An electrothermal microactuator with Z-shaped beams

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2010 J. Micromech. Microeng. 20 085014

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J. Micromech. Microeng. 20 (2010) 085014 (9pp) doi:10.1088/0960-1317/20/8/085014

An electrothermal microactuator withZ-shaped beamsChanghong Guan and Yong Zhu1

Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh,NC 26795, USA

E-mail: yong zhu@ncsu.edu

Received 16 March 2010, in final form 3 June 2010Published 8 July 2010Online at stacks.iop.org/JMM/20/085014

AbstractThis paper introduces a Z-shaped thermal microactuator for in-plane motion, which could becomplementary to the well-established comb drives and V-shaped thermal actuators. TheZ-shaped actuators share many features in common with the V-shaped ones, but offer certainadvantages such as smaller feature size and larger displacement. They also offer a large rangeof stiffness and output force that is between those of the V-shaped actuators and comb drives.In particular, they can achieve smaller stiffness without buckling, which renders them assimultaneous load sensors. The Z-shaped actuator was modeled analytically and verified bymultiphysics finite element analysis. Among all the design parameters, the beam width and thelength of the central beam were identified as the major ones in tuning the device displacement,stiffness, stability and output force. Experimental measurements of three arrays of Z-shapedthermal actuators agreed well with the finite element analysis. In addition, the quasi-static anddynamic performances of individual Z-shaped thermal actuators were measured. The averagetemperature in the device structure was estimated from the electric resistivity at each actuationvoltage. The bandwidth of the Z-shaped thermal actuators can be increased for devices with asubstrate underneath.

(Some figures in this article are in colour only in the electronic version)

Nomenclature

fij compliance coefficientsFx internal (reaction) axial forceP virtual unit forceM internal (reaction) momentα thermal expansion coefficient�T average temperature changeU deflection at the tipL length of the long arm beaml length of the central beamLtotal total length of a Z-shaped beam (=2L + l)w beam widthh beam thicknessE Young’s modulus of siliconA beam cross-sectional area (=wh)I beam moment of inertia (=w3h/12)k stiffness of a Z-shaped beam

1 Author to whom any correspondence should be addressed.

kp thermal conductivity of siliconV applied voltage across a Z-shaped beamρ resistivity of silicon

1. Introduction

In microelectromechnical systems (MEMS), electrostaticactuators [1] and electrothermal actuators [2–4] are the twomajor categories for achieving in-plane motion. Electrostaticactuators especially comb drives have an output force typicallyon the order of 1 μN at an actuation voltage more than 30 V [5],while thermal actuators can easily generate a force of 1 mN atan actuation voltage around 5–10 V [6]. Though comb driveshave many advantages such as low temperature sensitivityand essentially zero dc power consumption, they require highactuation voltage (>30 V), which is generally not compatiblewith microelectronic power supplies and even silicon substrate(e.g., causing breakdown of the dielectric layer) [7].

0960-1317/10/085014+09$30.00 1 © 2010 IOP Publishing Ltd Printed in the UK & the USA

J. Micromech. Microeng. 20 (2010) 085014 C Guan and Y Zhu

Thermal actuators, on the other hand, have attractedsignificant attention in recent years, as they have beendemonstrated to be compact, stable and high-force actuationapparatus [4]. The operating principle of thermal actuators isbased on thermal expansion. Thermal actuators in a varietyof configurations have been exploited for in-plane motion.One is the pseudobimorph (also called U-shaped) thermalactuator [8–11], which employs asymmetrical thermal beamswith different cross-sectional areas. The locus of motion isan arc. The other is the bent-beam (also called V-shaped)thermal actuator [6, 7, 12–14] utilizing thermal expansion ofsymmetric, slanted beams to generate rectilinear displacementof the central shuttle. The V-shaped thermal actuators havebeen employed in many applications including linear androtary microengines [6], nanoscale material testing systems[12, 13] and nanopositioners [15].

