Time-Efficient Quasi-Static Algorithm for Simulation of Complex Single

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Time-Efficient Quasi-Static Algorithm for Simulation of Complex Single-Sided Clamped Electrostatic Actuators Joachim Oberhammer, Member, IEEE, A.-Q. Liu, and Göran Stemme, Fellow, IEEE

Abstract—This paper reports on a very time- and resourceefficient numerical algorithm for quasi-static modeling of the static behavior and the “quasi-static movement” of highly nonlinear electrostatic actuators with single-side clamped moving elements. The algorithm is capable of simulating prestressed materials and multicontact touching surfaces with complex geometries, including distance-keeping stoppers and thickness and material inhomogeneities of the moving parts. Thus, it is very suitable for predicting the behavior of actuators such as laterally moving curved-electrode actuators or vertically moving touch-mode or zipper actuators. In contrast to conventional, very time- and memory-consuming simulation methods such as finite-element analysis, the proposed algorithm—even if implemented in the slow script-language of MATLAB—takes only a fraction of a second to solve a complex problem, which makes it a very powerful design tool for parameter optimization of the actuator geometry. The reason for the efficiency of this algorithm is that its core is based on the one-dimensional mathematical description of a two-dimensional model geometry and that the differential equation is solved by a simple triple-integration for each iteration step, which is a method very suitable for thin-film single-side clamped moving elements. This paper describes the algorithm, analyzes its accuracy and its limitations, and reports on its performance as compared to other methods such as simplified analytical models for very basic structures, finite-element method (FEM) simulations of complex structures, and measurements of fabricated devices, including laterally moving microelectromechanical systems (MEMS) switches and vertically closing prestressed thin-film zipper actuators. Furthermore, the efficiency of the algorithm as a design tool was evaluated for the parameter optimization of electrostatic curved-electrode actuators. The algorithm’s main application is seen in the fast determination of suitable parameter sets for MEMS electrostatic actuators, but it cannot substitute for a more accurate FEM analysis to investigate a final design in great detail. [1725]

Index Terms—Curved-electrode actuator, design optimization, electrostatic actuator, microelectromechanical systems (MEMS) design, MEMS simulation, quasi-static modeling, touch-mode actuator.

Manuscript received November 25, 2005; revised September 1, 2006. Subject Editor N. Aluru. J. Oberhammer and G. Stemme are with the Microsystem Technology Group, School of Electrical Engineering, Royal Institute of Technology, SE-100 44 Stockholm, Sweden. A.-Q. Liu is with Photonics Lab II, School of Electrical and Electronics Engineering, Nanyang Technological University, 639798 Singapore. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JMEMS.2007.892917

I. INTRODUCTION

E

LECTROSTATIC actuation is a concept of high interest for generating moving microsystems because of the highenergy densities and large forces due to the scaling laws in small dimensions and because of relatively simple fabrication [1]. However, the main disadvantage of electrostatic actuation is the limitation to small displacement. For practical designs (medium actuation voltage, electrode area, and reliable restoring spring force) of parallel-plate electrode arrangements and movement in the direction of the field lines, the maximum traveling distance of the moving electrode is limited in most practical designs to a few micrometers. Curved-electrode actuators, also known as touch-mode, moving-wedge, or rolling-contact actuators, are electrostatic actuators with single-side clamped moving structures and are based on an electrode geometry with the electrode distance gradually increasing from the clamped end to the free end of the moving structure. Due to the narrow gap in the beginning, high forces initialize the movement of successive parts of the flexible electrode and the short electrode distance, and thus the location of the large actuation force is moving along the fixed electrode in a zipper-like way [2], [3]. Such an actuator achieves very large deflection of the free end of the moving structure at relatively low actuation voltages. Another feature of electrostatic curved-electrode actuators, in contrast to electrothermal actuators that achieve a similar large displacement, is that the maximum force is created in the end-position of the movement, which makes them very suitable for electrical microswitches. The two most common types of such zipper actuators are a) vertically moving touch-mode actuators [2], as shown in Fig. 1(a), where the prebending of the moving part is usually created by a controlled, fabrication process–related stress gradient, and b) laterally moving curved-electrode actuators [4], as shown in Fig. 1(b), where the shape of the curved electrode is simply defined by a photolithographical mask whose pattern is vertically etched into the device layer. Among others, these actuators have found applications in MEMS electrical switches [5]–[8], tunable capacitors [9], [10], gas valves [11]–[14], and for microoptical applications [15], [16]. However, the increased academic and industrial research and development activities on such actuators lack good and fast design tools. The main requirements on such a design tool are to predict the static and dynamic behavior of the actuator and to allow for the optimization of the actuator parameters, such as

