Numerical algorithm for quasi-static nonlinear simulation of touch

1A1.1 Numerical Algorithm for Quasi-static Nonlinear Simulation of Touch-mode Actuators with Complex Geometries and Pre-stressed Materials Joachim Oberhammer(1,2), Ai Qun Liu(2), and Göran Stemme(1) Dept. of Signals, Sensors and Systems, Royal Institute of Technology (KTH), 100 44 Stockholm, Sweden (2) School of Electrical and Electronic Engineering, Nanyang Technological University (NTU), Singapore 639798 e-mail: [email protected] (1)

ABSTRACT This paper reports on a numerical algorithm for quasistatic dynamic modeling of highly nonlinear electrostatic actuators with single-side clamped moving elements, such as curved-electrode actuators or zipperlike touch-mode actuators. The algorithm is capable of simulating pre-stressed materials and touching surfaces with complex geometries of the moving and the rigid structures, including stoppers and thickness variations of the moving parts. In contrast to conventional, very time-consuming simulation methods, the proposed algorithm takes only a fraction of a second which makes it a very powerful design tool for parameteroptimization of the actuator geometry. The paper describes the algorithm, implemented in MATLAB, and reports on its performance evaluation, comparing its simulation results with those obtained by other methods such as simplified analytical models, FEM/BEM/FCM/BCM-simulations and measurements of fabricated structures, including laterally moving MEMS switches and vertically closing pre-stressed thin-film zipper-actuators. The algorithm shows good agreement with measurements and results obtained by these methods.

(a)

(b)

Figure 1: (a) vertically moving touch-mode actuator; (b) laterally moving curved-electrode actuator. Analytical descriptions are not available in a closed-form and are only suitable for predicting the force equilibrium to estimate the pull-in voltage of very basic, mathematically describable geometries. Simplified models based on the force equilibrium equation can be developed via Rayleigh-Ritz analysis [1], but only describe the state of equilibrium and are not capable of simulating the dynamic or touching behavior of the actuator. Finite-element/ boundary-element (FEM/BEM), finite-cloud/boundarycloud (FCM/BCM) or mixed-domain analysis based simulation software [8] can predict the highly nonlinear behavior of touching structures and complex geometries, but require complicated coupled electrostaticmechanical domain models with moving boundaries and the capability of solving contact problems. Simulating such models is very time-consuming and therefore 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 for quasi-static simulation of single-side clamped structures is capable of handling complex cantilever and electrode geometries including stoppers, isolating layers, multilayer cantilever structures and non-uniform material properties. It is also capable of simulating the pull-in and contact behavior of pre-stressed materials which typically determine the curvature of vertically closing zipper actuators.

Keywords: curved-electrode actuator, MEMS simulation, quasi-static modeling, touch-mode actuator. I. INTRODUCTION For conventional parallel-plate electrostatic actuators with movement in the direction of the field lines, the displacement at acceptable actuation voltages is limited because the attraction forces decrease with the square of the electrode distance [1]. Electrostatic touch-mode actuators with either curved rigid electrodes [2], as shown in Figure 1b, or flexible pre-bended moving elements [3], as shown in Figure 1a, are very attractive for MEMS device actuation because they result in large displacement and large forces at relatively low actuation voltages, achieved by the small effective electrode distance of the zipper-like movement of successive parts of the flexible actuator element [3]. Among other applications, such actuators have been used for MEMS switches [4, 5], valves [6] and optical fiber alignment [7]. The increased academic and industrial research and development activities on such actuators lack on good and fast design tools for optimizing the actuator geometry.

