An efficient DIPIE algorithm for CAD of electrostatically actuated MEMS devices

612 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 11, NO. 5, OCTOBER 2002

An Efficient DIPIE Algorithm for CAD ofElectrostatically Actuated MEMS Devices

Ofir Bochobza-Degani, Student Member, IEEE, David Elata, Member, IEEE, and Yael Nemirovsky, Fellow, IEEE

Abstract—Pull-in parameters are important properties ofelectrostatic actuators. Efficient and accurate analysis tools thatcan capture these parameters for different design geometries, aretherefore essential. Current simulation tools approach the Pull-Instate by iteratively adjusting the voltage applied across the actuatorelectrodes. The convergence rate of this scheme gradually deterio-rates as the Pull-In state is approached. Moreover, the convergenceis inconsistent and requires many mesh and accuracy refinementsto assure reliable predictions. As a result, the design procedure ofelectrostatically actuated MEMS devices can be time-consuming.In this paper a novel displacement iteration Pull-In extraction(DIPIE ) scheme is presented. The DIPIE scheme is shown to con-verge consistently and far more rapidly than the voltage iterations(VI ) scheme ( 100 times faster!). The DIPIE scheme requires sep-arate mechanical and electrostatic field solvers. Therefore, it can beeasily implemented in existing MOEMS CAD packages. Moreover,using the DIPIE scheme, the Pull-In parameters extraction can beperformed in a fully automated mode, and no user input for searchbounds is required. [782]

Index Terms—Actuator, computer-aided design (CAD), dis-placement iteration Pull-In extraction (DIPIE), electrostatic,Pull-In.

I. INTRODUCTION

E LECTROSTATIC actuation is widely used in MEMSdevices to deform elastic elements. The electromechanical

response of these actuators may exhibit an inherent instability,known as thePull-In phenomenon [1]–[11]. By applying avoltage difference across the electrodes of the actuator, anelectrostatic force is generated that tends to reduce the gapbetween the electrodes. For a sufficiently low voltage, theelectrostatic force is balanced by an elastic restoring force. Inthis stable state, the gap between the electrodes is inverselyproportional to the applied voltage. Above a certain voltage,the electrostatic force is larger than the restoring elastic forcefor any deformation. As a result, the actuator is unstable andthe gap between the two electrodes rapidly vanishes. Thevoltage and deformation at the onset of instability are termedPull-In voltageandPull-In deformation, or in short thePull-Inparametersof the actuator.

Manuscript received November 21, 2001; revised February 28, 2002. SubjectEditor G. K. Fedder.

O. Bochobza-Degani and Y. Nemirovsky are with the Kidron Microelec-tronics Research Center, Electrical Engineering Department, Technion—IsraelInstitute of Technology, Haifa, Israel, 32000 (e-mail: [email protected];[email protected]).

D. Elata is with the Mechanical Engineering Department, Technion—IsraelInstitute of Technology, Haifa 32000, Israel.

Digital Object Identifier 10.1109/JMEMS.2002.803280.

Characterization of the Pull-In parameters is important whendesigning electrostatically actuated micromachined devices. Inswitching applications, either optical [12] or electrical [13], thePull-In voltage is minimized to obtain optimal performance. Inanalog scanning micromirror applications [14], the travel rangeof the actuator is important and therefore the Pull-In deforma-tion should be maximized. Modeling tools that can simulate thenonlinear electromechanical response and extract thePull-Inparametersof electrostatic actuators are therefore of great im-portance. To enable an accurate determination of optimal ma-terial and geometrical parameters of actuators, these modelingtools should be based on accurate and efficient calculations.

Several approaches for extracting the Pull-In parameters havebeen reported in literature [1]–[11] and have been implementedin commercially available MEMS CAD tools [15], [16]. Ap-proximate analytical models have been suggested for electro-static actuators [1]–[5]. These models yield fast results but arelimited to actuators with very few degrees of freedom. To ac-curately calculate the Pull-In parameters of general deformableelements with infinite degrees of freedom, such as beam andplate actuators, a more general approach has been suggested[4]–[10], [15], [16]. In this approach, the electromechanical re-sponse of the actuator is numerically simulated by fixing theapplied voltage. The Pull-In parameters are calculated by itera-tively approaching the Pull-In voltage with decreasing voltageincrements [4]–[10], [15], [16], and henceforth this approach isreferred to as thevoltage-iteration(VI) scheme. This algorithmwas implemented in a finite-difference scheme [4], [5] and incoupled finite-elements (FEM) and boundary-elements (BEM)scheme [6]–[10], [15], [16].