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Fig. 1. The two most common implementations of zipper actuators: (a) vertically moving touch-mode actuator: flexible, initially bent curved electrode, pulled-in toward the fixed flat counterelectrode; (b) laterally moving curvedelectrode actuator: flexible, initially straight moving electrode bending along the curved, fixed electrode.

the geometry and the actuation voltage. The complexity of the type of actuators under discussion evolves many problems for the simulation model, mainly because of • the highly nonlinear nature of the deflection-dependent forces (requiring a nonlinear solver); • the large change in the electrode distance (requiring an adaptive mesh and moving boundaries); • touching surfaces for the zipping or rolling motion (multiple contact problem); • material prestress determining the initial bending of the moving structure, especially for vertically moving zipper actuators. Analytical solutions of nonparallel electrode problems are not available. Only for very simple structures describable by analytical functions, the energy equilibrium at the instable snap-in point can be described by a nonlinear differential equation [1]. However, an analytical closed-form solution of that differential equation cannot be found, but it is possible to construct an approximate solution by a simplified model based on the Rayleigh–Ritz method [17], which allows at least for an estimation of the necessary pull-in voltage [1]. Finite-element/boundary-element (FEM/BEM), finite-cloud/boundary-cloud (FCM/BCM), or mixed-domain analysis-based simulation software [18]–[20] can predict the highly nonlinear behavior of touching structures and complex geometries but require complicated coupled electrostatic-mechanical domain models with moving boundaries and the capability of solving multiple-contact problems. Furthermore, because of the large ratio between lateral and vertical dimensions of typical MEMS thin-film structures, the mesh of accurate FEM models has to consist of a large number of elements with an exploding number of degrees of freedom (DOFs), which results in memory-consuming matrices to be solved. Simulating such models is very time-consuming and often results in convergence problems. Therefore, they are less suitable for the optimization of design parameters such as the electrode shape, the cantilever geometry, or the stopper sizes and positions. The presented numerical algorithm allows for quasi-static simulation of the static basic dynamic behavior of two-dimensional single-side clamped structures and is capable of handling complex cantilever and electrode geometries including stoppers, isolation layers, multilayer cantilever structures, inhomogeneous material properties, and prestressed materials, which typically determine the curvature of vertically

Fig. 2. Geometrical model of the actuator with isolation layers and stoppers internally represented as small isolation layer bumps.

closing zipper actuators. The algorithm is faster by a few orders of magnitude than finite-element methods and therefore very suitable for design parameter optimization. II. DESCRIPTION OF THE ALGORITHM A. Geometrical Model The algorithm is based on a two-dimensional geometrical description of the two-electrode problem. The deflecting structure is mathematically modeled one-dimensionally and thus assumed to be very thin compared to its length. This description is appropriate for typical single-side clamped electrostatic actuators because the cantilever thickness is typically very small as compared to its length. The geometry of the fixed electrode as well as the geometry of an optional isolation layer on that electrode are described by numerical vectors and thus can have a discontinuous shape, which allows for complex electrode features such as stoppers and isolation layers of any shape, as shown in Fig. 2. Geometrical discontinuities and multiple layers of the flexible structure are modeled by variable material parameters over the cantilever length (Young’s modulus and moment of inertia ). The initial curvature of the flexible structure under an optional material prestress is modeled by a distributed torque or by an external force distribution additionally acting on the structure. A two-dimensional model is sufficient to describe most MEMS actuators of this type. The depth of the electrodes in the -axis does not affect the behavior of the actuator, since the electrostatic force is linearly increasing with the electrode depth and therefore compensating for the also linearly increasing counteracting mechanical spring force (neglecting fringe-field effects). Multilayer cantilevers, i.e., structures with a cross-section composed by a stack of homogeneous materials such as bimetallic strips, e.g., can be simulated by using the material parameters of an equivalent cross-section, i.e., an effective -product representing the equivalent stiffness of the beam. can be calculated by formulas The equivalent stiffness from textbooks [21, Section 8.2] or directly be derived from its definition

(1) with the total cantilever thickness and ture in the third dimension.

the width of the struc-

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the previous iteration step, i.e., the electrostatic force of a single element is calculated by

(5)

Fig. 3. Diagram illustrating how the forces and torques are calculated for a single iteration step of the space discretizised model (for simplicity reasons, without external forces and material prestress): (a) the electrodes are modeled by distributed discrete capacitors; (b) the forces between the plates of each capacitor are calculated; and (c) the torque at each single discretization point x is calculated by summing up all of the torques created by the forces on the side toward the free end in relation to the point x .

