2005 Ref12 EnikoV-shapedthermal

788 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 14, NO. 4, AUGUST 2005

Analytical Model for Analysis and Design ofV-Shaped Thermal MicroactuatorsEniko T. Enikov, Member, ASME, Shantanu S. Kedar, and Kalin V. Lazarov

AbstractAn analytical solution of the thermoelasticbending/buckling problem of thermal microactuators is pre-sented. V-shaped beam actuators are modeled using the theoryof beam-column buckling. Axial (longitudinal) deformationsincluding first-order nonlinear strain-displacement relations andthermal strains are included. The resulting nonlinear transcen-dental equations for the reaction forces are solved numerically andthe solutions are compared with a nonlinear finite element (FE)model. A test actuator has also been fabricated and characterized.The obtained accuracy of the prediction is within 1.1% of thenonlinear FE solution and agrees well with the experimentaldata. A corresponding one-dimensional (1-D) heat transfer modelhas also been developed and validated against experimental -measurements at various temperatures. The developed analyticalmodels are then used to analyze maximum stress and the heattransfer paths. It has been confirmed that the heat flux towardthe substrate is a dominant heat dissipation route in sacrificiallyreleased devices. [1311]

Index TermsBeam-column theory, buckling analysis, thermalV-shaped actuator.

I. INTRODUCTION

S INCE their initial inception more than 30 years ago,micromechanical actuators have become the hallmark ofmicroelectromechanical systems (MEMS). Initially, these weresimple electrostatically driven cantilevers, fabricated usingsemiconductor processing techniques [1]. With the maturationof this technology and the emergence of high-aspect-ratiomicromachining methods such as LIGA [2], HEXSIL [3],soft-LIGA [4], and deep reactive ion etching (DRIE) [5], thesedevices grew closer and closer to their macroscopic counter-parts such as gear trains or miniature dc motors by extendingin the third dimension. During this development, the use ofvirtually all known actuation mechanisms have been exploredin the construction of MEMS actuators. Most commonly, thesedevices utilized electrostatic [6], piezoelectric [7], electro-magnetic [8], electrothermal [9], [10], thermopneumatic [11],electrochemical [12], electro- and magnetostrictive [13], shapememory [14], and mass transport [15] effects. Among these,electrothermal actuation gained significant popularity, sincedisplacements in excess of 20 and forces as large as 40 mNhave been demonstrated [16]. The first electrothermal microac-tuators can be traced back to Henry Guckels research [17].

Manuscript received March 29, 2004; revised August 29, 2004. This work wassupported by the National Science Foundation by Grant DMI-0134585. SubjectEditor N. R. Aluru.

The authors are with the Department of Aerospace and MechanicalEngineering, University of Arizona, Tucson, AZ 85721 USA (e-mail:[email protected]).

Digital Object Identifier 10.1109/JMEMS.2005.845449

Since then, a large volume of papers have been published on thesubject [9], [18][31]. Many versions of the actuators have beendeveloped in single crystal Si [24][26], [29], in polysilicon[9], [19], [20], [27], [28], [30], [31], in GaAs [21], as well as incomposite devices (thermal bimorphs) [23], [32]. The analysisand design of these actuators require the solution of thermo-elastic and heat transfer problems. The heat transfer portion ofthe problem has been solved analytically by many in the MEMSliterature [9], [19], [24], [30], [31]. However, when dealingwith the thermoelastic deformation, the predominant approachhas been to use a nonlinear finite element (FE) model, or resortto a linear analytical analysis. A few authors have reportedanalytical and FE results, which agree for a relatively limitedrange of loading. For example, a recent work [30] reportedanalytical and FE results, that reach an 18% error at displace-ments exceeding 8.5 . This lack of a simple and yet accurateanalytical tool for analyzing the deflections results in a poorinitial guess of the desired geometry and in multiple designiterations, and cannot provide an insight into the deformationproblem. Thus when an optimal design is needed, a commonapproach is to follow somewhat ad hoc design variations [27].In an attempt to simplify the problem, a frequent approach is toneglect the effect of lateral bowing (buckling) of the hot arms,resulting in overestimation of the axial compressive force andhence the total deformation. The present work develops anexperimentally and numerically tested analytical model of thethermoelectric and thermoelastic problems, which accounts forthe lateral deflection (buckling) due to both thermal axial loadand transversely applied external load. Analysis of the modelshowed that for sacrificially released devices the majority ofthe heat is dissipated toward the substrate. It has been alsodemonstrated that a commonly used expression for the shapefactor (see discussion in Section II-B) breaks down for verythin air-gaps between the actuator and the substrate (presentcase). Finally, closed form expressions for the maximum stressin the actuator have also been developed to aid in the predictionof the onset of plastic deformation.