In this paper, a novel algorithm for extracting the Pull-In pa-rameters of general electrostatic actuators is suggested. The al-gorithm is based on iterating the displacement of a pre-chosendegree-of-freedom node of the actuator, rather than the appliedvoltage. In essence, the new displacement iteration Pull-In ex-traction (DIPIE) algorithm replaces the original problem thathas stable and unstable equilibrium states, with a series of equiv-alent problems for which the equilibrium solution isalwaysstable.

The different approaches of theVI algorithm and the newDIPIE algorithm are discussed in Section II. Section III presentsthe formulation of theDIPIE algorithm. Implementation of thenew algorithm within a finite-difference code for a clamped–clamped beam actuator is described in Section IV. A compar-ison, presented in Section V, between the performances of thetwo algorithms shows that theDIPIE scheme converges muchfaster than theVI scheme ( 100 time faster) and it is far moreconsistent and well behaved.

1057-7157/02$17.00 © 2002 IEEE

BOCHOBZA-DEGANI et al.: EFFICIENT DIPIE ALGORITHM FOR CAD OF ELECTROSTATICALLY ACTUATED MEMS DEVICES 613

Fig. 1. Equilibrium states of the electrostatic actuator. The solid line repre-sents the stable equilibrium states and the dashed line represents the unstableequilibrium states. The horizontal dotted lines represent the VI algorithm andthe vertical dotted lines the DIPIE algorithm.

II. DIPIE VERSUSVI SCHEMES

A typical static equilibrium curve of an electrostatic actuatoris schematically described in Fig. 1. The convex function de-scribes the applied voltage as function of a representative pa-rameter of the actuator deformation. Such a parameter may bethe displacement of the center of a clamped–clamped beam. Fordeformations smaller than the Pull-In deformation, the staticequilibrium state is stable (solid line). In contrast, for deforma-tions larger than the Pull-In deformation the static equilibriumstate is unstable (dashed line).

Two aspects of the physical response of electrostatic actuatorsare apparent in this figure: I) The voltage is a unique function ofthe deformation whereas deformation is not a unique function ofvoltage; II) The maximal deformation can be trivially estimatedas it is bounded by the gap between the electrodes. In contrast,the maximal voltage cannot bea priori estimated.

A. The VI Algorithm

In the VI algorithm, the Pull-In voltage is iteratively ap-proached. At each iteration, the static equilibrium deformationis calculated for the applied voltage. This calculation can be car-ried out by a relaxation method [6], [7], [9], Newton–Raphsonmethod, or a host of other numerical schemes [6], [9]. If thedeformation calculation converges, it is concluded that theapplied voltage is bellow the Pull-In value. On the other hand,if the calculated deformation fails to converge it is concludedthat the applied voltage is higher than the Pull-In value. Severalmethods have been employed to establish whether the defor-mation calculation converges [4]–[6], [9], [10]. The intervalbetween these two limits is continuously decreased until thevoltage interval is smaller than a predetermined accuracy. Theiterations are represented by the set of horizontal lines in Fig. 1.It can easily be seen that not all the horizontal lines cross theequilibrium curve, and therefore not all lines are associatedwith equilibrium states.

The main advantage of the VI algorithm is its simplicity andease of integration into commercial CAD tools. For any appliedvoltage, the electro-elastic problem is solved by iteratively

solving uncoupled electrostatic and elastic problems. It istherefore easy to implement this algorithm by sequentiallyemploying existing numerical codes for each of these problems[6], [10], [15], [16].

B. The DIPIE Algorithm

First, it is explained how an unstable equilibrium state of avoltage controlled problem, may be replaced by a stable equi-librium state of a displacement controlled problem that is equiv-alent in the sense that both are the same equilibrium solution. Tothis end, consider an electrostatic actuator where a pre-chosenpoint is displaced and fixed at its new location. This will giverise to a mechanical reaction force at the prechosen point. Next,a voltage is applied to the actuator inducing electrostatic forcesand modifying the reaction force. The voltage is then continu-ously modified until the reaction force at the pre-chosen pointvanishes. Because the reaction force is zero, the voltage at the re-sulting state is the same voltage that would induce the same dis-placement in the original voltage controlled problem. However,while in the original voltage controlled problem this state maybe unstable, in the displacement controlled problem the state isstable because the pre-chosen point is fixed.

