEM) has been developed to describe both the electrothermaland thermomechanical behaviors of a bimorph actuator withan embedded heater. Compared to previously reported models,the major advantages of this model include the use of only onepower source in both the electrical and thermal domains of theLEM, the reduced complexity of the derived equations, and theability to predict the bimorph mechanical response to electricalinputs (including current, voltage, and power).

The modeling strategy employed here is to first develop athermomechanical LEM (TM-LEM) of the bimorph actuatorand an ET-LEM of the heater separately and then integrate theminto a single ETM-LEM. The development of the LEMs followsthe methods described by Senturia [20]. In the next section,the electrothermal bimorph used in the model will be brieflyintroduced. In Section III, the TM-LEM is developed usingthe bimorph thermomechanical equations. In Section IV, theET-LEM is derived using a previously reported electrothermaltransducer model [16]. In Section V the ET-LEM and TM-LEMare combined to form an integrated ETM-LEM that is usedto derive equations for the bimorph tip angular rotation inresponse to power, current, and voltage. Section VI comparesthe model results to finite element model (FEM) simulations

1057-7157/$25.00 © 2008 IEEE

214 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2008

Fig. 1. Schematic of a cantilevered bimorph.

and experimental measurements of an electrothermal bimorphmicromirror.

II. ELECTROTHERMAL BIMORPH ACTUATOR

A thermal bimorph is composed of two material layers withdifferent CTEs, as shown in Fig. 1. Typically, the layers consistof one material with a high CTE, such as a metal like Al,and another material with a low CTE, such as a dielectric likeSiO2. 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 momentto change the bimorph curvature. This curvature change resultsin an angular rotation and a vertical displacement at the end ofthe bimorph, which can be used for actuation in many types ofdevices.

In electrothermal bimorph actuators, a resistive heater is usedto generate a temperature change. The heater can be externallyattached to the bimorph [7] or embedded in the bimorph [4].Since the embedded heater layer is typically much thinner thanthe two layers of the bimorph, its mechanical contributions areignored in the following analyses. Fig. 1 shows a schematicof a cantilevered bimorph actuator with a top layer materialof thickness t1 and width w1 and a bottom layer material ofthickness t2 and width w2. All parameters of the top and bottomlayers will be denoted by subscripts “1” and “2,” respectively.For simplicity, the widths of the material layers are assumedto be equal (i.e., wA = w1 = w2). The length and the totalthickness of the bimorph actuator are, respectively, LA andtA = t1 + t2. The radius of curvature of the bimorph ρA isdefined as positive when the bimorph curls in the positivez-direction, as shown in Fig. 1. In the next section, we willdevelop the TM-LEM of the bimorph actuator.

III. BIMORPH ACTUATOR TM-LEM

The thermomechanical equations which describe a bimorphdeformation in response to a temperature change are wellknown, and they were established a long time ago [10], [11].However, to the best of the authors’ knowledge, an analyticalmodel for the mechanical response of a bimorph to a tem-perature distribution has not been accurately established. Inthis section, it will be shown that the tip angular rotation ofa bimorph is linearly proportional to its average temperaturechange. A TM-LEM will be developed where this relationshipwill be implemented.

A. Development of the TM-LEM

Consider a bimorph actuator at an ambient temperature ofT0 that is uniform about the beam. An applied moment dueto an internal strain in the bimorph can result from residualstress present in the material layers at T0 or if the temperaturechanges. In this analysis, the residual stress is assumed to beuniform about the material layers. The curvature of a bimorphbeam can be represented in terms of both the applied momentcaused by the internal strain and the composite stiffness of thebimorph shown as [11], [22]

1ρA

=M0 + MT

EI(1)

where ρA is the radius of curvature, M0 and MT are themoments 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 thetransformed-section method [11], [12], [22] and is expressed as

EI =wA

12t41E

′21 + t42E

′22 + t1t2E

′1E

′2

(4t21 + 6t1t2 + 4t22

)t1E ′

1 + t2E ′2

(2)

where E′ is the biaxial elastic modulus of either layer, andall other parameters are defined in Fig. 1. The biaxial elasticmodulus is given by

E ′ =E

1 − ν(3)

where E is the elastic modulus of either layer, and ν is thePoisson ratio of either layer. The applied moment caused bythe residual stress in the bimorph can be derived by integratingthe stress across the thickness of the bimorph [22] and isgiven by

M0 = mA

(σ1

E ′1

− σ2

E ′2

)= mA∆ε0 (4)

where σ is the residual stress in either layer, mA is a parameterthat we call the moment coefficient, and ∆ε0 is the difference instrain in the material layers caused by the residual stress. In thisconvention, 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 notationin the model and is given by

mA =wAtAt1t2E

′1E

′2

2 (t1E ′1 + t2E ′

2). (5)

For a bimorph with an embedded heater, the temperaturechange resulting from actuation will be nonuniform about thebimorph length and is given by the temperature distributionT (s). This means that the strain due to the temperature dis-tribution is nonuniform about the bimorph and will depend onthe position along the length. The bimorph moment caused bythe temperature distribution will also be distributed along thelength 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 differencecaused by the temperature distribution, s is used to representthe position along the length of the bimorph, where s followsthe 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 thelength, the curvature will also be position-dependent. This iswhy it is difficult to analytically derive the tip displacement ofthe bimorph. However, it is possible to analyze the tangentialangle at the tip of the bimorph (tip angle) by integrating themoment distribution about the length. The tip angle of the bi-morph can be found by integrating the arc angles of differentialsections across the length of the bimorph, which yields [23]

