MEMS+ for Scanning Mirror

Rapidly Analyzing Parametric Resonance and Manufacturing Yield of MEMS 2D Scanning Mirrors using Hybrid Finite-Element / Behavioral Modeling Sandipan Maitya, Shuangqin Liub, Stephane Rouvilloisc, Gunar Lorenzc, Mattan Kamonb a Coventor Inc, 465 S Mathilda Ave., Sunnyvale, CA, USA 94086; b Coventor Inc, 625 Mount Auburn St., Cambridge, MA, USA 02138; c Coventor SARL, 3 avenue du Quebec, Zl de Courtaboeuf, 91140 Villebon sur Yvette, France ABSTRACT A new hybrid 3D finite-element/behavioral-modeling approach is presented that can be used to accurately predict the nonlinear dynamics (parametric resonance) in electrostatically driven 2D resonant MEMS scanning mirrors. We demonstrate new levels of accuracy and speed for thick SOI scanning mirrors with large scanning angles and validate the modeling approach against measurement on a previously fabricated scanning mirror. The modeling approach is fast and treats the design parameters as variables thus enabling rapid design iterations, automatic sensitivity and statistical yield analyses, and integration with system and circuit simulators for coupled MEMS-IC cosimulation. Keywords: Parametric Resonance, Scanning Mirrors, Electrostatic Actuation, MEMS Modeling, Manufacturing Yield, System Simulation

1. INTRODUCTION Silicon resonant scanning mirrors are being increasingly used in various applications like laser printers, barcode scanners, projection displays, tunneling microscopes, etc. Designers often rely on analytical techniques or FEM based reduced-order modeling techniques to characterize and optimize performance for these devices. Predicting performance of electrostatically actuated resonant 2D scanning mirrors with out of plane motion involves solving for the dynamic response from the nonlinear electrostatic and mechanical governing equations. Analytical solutions for the response exist only for overly simplified forms of the governing equations that neglect important effects such as the large angle fringing field effects along electrostatic comb fingers. Using a finite-element (FEM) or boundaryelement method (BEM) could solve for the full response, but because of the complexity of the analysis involved, the computation time is prohibitive and such an approach is rarely used. Another approach is to use FEM simulations to populate the coefficients of a reduced-order model. However, such an approach suffers from requiring re-extraction of the coefficients for even minor design changes, thus limiting its usefulness to late in the design cycle when the design is close to be finalized. This paper presents a hybrid modeling methodology that attains speeds similar to reduced-order modeling but with sufficient geometric and physical complexity to model non-linear dynamics associated with parametric resonance in scanning mirror designs. The approach combines high-order finite elements for mechanics with semi-analytic components for electrostatics forces including fringing fields. This methodology has been implemented in the commercial MEMS/IC co-design platform named MEMS+ [13] [18]. This new approach overcomes the limitations of traditional reduced-order models and classical FEM techniques by using a library of MEMS specific parameterized elements with built in non-linear mechanics and non-linear electrostatic behavior in conjunction with industry standard ODE solvers. Details and the background of the hybrid modeling methodology are given in Section 2. The methodology is then used in Section 3 to predict the parametric resonance behavior of a electrostatically actuated resonant scanning mirror. Accuracy of the model is verified using FEM and experimental results. Statistical calculations can be done to simulate yield based on wafer-level variations.

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2. APPROACH 2.1 Hybrid / FEA Approach In classical MEMS behavioral modeling methodologies [8-12], devices were composed of basic building blocks like beams, plates, and comb fingers represented as symbols in an electrical circuit editor. In MEMS+, the device is instead created and assembled in a 3D schematic editor with full 3D rendering of its actual geometry. Components are then automatically connected based on geometric location. The resulting 3D view differs from traditional 3D CAD modeling tools in that there is an underlying behavioral model associated with each MEMS+ component that compares in accuracy with FEM models. The MEMS+ component library consists of different classes of electromechanical elements: rigid plates, flexible plates, beams and suspensions. Each of these models have high-order finite element equations like beams based on Bernoulli and Timoshenko theories, rigid mass based on its 3D kinematic equations, flexible plates that are described by shells of Mixed Interpolation of Tensorial Components (MITC) [15], as well as plates described by full 3D brick FEM models. Electrostatic forces can be modeled with a simple parallel plate formula only in ideal cases. In MEMS+, conformal mapping techniques [17] enhanced with numerical quadrature are used to model electrostatic field patterns and forces in complicated multi conductor configurations. Results comparison between MEMS+, FEM and experimental measurements for an electrostatic contact RF Switch was already demonstrated [19]. Among the main advantages of this methodology is the complete parameterization over the design space, including the 3D model, its HDL representation, the 2D layout, materials and process. For instance, the materials editor provided, allows for every property to be defined in terms of variables, algebraic equations of other properties, environmental variables (e.g. temperature and atmospheric pressure) or even entirely abstract variables such as the equipment settings of a given fabrication process. The process editor can be used to define the sequence of MEMS fabrication steps including process stack order, material type, thickness and sidewall profile, etc. The separation of the materials and process data from the MEMS design and simulation environment allows the model to take into account effect of individual foundry vendors on performance. Similarly parameters of the MEMS building blocks (length, width, number of comb fingers, etc) can be set as variables and equations by the MEMS designer for design variation studies. As part of a specialized 3D environment, sophisticated elements are provided to represent a design which is accurate and fast to simulate. For instance, in Figure 1a, the multi-layer shell component that is needed to model complex effects like mirror surface warpage is a single component, but consists of multiple stacked mixed interpolation shell elements [15] whose vertically aligned nodes are constrained to move together to reduce the size of the system. Figure 1c illustrates warping of the cross section of a beam under torsion modeled traditionally with over 1000 finite-element bricks. A single beam component can model the warping of this beam [16] which is required for modeling high aspect ratio suspensions, as is common today for scanning mirrors fabricated with deep reactive ion etching (DRIE). Figure 1b shows beam deflection not normal to beam axis based on Timoshenko theory. Figure 1d shows the parameters to specify a beam of varying-width which can be used, for example, to model a fillet with a single component.