The V-shaped thermal actuators cannot achieve largemotion (typically up to a few μm) but can easily generatelarge force (on the order of mN). Two possible limitationsexist for the V-shaped actuators. The first limitation comesfrom the slanted beams. The slanted beams pose challengesfor fabricating small features with smooth sidewall surfaces,which deteriorate as the beam width gets close to the resolutionof photolithography (typically ∼2 μm). As will be shownlater, however, smaller beam width is conducive to largerdisplacement. The other limitation is due to the largestiffness of the V-shaped actuator (on the order of thousandsof N m−1 and above). As a result, the V-shaped actuatorscannot be used as load sensors and actuators simultaneously;thus, additional load sensors are required for applicationssuch as nanomechanical testing [13] and nanomanufacturing[16].

This paper introduces Z-shaped thermal actuators forin-plane motion, which can overcome the above-mentionedshortcomings in the V-shaped thermal actuators and offer alarge range of stiffness and output force that is complementaryto the comb drives and V-shaped actuators. The structure andoperating principle of the Z-shaped actuator is first described,similar to a previously reported out-of-plane actuator [17].Analytical models were derived to explicitly delineate thedependence of device characteristics (including displacement,stiffness and buckling load) on the device geometry. Finiteelement multiphysics simulations were performed to obtainthe field variables including the temperature distribution inthe device. The quasi-static and dynamic performancesof the device were measured in vacuum. A comparisonbetween simulations, analytical formulas and experimentswere pursued for interrogation and calibration of thedevice. The devices were fabricated using the SOI-MUMPs(silicon-on-insulator multi-user MEMS process) (MEMSCAP,Durham, NC) with 10 μm thick silicon as the structural layer.

2. Device concept and modeling

The schematic of the Z-shaped thermal actuator is shown infigure 1(a); for the purpose of comparison, the schematic of theV-shaped thermal actuator is shown in figure 1(b). It is seenthat the basic unit of a Z-shaped actuator is a pair of Z-shaped

(a)

(b)

(c)

Figure 1. Schematics of (a) a Z-shaped thermal actuator and (b) aV-shaped thermal actuator before and after motion. Drawn not toscale. (c) SEM image of the Z-shaped thermal actuator. The blackarea is an etched hole underneath. I is the current passing throughthermal beams, while dc is the power source.

beams and a shuttle in the middle. In principle, the Z-shapedactuators are similar to the V-shaped actuators; but the Z-shaped actuators rely on bending of the symmetric Z-shapedbeams induced by thermal expansion to achieve rectilineardisplacement of the central shuttle. More specifically, whena current is passed through the device, heat is generatedalong the beams due to Joule heating. The temperature riseleads to thermal expansion of all the beams, especially thelong beams (denoted with length L). The long beams cannotexpand straight due to symmetry (and thus constraint) ofthe structure; rather, they bend to accommodate the lengthincrease. As a result the shuttle is pushed forward. Figure 1(c)shows a scanning electron microscopy (SEM) image ofa Z-shaped thermal actuator with two pairs of Z-shapedbeams.

Analysis of a thermal actuator requires a coupledelectrothermal and thermomechanical investigation. Thethermomechanical response was analytically derived based onthe following assumptions: the central shuttle is rigid and itsthermal expansion is neglected; thermal expansion of the shortbeam (with length l in figure 1(a)) is neglected; small strainsand displacements are considered; average temperature rise ina Z-shaped beam is given [12].

The mechanical response of the structure in figure 1 canbe equivalently modeled by considering half of the structurewithout the shuttle, as shown in figure 2. In this section, deviceperformances such as displacement, stiffness and internal forcewill be derived as functions of the device dimensions. Detailson the derivation are given in the appendix. Following theenergy method, three reaction forces/moments, axial force

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J. Micromech. Microeng. 20 (2010) 085014 C Guan and Y Zhu

Figure 2. Free-body diagram of a single Z-shaped beam.