θ =LA

EI· 1LA

LA∫0

(M0 + MT (s)) ds

=LAmA

EI

[(σ1

E ′1

− σ2

E ′2

)− ∆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 canbe represented in terms of the average temperature change, i.e.,MT = −mA∆αT ∆T . Although the temperature distributionalong the bimorph beam is not uniform, it is the averagetemperature that determines the tip angular rotation. This isa convenient result because only a single parameter, which isthe average temperature change of the bimorph, is needed fromthermal analysis to predict the tip angular rotation. It shouldbe noted that (8) is valid for large deflections because it isbased on the moment–curvature relationship that is valid for anycurvature that causes stresses within the linear-elastic range ofthe beam materials [22]. The integration over ds in (8) followsthe contour of the deformed bimorph, and it does not involve anapproximation of the curvature.

Fig. 2. TM-LEM of the bimorph actuator.

The resultant TM-LEM is shown in Fig. 2, where the bi-morph moment is the effort variable and the tip angular rotationrate θ is the flow variable. Thus, the tip angle is simply theintegral of the flow variable. Therefore, the bimorph can bemodeled as an equivalent capacitor, and the charge of the capac-itor represents the output tip angle. In the TM-LEM, the effortand flow variables are conjugate power variables. The appliedmoment is divided into two voltage sources: a dc source, whichrepresents the initial moment applied by the residual stress,and a voltage-controlled voltage source, which represents theadditional moment applied by a change in temperature. Thecompliance of the bimorph is represented by a capacitor witha capacitance that is equal to the bimorph length-to-stiffnessratio. The analysis of the TM-LEM is trivial because there isonly one passive element in the circuit. It is possible to includethe transient behavior in the TM-LEM by inserting inductorsto represent the equivalent masses of the structures and thedissipative behavior by inserting resistors to represent damping.

Representing the curvature and tip angle in terms of the totalmoment and the stiffness is clumsy because these terms involvea complicated combination of variables. It is more convenientto rearrange the curvature in (1) and tip angle in (8) in terms ofthe total bimorph thickness and a parameter called the curvaturecoefficient. The curvature coefficient is given by [11], [24]

βρ = 6(1 + m)2

1/mn + m3n + 4m2 + 6m + 4(9)

where m = t1/t2 is the thickness ratio, and n = E ′1/E ′

2 is thebiaxial elastic modulus ratio. The curvature coefficient is aunitless parameter that varies from 0 to 1.5. The ratio betweenthe moment coefficient and the stiffness yields the curvature co-efficient to the total thickness ratio (mA/EI = βρ/tA). Thus,the position-dependent curvature and the tip angle can be,respectively, expressed as

1ρA(s)

=βρ

tA

[(σ1

E ′1

− σ2

E ′2

)− ∆αT ∆T (s)

](10)

θ=βρLA

tA

[(σ1

E ′1

− σ2

E ′2

)− ∆αT ∆T

]= θ0 − θT . (11)

These are much more convenient forms because they repre-sent the curvature and tip angle in terms of the bimorph dimen-sions and a unitless variable that can only be equal to a value


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 thatthe average temperature change is the variable which controlsthe tip angular rotation of the bimorph, we must determine theaverage temperature change due to the actuation of the heater.

IV. HEATER ET-LEM

In this section, we will develop an ET-LEM that can be usedto predict the average temperature change of an electrothermalheater. The ET-LEM can then be integrated with the TM-LEMto form the ETM-LEM. We are applying the ET-LEM to a bi-morph actuator with an embedded heater, although the ET-LEMcould be applied to many different types of electrothermaltransducers. The ET-LEM uses temperature change and poweras the effort and flow variables in the thermal domain, andvoltage and current as the effort and flow variables in theelectrical domain. Resistors are used to represent the thermalresistance in the thermal domain and the electrical resistancein the electrical domain. We will first develop the thermaldomain LEM and then later add the electrical domain to formthe ET-LEM.

A. Development of the ET-LEM

A schematic showing the structure of a bimorph actuatorwith an embedded heater used to generate the electrothermalmodel is shown in Fig. 3. The heater is actuated by applying avoltage or current to the embedded electrical resistor. Beforethe heater is actuated, the actuator temperature is equal tothe substrate and ambient temperature T0. Upon actuation, the

temperature distribution is assumed to vary only along thelength of the actuator and is given by T (s). The change intemperature at a position along the length of the actuator is∆T (s) = T (s) − T0. By ignoring convection and radiation onthe actuator surface, the temperature distribution of the actuatoris given by [16]

∆T (s) = P

[RTA

(− s2

2L2A

+ fs

LA

)+ fRTL

](12)

where P is the total power dissipated by the electrical resistor,RTAis the conduction thermal resistance of the actuator, f is aparameter called the balancing factor, and RTL is the equivalentexternal thermal resistance that the actuator sees at its left-side boundary. The actuator conduction thermal resistance isgiven by

RTA =LA

κAwAtA(13)

where κA is the composite combination of the thermal conduc-tivities of the layers in the actuator. The balancing factor is avery important unitless parameter that varies from zero to oneand is expressed as [16]

f =RTA/2 + RTR

RTL + RTA + RTR(14)

where RTR is the equivalent external thermal resistance that theactuator sees at its right-side boundary. The balancing factormeasures the relative importance of the actuator conductionthermal resistance and external thermal resistances.


