GENERAL CONTACT AND HYSTERESIS ANALYSIS OF MULTI-DIELECTRIC MEMS DEVICES UNDER THERMAL AND ELECTROSTATIC ACTUATION

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1 GENERAL CONTACT AND HYSTERESIS ANALYSIS OF MULTI-DIELECTRIC MEMS DEVICES UNDER THERMAL AND ELECTROSTATIC ACTUATION Yie He, James Marchetti, Carlos Gallegos IntelliSense Corporation 36 Jonspin Road Wilmington, MA USA ABSTRACT Many MEMS devices which are thermally and/or electrostatically actuated contain moveable components which may undergo contact. The behavior is further complicated when hysteresis is exhibited after contact [1,2]. Generally, MEMS devices consist of multiple materials with different mechanical and electrical properties. These factors pose a challenge for modeling highly nonlinear MEMS behavior. This paper presents an improved approach to modeling contact and hysteresis between any two bodies in a MEMS device. In particular, the three assumptions are relaxed. First, previous methods for modeling contact were able to model only a single dielectric layer. The new approach is capable of modeling multi-dielectric layers. Second, previous modeling methods required the specification of contact faces. This restriction has been generalized to specifying contact entities. Third, previously, modeling methods required an air stop gap assumption above a conductor to avoid numerical difficulties in electrostatic calculations. The new approach developed removes the air stop gap assumption. More realistic mechanical contact is simulated. Electrostatic and thermal driven actuation is considered. A number of case examples is presented to demonstrate the utility of the new contact and hysteresis approach. First a suspended test-beam structure is analyzed under different sets of boundary constraints and loading. Then, contact of an electrostatically actuated deformable mirror is analyzed. MODELING CHALLENGES AND SOLUTION APPROACH General contact and hysteresis analysis poses a challenge for MEMS modeling. Several Presented at the 1998 International Mechanical Engineering Conference and Exposition, Symposium on MEMS, November 15-20, Anaheim, CA USA. improvements on analysis techniques are presented in this paper. These improvements have been implemented in the [IntelliSuite TM ] MEMS analysis tool. [IntelliSuite TM ] utilizes an iterative approach to couple the electrostatic and thermomechanical solvers together. The result is an analysis tool which is capable of handling more general contact and hysteresis problems. Modeling of multiple dielectric layers Many commercial modeling software packages for MEMS rely on electrostatic boundary solvers such as FASTCAP [4]. Such solvers are capable of handling multiple dielectric layers. However, the difficulty arises in modeling the physical topology of multiple dielectric layers during contact. Before contact, a void separates the bodies, while after contact, the separation gap includes the void as well as the contact surfaces. Proprietary algorithms which take into account moving/deforming boundaries during contact have been implemented. Specifying contact entities vs. contact faces Many CAD software packages for MEMS require the user to specify the contact faces prior 1

2 to performing a contact analysis. The contact faces may be difficult to determine, depending on the topology of the model. The examples presented in this paper make use of an algorithm which allows the user to specify the contact entities. The algorithm automatically determines the contact faces. Simulation of true mechanical contact in coupled electro-mechanical problems Mechanical contact analysis requires the specification of a numerical tolerance in finding the contact body. In coupled electro-mechanical problems, this numerical tolerance often causes erroneous electrostatic results due to interpenetration of contact bodies. Previous modeling methods have required the definition of an artificial air stop gap between the contact bodies to prevent this problem. [3] The results presented in this paper are based on an algorithm developed which exactly locates the point of contact, while still preserving the true mechanical stress distribution. This allows an accurate calculation of electrostatic results at the point of true mechanical contact. Stiction and friction effects are not considered in the models. RESULTS With this improved analysis method, the following simulations can now be successfully conducted in a general manner: (I) Electrostatically driven contact and hysteresis analysis with multidielectric materials (II) Thermally driven contact analysis (III) Combined electrostatically and thermally driven contact and hysteresis These simulations are demonstrated in the following two examples. The first example considers a suspended test-beam structure with different sets of material properties. The second example considers an electrostatically actuated deformable mirror structure. Example 1: Suspended beam under thermal and/or electrostatic loads Figure 1: [IntelliSuite TM ] model of beam structure suspended over a layer deposited ground plane The solution approach presented in this paper is used to analyze contact of a suspended silicon beam (fixed at one or both ends) with a ground plane as shown in Figure 1. The ground plane is covered by a layer which is a dielectric/insulator. The beam length, width, and thickness are 80 µm, 10 µm, and 0.5 µm, respectively. The gap between the suspended beam and the dielectric layer is 0.6 µm. The beam material properties are E=169 GPa, ν = 0.25, and α = * 10-7 / o C. For comparison, the geometric layout in this example matches previously published results [3]. Voltage loads are applied to the silicon ground plane and the polysilicon beam. For one of the analyses performed with this beam model, a temperature gradient is applied across the top and bottom faces of the suspended beam in order 2