Fx , virtual force P and moment M , can be obtained by solvingthe following set of equations [8]:⎡

⎣f11 f12 f13

f21 f22 f23

f31 f32 f33

⎤⎦

⎡⎣F

P

M

⎤⎦ =

⎡⎣2α�T L

U

0

⎤⎦ (1)

where

f11 = 2L

EA+

l3

3EI+

Ll2

EI, f12 = 3L2l

2EI+

Ll2

2EI,

f13 = − l2

2EI− Ll

EI,

f21 = Ll2

2EI+

3L2l

2EI, f22 = l

EA+

L2l

EI+

8L3

3EI,

f23 = −2L2

EI− Ll

EI,

f31 = − l2

2EI− Ll

EI, f32 = −2L2

EI− Ll

EI,

f33 = 2L

EI+

l

EI.

In order to obtain deflection in the y direction, let thevirtual force P equal to zero. The deflection in the tip can bewritten as

U = 12α�T L3

l2 + 6L(l + w2

3l

) . (2)

The stiffness of a Z-shaped beam is given by

k = Ew3h(l3 + 2Lw2 + 6Ll2)

(8L3l3 + w2l4 + 16w2L4 + 2w4Ll + 12L4l2 + 6w2Ll3).

(3)

The stiffness of a Z-shaped actuator with a pair of beamsis 2k. The internal force is given by

Fx = 24α�T EIL

l3 + Ll2 + 2Lw2. (4)

The output force f is given by the product of displacementand stiffness,

f = kU. (5)

Thus the output force of a Z-shaped actuator with apair of beams is 2f . In terms of heat transfer, convectionand radiation are generally neglected in MEMS structures;conduction through the air layer between the device layer andthe substrate is a major heat transfer mechanism for surfacemicromachined devices, as the air layer is typically very thin(on the order of a few μm) [8, 11–13]. But in our SOIdevices, the underneath silicon substrate is totally etched as

shown in figure 1(c); as a result, heat conduction throughthe underneath air layer does not occur either. The only heattransfer mechanism is therefore heat conduction to the anchorsacross the beams. The average temperature increase in the Z-shaped beam (�T ) can be estimated by a parabolic temperaturedistribution [10], namely

�T = 1

Ltotal

∫ Ltotal

0T (x) dx = V 2

12kpρ. (6)

It is apparent that the device performance is geometrydependent. The peak displacement of one single Z-shapedbeam at a given temperature can be increased by simplyincreasing the length of the long arm (L). The device thickness(h) is not related to the displacement, but affects the stiffnessand output force in direct proportion. Higher force can alsobe generated by setting multiple pairs of Z-shaped beams inparallel, which increases the structural stiffness but not thedisplacement. Since the beam length is limited by the SOI-MUMPs design rules [18], the length of all the long beams(L) in our design was selected as 88 μm and the width of thecentral shuttle was 60 μm. Again the thickness of the entiredevice is 10 μm as specified in the SOI-MUMPs. To simplifythe parametric study, widths of all the long beams and centralbeams are taken as the same. The central beam length (l) andbeam width (w) are design parameters whose effects will befurther investigated.

Thermomechanical finite element analysis (FEA) wasperformed in comparison to the analytical modeling presentedabove in order to evaluate the assumptions adopted abovein analytical modeling. A 2D multi-field plane elementPLANE223 was used in the FEA (ANSYS version 11.0). Thesimulated beam had exactly the same dimensions as that usedin analytical modeling. A constant temperature increase of400 K was applied to the entire Z-shaped beam as shownin figure 2. Note that all the thermal properties of singlecrystalline silicon (SCS) used in the simulation are constantsat room temperature as used in analytical modeling. However,all the thermal properties of SCS are temperature dependentas listed in table 1. In the multiphysics simulation of the realdevice to be discussed in section 3, the temperature-dependentproperties were used so as to compare with experiments.Figure 3(a) shows an excellent agreement for displacementsbetween the FEA and analytical solution, given byequation (2), for the Z-shaped beam. Here the displacementis plotted as a dimensionless quantity (U/2α�TL). Themaximum displacements are 1.76 μm, 3.34 μm and 6.68 μmfor the beam widths of 8 μm, 4 μm and 2 μm, respectively.This agreement confirms the validity of the analyticalmodel.