This paper is organized as follows: The analytical deforma-tion and thermal models are described in Section II, followedby an experimental Section III, describing the fabrication andtesting of electroplated Ni actuators. In Section IV, the model isused to determine maximum stress, the heat conduction routesand to analyze the effect of beam buckling.

II. V-SHAPED ACTUATOR DISPLACEMENT MODELV-shaped actuators have distinct advantages over the other

commonly used folded-beam actuators in that they provide rec-tilinear motion and allow stacking. Further, since the role of the

1057-7157/$20.00 2005 IEEE

ENIKOV et al.: ANALYSIS AND DESIGN OF V-SHAPED THERMAL MICROACTUATORS 789

Fig. 1. V-shaped beam actuator: geometry and loads.

cold-arm is taken by the substrate, no heat is generated in it re-sulting in a more efficient device. Multiple applications havebeen developed using this type of actuators some of which aremicrorelays [33], [34] and the in-package microaligner (IPMA)[16] used to align optical fibers to semiconductor lasers.

A. Thermoelastic Buckling Model of V-Shaped Actuators

A typical V-shaped actuator is shown in the inset of Fig. 1.This type of actuator utilizes the thermal expansion of the beamresulting from the heat generated by the current passing throughit. The constrained beam is subject to both compression andlateral bending moment, resulting in lateral displacement. Suchbeams are called beam-columns, since they support both axialand lateral loads, and have been studied by Euler, Timoshenko,and others [35], [36]. Fig. 1 shows the half span of a V-shapedthermal actuator with reaction forces replacing the action of themissing half. Using geometric symmetry, force, and momentequilibrium conditions, the reaction forces acting at the anchorof the beam can be expressed as

(1)

where is the vertical load applied to the actuator, andare the horizontal force and moment, respectively, transmittedfrom the missing half of the actuator. The initial angle of thebeam is , its half-length is , and the transversal deflectionof the mid cross section is . Assuming that the deformedshape of the beam can be described by a longitudinal displace-ment and transversal displacement , the strain in anycross section of the beam is given by

(2)(3)

where is the extensional (average) strain and is the in-dependent coordinate along the axis (see Fig. 1.) Applyingthe beam-column theory [37] with the addition of the thermalstrains results in the following set of differential equations for

and

(4)(5)

where is the cross-sectional area, is the local temper-ature, and is the substrate temperature. Equations (4)(5)require three boundary conditions. Since the internal forceand the reaction moment are not known, an additional twoboundary conditions are needed to solve the bending problem.These five conditions are

(6)(7)(8)(9)

(10)

Conditions (6)(9) are self-explanatory and (10) expresses thefact that the center of the beam can only move in the verticaldirection, as shown in Fig. 1. The above boundary conditionsconstrain the actuator to symmetric buckling modes only, whichresults in a closed-form solution for the tip displacement and re-action forces. Phenomena such as snap-through, which includestransition through asymmetric modes [38], are not included inthe present model. Integrating (4) along the half-span andupon applying (9)(10), one has

(11)

where is the average temperatureincrease of the beam. In passing it is worth mentioning that when


Fig. 2. Tip displacement simulations: Analytical and ANSYS without external load.

the beam is buckled ( ), the axial force and subse-quently the tip displacement are reduced as evident from the op-posite signs of the last two terms in (11) (see also Section III-B).Solving the second-order (5) with (6)(9) results in

(12)

where and the relations (1) have been used toshow that

(13)

Upon substitution of (12) into (11) and completing the integra-tion, a transcendental equation containing the externally appliedforce , the average temperature increase of the beam and theunknown eigenvalue is obtained

(14)

where

(15)

Thus for each value of the external load and temperature rise, (14) is solved numerically to determine the eigenvalue .