The thermal LEM will be developed by using the temperaturedistribution equation shown in (12) in solving for the maxi-mum, average, and boundary temperature changes. Althoughthe model ignores the surface convection on the actuator, itcan include the surface convection that is present in the regionsadjacent to the actuator, represented by the equivalent thermalresistances at the boundaries (RTL and RTR). The equivalentthermal resistance between two temperature nodes is foundby dividing the temperature difference between the nodes bythe total power dissipated between the nodes. The maximumtemperature is found by determining the position of the max-imum temperature s and then evaluating ∆T = ∆T (s = s).The position of the maximum temperature can easily be foundby taking the derivative of (12) and setting it to zero, whichyields s = fLA. Thus, the maximum temperature change isgiven by

∆T = P (f2RTA/2 + fRTL). (15)

At the position of maximum temperature, the slope of thetemperature distribution will be zero, and the temperature willdecrease directly on each side. This means that all of the powerdissipated by the heater to the left of the maximum temperaturenode ∆T will flow to the left toward RTL. Similarly, all of thepower dissipated by the heater to the right of ∆T will flow tothe right toward RTR. The total power dissipated to the left of∆T is equal to the left-side actuator volume times the powerdensity which yields fP , and the total power dissipated to theright of ∆T is equal to the right-side actuator volume timesthe power density which yields (1 − f)P . The current sourcethat represents the total power dissipated is placed at ∆T . Thetemperatures at the left and right boundaries of the actuator canbe found by, respectively, evaluating ∆TL = ∆T (s = 0) and∆TR = ∆T (s = LA) using (12), which yield

∆TL = fPRTL (16)

∆TR = (1 − f)PRTR. (17)

Now, the question becomes how to split the actuator conduc-tion thermal resistance on the left and right sides of ∆T . Theactuator conduction thermal resistances on each side of ∆T aredenoted as RAL and RAR. It was shown earlier that the totalpower flow at the left and right boundaries of the actuator arefP and (1 − f)P , respectively. Thus, we have

RAL = (∆T − ∆TL)/(fP ). (18)

Substituting (15) and (16) into (18) yields

RAL = fRTA/2. (19)

Similarly, we have

RAR = (∆T − ∆TR)/ [(1 − f)P ] = (1 − f)RTA/2. (20)

Note that (14) is needed to obtain (20). We now have enoughinformation to construct a basic thermal LEM using a singlepower source at the maximum temperature node and equivalentthermal resistances between the maximum temperature andboundary nodes. The basic thermal LEM is shown in Fig. 4,

Fig. 4. Basic thermal LEM.

where the balancing factor plays a crucial role in the thermalbehavior of the heater. The balancing factor f is given its namebecause it determines where the maximum temperature is lo-cated and then balances the power flow and actuator conductionthermal resistance on each side of the maximum temperaturenode. Notice that half of the total actuator conduction thermalresistance (RTA/2) is split on each side of the maximumtemperature node by f . The quadratic temperature distributionacross the actuator causes the actuator conduction thermalresistance to be divided by two.

The thermal LEM in Fig. 4 does not include a node for theaverage temperature. As shown in (8) and (11), an averagetemperature node is needed to determine the tip angle of thebimorph actuator. The average temperature change is found byevaluating ∆T = (1/LA)

∫ LA

0 ∆T (s)ds of (12), which yields

∆T = P [(f − 1/3)RTA/2 + fRTL] . (21)

The location of the average temperature node ∆T in theLEM depends on the relative values of the actuator conductionthermal resistance and external thermal resistances, as repre-sented by the balancing factor. If f ≥ 2/3, ∆T will exist tothe left of ∆T . If f ≤ 1/3, ∆T will exist to the right of ∆T . If1/3 < f < 2/3, ∆T can exist on either side of ∆T . In the LEMconsidered for a cantilevered bimorph actuator, it is assumedthat the left-side external thermal resistance RTL is less thanthe right-side external thermal resistance RTR because RTL

connects directly to the substrate, whereas RTR depends onconvection to dissipate heat (see Fig. 3). This places ∆T to theleft of ∆T .

To insert ∆T into the thermal LEM, we must find theequivalent thermal resistance between ∆T and ∆T as well asthe equivalent thermal resistance between ∆T and ∆TL. Theequivalent thermal resistance between ∆T and ∆T is given by

(∆T − ∆T )/(fP ) = (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 ∆Tand ∆TL is obtained as

(∆T − ∆TL)/(fP ) = (1 − 1/3f)RTA/2. (23)


Fig. 5. ET-LEM, including the average temperature node.

Notice that the sum of these equivalent thermal resistancesis equal to the total equivalent thermal resistance that existsbetween ∆T and ∆TL which is fRTA/2.