A systematic comparison between the Z-shaped thermalactuators and the V-shaped thermal actuators was carried outto further illustrate their characteristics, as shown in figure 3.For the V-shaped actuators, the inclined angle varied from 0◦

to 10◦, while the beam width and total length were the sameas those in the Z-shaped actuators. Figures 3(a) and (b) showthat both actuators can achieve a similar displacement range(on the order of μm). For both actuators, the smaller thebeam width the larger the displacement. However, fabricating

3

J. Micromech. Microeng. 20 (2010) 085014 C Guan and Y Zhu

0 2 4 6 8 100

10

20

30

40

50

Central beam length (μm)

Nor

mal

ized

tip

def

lect

ion

U/2

αΔT

L

Width 8μm -- AnalyticalWidth 4μm -- AnalyticalWidth 2μm -- AnalyticalWidth 8μm -- FEAWidth 4μm -- FEAWidth 2μm -- FEA

0 2 4 6 8 100

10

20

30

40

50

Inclined angle θ

Nor

mal

ized

tip

def

lect

ion

U/2

αΔT

L Width 8μmWidth 4μmWidth 2μm

(a)

0 2 4 6 8 100

0.001

0.002

0.003

0.004

0.005

Central beam length (μm)

Nor

mal

ized

sti

ffne

ss k

/(E

A/2

L) Width 8μm

Width 4μmWidth 2μm

0 2 4 6 8 100

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Inclined angle θ

Nor

mal

ized

sti

ffne

ss k

/(E

A/2

L)

Width 8μmWidth 4μmWidth 2μm

0 2 4 6 8 100

2

4

6

8

10

12

14

Central beam length (μm)

Axi

al in

tern

al f

orce

(m

N)

Width 8μmWidth 4μmWidth 2μm

Critical Buckling

Critical Buckling

0 2 4 6 8 100

2

4

6

8

10

12

14

Inclined angle θ

Axi

al in

tern

al f

orce

(m

N) Width 8μm

Width 4μmWidth 2μm

Critical Buckling

Critical Buckling

(b)

(c) (d )

(e) ( f )

Figure 3. Comparisons between the (a, c, e) Z-shaped and (b, d, f ) V-shaped thermal beams. Set the length of the V-shaped beam equal to2L as in the Z-shaped beam. For given beam widths, the variable for the Z-shaped beam is the central beam length, while the variable for theV-shaped beam is the inclined angle. (a) and (b) Dimensionless displacement, in which L = 88 μm and 2α�TL = 0.176 μm; (c) and(d) dimensionless stiffness, in which EA/2L = 7.27 × 104 N m−1 (for the beam width of 8 μm); (e) and (f ) internal axial force with acritical buckling criterion.

Table 1. Silicon properties used in simulations of Z-shaped thermal actuators.

Material properties Unit Value Reference

Young’s modulus GPa 160 [20]Poisson’s ratio – 0.28 [20]Thermal conductivity (constant) W m−1 K−1 146 [4]Thermal conductivity (temperature dependent) Wm−1 K−1 kt (T ) = 210 658T −1.2747 [20]Resistivity (constant) � m 5.1 × 10−5 MeasuredResistivity (temperature dependent) � m ρ (T ) = 5.1 × 10−5

[1 + 3 × 10−3 (T − 273)

][20]

Thermal expansion coefficient (constant) K−1 2.5 × 10−6 [4]Thermal expansion coefficient (temperature dependent) K−1 α(T ) = −4 × 10−12 T 2 + 8 × 10−9 T + 4 × 10−7 [4]

4

J. Micromech. Microeng. 20 (2010) 085014 C Guan and Y Zhu

a small beam width (especially �2 μm) is more challengingfor inclined beams. In this regard, the Z-shaped actuators canachieve relatively larger displacement.