This value is then used to determine the reaction forces , ,the moment , and the tip displacement through

(16)(17)

(18)

(19)

respectively. Equations (14) and (19) have been programmedin MATLAB and solved for various input values of and av-erage temperatures . The geometricand material parameters used in these simulations are listed inTable I. Fig. 2 shows the predicted tip displacement for variousangles ( , 0.1, 0.2, and 0.3 rad), and no applied ex-ternal force. Fig. 3 shows the model predictions for nonzero ap-plied force . Both figures contain results of a FEmodel in ANSYS, using SOLID95 elements with three transla-tional degrees of freedom per node and geometric nonlinearityoption turned on. The apparent maximum error at 300 is1.1%, indicating very good agreement between analytical andFE solutions.

B. Thermal Model and Steady-State Temperature DistributionOne-dimensional (1-D) models of the heat generation and

dissipation are well-known and have already been published [9],[19], [28], [31]. A good analysis of the relative importance ofconvective versus conductive and radiative heat flux is providedin [31]. Here, a similar 1-D thermal problem has been solvedfor the sake of completeness of the analysis and to allow for in-terpretation of experimental data in light of the predictions ofthe overall model. The 1-D differential equation describing the


Fig. 3. Tip displacement simulations: analytical and ANSYS with applied external load.

TABLE IGEOMETRIC AND MATERIAL DATA OF V-SHAPED ACTUATOR

steady-state temperature distribution can be derived from energybalance considerations as reported by [28]

(20)

where is the specific resistivity of Ni, is the total currentpassing through the actuator, and are the heat conduc-tion coefficients of Ni and air, respectively, is the tempera-ture coefficient of resistance (TCR), and is a shape factor ac-counting for the additional heat flux conducted to the substrateacross the side walls of the actuator. The three terms in (20)correspond to the diffusive heat flux along the beam, the Jouleheating, and the heat flux conducted to the substrate. The valueof factor is not well established. An empirical expression forhas been developed for cases when the actuator is submerged in

water and the aspect ratio (beam height/beam width) is one-half[39].

(21)

The solution of (20) requires two boundary conditions,and , and depends on the relative magnitude of

the Joule heating versus heat conduction to the substrate. Intro-ducing a coefficient as

(22)

the differential (20) takes the form

(23)

Depending on the sign of , the solution is comprised of har-monic functions ( ) or hyperbolic functions ( )

for

for .(24)

The parameter measures the relative magnitude of the self-heating increase versus the heat flux increase toward the sub-strate for a given temperature rise. For example, for moreheat is generated due increase in the resistivity than is dissipateddue to an increase of heat flux toward the substrate for a givenrise of the local temperature. The converse is true for negative

. The average temperature increase can then be calculatedthrough integration along the length of the beam

for

for(25)


Fig. 4. Fabrication sequence.

To compare the model predictions with - experimental mea-surements, an expression for the total resistance and voltagedrop of the actuator were developed. Using the entire span ( )of the actuator these are

(26)

(27)

III. EXPERIMENTAL VALIDATION

A. Actuator FabricationElectroplated thermal actuators are very easy to fabricate

through a robust single mask fabrication process [10]. Thedevices described here were fabricated via Ni electroforming inpositive photoresist AZ4903. A seed layer of titanium/copperwas also used as a sacrificial release layer. The entire processflow is shown in Fig. 4. The starting substrate is silicon with0.4- -thick thermally grown oxide. A thin seed layer of5 nm Ti/100 nm Cu was deposited via e-beam evaporation.Photoresist with a thickness in the range of 2530 wasspun and patterned in order to create an electroplating moldfor the devices. Approximately 20 of nickel was thenelectroplated using a commercially available bath (Microfab Ni100, Enthone-OMI, Inc.). After removal of the photoresist, theactuators were released via wet etching in an aqueous solutionof ammonium persulfate. The lateral undercut of the narrower

Ni structures releases them completely from the substrate whilethe wider features, such as bonding pads, remain anchored tothe substrate. This process is analogous to the surface microc-machining of polysilicon structures, where a sacrificial oxide isused instead of the copper seed layer. An SEM micrograph ofthe completed device is shown in Fig. 5. The center of the beamhas an H-shaped structure, which is used as an optical vernier(see inset in Fig. 5). The actual dimensions of the devices wereverified using a Tencor Alpha Step profilometer and are listedin Table I.