The ET-LEM, which includes the average temperaturenode and the power supplied to the thermal domain by theelectrical domain, is shown in Fig. 5. The power source inthe thermal domain is replaced by a voltage- or current-controlled current source that represents the heat supplied tothe thermal domain from the power dissipated by the elec-trical resistor. Notice that a temperature-dependent electricalresistor exists in the electrical domain. The total electricalresistance depends on the average temperature change and isgiven by

RE =ρE0LA

wEtE(1 + ξ∆T ) = RE0(1 + ξ∆T ) (24)

where wE is the width of the electrical resistor, tE is thethickness of the electrical resistor, ρE0 and RE0 are, respec-tively, the initial electrical resistivity and resistance at T0, andξ is the TCR. This temperature-dependent resistor representsthe thermal feedback to the electrical resistance that existswhen the heater temperature rises. It is possible to expand theET-LEM to include transient behavior by inserting capacitorswhich represent the capacitance in the electrical domain andheat capacity in the thermal domain.

B. Electrothermal Response

Now that we have a complete ET-LEM, the relationshipsbetween temperature changes and electrical inputs can be ob-tained using circuit analysis. We will limit this analysis to solvefor the average temperature change since it is the parameterthat determines the bimorph tip angular rotation, as shown inSection III. Solving for the average temperature change in termsof power yields

∆T = PRT1 (25)

where RT1 is defined as the equivalent average thermal resis-tance when convection is ignored on the surface of the actuatorand is given by

RT1 = (f − 1/3)RTA/2 + fRTL. (26)

Solving for ∆T in terms of current and voltage, respectively,yields

∆T (I) =I2RE0RT1

1 − I2RE0ξRT1

(27)

∆T (V ) =12ξ

√4ξRT1

RE0V 2 + 1 − 1

=2V 2RT1/RE0

1 +√(

4ξRT1/RE0

)V 2 + 1

. (28)

The positive TCR of the electrical resistor causes the electri-cal resistance to increase when the temperature increases. Thetemperature-dependent electrical resistance causes the currentto provide positive feedback and the voltage to provide negativefeedback to the power and temperature as shown in (27) and(28). At small current and voltage, the power and temperatureare quadratically related to the current and voltage as shown inthe numerators of (27) and (28). At larger current and voltage,the temperature increases enough to appreciably change theelectrical resistance, causing the power and temperature to benonquadratically related to the current and voltage as shownby the feedback terms in the denominators of (27) and (28).The positive feedback from the TCR to the current causesthe average temperature change to approach infinity when thecurrent approaches the critical current, which is given by IC =√

1/RE0ξRT1. The negative feedback from the TCR to theapplied voltage causes the average temperature change to beapproximately linear with the voltage when half of the criticalvoltage is passed, where the critical voltage is given by VC =√

RE0/ξRT1. From the way we have defined the critical cur-rent and critical voltage, the initial electrical resistance is equal


Fig. 6. 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 changeand applied voltage is expressed as

∆T (V � VC/2) ≈√

RT1/ξRE0 · V. (29)

The linear actuation range of the heater can be very beneficialto many types of devices where linear actuation is important. Byequating (27) and (28), we obtain the I–V characteristic of theheater, which is given by

I =2V/RE0

1 +√(

4ξRT1/RE0

)V 2 + 1

. (30)

This equation is a much simpler form of the heater I–Vcharacteristic compared with other models reported in the lit-erature [13]–[15], [18]. The I–V characteristic demonstratesthat voltage provides negative feedback to current. At smallvoltage, the current is linearly related to voltage by Ohm’sLaw as shown in the numerator of (30). At larger voltage,the increased electrical resistance causes the current to becomenonlinear with voltage as shown by the feedback term in thedenominator of (30).

C. Electrothermal Model With Actuator Convection

Although the ET-LEM ignored the convection on the surfaceof the actuator, the aforementioned equations can representactuator convection by using a more general expression for theequivalent average thermal resistance. The actuator convectionthermal resistance is defined as

RCA =1

2hLA(wA + tA)(31)

where h is the average convection coefficient on the surfaceof the actuator. The factor of two exists because convectiondissipates heat on all four sides of the actuator. In general, weassume that the convection coefficient is constant and uniformon the actuator and on the regions adjacent to the actuator.

The average convection coefficient could represent nonuniformconvective heat transfer on a cross section of the actuator byincluding a shape factor, as discussed by Lin and Chiao [13]and Huang and Lee [18]. The convection thermal resistancecould include a different-shaped cross section by modifying thearea dimensions used in (31). It could also include conductiveheat transfer normal to the actuator surface by modifying (31)to include a conduction term. If convection is considered onthe surface of the actuator, the equivalent average thermalresistance is given by [16]

RT2

=RCA

[2 [1−cosh(a)]−a(rL+rR) sinh(a)

a2(rL+rR) cosh(a)+a (1+a2rLrR) sinh(a)+1

](32)

where a =√

RTA/RCA, rL = RTL/RTA, and rR =RTR/RTA. The relationships derived in (25)–(30) caninclude actuator convection by replacing RT1 with RT2.Equation (32) was derived from an electrothermal transducermodel that considered convection on the surface of theactuator [16].