Figures 3(c) and (d) show that the stiffness of the Z-shapedactuators is about one order of magnitude smaller than that ofthe V-shaped actuators, when the beam width ranges from 2 to8 μm. It is noted that for V-shaped actuators the normalizedstiffness almost does not change with the beam width; incontrast, for Z-shaped actuators the normalized stiffness scalesapproximately with the square of the beam width. This isessentially due to the fact that V-shaped actuators are mainlybased on beam extension while Z-shaped actuators are mainlybased on beam bending. It is known that for beam bendingthe stiffness is proportional to the cube of the beam width.Hence, Z-shaped actuators possess a large stiffness range for agiven range of beam width. Furthermore, the typical stiffnessrange offered by Z-shaped actuators (about 50–103 N m−1) isan excellent complement to those offered by the comb drives[19] and V-shaped actuators [12].

Upon operation, the beams in both actuators are undercompression that might lead to buckling instability. Thecritical buckling force of a beam is given by

Pcr = π2EI

(KLb)2. (7)

where K is the column effective length factor and Lb = 2 Lis the length from the anchor to the central shuttle. For bothV-shaped and Z-shaped actuators, the boundary conditions areas follows: the anchor is fixed and the other end is sliding,which gives K = 1. Buckling simulations were performedin ABAQUS to verify the effective length factor K. Thus, thecritical buckling forces of Z-shaped and V-shaped thermalactuators are the same, as shown in figures 3(e) and (f ). ForLb = 176 μm, no buckling is expected in both Z-shape andV-shape thermal actuators when the beam width is 8 μm.However, the critical central beam lengths are 2.6 μm and3.2 μm for the Z-shape actuators with beam widths of 4 μm and2 μm, respectively, while the critical inclined angles are at 1.5◦

and 2◦ in the V-shape actuators. Note that the displacement,stiffness and axial internal force of the V-shaped actuatorswere calculated using equations (1) and (2) in [12]. For theV-shaped actuators, smaller stiffness can be achieved with asmaller inclined angle, which, however, might lead to buckling(see figures 3(d) and (f )). For the Z-shaped actuators, smallerstiffness can be achieved with the combination of a smallerbeam width and smaller central beam length, which can betailored to prevent buckling (see figures 3(c) and (e)).

The output force of a Z-shaped thermal beam accordingto equation (5) is shown in figure 4. The peak output force isin the range of 30 to 490 μN depending on the beam width.Apparently, the output force of the Z-shaped actuators is inbetween those offered by the comb drives [1, 19] and theV-shaped thermal actuators [6, 12].

3. Multiphysics FEA

The performance of Z-shaped thermal actuators was simulatedusing ANSYS multiphysics, version 11.0. The simulation

0 2 4 6 8 100

100

200

300

400

500

Central beam length (µm)

Loa

ding

for

ce (

µN)

Beam width 8µmBeam width 4µmBeam width 2µm

Figure 4. The output force of a Z-shaped thermal beam.

(a)

(b)

Figure 5. (a) Temperature increase field (K) and (b) displacementfield (nm) in a Z-shaped thermal actuator (L = 88 μm, w = 4 μmand l = 20 μm). The displacement component in the plot is in theshuttle axial direction.

is a coupled-field analysis involving electric, thermal andmechanical fields. The input was the actuation voltageacross the anchors and the output included the actuatortemperature and displacement distributions. The thermalboundary conditions were the zero temperature change atthe anchors. The mechanical boundary conditions werethe fixed displacements at the anchors. The temperaturematerial parameters for SCS listed in table 1 were usedin the simulations. The displacement of the actuatorscan be obtained experimentally; however, it is difficult tomeasure the temperature distribution. Therefore, the coupled-field simulation is particularly relevant to provide suchinformation.

Figure 5 shows the temperature distribution and thedisplacement in the thermal actuator for an actuation voltage of5 V. As previously stated, the only heat dissipation path in theZ-shaped actuators (without substrate underneath) is throughthe anchors. Since the shuttle is farthest from the anchors,the highest temperature occurs in the shuttle, as shown in

5

J. Micromech. Microeng. 20 (2010) 085014 C Guan and Y Zhu

Figure 6. An array of Z-shaped thermal actuators with the samebeam width of 4 μm.

figure 5(a). The displacement distribution is shown infigure 5(b). It is noted that the displacement in the shuttleis also not uniform because of the thermal expansion along theshuttle length. A series of simulations were performed withthe constant long beam length (88 μm), three beam widths (2,4 and 8 μm) and a large number of central beam lengths (from0 to 20 μm).