The DIPIE algorithm is based on this inverse approach inwhich all calculations converge. At each iteration, the displace-ment of a pre-chosen degree-of-freedom (DOF) node of theactuator is postulated. A set of reduced (voltage-free) electro-elastic coupled equations, discussed in the next section, is thensolved to yield the deformation of the actuator while nullifyingthe reaction force applied to the pre-chosen DOF node. Next,the applied voltage that is required to induce the given deforma-tion is calculated. A simple local-maximum search is then em-ployed to iteratively approach the Pull-In state where the voltageis maximal. The iterations are represented by the set of verticallines in Fig. 1. Each of these vertical lines crosses the equilib-rium curve and is therefore associated with an equilibrium stateof the actuator.

Like the VI algorithm, the DIPIE algorithm can be easily inte-grated into commercial CAD tools, using separate electrostaticand mechanical field solvers, as will be shown in the following.

III. FORMULATION OF THE DIPIE ALGORITHM

Consider a general electrostatic actuator formed from anelastic conducting body with a general shape and a counterelectrode, shown in Fig. 2. The surfaces of the electrode andof the conducting body form a free space capacitor, with avariable gap. An energy source applies a voltage differenceacross the capacitor inducing an electrostatic force. In response,the body deforms, its mechanical strain energy is increasedand generalized mechanical restoring forces develop. Externalforces such as the gravity and external pressure may also apply.The free space capacitor,, is assumed to be a linear electricalelement, for which the charge is proportional to the voltage

, but the capacitance may be a nonlinear function of thedeformation.

The deformation of the actuator is discretized usingde-grees of freedom , , , . This discretization must be it-eratively refined to validate the convergence of the calculations


Fig. 2. A general configuration of an electrostatic actuator.

toward the continuum limit. While the mechanical energy de-pends on all degrees of freedom, the electrostatic energy maydepend only on of these degrees of freedom, where(e.g., the electrostatic force depends only on the surface degreesof freedom). Without restricting the generality of the formula-tion, it is assumed that the degrees of freedom are, , ,

. The total co-energy of the actuator , can be then writtenas a linear combination of the electrical co-energy and the me-chanical energy

(1)

where the first term on the right-hand side (RHS) of (1) is theelectrical co-energy.

The equilibrium states of the actuator are those states forwhich the total co-energy has a local extremum. In these statesof equilibrium, the derivatives of the total co-energy with respectto the degrees of freedom vanish, resulting in the following equi-librium equations

(2a)

(2b)

Here the left-hand side (LHS) of the equations denote the gen-eralized mechanical force associated with theth degree offreedom. The RHS of (2a) denotes the electrostatic forceas-sociated with the th degree of freedom.

We define the applied voltage, , associated with thethdegree of freedom as follows:

(3)

The mean square of these applied voltages of thedegrees offreedom is therefore

(4)

Fig. 3. The displacement iterations Pull-In extraction (DIPIE) algorithm.

The voltage in (2a) can be eliminated by substituting (4), leadingto the following reduced equations

(5a)

(5b)

The resulting equations (5a) are linearly dependent, andtherefore one of them (denoted byth equation) should beomitted. Equations (5a) and (5b) are reduced to equa-tions. As can be obviously seen from (5a) and (5b) two sets ofvectors should be calculated, the generalized mechanical forceassociated with each degree of freedom and the electrostaticforce associated with each degree of freedom for a unit appliedvoltage.

Equations (5a) and (5b) are the basis of the displacement iter-ations algorithm. For a given deflection deviating from the solu-tion deflection, the LHS of (5a) can be interpreted as the voltagedeviation at the th degree of freedom from the average voltagegiven by the mean square in (4). According to this interpreta-tion, (5a) requires that all deviations vanish.

If a physical solution of the original problem, stable or un-stable, exists for the postulated deflection of the prechosen de-gree of freedom, and a solver that reduces these deviations isemployed, the solver will necessarily converge to this physicalsolution. Moreover, the voltage mean square will converge tothe voltage required to achieve the postulated deflection of thepre-chosen degree of freedom, while eliminating all reactionforces along the structureincludingthe reaction at the postulated


(a) (b)

Fig. 4. (a) The DIPIE relaxation scheme. (b) The DIPIE Gradient-based scheme.

degree of freedom. Thus, while the original problem had stableequilibrium states and unstable equilibrium states (see Fig. 1),in the displacement iteration scheme, it is replaced with a seriesof equivalent problems for which the equilibrium solution is al-ways stable.