with the width of the cantilever and thus the electrodes in and the the third dimension, the actuation voltage, local thickness and dielectric constant of the dielectric layer, if the description of the curved electrode’s shape, existing, and the deflection of the cantilever calculated during the last iteration step at the position . The deflection of the flexible structure is calculated by double numerical integration of (2), which is for the space-discretized model

B. Quasi-Static Iteration Solver The two main steps of a single quasi-static iteration step of the electrostatic/elastomechanical coupled-domain analysis are [22, Section 7.3.6]: • electrostatic domain: determine the electrostatic forces imposed on the current geometry; • elastomechanical domain: determine the deformation ( new geometry) due to these forces. The elastomechanical description of the deflection in a single iteration step is based on the classical Euler–Bernoulli beam differential equation [23]

(2) describing the local curvature of a structure with the moment of inertia and Young’s modulus bending due to the torque distribution . This equation is valid for “small” deflections in of infinitesimal thin structures. The sum of moments each position is numerically derived from the electrostatic , as shown in Fig. 3, and optionally inforce distribution fluenced by mechanical prestress and an external force distribution (3) or for the space-discretized model

(4) The electrostatic force is the attracting force between the plates of—for the space-continuous model infinitesimal small—parallel-plate capacitors distributed along the electrodes. For the space-discretized model, the number of capacitors is given by the number of nodes on the cantilever and the width of each capacitor is given by the cantilever length divided by . The distance between the two electrodes of each capacitor is determined by the cantilever’s deflection of

(6) with from (4) and the moment of inertia optionally variand are the integration able over the cantilever length. constants representing the -offset and initial slope of the cantilever, which are especially necessary to be taken into account when continuing the simulation after a contact has occurred (see Section II-C). The new deflection is used for calculating the new electrostatic force distribution in the subsequent iteration step. Thus, the two main mathematical steps of the algorithm for solving a single iteration are the calculation of the electrostatic force distribution and the triple-integration thereof according to (2) and (3). This straightforward approach without solving any nonlinear differential equation systems is the reason for the extremely short simulation times and the negligible memory consumption of the presented algorithm, with all its limitations and inaccuracies resulting from the simplifications as analyzed and discussed in Section III. C. Simulation of Static Nonlinear Behavior and “Quasi-Static Movement” For calculating the deflection of a cantilever structure due to a constant external force distribution, basically only one single iteration step as described in Section II-B would be necessary (linear quasi-static analysis). However, since the electrostatic force depends on the deflection of the cantilever and is highly nonlinear, sequential iteration steps with slowly increasing deflection have to be carried out until a steady state is reached (nonlinear quasi-static analysis). Especially when determining “static” parameters such as the pull-in voltage of an electrostatic actuator, a quasi-static approach is sufficient and a full-dynamic analysis is not necessary. Beyond that, the deflection of sequential iteration steps mimics the dynamic movement of the cantilever. Thus, this algorithm is, for example, capable of predicting the pull-in behavior of an electrostatically actuated cantilever and can be used to determine the pull-in position or to simulate the rolling behavior of a zipper-actuator. It should be kept in mind, though, that the presented quasi-static algorithm is

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Fig. 5. Demonstration of simulating “quasi-static” movements: Tip deflection and dynamic step-size over the iteration steps of a cantilever pulling in along three stoppers. The cantilever shape of a few selected steps of the pull-in history are displayed: A) critical pull-in position to the first stopper; B) contact with the first stopper; C) contact with the second stopper; D) critical pull-in position to the third stopper; and E) contact with the third stopper and final steady state. m, y m, y m, (Actuator configuration: L n : ,n ,a : ,a : ,t m, E GPa; ,( y : nm, n , actuation simulation parameters: N voltage : V.).

= 25 = 3 = 104 2

Fig. 4. Basic flowchart of the algorithm. The torques M shown in Fig. 3.

(x ; t) are derived as

not thought to replace a full-dynamic analysis of dynamic processes, which includes the effects of kinetic energy and inertia of the masses [24]. The basic flowchart of the algorithm is shown in Fig. 4. For simulating static nonlinear processes and “quasi-static movements” such as the pull-in behavior of electrostatic actuators, the algorithm is endowed with the following features. • (Spatial) step-size adaptation for quasi-static modeling: The maximum change in the deflection between succesto sive iteration steps is limited to a parameter ensure the sufficient quantization of the highly nonlinear electrostatic force. • Contact detection: A contact is detected when the distance between the flexible structure and the isolation layer on the rigid electrode gets below a certain limit. The position and the slope of the contact point determine the constrains and for solving (5) in the and integration constants subsequent iteration. The algorithm is capable of detecting multiple contact points in different iteration steps. Thus, it is capable of simulating the pull-in and rolling behavior of a zipper-actuator.