4. Experimental results

The Z-shaped thermal actuators were fabricated by the SOI-MUMPs process in run 27. The design rules of SOI-MUMPsrestrict the total length of a suspended structure within 500 μmto minimize out-of-plane deformation. In our design, all theZ-shaped thermal devices have the same anchor–anchordistance (412 μm); the design parameters are the central beamlength and the beam width. Of course, the length of the twolong beams and the width of all three beams are not necessarilyto be the same, as long as they are symmetric about the shuttle.This in fact offers a larger design space to optimize the deviceperformance for different applications.

Three arrays of Z-shaped thermal actuators with differentbeam widths (2 μm, 4 μm and 8 μm) were fabricated forthe parametric study and device optimization. Within eacharray, the length of the central beam varies from 1 to 20 μm.One such array of the Z-shaped thermal actuators is shownin figure 6. The displacement was measured using an opticalmicroscope with the edge detection method [20]; the resolutionwas calibrated as 81.5 nm per pixel. Figure 7 shows thedisplacement of the Z-shaped thermal actuators of two arrays(the third array shows the similar trend and is not plotted). Thearray with a 2 μm beam width was actuated at 2 V and the arraywith a 8 μm beam width was actuated at 3 V. Multiphysics FEAresults for both arrays with the corresponding applied voltages

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

Central beam length (μm)

Dis

plac

emen

t (n

m)

Width 2μm -- FEAWidth 2μm -- ExperimentWidth 8μm -- FEAWidth 8μm -- Experiment

Figure 7. Measured and FEA simulated displacements of Z-shapedthermal actuator arrays with two different beam widths (2 μm and8 μm). The actuation voltage on the array with 2 μm beam width is2 V and that on the array with 8 μm beam width is 3V.

are also plotted in the figure. The displacement results agreedquite well between experiments and FEA, which are also inline with the analytical modeling as shown in figure 3(a). Itshould be noted that no thermal crosstalk between actuatorswas observed [14].

The following tests were carried out inside an SEM (JEOL6400F) on a particular Z-shaped thermal actuator with a 4 μmbeam width and 20 μm central beam length. Displacementwas measured with the actuation voltage from 0 to 6 V. Themeasured displacement is plotted with respect to the inputcurrent, as shown in figure 8(a). The maximum output force,necessary to reduce the actuator displacement to zero, is alsoplotted, as calculated by equation (6). The stiffness of thestructure was calculated to be 273.4 N m−1 based on themeasured dimensions. Figure 8(b) shows the dependenceof electric resistance of the structure on the input current.Assuming the linear dependence of resistivity on temperatureas listed in table 1, the average temperature in the device wasestimated [8, 9], and also plotted in figure 8(b).

Thermal actuators generally respond more slowly thanelectrostatic ones because the time constants involvedin the thermal dissipation are larger than those in theelectromechanical actuators. The bandwidth of V-shapedactuators was found to be ∼700 Hz, which is relatively high asa result of the very small thermal mass in the MEMS structure[9, 21]. Dynamic tests of the Z-shaped thermal actuator wereperformed in the vacuum to interrogate their bandwidth. Thedevice was actuated at a square-wave ac voltage. The rasterimaging mode of SEM was used to capture the oscillation ofa straight (vertical) edge (figure 9(a)), following the methoddeveloped in [20]. The measured −3 dB bandwidth was about70 Hz, as shown in figure 9(b).

As discussed earlier, our devices fabricated by SOI-MUMPs in this study had no substrate underneath. Theonly heat dissipation mechanism in our Z-shaped thermalactuators is heat conduction to the two anchors across theentire beams. This led to the relatively low bandwidth(∼70 Hz) of these devices. For devices with a substrate, amajor heat dissipation mechanism, conduction to the substratethrough the air underneath the devices [14], will be activated.