The DIPIE algorithm is shown schematically in Fig. 3. Ateach iteration, a pre-chosen degree of freedomis postulatedand (5a) and (5b) are iteratively solved. Next, the voltage as-sociated with the resulting equilibrium state is calculated. Themaximal voltage is then searched to a predefined accuracy ofthe Pull-In displacement.

The iterative solution of (5a) and (5b) can be carried out usingdifferent solution schemes. Two such schemes are described inthe following.

A. Relaxation Scheme

In this scheme, shown in Fig. 4(a), the applied electrostaticforces for aunit voltageare calculated, for the current deforma-tion using an electrostatic solver. A voltage associated with eachof the degrees of freedom can be calculated from (3) using thelast estimate of the mechanical forces. For a deformation other


than the equilibrium deformation, thesevalues of voltage areunequal. In the current configuration, the actual applied voltageis estimated as the mean square of these values and the actualapplied electrostatic forces are estimated using theunit voltageelectrostatic forces and the mean square voltage. Then a new es-timation for the mechanical forces is calculated

(6)

A new deflection is then calculated using these new estimatedmechanical forces while the prechosen DOF is fixed, using amechanical solver. Therefore, the relaxation scheme requiresseparate electrostatic and mechanical solvers.

The forces applied at degrees of freedom where the voltagewas larger than the mean square are reduced and thus tend todecrease the local deflection, and vise versa. Accordingly, thenew voltage deviations diminish.

B. Gradient Based Scheme

In this scheme, shown in Fig. 4(b), for a current deflection,the applied electrostatic forces for aunit voltageand the me-chanical forces and their gradients are calculated. The gradientsof equations (5a) are given by

(7)

where , ; , .Thus, the calculations can be carried out using separate

mechanical and electrostatic field solvers. The deflectioncorrections can be calculated using the current value of thefunctions and their gradients, in standard Newton method orsimilar schemes.

C. The Initial Deformation

The initial deformation at the first iteration is chosen as theelastic solution of the mechanical problem where the prechosenDOF is fixed with no electrostatic forces applied. At any of thefollowing Pull-In search iterations the initial deformation is pro-portional to the solution of the previous deformation of the pre-chosen DOF. The proportionality factor is the ratio between thedeformations of the prechosen DOF at the current and previousiterations.

IV. NUMERICAL EXAMPLE—THE CLAMPED–CLAMPED BEAM

To demonstrate the capabilities of the new scheme, the typicalproblem of the clamped–clamped beam is chosen [4]–[6]. Thegeometry of the problem is shown in Fig. 5. To generalize theproblem and to illustrate a case , the electrode is limited

Fig. 5. The clamped–clamped beam configuration, whereL is the length of thebeam,d is the nominal gap, and� is the portion of the bottom electrode-freearea.

to a portion of the upper beam [ , whereis the beam length and ]. The equilibrium equations ofthe elastic beam are

(8)

where the effective elastic modulus, , is equal to the Youngmodulus if the beam thickness is of the order of the beamwidth, and is equal to the plate modulus if the beamwidth is much larger than its thickness [5]. Also,is the secondmoment of the beam cross-section,is the cross-section area,

is the effective residual stress, is the permitivity of free-space, the initial gap between the electrodes,the width ofthe beam and are the Cartesian coordinates.

In the above equation the first term on the LHS is the me-chanical bending force, the second term is the mechanical forcedue to residual stress and the third term is the mechanical forcedue to stress stiffening. The electrostatic force on the RHS ofthe equation includes the fringing field effect [17].

Equation (8) can be presented in the following normalizedform:

(9)

where

and

Equations (9) are iteratively solved using finite-differencesin the VI scheme [4], [5] and the DIPIE scheme. In each of thePull-In search points in the DIPIE scheme the elastic deflectionis iteratively solved using the relaxation method described in


Fig. 6. Comparison between the convergence of the deformation in the VI and DIPIE algorithms (for� = 0:25 and a center deflection of~y = 0:37).

Fig. 7. Comparison between the convergence rates and the number of iterations required to converge of the VI and DIPIE schemes (for� = 0:25). When thePull-In state is approach, the convergence rate of the VI scheme vanishes and the required number of iterations to converge is unbounded.