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Figure 1. Specialized FEM Models for MEMS.

Comb-finger models included in MEMS+ calculate electrostatic fringing fields and forces using conformal mapping techniques coupled with numerical quadrature. Fig 2a shows examples of such mapping configurations while Fig 2b shows comb cell integration techniques employed to reduce time of simulation for large number of combs.

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(b) Figure 2. Electrostatic Field Computation in Comb Fingers

One of the key challenges in modeling an out-of-plane comb-actuated resonant scanning mirror is accurate modeling of the capacitive forces when the moving comb finger has completely disengaged from the stator. For large angle deflections, this affects a majority of the actuating fingers. We present here in Fig 3, a comparison of out of plane vertically disengaged electrostatic forces and moments between MEMS+ and 3D BEM based electrostatic solver included in CoventorWare [14]. Analytical solution [4] is also presented here which is valid only in the range where there is some overlap between the combs.

Figure 3. Electrostatic Fringing Force & Moment in a Single Comb Cell with Comb Finger Translation & Rotation

The CoventorWare 3D BEM Stokes solver can be used to estimate the rotational damping associated with fingers sliding. Appropriate Knudsen number corrections can be used to take into account slip and transition flow effects. The results obtained can be used within MEMS+ platform.

The entire 3D component-based model can be placed as a single symbol in an electrical schematic for MEMS-IC cosimulation in Cadence-Spectre for transistor level modeling, or in Simulink for block level control simulation. The schematic can also be imported into MATLAB scripting environment for fast MEMS analysis as demonstrated in this

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paper. Finally Verilog-A reduced order model export option exists for simulation in any Verilog compatible circuit simulator.

3. RESULTS 3.1 Sample Device The 2D scanning mirror is based on the HKASTRI device [7]. The fabrication process is described in [6].

Figure 4. Optical pictures of the HKASTRI 2D Scanning Mirror

The device used in this study is the dual axis circular mirror as shown in Fig 4. It has two by two sets of comb fingers, a pair set for center mirror rotation about the Y axis while another pair for outer gimbal rotation about the X axis. The MEMS+ model representation of the device was built using a total of 2 MEMS+ plates, 4 beams and 8 comb electrostatic elements as shown in Fig 5.

Figure 5. MEMS+ Model of the Circular Dual Axis Mirror

The model has a total of 12 mechanical degrees of freedom and 76 numerical integration points of the conformal mapping based electrostatic forces. To validate the model, we check the linear mechanics by comparing to FEA modes simulated in the commercial FEM software package CoventorWare. We demonstrate that the MEMS+ model is valuable not only for evaluating device performance but and also coupled circuit and device performance.

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3.2 Verification of Mechanics The mechanics of the mirror model is first validated against FEM. Shown here in Table 1 is the comparison of resonant frequencies between MEMS+ and FEM as well as experimental measurements. The results show good agreement.

Figure 6. Simulated Modes of Vibration, Gimbal Rotation RX Mode & Mirror Rotation RY Mode

Table 1. Comparison between MEMS+, FEM & Experiments Gimbal RX Mirror RY

Experimental Meas. 1.49 kHz 18.8 kHz

FEM 1.50 kHz 18.4 kHz

MEMS+ 1.50 kHz 20.4 kHz

3.3 Damping CoventorWare 3D BEM Stokes solver is used to estimate the rotational gas damping associated with fingers sliding. Fig 7(a) shows the damping coefficients obtained as a function of scanning angle position of the gimbal. The damping force distribution on comb forces is shown in Fig 7(b).