6

J. Micromech. Microeng. 20 (2010) 085014 C Guan and Y Zhu

0 2 4 6 8 10 120

200

400

600

800

1000

1200M

easu

red

Dis

plac

emen

t (n

m)

Measured current (mA)0 2 4 6 8 10 12

0

50

100

150

200

250

300

Est

imat

ed f

orce

( μN

)

(a) (b)

0 2 4 6 8 10 12100

150

200

250

300

Mea

sure

dre

sis t

anc e

(Ohm

)

Measured current (mA)0 2 4 6 8 10 12

300

400

500

600

700

Est

imat

ed t

empe

rat u

re(K

)

Figure 8. (a) Measured displacement and corresponding (calculated) force as functions of the input current. (b) Measured resistance andcorresponding (estimated) average temperature change as functions of the input current.

10-1 100 101 102 1030

200

400

600

800

1000

1200

Frequency (Hz)

Dis

plac

emen

t (n

m)

4V actuation6V actuation

(b) (a)

Figure 9. (a) SEM image of the device at ac voltage with 10 Hz frequency and 6 V amplitude. During image scanning, the device actuationunderwent on/off cycles for six times. (b) Displacement as a function of the applied frequency at constant applied voltages.

This additional dissipation mechanism significantly increasesthe rate of heat dissipation as characterized by a large decreasein the time constant of thermal dissipation [9]. As a resultthe bandwidth of the Z-shaped thermal actuators will beincreased.

5. Conclusions

This paper introduces a simple class of electrothermalactuators with symmetric Z-shaped beams. The Z-shapedactuators share many features in common with the V-shapedones. But they offer some advantages such as smallerfeature size and larger displacement. They possess a largerange of stiffness and output force that is in between thoseof the comb drives and V-shaped thermal actuators, thusfill the gap between these two well-established actuators.In particular, they can achieve smaller stiffness withoutbuckling; Z-shaped actuator with smaller stiffness could beused as simultaneous load sensors. The Z-shaped thermalactuator was modeled analytically and verified by coupled-field FEA. Among all of the design parameters, the lengthof the central beam and beam width were identified asthe major ones in tuning the device displacement, stiffness,stability (buckling force) and output force. Experimentalmeasurements of three arrays of Z-shaped thermal actuators

agreed well with the FEA predictions. In addition, thequasi-static and dynamic displacements of individual Z-shapedthermal actuators were measured. The average temperature inthe device structure was estimated from the electric resistivityat each actuation voltage. The bandwidth of the Z-shapedthermal actuators can be increased for devices with a substrateunderneath.

Acknowledgments

This work was supported by the National Science Foundationunder award no CMMI-0826341 and the start-up fund fromNorth Carolina State University. The in situ SEM experimentswere performed in the Analytical Instrumentation Facility atNorth Carolina State University. The authors would liketo thank one of the reviewers for pointing out an importantreference.

Appendix.

A Z-shaped thermal actuator consists of two Z-shaped beamsand a central shuttle; each Z-shaped beam includes two longbeams with length L and a central short beam with length l,with a uniform beam width w and a constant beam thickness

7

J. Micromech. Microeng. 20 (2010) 085014 C Guan and Y Zhu

(h = 10 μm). Below we analyze only one Z-shaped beamas shown in figure 2. The x-axis is defined on each beamseparately, 0 < x1, x3 < L and 0 < x2 < l. Because theshuttle is constrained in the x direction but free to move inthe y direction due to thermal expansion, an axial reactionforce Fx and a reaction moment M are present. To calculatedisplacement in the y direction, a virtual force P in they direction is provided. The moments in three segments,respectively, are given as⎧⎪⎨

⎪⎩M1(x1) = Px1 − M 0 < x1 < L

M2(x2) = PL − M + x2Fx 0 < x2 < l

M3(x3) = P(L + x3) − M + lFx 0 < x3 < L.