II. DESCRIPTION OF THE ALGORITHM 2.1. Geometrical Description The algorithm is based on a two-dimensional geometrical description with a deflecting structure which is assumed to be infinitively thin compared to its length, as shown in Figure 2. This description is

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electrode geometry is smaller than a given1A1.1 limit; this limit can also be used to describe an isolation layer coating the rigid electrode. These features are demonstrated in Figure 2, showing the simulation of a cantilever successively bending along the stoppers of a curved electrode.

appropriate for single-side clamped touch-mode actuators, because the cantilever thicknesses and displacements are typically very small as compared to their length. Also, the moment of inertia and the Young’s Modulus can be defined as parameters variable over the cantilever length, to take thickness variations and varying material properties of the cantilever into account, as shown in Figure 5. The rigid electrode can take on any shape describable by a vector defining the y-values at each position of the quantisized x-axis. A two-dimensional model is sufficient to describe typical MEMS actuators, and the depth of the electrodes in the z-axis does not affect the behavior of the actuator (neglecting fringe-field effects). y / µm 20 15 10 5 0 0

calculate the initial shape y(x,t=0), determined by pre-stress M0(x) and external forces FEXT(x)

calculate FEL(x,t) as a function of y(x,t)

calculate M(x,t) as a function of the current beam shape y(x,t), pre-stress M0(x), el.-stat. forces F EL(x,t) and external forces F EXT(x)

L calculate beam deflection M(x,t) y(x,t+1) = E I(x)

100

200

300

x / µm

time-step adaptation to max( y(x,t+1) - y(x,t) ) < ∆yMAX

400

Figure 2: Quasi-static deflection states of a cantilever successively bending along the stoppers on a curved electrode (only a few iteration steps are shown).

stable stable state detection not stable no contact

2.2. Quasi-static Modeling

contact detection

The mathematical description of the deflection is based on the differential equation [9]

y" ( x ) =

M ( x) E ⋅ I ( x)

contact occured calculate new constrains x0, y(x0), y’(x0)

(1) t = t+1

where M(x) is numerically determined by the electrostatic force distribution FEL(x), and optional by mechanical pre-stress M0(x) and by an additional external force distribution FEXT(x). The electrostatic force FEL(x) is the attracting force between the plates of distributed parallel-plate capacitors of the length of the x-axis discretization unit and with the plate separation defined by the distance between the rigid electrode and its flexible counterpart. The deflection of the flexible structure is calculated by double integration of equation (1). A flowchart of the algorithm is shown in Figure 3. For dynamic simulation, the algorithm is endowed with: - time-step adaptation for quasi-static modeling: the maximum change in deflection between each iteration step is limited to ∆yMAX, to ensure a good quantization of the highly nonlinear electrostatic force; the change of deflection at each iteration step is linearly scaled by ∆yMAX, if necessary. - detection of stable states: the algorithm terminates if the change in deflection between a number of iteration steps is below a certain limit ∆yMIN. - contact detection: the distance between at least one point of the moving structure and the rigid

display results

Figure 3: Basic flowchart of the quasi-static algorithm. III. PERFORMANCE COMPARISON 3.1. Comparison of pull-in results The performance of the algorithm in terms of predicting the dynamic pull-in behavior of typical micromechanical structures was compared to: - implicit, Rayleigh-Ritz based analytical descriptions of lateral curved electrode structures of different order as reported in the literature [1]. The deviation of the results obtained by the presented algorithm from results by the analytical description varies between -0.5% and -7.5% (Figure 4, summarized in Table 1). - vertically moving cantilevers modeled by FEM/BEM and FCM/BCM [8]: The pull-in voltages determined by the presented algorithm deviate from the simulated values by less than ±1.3% (Table 2).