= 400 =2 = 1 2 = 0 73 = 4 =2 1 =01

= 20 = 150 = 100

• Detection of stable states: The algorithm terminates if the for a change in deflection is below a critical limit of subsequent iteration steps. certain number These features are demonstrated in Fig. 5, showing the simulation history of a cantilever pulling in along a curved-electrode actuator, by displaying the cantilever deflection at a few selected quasi-static iteration steps. The pull-in voltage of an electrostatic actuator can be determined by running a simulation series with different actuation voltages. The pull-in position can easily be detected by identifying the abrupt step in the plot of the final steady-state tip deflection over the actuation voltage. A fast algorithm to accurately determine the pull-in voltage can be built around the algorithm presented in this paper by simulating a small set of actuation voltages whose values is refined after every iteration step according to the threshold voltage found in the previous step. Fig. 6 shows how such an algorithm was used to determine the pull-in voltages of a cantilever successively snapping in to three stoppers when the actuation voltage is gradually increased. III. INACCURACIES AND LIMITATIONS OF THE ALGORITHM A. Applicable Class of Problems Because this algorithm is specially written for a certain class of problems, it is only applicable for the following categories of geometrical problems: • single-sided clamped electrostatic actuators; • two-dimensional geometry, i.e., homogeneous in the third dimension;

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Fig. 6. Determination of the pull-in voltages of a cantilever successively snapping-in on three stoppers: A) actuation voltage just below the pull-in voltage for snapping in to the first stopper; B) and C) before pulling-in to the second and third stopper; and D) final deflection. Fig. 8. Simulation error of the pull-in voltage of a mass-spring electrostatic actuator as compared to a full-dynamic model: (a) deflection over the iteration steps for the quasi-static model; (b) deflection over the time of a full-dynamic model set-up in Simulink, with Q = 10; and (c) comparison of the pull-in voltages of the two models. Only for undercritical damping, the deviation of the quasi-static model is significant. Model data: mass-spring system of a silicon cantilever; length, width, thickness = 100; 60; 4 m; spring constant = 10 Nm , resulting in mass = 48 ng, resonance frequency= 72:6 kHz; actuation voltage in (a) and (b) is 50 V.

Fig. 7. Nonlinearity error of the Euler–Bernoulli beam equation based algorithm as compared to FEM analysis. The error for a cantilever tip deflection up to 10% of the cantilever’s length is less than 1.5% and nearly independent of the cantilever thickness. (Configuration: L = 300 m, E = 150 GPa, deflection due to a variable force at the tip).

• thin moving films/cantilevers (as compared to the length); • small deflections (as compared to the length). These conditions, however, are typical for most MEMS electrostatic actuators, which consist of thin films and are characterized by small deflections. For these type of structures, the algorithm supports complex geometries of the flexible and moving electrodes, including stoppers, isolation layers, thickness inhomogeneities, and composite multilayer cantilevers. B. Inaccuracies of the Algorithm According to the limitations listed in Section III-A, the inaccuracies of the simulation results of the presented algorithm have been investigated. Fig. 7 shows the nonlinearity error of the algorithm for large tip deflections. The nonlinearity error for a typical MEMS actuator with a deflection of 10% of the cantilever’s length is less than 1.5% and nearly independent of the cantilever thickness. A FEM model with large deflection elements and deformable mesh, simulated in COMSOL MultiPhysics, has been used as the reference model.

The algorithm is based on a quasi-static approach neglecting the effects of inertia of the moving masses. Kinetic energy, however, might even have an influence on apparently “static” parameters, such as the pull-in voltage. Fig. 8(a) shows the deflection of a cantilever mass-spring system over the iteration steps, modeled by the quasi-static approach for an actuation voltage below the pull-in threshold. Fig. 8(b) displays the time response of a full dynamic model, clearly showing oscillating behavior for an undercritical damped system. Due to the oscillation, the cantilever in the full-dynamic simulation gets much closer to the electrode than the final steady-state deflection in the quasi-static approach, which results in a lower pull-in voltage. Fig. 8(c) plots the error of the pull-in voltage of the quasi-static model over the quality factor of the system. For a critical or overcritical damped system, the error of the quasi-static approach is less than 0.5%. For an undercritical damped system, the error reaches up to 8.5% for a of 1000. When comparing the performance of this quasi-static algorithm with a full-dynamic analysis, it has to be considered that if the rise-time of the actuation voltage is larger than the risetime of the system’s step response, the dynamic behavior of the system is fully equivalent to the quasi-statically simulated behavior. This point is quite relevant from a practical perspective, since most MEMS electrostatic actuators have rather short system response times. The rise-time of the step-response of the typical cantilever curved-electrode actuator investigated in Fig. 8, for example, is only 4.82 s long. C. Choice of Suitable Algorithm Parameters The most important parameters of the algorithm are: • number of nodes ( -axis discretization); • maximum step-size per iteration step;