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Figure 7. Rotational Damping Coefficient with Scanning Angle & Damping Force Distribution on the Comb Fingers

3.4 Parametric Resonance Scanning mirrors of this type exhibit parametric resonance due to the particular dependence of the electrostatic drive on both position and time [2]. A periodic excitation at frequencies f=2f0/n for n=1,2,... is capable of exciting the resonance frequency, f 0. For simplifications to one degree-of-freedom and linear electrostatic force, theory for the Mathieu equation

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as described in [2] can be used to predict the behavior. Unfortunately, such analytic solutions do not exist for the general case of multiple degrees of freedom, arbitrary periodic input, and strongly nonlinear position dependence as is the case with this scanning mirror application. To determine the frequency response for this mirror, the full MEMS+ model described previously with 12 degrees-of-freedom and electrostatic combs with angle dependence as shown in Fig 3 was simulated with a sinusoidal voltage input of varying frequency. A transient simulation with upward and then downward frequency sweep was used to simulate parametric resonance and its hysteresis behavior. Fig 8 below shows the angle response near the resonance frequency for a periodic source of double the frequency (n = 1). The period source was applied to the stator combs of the gimbal (left) and mirror (right).

Figure 8. Simulated Parametric Resonance Response for Gimbal RX Mode & Mirror RY Mode & Time Domain response

3.5 Closed-loop control Appropriate control of the mirror drive frequency is necessary to achieve maximum scanning angle given the hysteretic frequency response. To design a closed-loop control system, an accurate model of the nonlinear dynamics must be included in system simulators such as Mathworks Simulink or circuit simulators such as Cadence Spectre. Typically, however, simplified models are used which miss aspects of the dynamics, or extracted models are used which require reextraction whenever the design dimensions change. The MEMS+ model described in the previous section suffers from neither of these issues. It is capable of capturing the nonlinear dynamics as shown in the previous section on parametric resonance and requires no extraction at all since the model definition is in terms of design dimensions, not extracted parameters. To demonstrate, a MEMS+ instance was inserted into a Mathworks Simulink model containing a digital phase-lock loop (dPLL) as shown in Fig 9a. The MEMS+ instance is an automatically generated symbol with

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appropriate inputs, outputs, and design variables as specified by the model creator when the model was built. The underlying representation is a Simulink s-function that calls into the native MEMS+ library. The s-function presents to Simulink a system of nonlinear ordinary differential equations representing the mirror. For simulation, the numerically controlled oscillator (NCO) of the dPLL was set with a center frequency of 25kHz which corresponds to a 50kHz drive signal. The results are shown in Fig 9b. Since 50kHz is well above the peak in the parametric resonance response shown previously, the dPLL adjusts the frequency down to near twice the resonance to achieve peak amplitude. Note that the dPLL required roughly a 25% duty cycle so the amplitude of the angle response is less than shown in the previous section.

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Figure 9. MEMS+ mirror model controlled by dPLL. (a) Simulink Model with MEMS+ mirror model and dPLL. (b) Results for an NCO center frequency of 25kHz. The top graph is the "ctrl" input to the NCO which results in a drive frequency shown in the middle graph which drives the mirror to the amplitude shown.

3.6 Rapid Design Iterations Rapid design optimization can be key to reducing time to market and meeting performance specifications. Because geometry and material parameterization are implemented within the MEMS+ structure, design iterations can be performed rapidly to explore design space and optimize a given design. Shown here is the effect of the suspension length and width variations on the resonance frequencies.

Figure 10. Simulated Effect of Geometrical Parameters on Resonance Frequency

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3.7 Sensitivity and Yield Similarly with available fabrication wafer level variance, a typical statistical analysis can be run to evaluate cross wafer variations leading to yield studies. In Fig 11, a simulation of two parameters hypothetical variation across a wafer is run to show how that affects the resonance variation of the mirror and gimbal across the wafer. This entire study can be finished in an hour in a standard laptop leading to significant cost savings during manufacturing.

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Figure 11. (a) Hypothetical Wafer Map of % change in Silicon Layer Thickness, (b) Hypothetical Wafer Map of % change in Young's Modulus, (c) Simulated Wafer Map of Gimbal RX Mode Resonant Frequency, (d) Simulated Wafer Map of Mirror RY Mode Resonant Frequency

4. CONCLUSIONS & FUTURE WORK It has been shown that a novel hybrid FEM behavioral modeling method can accurately capture the highly complex non linear behavior associated with electrostatically actuated resonant micro mirrors. Easy model creation in 3D along with seamless circuit and script level integration allow for a truly multi-physics multi-domain analysis. Very fast parameter variation allows for optimization studies and statistical analysis to be employed to predict yield. Our plans to enhance this software for scanning mirror modeling include introducing sliding gas damping models for comb fingers as well as models to calculate electrostatic actuation between two moving sets of comb fingers.

5. ACKNOWLEDGEMENTS The authors would like to thank Professor N. Wong, Dr. Y. Zhang, and their students at The University of Hong Kong for the PLL design and Dr. W. Ma formerly from Hong Kong Applied Science and Technology Research Institute Company Limited for the mirror design and measurement data. The developments presented in this article have been

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partly obtained thanks to the project SMAC (SMART systems Co-design) funded by the European Community's Seventh Framework Program under grant agreement no. 288827.

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