B. Temperature and Resistance Measurements

To validate the temperature-deflection predictions, an inde-pendent measurement of the average temperature is needed. Theaverage temperature increase has been determined indirectly bymeasuring the voltage drop across the actuator for a given exci-tation current [10]. This technique requires accurate knowledgeof the resistance of electroplated Ni as a function of tempera-ture. To establish such calibration plot, a silicon die containingthe unreleased actuators was mounted in the center of a 2 in 2in copper plate along with two thermocouples (OMEGA 5TCseries, type K) straddling the die approximately 10 mm apart.The die was installed between the two thermocouples by meansof silicone thermal grease. The entire stack was then placed ona laboratory hot plate and heated to temperatures ranging from55 to 315 . A current source (Hewlett Packard 6612C


Fig. 5. Fabricated V-shaped actuator.

Fig. 6. Experimental nondimensional resistance versus temperature curve of electroplated nickel.

DC Power Supply System) was used to pass constant currentthrough the actuator, and the voltage drop was recorded usinga multimeter (FLUKE 77III). The temperature of the thermo-couples was measured using a thermocouple reader (OMEGAThermometer Model HH21). To verify that the self-heating ef-fect is minimal, the resistance was determined using two cur-rent levels, 25 mA and 50 mA. From these measurements, acalibration plot of nondimensional resistance was established,as shown in Fig. 6. The resistance was scaled with respect tothe room temperature (18 ) resistance . Theerror bars in the figure indicate error estimates based on theuncertainty in the constant current source (HP6612C) and thevoltage drop measurement error (Fluke77III). A least squareslinear fit to the measured data was used to determine the ap-parent specific resistance and temperature coefficient of re-sistance according to (26). The linear fit was constrained toprovide at room temperature. The numerical values arelisted in Table I. As evident from the experimental data, the re-sistance is not a linear function of temperature, which should beconsidered in high-fidelity models.

C. Displacement Measurements and Model ValidationThe released actuators were energized using a constant cur-

rent supply (HP 6612C). The voltage drop across the actuator

was measured using a voltmeter (FLUKE 77III Multimeter),and the actuator displacement was recorded using a CCDcamera attached to the microscope. Current levels ranging from100 to 405 mA were used. For each data point, the voltagereading was converted into an equivalent average temperatureusing the calibration curve described in the previous section.The resulting displacement versus average temperature plot isshown in Fig. 7. Also shown in the figure are the predictionsof the analytical model described by (19) and predictions ofa nonlinear ANSYS model for the same geometry along witherror estimates based on the standard deviation of a series ofmeasurements for each data point. A remarkable agreementbetween both simulations and the measured displacement datawas observed, despite the fact that a constant thermal expansioncoefficient was used.

Figs. 8 and 9 show the predictions of the analytical expres-sions (25) and (27) using the resistance parameters extractedfrom Fig. 6 for two values of the parameter . This parametermeasures the ratio between the total flux carried to the substrateand the portion of it conducted across the air gap. Intuitively,when the a small aspect-ratio actuator is surrounded by a goodthermal insulator and is in close proximity to the substrate, onewould expect that the majority of the heat flux is carried acrossthe thin air-gap and relatively small portion of it will be con-


Fig. 7. Experimental and analytical displacement versus temperature plots.

Fig. 8. Experimental and analytical voltage drop versus driving current plots.

ducted across the sidewalls. In this case the value of shouldbe close to unity or slightly exceeding this value. In the presentcase, the thickness of the sacrificial layer and the underlyingsilicon dioxide layer is 0.3 and 0.4 , respectively. The esti-mated air gap thickness therefore is , resulting in

according to (21). As evident in Figs. 8 and 9, the sug-gested value from (21) overestimates the real value of and re-sults in excessive cooling. When the parameter is fitted to theactual experimental data, however, an optimal value ofis found, which is close to the physically intuitive value of one.This apparent breakdown of (21) was traced back to its origins[39], where its use resulted in the estimation of positive TCRcoefficient for attached polysilicon actuators and negative TCRcoefficient for suspended devices. While it is clear that thereare cases when (21) holds, this analysis indicates that for thepresent geometry this is not true. Fig. 9 also indicates that for

the highest input currents used, the presented model still under-estimates the average temperature rise, which can be attributedto the nonlinear character of the - curve (see Fig. 6).