V. INTEGRATED ETM-LEM

The TM-LEM and ET-LEM given in Sections III and IV canbe integrated to form the complete ETM-LEM shown in Fig. 6.The ETM-LEM demonstrates that, in the electrical domain, aninput voltage or current causes the electrical resistor to dissipatepower and raise the actuator temperature in the thermal domain,delivering an applied moment and angular rotation to the actu-ator in the mechanical domain. The link between the electricaland thermal domains is a voltage- or current-controlled currentsource, which represents the dissipated power delivered to thethermal domain from the electrical domain. The feedback fromthe thermal domain to the electrical domain is representedby the temperature-dependent electrical resistor. The link be-tween the thermal and mechanical domains is a voltage-controlled voltage source that represents the applied momentdue to the expansion of the materials in response to a rise in


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 theLEM, one obtains the tip angular rotation in terms of power,current, and voltage, which are given, respectively, by

θT(P ) =βρLA

tA∆αTRT1P (33)

θT(I) =βρLA

tA∆αT

I2RE0RT1

1 − I2RE0ξRT1

(34)

θT(V ) =βρLA∆αT

2tAξ

√4ξRT1

RE0V 2 + 1 − 1

. (35)

These equations can also include convection on the surfaceof the actuator by replacing RT1 with RT2, as discussed inSection IV. It should also be noted that we only chose toderive the tip angular rotation due to the temperature changeand not the total tip angle, which includes the initial angledue to the residual stress. This was done because we will onlyreport experimental results which measure the angular rotationin the next section. However, as shown in (11) and in the LEMin Fig. 6, the model can measure the total angle due to boththe residual stress and the temperature change represented byθ = θ0 − θT.

Fig. 8. SEM photograph of the micromirror device.

VI. ELECTROTHERMAL BIMORPH MICROMIRROR

The ETM-LEM described earlier is used to model the behav-ior of a previously reported electrothermal bimorph micromir-ror [4]. The micromirror design consists of four regions—abimorph-actuator region, a mirror plate region, a substratethermal isolation region, and a mirror thermal isolation region,as shown in Fig. 7. An SEM photograph of the micromirror


TABLE IGEOMETRIC AND MATERIAL PARAMETERS OF ONE TWO-BEAM SECTION OF THE MICROMIRROR DEVICE

device is shown in Fig. 8. The geometry and parameters of thedesign are listed in Table I.

A. Device Design

The bimorph-actuator region is composed of an array of72 bimorph beams that attach the mirror plate to the substrate.Each bimorph beam comprises an SiO2 bottom layer, an Al toplayer, and a polysilicon layer embedded in the SiO2. The em-bedded polysilicon layer serves as an electrical resistor for heatdissipation. The mirror plate is composed of an Al reflectivesurface and a thick single-crystal-silicon bottom layer to ensuremirror flatness. The thermal isolation regions are compositestructures of Al and SiO2.

The mirror plate tilts at an angle equal to the bimorph tipangle. Thus, the ETM-LEM can be used to predict the mirrorrotation angle in response to an applied electrical input. Sincethe device is symmetric about the bimorph-actuator array, onetwo-beam section [shown in Fig. 7(a)] can be used in the mod-els. A cross-sectional side view of a two-beam section is shownin Fig. 7(b). The material properties used in the analyticalmodels were found from published data. The geometric andmaterial properties of one two-beam section of the micromirrordevice are given in Table I.

The regions adjacent to the bimorph-actuator array dissipateheat through conduction and convection. The external thermalresistance to the left of the bimorph RTL is due to the conduc-tion and convection in the substrate thermal isolation region andis approximately equal to

RTL ≈ 2RTisRCis

RTis + 2RCis(36)

where RTis is the conduction thermal resistance of the substratethermal isolation region, and RCis is the convection thermalresistance of the substrate thermal isolation region. The externalthermal resistance to the right of the bimorph RTR is due to theconduction and convection in the mirror thermal isolation andis approximately equal to

RTR ≈ RTim + RTm/2 + RCm (37)

where RTim and RTm are, respectively, the conduction thermalresistances of the mirror thermal isolation and mirror plateregions, and RCm is the convection thermal resistance of themirror plate region. The more accurate forms of RTL and RTR

were given previously in [16]. The expressions for the bimorphactuator and external thermal resistances complete the modelsof the electrothermal micromirror.


TABLE IITEMPERATURE-DEPENDENT MATERIAL PROPERTIES USED IN FEM SIMULATIONS

B. Models Compared to FEM Simulations andExperimental Results

DC characterization of the micromirror was conducted inan open-air environment by measuring the I–V characteristicand mirror rotation angle over a range of applied voltages.The mirror was positioned on a rotation stage. The electricalresistance was determined at each actuation point by taking theratio of the applied voltage to the measured current. Althoughthe average temperature change could not be measured directly,it was calculated from the measured electrical resistance byusing the polysilicon TCR with (24). The polysilicon TCR wasfound to be 5.85 × 10−3 K−1 in a previous experiment [23].The micromirror tilts at an angle of approximately 32◦ at roomtemperature due to the residual stress present in the materiallayers of the bimorph. We did not make precise measurementsof the residual stress present in the material layers; therefore,we did not model the total angle that includes both the initial tipangle due to the residual stress and the tip angular rotation dueto a temperature change. Thus, the following analysis focusesonly on the tip angular rotation resulting from actuation.

In the following analyses, the experimental data are com-pared to the models and FEM simulations. Both the LEMand the model which includes actuator convection (using RT1

and RT2, respectively) are considered. The model that consid-ers the actuator convection is labeled as “actuator convectionmodel” in the following figures. FEM simulations of the elec-trothermomechanical behavior of the device were conducted inCoventorWare. In the models, we assumed that most mate-rial properties were independent of temperature. This assump-tion is not accurate in reality because many of the thermaland mechanical properties of Al, SiO2, and polysilicon aretemperature-dependent. To simulate the effect of temperature-dependent material properties on the behavior of the de-vice, we conducted one set of FEM simulations with mostlytemperature-independent material properties (all material prop-erties being temperature-independent except for polysiliconelectrical resistivity) and another set which included first-ordertemperature dependence of selected material properties. Inthe following figures, we refer to the temperature-independentFEM 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-ordertemperature-dependent terms used in the FEM simulations arelisted in Table II.