Section III-A. In the VI scheme the elastic deflection is solvedusing the standard relaxation [4]–[7].

In the following case study: , ,and . The convergence of the elastic deflection in bothschemes is described in Fig. 6. A voltage point was chosen forrunning the VI relaxation scheme. The norm of the relativeerror of the deflection is calculated at each of the iterationsand is shown in the figure. The resulting displacement at thecenter of the beam is used for running the DIPIE relaxationscheme. The norm of the relative error of the deflection is

presented in logarithmic scale against the number of iterations.It is seen that the convergence of the DIPIE scheme is muchfaster than the convergence of the VI scheme. Moreover, theconvergence rate in the DIPIE scheme is constant, whereas theconvergence rate in the VI scheme varies and declines witheach iteration. It is therefore concluded that the convergenceof the DIPIE scheme can be further improved using numericalacceleration methods [18]. In contrast, since the convergenceof the VI scheme is not constant, it cannot be as easilyaccelerated.


Fig. 8. Comparison between the Pull-In parameters derived by the VI and DIPIE schemes for mesh and accuracy refinements exhibiting the better consistency ofthe DIPIE scheme, (a), (b) the normalized Pull-In displacement at the center of the beam for the DIPIE and VI schemes, respectively, and (c), (d) the normalizedPull-In voltage displacement for the DIPIE and VI schemes, respectively.

The convergence rates and the total number of iterationsrequired to converge for both schemes at different deflections ofthe beam center point are described in Fig. 7. The relative error,

, described in Fig. 6 is approximated by ,where is a constant, is the convergence rate andisthe iteration number. This approximation is motivated by thelinear convergences (semi logarithmic scale in Fig. 6) of theDIPIE scheme. Also, the convergence of the VI scheme tendsto linearity after many iterations. It can clearly be seen thatthe convergence of theVI scheme rapidly deteriorates as thePull-In point is approached, and that at the Pull-In point theconvergence vanishes. In contrast, the convergence rate of theDIPIE scheme is high and remains high even beyond the Pull-Inpoint.

In order to examine the consistency of both schemes, aspecific problem is solved using increasingly refined meshesfor several convergence accuracies. To assure the convergenceof the inner equilibrium loop, its accuracy (defined on thenorm of the deflection errors) is set to two orders of magnitudehigher than the required accuracy of the outer Pull-In searchloop. Fig. 8 presents the Pull-In parameters calculated by both

schemes against the inverse of the number of nodes. For anygiven accuracy, the DIPIE scheme shows a similar consistentconvergence as the mesh is refined, making it easy to predict avalue at the limit of continuum. This predicted limit convergeswith increasing accuracy. In contrast, the VI scheme shows aninconsistent behavior, which is less pronounced for the Pull-Involtage at high accuracies. It is therefore concluded that itis impractical to extract a reliable estimation of the Pull-Indeflection. The voltage near the Pull-In state is insensitiveto the deflection errors and therefore for a sufficiently highaccuracy, the VI scheme yields reasonable estimations for thePull-In voltage.

Fig. 9 illustrates the difference in numerical effort requiredby each scheme to converge to the Pull-In state within a givenaccuracy. The numerical effort is measured by the CPU runtimerequired to approach the Pull-In state. The great advantage ofthe DIPIE scheme, in terms of runtime (20–120 faster), is triv-ially seen in this figure. In practice, due to the consistency of theDIPIE scheme, a lower accuracy and a coarser mesh are suffi-cient to extract a reliable estimation of the Pull-In parameters.On the other hand, in the VI scheme a higher accuracy and a


Fig. 9. Comparison of the CPU run-time required to obtain the Pull-In parameters within a given accuracy, by the VI and DIPIE schemes.

finer mesh are required to reach a reliable estimation. Therefore,the advantage of the DIPIE scheme is even higher than appearsfrom Fig. 9 ( 100 times faster).

To demonstrate the capability of the DIPIE scheme to solvemore general problems, including nonlinear stress stiffening, acase study presented in the literature is chosen. Following [19]an actuator with the specific parameters [ m],[ m], [ m], is solved.

Hung and Senturia used a finite difference VI scheme to de-rive the Pull-In parameters and found that [V] and acenter beam deflection of [ m], while the parametersderived using theDIPIE algorithm were [V] and acenter beam deflection of [ m]. This good agree-ment has been verified in other cases as well thus confirmingthe capability of the DIPIE algorithm to extract the Pull-In pa-rameters of stress-stiffened actuators.