IV. MODEL ANALYSIS

A. Temperature Limitations

The beam-column solution developed above allows for anal-ysis of the failure modes of the actuators. In most applications,the repeatability of the actuation is required, therefore any kindof plastic deformation leading to hysteresis should be avoided.In a uniaxial stress condition, the elastic limit is reached whenthe stress reaches the yield stress of the material. Thus, to avoidplastic deformation, it is necessary to maintain the magnitudeof the maximum stress below the yield stress of the material.


Fig. 9. Average temperature T versus driving current: experiment and analysis.

Fig. 10. Peak compressive stress for = 0:1 and 0.2 rad.

Using (12), one can show [40] that the maximum stress in thebeam (compressive or tensile) is given by

(28)

where is the width of the actuator and is its cross-sectionalarea (see values in Table I.) Numerical results for two angles ofinclination ( and 0.2) and two half spans ( and752) are shown in Fig. 10. The dashed line in the figure indi-cates one common estimate of the yield stress of electroplatedNi, [41]. Experimental measurements of thedisplacement for currents up to 405 mA did not show any plasticdeformation. This is in agreement with the stress predictions

shown in Fig. 10, since the corresponding average temperatureis 264 and stress is 250 MPa. Surprisingly, when the samemodel is used on a shorter beam ( , ), theresulting stresses indicate that even a modest temperature differ-ence of 150 will induce plastic deformation. This apparentincrease of the stress in shorter beams is due to the increasedbuckling resistance.

B. Effect of Buckling on Tip DisplacementAs aforementioned, the buckling of the beam (defined here as

deviation from its straight initial shape) reduces the amount oftip displacement. To quantify this, FE and analytical simulationshave been performed for temperatures in the interval 18 upto 700 with and without the term in (11), accounting for


Fig. 11. Linear and nonlinear displacement models.

Fig. 12. Nondimensional heat flux conducted across the air gap (P =P ).

the reduction of the axial force due to lateral buckling. The cor-responding curves called nonlinear analytical and linear an-alytical, respectively, are shown in Fig. 11 along with FE sim-ulation. Also included in the figure are the experimental datacollected from the fabricated device in the temperature intervalof 18 260 . The comparison between the linear and non-linear models indicate that 2 error is reached at 200 con-stituting 12.5% difference between linear and nonlinear models.Unfortunately experimental data beyond 261 is not avail-able due to the detected plastic deformation for currents above405 mA. Despite this, from the comparison between the linearand nonlinear models it is clear that for average temperaturesexceeding 300 , the linear analysis will results in increasingover estimation of the tip displacement. This effect might be

significant in polysilicon devices, where the useful temperaturerange reaches 800 .

C. Heat Dissipation RoutesThe heat dissipation rate determines the steady-state power

consumption and the actuation bandwidth. A great deal of atten-tion has been attributed to the relative role of heat flux dissipatedacross the air gap in comparison to the flux exiting through thebonding pads and radiative losses. Indeed, the amount of heatdissipated across the air gap is strongly dependent on the thick-ness of the air gap, as described by [19], [31]. This flux is largelyresponsible for the reduction of the cooling time constant andthe increase in attainable bandwidth. While the transient thermalanalysis is beyond the scope of this work, the developed model


can be used to investigate the magnitude of these two heat fluxes.The total heat input delivered to the actuator at steady-state con-duction is

(29)

while the heat dissipated through the air gap can be found bysubtracting the heat conducted through the two bonding pads(neglecting radiative cooling)

(30)The ratio of has been plotted in Fig. 12 for currentsin the 100415 mA range. As evident, the heat flux across theair gap remains greater than 72% over the entire useful range ofdriving currents.

V. SUMMARY AND CONCLUSION

A coupled nonlinear thermomechanical model of bendingof V-shaped actuators has been described. The model includesnonlinear extensional strain and explicitly accounts for externalloads. Excellent agreement has been found among analyticalpredictions, FE models and experimental observations.