Fig. 9 shows the thermomechanical data where the mir-ror rotation angle (equivalently, the bimorph tip angular ro-tation) is plotted against the average temperature change formodel, FEM simulation, and experimental results. The LEMand temperature-independent FEM simulation curves matchvery closely, showing that the analytical model is very accu-rate when the material properties are temperature-independent.The temperature-dependent FEM simulation curve lies slightlyabove the other curves. This is largely due to the increase of theAl CTE with temperature. The experimental curve lies betweenthe temperature-dependent FEM simulation curve and the LEMcurve, indicating that the temperature dependence of the AlCTE may have been overestimated in the FEM simulation.

Fig. 10 shows the electrothermal data where the averagetemperature change is plotted versus the following: 1) power;2) voltage; and 3) current for model, FEM simulation, andexperimental results. Before the experiments were conducted,the convection coefficient on the surface of the device was


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 co-efficient was the same in all regions of the device. To estimatethe convection coefficient, the experimental data of the average

temperature change versus the total power were fitted usinga quadratic equation, and the convection coefficient used inthe equivalent average thermal resistance RT2 in (32) wasadjusted until RT2 equaled the linear term of the quadraticexperimental fit. This approach yielded a convection coefficientof approximately 133 W/(K · m2) on the surface of the device.

The LEM curve in Fig. 10(a) produces a slightly higheraverage temperature change compared to the actuator con-vection model curve. This is due to the fact that the LEMignores the heat dissipation caused by convection on the surfaceof the actuator. Notice that there is only a slight differencebetween these curves in Fig. 10(a). This result suggests that theET-LEM can accurately predict the electrothermal behavior ofa bimorph actuator despite the fact that it ignores the convectionon the surface of the actuator. Remember that, although theLEM ignores the convection on the actuator, it includes theconvection in the regions adjacent to the actuator. Therefore,as long as much more heat is dissipated in the actuator byconduction compared to convection, the LEM can provide areasonable approximation of the electrothermal behavior ofthe device. In other words, the LEM is a good approximationas long as the actuator convection thermal resistance is muchgreater than the actuator conduction thermal resistance (i.e.,RCA � RTA). The experimental data in Fig. 10(a) show thatthe average temperature change was nonlinear versus the power.At higher power, the slope of the average temperature changeversus the power decreased. The temperature-dependent FEMsimulation data also show this trend, but not to the same extentas the experimental data. This could mean that the thermal con-ductivities in the real device were more temperature-dependentthan the thermal conductivities used in the simulation. It couldalso mean that the tendency of the slope to decrease at higherpower was caused by a larger heat dissipation resulting fromthe following: 1) a convection coefficient that increased withtemperature; 2) radiation; and/or 3) the Thompson effect [14].

The plots of the average temperature change versus thecurrent and voltage shown in Fig. 10(b) and (c), respectively,demonstrate the effect of the polysilicon TCR on the elec-trothermal behavior of the device. Fig. 10(b) shows that theaverage temperature change increases dramatically as the cur-rent approaches the critical current. This behavior demonstratesthe positive feedback of the increasing electrical resistance tothe applied current. Fig. 10(c) shows that, for voltages greaterthan half the critical voltage, the average temperature changebecomes approximately linear with the voltage. This behaviordemonstrates the negative feedback of the increasing electricalresistance to the applied voltage.

Model, FEM simulation, and experimental results of the I–Vcharacteristic of the device are shown in Fig. 11. The slope ofthe initial electrical conductance is also included in Fig. 11.As was mentioned previously, the initial electrical resistance isequal to the ratio of the critical voltage to the critical current,which is given by RE0 = VC/IC. This relationship is shown inFig. 11. Notice that, for small voltages, the slope of the I–Vcharacteristic is approximately linear and equal to the initialelectrical conductance. This occurs because, at small voltages,the temperature change is too small to cause an appreciableincrease in the electrical resistance. At larger voltages, the


Fig. 11. Plots of the I–V characteristic of the electrothermal bimorphactuator.

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 slopebecomes more significant as the voltage becomes larger becausethe temperature change becomes large enough to sufficientlyincrease the electrical resistance. As the voltage becomes verylarge, the slope of the I–V characteristic approaches zero,demonstrating that the current of the device is always less thanthe critical current. The critical current is labeled in Fig. 11 todemonstrate this phenomenon.

Fig. 12 shows the electrothermomechanical data where themirror rotation angle is plotted against the voltage for model,FEM simulation, and experimental results. The experimentaldata in Fig. 12 demonstrate that the mirror rotation angle isapproximately linear with the voltage for voltages greater thanhalf the critical voltage. This experimental result verifies that anelectrothermal bimorph actuator can be linearly actuated withthe voltage when operated in a certain range. This property wasused in a multidegree-of-freedom micromirror design, wherethe application of equal and opposite voltages to bimorphactuators on either side of the mirror plate caused the mirror

to rotate about a fixed axis [34]. The experimental data matchthe model and FEM simulation results within 15% for all of thereported measurements on the micromirror device.