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-

-

FEM simulation and measurement results of a laterally moving multilayer MEMS switch etched in the device layer of an SOI wafer and sputtercoated with an aluminum layer of a measured thickness of 600 nm, fabricated at NTU (Figure 5, results summarized in Table 3). The result of the proposed algorithm shows a much better agreement with the measured data (-2.5% deviation) as compared to the results from FEMsimulations (+25% deviation from the measured value) [10] measurement results of a vertically moving, prestressed thin-film zipper actuator fabricated for a MEMS switch at KTH (Figure 6): the pre-stress bending was modeled by using the measured total stress gradient of 39 MPa/µm of the 1 µm thick silicon nitride membrane coated with a 190 nm thick Cr/Au layer [5]. The pull-in voltage of the film was determined by the presented algorithm to be 49.6 V as compared to the measured value of 55 V (deviation of -9.8%). Also, the algorithm was used to predict the dynamic behavior of the film pulling-in and rolling over the substrate electrode, as shown in Figure 7. The etched pattern in the fabricated structure is not taken into account by the two-dimensional geometry of the algorithm which assumes a homogeneous film; however, this does not affect the actuator’s behavior since the membrane areas coated with electrode metal are fully congruent to the electrodes on the substrate.

Order of electrode shape N=2.0 N=1.5 N=1.0 N=0.5 N=0.0

analyt. model [1] 40 V 59.5 V 89.5 V 115.2 V 141.7 V

n=1

1A1.1 this algorithm 2.36 V

Table 2: Pull-in voltages of a vertically moving cantilever, determined by FEM/BEM and FCM/BCM [8]: beam length/width/thickness/initial gap = 80/10/ 0.5/0.7 µm, respectively; E=169 GPa.

275 µm

2.5 µm

(a)

4.8 µm

V

175 µm 2.8 µm

165 µm

(b) Figure 5: (a) SEM-picture and (b) schematic drawing of a laterally moving silicon MEMS switch, coated with 600 nm of aluminum on the side-walls, fabricated at NTU (all dimensions without the coating, [10]).

Pull-in voltages This Deviation algorithm 37 V -7.5% 56.6 V -4.9% 85.3 V -4.5% 114.6 V -0.5% 140.2 V -1.1%

Method measured [10] FEM-analysis [10] this algorithm

Pull-in voltage 19.2 V 24.0 V 18.5 V

Deviation to measurement N/A +25% -2.5%

Table 3: Measured values and FEM-simulation data of the switch shown in Figure 4, as compared to the simulation results of the presented algorithm.

Table 1: Simulation of laterally moving curved electrode actuators with different electrode shapes (see Figure 4), as compared to an analytical model [1]: beam length/width/thickness/electrode deflection/initial gap=500/2/5/30/2 µm, respectively; E=150 GPa.

n=0.5

Pull-in voltages FEM/BEM [8] 2.39 V

FCM/BCM [8] 2.35 V

n=1.5

n=2.0

Figure 4: Shapes of curved electrodes of different order (n=0 is the parallel electrode configuration).

Figure 6: SEM-picture of the zipper-actuator fabricated for a MEMS switch at KTH [5].

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single-side fixation of the film

35

initial shape, V=0 200 150 100

25 20

pull-in and rolling of the film along the substrate

50 0 0

20 18 16 14 12 10 8 6 4 2

30

200

400

600

800 1000 x in µm

15 10

rigid counter electrode on flat substrate

5

Figure 7: Quasi-static dynamic pull-in and rolling movement of the KTH zipper actuator as shown in Figure 6, simulated with the presented algorithm.

1.5 2

2.5 3 3.5 4 4.5 5 electrode shape order n

1A1.1 20 15 10 5 0 35

30

25

20

15

10

5 1.5

2.5

3.5

4.5

Figure 8: Two-parameter variation for optimizing the geometry of a curved electrode to achieve maximum deflection of a 2 µm thick, 300 µm long cantilever at an actuation voltage of 50 V.

3.2. Performance evaluation as a MEMS design tool agreement with measurements and with results obtained by other simulation methods. In contrast to FEM simulations which are very time-consuming when solving highly nonlinear contact problems, this algorithm typically takes only a fraction of a second for solving models with complex moving structures. Thus, this algorithm was found to be a very fast, versatile and powerful design tool for parameter-optimization of electrostatic actuator geometries.