The electrothermal model indicated that for air gaps muchsmaller than the width of the actuator, the shape factor is ap-proximately unity. In this regime, the heat flux dissipated to-ward the substrate is significant and should be accounted for inenergy balances and transient heat analysis problems. At highercurrents, the nonlinear nature of the thermal resistance also be-comes noticeable, and an increasing discrepancy between ex-perimental and analytical - and - curves is observed. Sim-ilarly, at higher thermal loading nonlinear effects such as lateralbuckling (deviation from the initial straight shape of the beam)results in reduced tip displacements. In the presented models(analytical and finite element), an average (constant) thermal ex-pansion coefficient has been used over the working temperaturerange, indicating that, for the material under study, the use of anaverage (constant) thermal expansion coefficient is acceptable.

The developed model can be used as a simple fidelity analysistool when building more complex FE models. It has also beendemonstrated that the developed closed-form solutions allowquick optimizations of the geometry, for example, to avoid theonset of plastic deformation.

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[41] T. Buchheit, T. R. Christenson, D. Schmale, and D. Lavan, Under-standing and tailoring the mechanical properties of liga fabricated mate-rials, in Materials Science of Microelectromechanical Systems (MEMS)Devices, Materials Res. Soc. Symp., vol. 546, 1999, pp. 121126.

Eniko T. Enikov received the M.S. degree fromthe Technical University of Budapest in 1993 andthe Ph.D. degree from the University of Illinois atChicago in 1998.

He then began a two-year Postdoctoral Fellowshipwith the Advanced Microsystems Laboratory, Uni-versity of Minnesota, during 19982000. Currently,he is an Assistant Professor with the Department ofAerospace and Mechanical Engineering, Universityof Arizona, Tucson. His current research is focusedon the design and fabrication of microelectromechan-

ical systems (MEMS), the development of theoretical models of active actuatormaterials used in MEMS and development of relevant applications of these. Hisgroup at the University of Arizona has an ongoing research and developmentprogram on tactile displays, electrostatic microgrippers for assembly of MEMS,and nanoassembly of macromolecules using electrostatic fields. His group isalso working on the development of MEMS-compatible wireless sensing plat-forms with biomedical applications.

Dr. Enikov is a member of the American Society of Mechanical Engineers(ASME), SPIE, ASEE, and the Society of Experimental Mechanics (SEM).

Shantanu S. Kedar received the B.S. degree in me-chanical engineering from the Visvesvaraya RegionalCollege of Engineering, Nagpur, India, in 2001 andthe M.S. degree, also in mechanical engineering,from The Aerospace and Mechanical EngineeringDepartment, University of Arizona, Tucson, in 2004.During his masters degree program, he worked onV- shaped thermal microactuators in the AdvancedMicrosystems Laboratory, University of Arizona,under Dr. Enikovs supervision.

Mr. Kedar is a member of the Society of Experi-mental Mechanics (SEM).

Kalin V. Lazarov received the M.S. degree fromSofia University in 2000 and the Ph.D. degree fromthe University of Arizona, Tucson, in 2004.

He is currently a Research Assistant with the De-partment of Aerospace and Mechanical Engineering,University of Arizona. During his graduate studies inthe Advanced Microsystems Laboratory, Universityof Arizona, he worked on a hybrid thermal/piezoelec-tric MEMS tactile display. His research is focused onthe design, fabrication, and modeling of MEMS sen-sors and actuators, and the integration of MEMS sen-

sors in wireless data acquisition systems.Dr. Lazarov is a member of the Society of Experimental Mechanics (SEM).

tocAnalytical Model for Analysis and Design of V-Shaped Thermal MicEniko T. Enikov, Member, ASME, Shantanu S. Kedar, and Kalin V. LI. I NTRODUCTIONII. V-S HAPED A CTUATOR D ISPLACEMENT M ODEL

Fig.1. V-shaped beam actuator: geometry and loads.A. Thermoelastic Buckling Model of V-Shaped ActuatorsFig.2. Tip displacement simulations: Analytical and ANSYS witho

B. Thermal Model and Steady-State Temperature Distribution

Fig.3. Tip displacement simulations: analytical and ANSYS with TABLE I G EOMETRIC AND M ATERIAL D ATA OF V-S HAPED A CTUATORFig.4. Fabrication sequence.III. E XPERIMENTAL V ALIDATIONA. Actuator FabricationB. Temperature and Resistance Measurements