VII. CONCLUSION

ET-LEM and TM-LEM of an electrothermal bimorph actua-tor have been developed. The models were integrated to forman ETM-LEM which predicts the bimorph tip angular rotationin response to an applied power, current, and voltage. Thethermal LEM demonstrated the importance of the balancingfactor in predicting the location of the maximum temperatureand the division of heat flow and conduction thermal resistanceon each side of the maximum temperature node. The ET-LEMdemonstrates how the TCR provides feedback from the thermaldomain to the electrical domain. The ET-LEM was appliedto a bimorph actuator and could also be applied to a widevariety of electrothermal transducers. It was analytically andexperimentally verified that the bimorph tip angular rotationis linearly proportional to the average temperature change.It was also verified that the average temperature change andtip angular rotation of an electrothermal bimorph actuator arelinear with respect to the voltage when actuated past half thecritical voltage. All of the experimental measurements agreewith the model and FEM simulation results within 15%.

ACKNOWLEDGMENT

The authors would like to thank A. Jain for helping incollecting the dc characterization data.

REFERENCES

[1] J. Buhler, J. Funk, O. Paul, F. P. Steiner, and H. Baltes, “Thermallyactuated CMOS micromirrors,” Sens. Actuators A, Phys., vol. 47, no. 1–3,pp. 572–575, Mar./Apr. 1995.

[2] G. Lammel, S. Schweizer, and P. Renaud, Optical Microscanners andMicrospectrometers Using Thermal Bimorph Actuators. Boston, MA:Kluwer, 2002.

[3] L. A. Liew, A. Tuantranont, and V. M. Bright, “Modeling of thermalactuation in a bulk-micromachined CMOS micromirror,” Microelectron.J., vol. 31, no. 9/10, pp. 791–801, Oct. 2000.

[4] H. Xie, A. Jain, T. Xie, Y. Pan, and G. K. Fedder, “A single-crystal silicon-based micromirror with large scanning angle for biomedical applications,”in Proc. Tech. Dig. CLEO, Baltimore, MD, 2003, pp. 858–860.

[5] H. Xie, Y. T. Pan, and G. K. Fedder, “Endoscopic optical coherence tomo-graphic imaging with a CMOS-MEMS micromirror,” Sens. Actuators A,Phys., vol. 103, no. 1/2, pp. 237–241, Jan. 2003.

[6] P. Robert, D. Saias, C. Billard, S. Boret, N. Sillon, C. Maeder-Pachurka,P. L. Charvet, G. Bouche, P. Ancey, and P. Berruyer, “IntegratedRF-MEMS switch based on a combination of thermal and electrostaticactuation,” in Proc. 12th Int. Conf. Solid-State Sens., Actuators, Microsyst.Dig. Tech. Papers, 2003, vol. 2, pp. 1714–1717.

[7] X. F. Wang, D. A. Bullen, J. Zou, C. Liu, and C. A. Mirkin, “Thermallyactuated probe array for parallel dip-pen nanolithography,” J. Vac. Sci.Technol. B, Microelectron. Process. Phenom., vol. 22, no. 6, pp. 2563–2567, Nov. 2004.

[8] S.-H. Lim, J. Choi, R. Horowitz, and A. Majumdar, “Design and fabri-cation of a novel bimorph microoptomechanical sensor,” J. Microelectro-mech. Syst., vol. 14, no. 4, pp. 683–690, Aug. 2005.

[9] P. Vettiger, G. Cross, M. Despont, U. Drechsler, U. Durig, B. Gotsmann,W. Haberle, M. A. Lantz, H. E. Rothuizen, R. Stutz, and G. K. Binnig,“The ‘millipede’—Nanotechnology entering data storage,” IEEE Trans.Nanotechnol., vol. 1, no. 1, pp. 39–55, Mar. 2002.

[10] A.-J. Villarceau, “Recherches sur le mouvement et la compensation deschronometers,” Annales de l’Observatoire Imperial de Paris, 1863.

[11] S. Timoshenko, “Analysis of bi-metal thermostats,” J. Opt. Soc. Amer.,vol. 11, no. 3, pp. 233–255, 1925.


[12] L. B. Freund and S. Suresh, Thin Film Materials. Cambridge, U.K.:Cambridge Univ. Press, 2003.

[13] L. W. Lin and M. Chiao, “Electrothermal responses of lineshape mi-crostructures,” Sens. Actuators A, Phys., vol. 55, no. 1, pp. 35–41,Jul. 1996.

[14] C. H. Mastrangelo, “Thermal applications of microbridges,” Ph.D. disser-tation, Univ. California, Berkeley, 1991.

[15] Y. C. Tai, C. H. Mastrangelo, and R. S. Muller, “Thermal conductivityof heavily doped low-pressure chemical vapor-deposited polycrystallinesilicon films,” J. Appl. Phys., vol. 63, no. 5, pp. 1442–1447, Mar. 1988.

[16] S. T. Todd and H. Xie, “Steady-state 1D electrothermal modeling of anelectrothermal transducer,” J. Micromech. Microeng., vol. 15, no. 12,pp. 2264–2276, Dec. 2005.

[17] S. T. Todd and H. Xie, “An analytical electrothermal model of a 1-Delectrothermal MEMS micromirror,” Proc. SPIE, vol. 5649, pp. 344–353,2004.