V. SUMMARY

The qualities of the DIPIE scheme result from the fact thatthe original inherently unstable physical problem of the voltagecontrolled electrostatic actuator is replaced by an equivalentproblem that is inherently stable.

To conclude, the main advantages of the DIPIE scheme overthe VI scheme are:

1) runtime—over a 100 times faster;2) accuracy—consistent convergence in both accuracy and

mesh refinements whereas the VI scheme is inconsistentand the Pull-In deformation is impractical to extract;

3) fully automated—the displacement is naturally boundedby the geometry of the actuator whereas the voltage upperbound is unknown and requires a user input.

As mentioned in Section III-B, gradient-based schemes (e.g.,Newton method) can also be used to solve the equilibrium equa-tionsandextract thePull-Inparameters.Although thismayaccel-erate the computation of both the VI and DIPIE schemes, it will

not affect the lack of convergence deterioration of the Voltage-It-erations at Pull-In (since above the Pull-In voltage the system hasno equilibrium points and therefore the calculations can not con-verge). Therefore, the DIPIE scheme will be faster than VI-basedschemes as it always converges at the Pull-In state.

The advantage of the relaxation scheme is that it only requiresseparate electrostatic and mechanical fields solvers and does notrequire the calculation of the electromechanical coupled forcegradients. Therefore, the relaxation DIPIE scheme can be easilyimplemented in existing MEMS CAD tools with separate fieldsolvers.

REFERENCES

[1] S. D. Senturia,Microsystem Design. Boston, MA: Kluwer Academic,2001.

[2] O. Bochobza-Degani, “Investigation of motion sensing and actuationmechanisms in micro-electro-opto-mechanical systems (MEOMS) andtheir applications,” Ph.D. dissertation, Supervised by Y. Nemirovsky,Technion—Israel Institute of Technology, in progress.

[3] Y. Nemirovsky and O. Degani, “A methodology and model for thePull-In parameters of electrostatic actuators,”J. Microelectromech.Syst., vol. 10, no. 4, pp. 601–615, Dec. 2001.

[4] P. M. Osterberg, “Electrostatically actuated microelectromechanicaltest structures for material property measurements,” Ph.D. dissertation,Supervised by S. D. Senturia, Massachusetts Institute of Technology(MIT), Sept. 1995.

[5] R. K. Gupta, “Electrostatic Pull-In test structure design forin-situmechanical property measurements of microelectromechanical systems(MEMS),” Ph.D. dissertation, Supervised by S. D. Senturia, Massachu-setts Institute of Technology (MIT), June 1997.

[6] P. Osterberg, H. Yie, X. Cai, J. White, and S. Senturia, “Self-consistentsimulation and modeling of electrostatically deformed diaphragms,” inProc. IEEE MEMS 94, Oiso, France, Jan. 1994, pp. 28–32.

[7] M. Fischer, M. Giousouf, J. Schaepperle, D. Eichner, M. Weinmann, W.von Munch, and F. Assmus, “Electrostatically deflectable polysiliconmicromirrors—Dynamic behavior and comparison with results fromFEM modeling with ANSYS,”Sens. Actuators, Phys. A, vol. 67, pp.89–95, 1998.

[8] E. K. Chan, K. Garikipati, and R. W. Dutton, “Characterization of con-tact electromechanics through capacitance–voltage measurements andsimulations,”J. Microelectromech. Syst., vol. 8, pp. 208–217, June 1999.

[9] S. D. Senturia, “CAD challenges for microsensors, microactuators, andmicrosystems,”Proc. IEEE, vol. 86, no. 8, pp. 1611–1626, 1998.


[10] D. S. Long, M. A. Shannon, and N. R. Aluru, “A novel approachfor determining Pull-In voltages in micro-electro-mechanical systems(MEMS),” in Proc. MSM’2000, US Grant Hotel, San Diego, CA, Mar.27–29, 2000.

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[12] L. J. Hornbeck, “Spatial light modulator and method,” U.S. Patent5 061 049, Oct. 1991.

[13] C. T. C. Nguyen, L. P. B. Katehi, and G. M. Rebeiz, “Micromachineddevices for wireless communications,”Proc. IEEE, vol. 86, no. 8, pp.1756–1768, Aug. 1998.

[14] D. L. Dickensheets and R. G. Kino, “Silicon-micromachined scanningconfocal optical microscope,”J. Microelectromech. Syst., vol. 7, no. 1,pp. 38–47, Mar. 1998.