Fig.5. Fabricated V-shaped actuator.Fig.6. Experimental nondimensional resistance versus temperaturC. Displacement Measurements and Model Validation

Fig.7. Experimental and analytical displacement versus temperatFig.8. Experimental and analytical voltage drop versus driving IV. M ODEL A NALYSISA. Temperature Limitations

Fig.9. Average temperature $\bar T$ versus driving current: expFig.10. Peak compressive stress for $\theta = 0.1$ and 0.2 rad.B. Effect of Buckling on Tip Displacement

Fig.11. Linear and nonlinear displacement models.Fig.12. Nondimensional heat flux conducted across the air gap (C. Heat Dissipation RoutesV. S UMMARY AND C ONCLUSIONK. Peterson, Silicon as a mechanical material, in Proc. SPIE, vW. Menz, Microactuators in liga technique, Int. J. Appl. AlectroC. Keller and R. Howe, Nickel-filled HEXSIL thermally actuated tD. Sadler, S. Gupta, and C. Ahn, Transformers, inductors, power H. Jansen, M. de Boer, H. Wensink, B. Kloeck, and M. Elwenspoek,M. Baltzer, T. Kraus, and E. Obermeier, A linear stepping actuatD. Damjanovic and R. Newnham, Electrostrictive and piezoelectricH. Tilmans, E. Fullin, H. Ziad, M. van de Peer, J. Kesters, E. vJ. Butler, V. Bright, and W. Cowan, Average power control and poE. Enikov and K. Lazarov, PCB-integrated metallic thermal micro-O. Jeong and S. Yang, Fabrication of a thermopneumatic microactuC. Neagu, J. E. Gardeniers, M. Elwenspoek, and J. Kelly, An elecE. Quandt and A. Ludwig, Magnetostrictive actuation in microsystC. Ray, C. Sloan, A. Johnson, J. Busch, and B. Petty, A silicon-E. T. Enikov and G. S. Seo, Large deformation model of ion-exchaJ. Haake, R. Wood, and V. Duhler, In package micro aligner for fH. Guckel, J. Klein, T. Christenson, K. Skrobis, M. Laudon, and R. Wood, R. Mahadevan, V. Dhuler, B. Dudley, A. Cowen, E. Hill, Q.-A. Huang and N. K. S. Lee, Analysis and design of polysiliconH. Kapels, R. Aigner, and J. Binder, Fracture strength and fatigT. Lalinsky, E. Burian, M. Drzik, S. Hascik, Z. Mozolova, and J.N. Mankame and G. Ananthasuresh, Comprehensive thermal modeling H. Sehr, A. G. Evans, A. Brunnschweiler, G. J. Ensell, and T. E.L. Que, J.-S. Park, and Y. Gianchandani, Bent-beam electrothermaR. Syms, Long-travel electrothermally driven resonant cantileverT. Akiyama, U. Staufer, and N. de Rooij, Fast driving technique R. S. Chen, C. Kung, and G.-B. Lee, Analysis of the optimal dimeC. Lott, T. McLain, J. Harb, and L. Howell, Modeling of the therW. Chen, C. Chu, J. Hsieh, and W. Fang, A reliable single-layer D. Yan, A. Khajepour, and R. Mansour, Modeling of two-hot-arm hoR. Hickey, D. Sameoto, T. Hubbard, and M. Kujath, Time and frequE. Enikov and K. Lazarov, Composite themal micro-actuator array T. Gomm, L. Howell, and R. Selfridge, In-plane linear displacemeY. Wang, Z. Li, D. T. McCormick, and N. C. Tien, A micromachinedS. P. Timoshenko, History of Strength of Materials . New York: MS. P. Timoshenko and J. M. Gere, Theory of Elastic Stability . NG. J. Simitses, An Introduction to Elastic Stability of StructurM. Vengbo, An analytical analysis of a compressed bistable bucklL. Lin and M. Chiao, Electrothermal responses of lineshape microK. Shantanu, Modeling of thermal microactuators for MEMS, MasterT. Buchheit, T. R. Christenson, D. Schmale, and D. Lavan, Unders

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