(A.1)

The thermal actuator is statically indetermined; the forcesand moment can be found by the Castigliano energy (orvirtual work) method [8, 22]. The strain energy within thesystem consists of contributions from both axial force andbending moment. Thus, the displacement in the x directionis

2α�T L = 2∫ L

0

Fx

EAdx +

∫ L

0

M1(x)

EI

∂M1

∂Fx

dx

+∫ l

0

M2 (x)

EI

∂M2

∂Fx

dx +∫ L

0

M3(x)

EI

∂M3

∂Fx

dx

=(

2L

EA+

l3

3EI+

Ll2

EI

)Fx +

(3L2l

2EI+

Ll2

2EI

)P

+

(− l2

2EI− Ll

EI

)M. (A.2)

The displacement in the y direction is

δy =∫ L

0

M1(x)

EI

∂M1

∂Pdx +

∫ l

0

M2(x)

EI

∂M2

∂Pdx

+∫ L

0

M3(x)

EI

∂M3

∂Pdx

=(

Ll2

2EI+

3L2l

2EI

)Fx +

(l

EA+

L2l

EI+

8L3

3EI

)P

+

(−2L2

EI− Ll

EI

)M. (A.3)

And the rotation is

0 =∫ L

0

M1 (x)

EI

∂M1

∂Mdx +

∫ l

0

M2(x)

EI

∂M2

∂Mdx

+∫ L

0

M3(x)

EI

∂M3

∂Mdx

=(

− l2

2EI− Ll

EI

)Fx +

(−2L2

EI− Ll

EI

)P

+

(2L

EI+

l

EI

)M, (A.4)

where E is the Young modulus for single-crystalline silicon,α is the thermal expansion coefficient, �T is the averagetemperature change in beams, and A is the cross-sectionalarea and I is the beam moment of inertia.

Equations (A.2–A.4) can be rewritten in the matrix form.Let the virtual force P = 0, one obtains⎡⎣2α�T L

δy

0

⎤⎦

=

⎡⎢⎢⎣(

2LEA

+ l3

3EI+ Ll2

EI

) (3L2l2EI

+ Ll2

2EI

) (− l2

2EI− Ll

EI

)(

Ll2

2EI+ 3L2l

2EI

) (l

EA+ L2l

EI+ 8L3

3EI

) (− 2L2

EI− Ll

EI

)(− l2

2EI− Ll

EI

) (− 2L2

EI− Ll

EI

) (2LEI

+ lEI

)⎤⎥⎥⎦

×⎡⎣ Fx

P |P=0

M

⎤⎦ . (A.5)

Solving equation (A.5) yields the vertical deflection δy:

δy =[(

Ll2

2EI+ 3L2l

2EI

) (− 2L2

EI− Ll

EI

)]

×[(

2LEA

+ l3

EI+ Ll2

EI

) (− l2

2EI− Ll

EI

)(− l2

2EI− Ll

EI

) (2LEI

+ lEI

)]−1 [

2α�T L

0

]

= 6AL2 × 2α�T L

Al2 + 6L(Al + 4I

l

) = 12α�T L3

l2 + 6L(l + w2

3l

) . (A.6)

When �T , L and w are constant, the Z-shaped beam reachesthe maximum deflection if l = w/

√3, which is consistent

with what is plotted in figure 3(a). The maximum deflectionis

δymax = 36α�T L3

w(w + 4√

3L). (A.7)

When thermal expansion is not applied, equation (A.5)becomes⎡⎣Fx

P

M

⎤⎦

=

⎡⎢⎢⎣(

2LEA

+ l3

3EI+ Ll2

EI

) (3L2l2EI

+ Ll2

2EI

) (− l2

2EI− Ll

EI

)(

Ll2

2EI+ 3L2l

2EI

) (l

EA+ L2l

EI+ 8L3

3EI

) (− 2L2

EI− Ll

EI

)(− l2

2EI− Ll

EI

) (− 2L2

EI− Ll

EI

) (2LEI

+ lEI

)⎤⎥⎥⎦

−1

×⎡⎣ 0

δy

0

⎤⎦ . (A.8a)

Then the structural stiffness k is given as

k = P

δy

= 3EIA(Al3 +24LI +6ALl2)

2A2L3l3+3AIl4 +48AIL4 +72I 2Ll+3A2L4l2+18AILl3.

(A.8b)

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