[15] Coventor, Inc.. Coventorware. [Online]. Available: www.coventor.com.[16] Intellisense Corporation. Intellisuite. [Online]. Available: www.intel-

lisense.com.[17] R. S. Elliot,Electromagnetics. New York: IEEE Press, 1993.[18] G. D. Smith,Numerical Solution of Partial Differential Equations: Fi-

nite Difference Methods, 3rd ed. Oxford, U.K.: Clarendon, 1993.[19] E. S. Hung and S.D. Senturia, “Extending the travel range of analog-

tuned electrostatic actuators,”J. Microelectromech. Syst., vol. 8, no. 4,pp. 497–505, Dec. 1999.

Ofir Bochobza-Degani(S’97) was born in Ashkelon,Israel, on April 17, 1974. He received the B.Sc. de-gree in electrical engineering and the B.A. degreein physics (bothsumma cum laude) in 1996 and theM.Sc. degree in electrical engineering in 1999, allfrom the Technion—Israel Institute of Technology,Haifa. He is currently working toward the Ph.D.degree in electrical engineering at the Technion.

He investigates motion sensing and actuationmechanisms in microoptoelectromechanical systems(MOEMS). His research focuses on the coupled

energy domain modeling as well as noise modeling of MOEMS including:electrostatic actuation, magnetostatic actuation and optical motion sensing. Heis involved in the development of micromachined inertial (acceleration andrate) sensors employing integrated optical sensing. His other fields of interestare analog readout and control interfaces, silicon optical benches, and thermalsensors.

David Elata (M’01) received the B.Sc. degree fromBen-Gurion University, Israel, in 1986 and the M.Sc.and D.Sc. degrees from the Technion—Israel Insti-tute of Technology, Haifa, in 1989 and 1993, respec-tively, all in mechanical engineering.

In 1996, he began teaching at the Faculty ofMechanical Engineering, Technion. From 1993through 1996, he was a Postdoctoral ResearchStaff Member at Lawrence Livermore NationalLaboratory and a Visiting Postdoc at the GeophysicsDepartment of Stanford University, Stanford, CA.

His current research interests are in solid mechanics, computational methods,and MEMS technology.

Yael Nemirovsky (SM’84–F’99) received the B.Sc.and 1966, D.Sc. degrees from the Technion—IsraelInstitute of Technology, Haifa, in 1966 and 1971, re-spectively.

In 1980, she joined the Department of ElectricalEngineering at the Technion. Prior to that she wasa research scientist specializing in microelectronicsin Rafael, a national R&D organization. She gradu-ated from Technion in chemistry and her D.Sc. thesiswas in electrochemistry. For over 20 years, she hasbeen active in electrooptical devices in II–VI com-

pound semiconductors and additional advanced semiconductor materials as wellas infrared focal plane arrays. She has been involved in growth, processing,device design, and modeling of detectors as well as VLSI circuits. She has awell-equipped MOCVD laboratory for growth of heterostructures, extensive fa-cilities for device and interfaces processing and characterization. She has been aprincipal investigator in large funded research programs that ended in prototypeinfrared detectors and systems that were transferred to industry. Twice she wasthe head of the microelectronics research center of the Department of ElectricalEngineering at the Technion. Currently, her research focuses on microoptoelec-tromechanical systems (MOEMS), CMOS compatible micromachining and mi-crosystems implemented in CMOS technology and integrated with silicon de-vices. She has published over 130 papers in the open literature, has filed severalpatents and a large number of classified reports. She has collaborated with themicroelectronics industry as a consultant in sensors and VLSI technology andhas been quite active in national and international conferences. She has super-vised over 40 graduate students for M.Sc and D.Sc.

Dr. Nemirovsky is a Fellow of the Institution of Electrical Engineers (IEE)and has been the Chairperson of the Israeli Association for Crystal Growth.Currently, she is the Chairperson of the microelectronics and photonics sectionof URSI. She has received awards as a “Best Teacher” at Technion; a nationalaward of high esteem—“The Award for the Security of Israel”; and a Technionaward for “Novel Applied Research.” She has received The Kidron Foundationaward for “Innovative Applied Research” (a $100 000 grant for research pro-gram). She is a Distinguished Lecturer of the Electron Device Society of IEEE.

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An efficient DIPIE algorithm for CAD of electrostatically actuated MEMS devices