[18] Q. A. Huang and N. K. S. Lee, “Analysis and design of polysilicon thermalflexure actuator,” J. Micromech. Microeng., vol. 9, no. 1, pp. 64–70,Mar. 1999.

[19] D. Yan, A. Khajepour, and R. Mansour, “Design and modeling of a MEMSbidirectional vertical thermal actuator,” J. Micromech. Microeng., vol. 14,no. 7, pp. 841–850, Jul. 2004.

[20] S. Senturia, Microsystem Design. Boston, MA: Kluwer, 2001.[21] R. P. Manginell, J. H. Smith, and A. J. Ricco, “An overview of micro-

machined platforms for thermal sensing and gas detection,” Proc. SPIE,vol. 3046, pp. 273–284, 1997.

[22] R. C. Hibbeler, Mechanics of Materials, 4th ed. Upper Saddle River, NJ:Prentice-Hall, 2000.

[23] S. T. Todd, “Electrothermomechanical modeling of a 1-D electrothermalMEMS micromirror,” M.S. thesis, Univ. Florida, Gainesville, 2005.

[24] W. Peng, Z. Xiao, and K. R. Farmer, “Optimization of thermally actuatedbimorph cantilevers for maximum deflection,” in Proc. NSTI Nanotech-nol. Conf. Trade Show, 2003, pp. 376–379.

[25] J. E. Hatch, Aluminum: Properties and Physical Metallurgy. MetalsPark, OH: Amer. Soc. Metals, 1984.

[26] S. M. Lee and D. G. Cahill, “Heat transport in thin dielectric films,”J. Appl. Phys., vol. 81, no. 6, pp. 2590–2595, Mar. 1997.

[27] M. von Arx, O. Paul, and H. Baltes, “Process-dependent thin-film thermalconductivities for thermal CMOS MEMS,” J. Microelectromech. Syst.,vol. 9, no. 1, pp. 136–145, Mar. 2000.

[28] H. L. Dodge, “The change in the elasticity of aluminum wire with currentand external heating,” Phys. Rev., vol. 6, no. 4, pp. 312–318, Oct. 1915.

[29] H. J. McSkimin, “Measurement of elastic constants at low temperaturesby means of ultrasonic waves—Data for silicon and germanium singlecrystals, and for fused silica,” J. Appl. Phys., vol. 24, no. 8, pp. 988–997,Aug. 1953.

[30] J. H. Jeong, S. H. Chung, S. H. Lee, and D. Kwon, “Evaluation of elasticproperties and temperature effects in Si thin films using an electrostaticmicroresonator,” J. Microelectromech. Syst., vol. 12, no. 4, pp. 524–530,Aug. 2003.

[31] H. E. Nassini, M. Moreno, and C. G. Oliver, “Thermal expansion behaviorof aluminum alloys reinforced with alumina planar random short fibers,”J. Mater. Sci., vol. 36, no. 11, pp. 2759–2772, Jun. 2001.

[32] J. H. Zhao, T. Ryan, P. S. Ho, A. J. McKerrow, and W. Y. Shih, “Mea-surement of elastic modulus, Poisson ratio, and coefficient of thermalexpansion of on-wafer submicron films,” J. Appl. Phys., vol. 85, no. 9,pp. 6421–6424, May 1999.

[33] A. A. Geisberger, N. Sarkar, M. Ellis, and G. D. Skidmore, “Electrother-mal properties and modeling of polysilicon microthermal actuators,”J. Microelectromech. Syst., vol. 12, no. 4, pp. 513–523, Aug. 2003.

[34] S. T. Todd, A. Jain, H. W. Qu, and H. K. Xie, “A multi-degree-of-freedommicromirror utilizing inverted-series-connected bimorph actuators,”J. Opt. A, Pure Appl. Opt., vol. 8, no. 7, pp. S352–S359, Jul. 2006.

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 theUniversity of Florida, Gainesville, in 2003 and 2005,respectively. He is currently working toward thePh.D. degree in electrical and computer engineeringat the University of California, Santa Barbara. Hisgraduate study is supported by a National DefenseScience and Engineering Graduate Fellowship.

He has coauthored three journal papers and fiveconference papers and has one pending patent. His

current research interests include design, modeling, fabrication, and character-ization 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 fromBeijing Institute of Technology, Beijing, China, theM.S. degree in electrical engineering from TuftsUniversity, Medford, MA, in 1998, and the Ph.D.degree in electrical and computer engineering fromCarnegie Mellon University, Pittsburgh, PA, in 2002.

From 1992 to 1996, he was a Research FacultyMember and Lecturer with the Tsinghua University,Beijing, working on various silicon-based chemicaland mechanical sensors. He spent summer 2001 at

Robert Bosch Corporation, Broadview, IL, designing 6-DOF inertial mea-surement units. He is currently an Associate Professor with the Departmentof 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 includemicro/nanofabrication, integrated inertial sensors, microactuators, integratedpower converters, optical MEMS, optical imaging and fiber-optic sensors.

Dr. Xie received the 1996 Tsinghua University Motorola Education Awardand the 1996 Best Paper Award from Chinese Journal of Semiconductors. Hewas named the 2006 Small Times magazine Best of Small Tech Researcher ofthe Year Finalist.

An Electrothermomechanical Lumped Element …JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 17, NO....

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