266 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 6, NO. 3, SEPTEMBER 1997Modeling and Optimal Design of PiezoelectricCantilever MicroactuatorsDon L. DeVoe and Albert P. PisanoAbstract—A novel model is described for predicting the staticbehavior of a piezoelectric cantilever actuator with an arbitraryconfiguration of elastic and piezoelectric layers. The model iscompared to deflection measurements obtained from 500-m-longZnO cantilever actuators fabricated by surface micromachining.Modeled and experimental results demonstrate the utility of themodel for optimizing device design. A discussion of design con-siderations and optimization of device performance is presented.[233]Index Terms—Actuator, bimorph, cantilever, model, piezoelec-tric, ZnO.I. INTRODUCTIONTHE STATIC analysis of piezoelectric cantilever actuatorsis typically performed using an approach employed byTimoshenko for calculating the deflection of a thermal bimorph[1]. In Timoshenko’s analysis, the principal of strain compati-bility is employed between two cantilever beams joined alongthe bending axis. The deflection of the two-layer structuredue to forces generated by one or both of the layers isthen determined from static equilibrium. For the case of apiezoelectric heterogeneous bimorph, the structure of interestconsists of a piezoelectric layer bonded to a purely elasticlayer. The purpose of the elastic layer is, in essence, to offsetthe neutral axis of the two-layer system so that a lateral strainproduced by piezoelectric effect is translated into an appliedmoment on the bimorph. Such structures are commonly usedin macroscale applications such as active structural dampingand precision positioning systems. For these macrodevices, thetwo-layer Timoshenko model (bimorph model) is sufficientfor determining the quasi-static behavior of the system sinceany additional (e.g., bonding) layers are relatively thin andcan be ignored. However, for micromachined devices, thebimorph assumption is quite often invalid since such devicesmay consist of additional layers with thicknesses on the sameorder as the piezoelectric film itself.The motivation of this work is to develop a model formultiple-layer cantilever devices, or multimorphs, which isapplicable to thin-film devices. This model can then be usedto investigate optimal film thicknesses for these devices.To avoid confusion, the term multimorph is used here toManuscript received September 17, 1996; revised May 20, 1997. SubjectEditor, R. O. Warrington.D. L. DeVoe is with the Department of Mechanical Engineering, Universityof Maryland at College Park, College Park, MD 20742 USA.A. P. Pisano is with the Berkeley Sensor and Actuator Center and De-partment of Mechanical Engineering, University of California at Berkeley,Berkeley, CA 94720 USA.Publisher Item Identifier S 1057-7157(97)06329-4.describe the full class of piezoelectric cantilever devices withan arbitrary configuration of piezoelectric and elastic layers,including bimorphs. Similar devices have been analyzed byseveral authors. Chu [2] presents a model based directly onTimoshenko’s approach, which predicts the static deflectionof a micromachined thermal bimorph, the results of which arealso applicable to piezoelectric actuators. Smits provides anelegant derivation of the static [3] and dynamic [4] behaviorof bimorph structures from basic thermodynamic principlesalthough the analysis does not extend directly to more complexmultimorph structures. In this work, a multimorph model isderived, and theoretical results from the model are comparedto experimental measurements from 500-m-long surface-micromachined multimorphs with varying ZnO thicknesses.Based on experimental results, the utility of the model asa design tool for specifying the optimal piezoelectric filmthickness is demonstrated.II. PIEZOELECTRIC MULTIMORPH MODELA model describing the deflection of a piezoelectric mul-timorph structure can be derived by appealing to the basicmechanics principles of: 1) static equilibrium and 2) straincompatibility between successive layers in the device. In thissense, the approach is similar to Timoshenko’s well-knownderivation for thermal bimorph deflections, but extended tobe applicable to an-layer multimorph rather than a simplebimorph.The basic geometry of an-layer multimorph is shownin Fig. 1. In this figure, the individual layers may be eitherpiezoelectric or purely elastic. In the formulation of thismodel, it is assumed that shear effects are negligible, residualstress-induced curvature may be ignored, beam thickness ismuch less than the piezoelectric-induced curvature, second-order effects such as electrostriction are negligible, and-plane strain and-plane stress are enforced. Under theassumptions of-plane stress and -plane strain, whichare reasonable for a wide flat beam, it can be shown that(1) and (2) hold. The effective modulus for theth layerand transverse-piezoelectric-coupling coefficient givenby these equations assume an isotropic Poisson’s ratioand are implicitly employed in the derivation of this model(1)(2)With the above assumptions outlined, the derivation pro-ceeds as follows. First, it is noted that both axial forces andmoments at any cross section of the-layer beam shown in1057–7157/97$10.00  1997 IEEEDEVOE AND PISANO: MODELING AND DESIGN OF PIEZOELECTRIC CANTILEVER MICROACTUATORS 267Fig. 1. Generic multimorph geometry and cross section.Fig. 1 must sum to zero at equilibrium, as expressed in (3)and (4)(3)(4)The individual moments are related to the curvature,,by (5), whereis defined as the flexural rigidity of the thlayer in the beam relative to the beam’s neutral axis. Note thatthe radius of curvatureis approximately equal for each layerin the structure since it was assumed that the beam thicknessis much less than the overall beam curvatureor (5)Equations (4) and (5) may be combined to express the beamequilibrium in terms of axial forces and curvature only asshown in(6)Equation (6) can be solved for the curvature as a functionof the unknown’s and rewritten in matrix form as(7)In (7),is the column vector of axial forces, while isthe premultiplication row vector operating on, which is afunction of beam geometry and flexural rigidity.An additional set ofequations constraining andtheaxial force vector follow from the requirement ofstrain compatibility at theinterfaces between each setof adjacent layers. The total strain at the surface of each layeris given by superposition of the strain