Analytical descriptions for touch-mode or curvedelectrode actuators are very simplified and thus not suitable to be used as design tools. FEM/BEM/ FCM/BCM simulations typically take minutes to hours for solving a highly nonlinear contact problem with complex geometries. Thus, they are not practicable either for being used as a brute-force design tool with multi-parameter variation to optimize the geometry of the actuator. The algorithm presented in this paper takes only a fraction of a second to solve a model as shown in the Figures 2 or 7, even though it was implemented in the slow script-language of MATLAB executed on a standard personal computer. The capabilities of the algorithm are demonstrated on a two-dimensional design parameter variation. The electrode shape of a curved electrode actuator was optimized to achieve a maximum deflection of a 2 µm thick and 300 µm long cantilever at a given actuation voltage of 50 V. The shape of the electrode, covered with a 0.2 µm thick isolation layer and having a minimum distance of 2 µm to the rigid electrode, is described by the formula y(x)=yel,max·(x/L)n with yel,max and n the parameters to be optimized. For this example, the largest deflection of 20.3 µm was found at an electrode shape order of n=2.67 and a maximum electrode gap of yel,max=22.5 µm. Each parameter was varied by 15 values, and the solving of the 225 different designvariations of the curved-electrode actuator took no more than a few seconds on a computer based on a Pentium Centrino processor with 1.3 GHz. The achieved deflection of each configuration plotted over the two parameters is shown in Figure 8.

References [1] R. Lengtenberg, J. Gilbert, S. D. Senturia, and M. Elvenspoek, “Electrostatic curved electrode actuators,” IEEE JMEMS, vol. 6, no. 3, pp. 257-265, Sept. 1997. [2] Y. Hirai, M. Shindo, Y. Tanaka, “Study of large bending and low voltage drive electrostatic actuator with novel shaped cantilever and electrode”, Proc. Micromechatronics and Human Science, Nov. 25-28, pp. 161-162. [3] C. C. Cabuz et al, “Factors enhancing the reliability of touch-mode electrostatic actuators,” Sensors and Actuators A, vol. 79, no. 3, pp. 245-250, Feb. 2000. [4] I. Schiele, B. Hillerich, F. Kozlowski, C. Evers, „Micromechanical relay with electrostatic actuation”, Proc. Transducers ‘97, June 16-19, 1997, pp. 1165-1168. [5] J. Oberhammer and G. Stemme, “Design and Fabrication Aspects of a S-shaped Film Actuator Based DC to RF MEMS Switch”, IEEE Journal of Microelectromechanical Systems, vol. 13, no. 3, pp. 421-428, June 2004. [6] M. Shikida, K. Sato, and T. Harada, “Fabrication of an Sshaped microactuator,” IEEE Journal of Microelectromechanical Systems, vol. 6, no. 18, pp. 18-24, March 1997. [7] R. Jebens, W. Trimmer, and J. Walker, “Microactuators for aligning optical fibers,” Sensors and Actuators A, vol. 20, pp. 65–73, 1989. [8] G. Li and N. R. Aluru, “Efficient Mixed-Domain Analysis of Electrostatic MEMS,” Trans. ComputerAided Design of Integrated Circuits and Systems, vol. 22, no. 9, pp. 1228-1242, 2003. [9] J. Li et al, “DRIE-fabricated curved-electrode zipping actuators with low pull-in voltage,” Proc. Transducers 2003, Boston, MA, June 8-12, 2003, pp. 480-483. [10] A. Q. Liu, M. Tang, A. Agrarwal and A. Alphones, „Low-loss lateral micromachined switches for high frequency applications,“ J. Micromechanics and Microengineering, vol. 15, pp. 157-167, 2005.

IV. CONCLUSIONS The presented algorithm is capable of predicting the static and dynamic behavior of electrostatically actuated, single-side clamped structures. Since it can handle highly nonlinear models and contact problems, it is very suitable for simulating curved-electrode and touch-mode actuators. The algorithm shows good

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