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Fig. 9. Investigation of the influence of the parameters of the stability criterion on the accuracy of the pull-in voltage.

Fig. 11. Dynamic, spatial step size over the iteration steps of a cantilever pulling in, for an actuation voltage just above the pull-in voltage. The step size reaches a minimum at the critical pull-in point, which implies that the pull-in voltage is independent on the maximum step-size limit y . (Actuator configuration: L m, y m, n : , m, y : V.) t m, E V, V GPa, V

=3

= 400 = 150

=2 = 51

(1 ) = 20 = 50 323

=25

results, since the dynamic step-size reaches a minimum around the critical pull-in point or the final stable point, as shown in the figure. IV. PRACTICAL PERFORMANCE EVALUATION A. Comparison to Other Simulation Methods

Fig. 10. Simulation error off the pull-in voltage depending on the x-axis discretization of the cantilever (number of nodes and number of discrete capacitors) for different stability criteria parameter sets: (a) N ,( y nm; (b) N ,( y nm; (c) N ,( y nm; and (d) N ,( y nm. (Actuator configuration: L m, y m, y m, n : ,t m, E GPa.)

1 10

= 400 = 150

= 10 1 = 1 = 10 1 = 100 =2 = 20

= 100 1 = = 10 1 = = 25 = 3

• the two parameter of the stability criterion: number of successive iteration steps with step-size smaller than to determine a stable state. Fig. 9 displays the dependency of the simulated pull-in voltage on the two parameters of the stability criterion for a typical MEMS curved-electrode actuator. For a threshold level nm, the error of the pull-in voltage is less than 0.1%, nearly independent of the second parameter . Even nm with an extremely coarse parameter set of , the error created by the stability criterion is and still less than 1%. The influence of the -axis discretization ( ) is illustrated in Fig. 10. For suitable parameter sets (a), (b), and (c) of the stability criterion, the error due to the -axis quantization is less than 1% for a number of nodes as low as 30, and is less than 0.25% for more than 100 nodes. has been investigated with Also the step-size limit the results plotted in Fig. 11. Interestingly, for simulating the pull-in voltage or the steady state of a not pulling-in cantilever, the step-size limit has basically no influence on the simulation

Besides the evaluation in Section III, the performance of the algorithm in terms of predicting the pull-in voltage of typical micromechanical structures was compared to the following. • Implicit, Rayleigh–Ritz method based simplified analytical descriptions of lateral curved-electrode structures with electrode shape of different order as reported in the literature [1]. The electrode shape is described by (7) the initial with the cantilever length of 500 m, electrode gap of 30 m, the maximum electrode gap of 30 m, and the order of the electrode shape, which is varied from 0 to 2.0. Further simulation parameters are GPa, cantilever width 2 m, and cantilever thickness 5 m. The results obtained by the presented algorithm deviate from the results of the simplified analytical model between 0.5% and 7.5%, as summarized in Fig. 12. • FEM simulations, a very simplified analytical model, and measurements of a laterally moving multilayer MEMS switch, as shown in Fig. 13(a), etched in the device layer of a silicon-on-insulator wafer and sputter-coated with an aluminum layer of a measured thickness of about 600 nm, fabricated at Nanyang Technological University, Singapore [25]. All of the simulations and calculations including the FEM analysis are based on the geometry model shown in Fig. 13(b), which is simplified at the thicker end of the cantilever as compared to the device structure. A FEM analysis reveals that this part of the cantilever of the real structure is by a factor of 1.8 times stiffer than the corresponding part of the simplified geometrical model

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Fig. 12. Simulation of laterally moving curved electrode actuators with variation of the electrode shape order , as compared to a simplified analytical model based on the Rayligh–Ritz method [1]. = 0 is the special case of initially parallel electrodes, and orders of 1 are not of any practical significance.

n n n