3 to actuate the structure. Figure 2 shows a schematic representation of the cross-section of the fixed-end beam being electrostatically actuated into contact Dielectr ic Const. = 1.0 Dielectr ic Const. = Applied Voltage (V) Figure 2: Schematic of electrostatically actuated contact phenomena. Voltage increases from V0 to V3 The following analysis were performed using the suspended beam model: Case (I) Electrostatic contact and hysteresis with multi-dielectric materials. For case (I), the beam was fixed at both ends. Two different analyses were performed, each with a different dielectric materials as the layer deposited on the silicon ground plane. The first analysis assumes that the layer deposited on the ground plane is equivalent to air from an electrostatic point-of-view; that is, the dielectric constant = 1, and the analysis considers only this single dielectric layer. This set-up parallels previously published results [3]. The second analysis assumes that the layer deposited on the ground plane is silicon oxide with a dielectric constant of 3.9. Air is modeled with a dielectric constant = 1. In the analysis, the voltage of the ground plane is set to 0V and the voltage of the polysilicon beam is ramped up from 0V to 20V. Figure 3: Capacitance between clamped beam and silicon ground plane due to applied voltage loads Figure 3 shows the variation of capacitance versus applied voltage to the beam. The sharp increase in capacitance occurs when the beam is pulled down to the layer covering the silicon ground plane. Clear differences can be seen between the behavior of the structure when the dielectric constant of the insulating layer changes. For example, the pull-in voltage is approximately 14.2V when the dielectric constant is 3.9 versus approximately 17.0V when the dielectric constant is 1.0. In addition, differences of approximately 300% can be observed in capacitance after contact. Results indicate that accounting for dielectric materials can play a key role in determining the contact behavior. Previous approaches which modeled a dielectric device with a dielectric constant = 1 cannot accurately predict the contact behavior of multidielectric devices. 3

4 Case (II) Thermally driven contact In the thermally driven contact analysis example a single fixed-end cantilever beam was considered. The beam was actuated by applying a temperature gradient through its thickness. A temperature of 200 o C and 27 o C was applied to the upper and lower surfaces of the beam, respectively. A linear temperature gradient was assumed. Figure 4 shows a plot of the x-location of the beam versus z-displacement. The first 30 µm of the beam have contacted the layer deposited over the ground plane. Z-displacement (microns) X-location on beam (micron) Figure 4: Location of the bottom of the deformed beam due to thermal actuation the polysilicon beam was the same as that in Case (II), the applied voltage to the ground plane was 0V, and the applied voltage to the polysilicon beam was ramped up from 0V to 12V and back down to 0V. Figure 5 shows the hysteresis curves resulting from actuating the beam fixed at both ends. As the voltage is ramped up, the beam is gradually attracted to the ground plane until a instability point is reached. At this point (between 11 and 12 volts), the pull-in voltage is reached and the beam snaps down to the layer insulating the silicon ground plane. When the voltage is then decreased, the beam remains in contact due to the electrostatic attraction with the ground plane until another instability point is reached when the electrostatic forces of attraction equal the mechanical force of the deformed beam. At this point (between 4.5V and 5V), the beam releases back to its pre-contact position. Z-displacement (microns) V -V Case (III) Combined electrostatic and thermal contact and hysteresis For this case, the suspended beam was fixed at both ends. The dielectric constant of the layer above the ground plane was taken as 3.9. The actuation resulted from both thermal and electrostatic loads. The temperature gradient in Applied Voltage (V) Figure 5: ThermoElectroMechanical hysteresis due to electrostatic and thermal actuation 4

5 Example 2: Deformable membrane under electrostatic and mechanical loads A circular polysilicon membrane suspended over a dielectric-covered silicon ground plane is considered. The mirror membrane has a diameter and thickness of 100 µm and 10 µm, respectively. The gap between the suspended beam and the dielectric layer is 2 µm. The dielectric layer thickness is 2 µm with a dielectric constant of 3.0. The membrane material properties are E=169 GPa and ν = Voltage loads of 100V and 0V were applied to the membrane and silicon substrate, respectively. In addition, a pressure load of 3.0 MPa was applied to the top face of the membrane. The number of iterations was set to 15 in the electro-mechanical solver. Figure 6 shows a 3D visualization of the electrostatic charge density on the membrane surface after contact. The maximum charge density is 651 Coulomb/m 2 which occurs at the center of the membrane where contact is made with the dielectric layer. Figure 6: [IntelliSuite TM ] results showing electrostatic charge density on the membrane surface Figure 7 shows the convergence curve for this analysis. The results (capacitance shown) are converged after eight mechanical-electrostatic iterations. Figure 7: ElectroMechanical convergence curve showing normalized capacitance between the silicon ground plane and mirror membrane SUMMARY This paper demonstrate the progress made in CAD tools used to model the behavior of physically complex MEMS. The ability to accurately model contact of entities in MEMS devices has been increased, and novel types of simulations can now be performed including fully coupled thermo-electro-mechanical contact analysis with full incorporation of multidielectric effects. The air stop gap assumption has been removed by means of an algorithm which deals with the numerical tolerance at the contact point. Simulations can be performed on models which more closely emulate physical reality. Improvements in accuracy of up to 300 % are observed in test case examples. ACKNOWLEDGMENTS 5

6 This work was supported by the Air Force Research Laboratory, Rome Research Site. REFERENCES [1] X. Sun, X. Gu, and W. Carr, "Lateral In-plane displacement microactuators with combined thermal and electrostatic drive", Tech. Digest 1996 Solid-State Sensor and Actuator Workshop, Hilton Head, SC., June 1996, pp [2] K. Sato and M. Shikida, "Electrostatic Film Actuator with a large vertical Displacement", Proc. MEMS 92, Travemunde, Germany, pp 1-5 [3] J.R.Gilbert, G.K. Ananthasuresh, and S.D.Senturia, "3D Modeling of contact problems and hysteresis in coupled electromechanics", Proc. MEMS'96, pp [4] K. Nabors, S. Kim, J. White, S. Senturia, FastCap USER s GUIDE, Department of Electrical Engineering and Computer Science, September