Thermo-Mechanical Analysis of a Multi-Layer MEMS Membrane

Transcription

1 Thermo-Mechanical Analysis of a Multi-Layer MEMS Membrane Heiko Fettig, PhD James Wylde, PhD Nortel Networks - Optical Components Ottawa ON K2H 8E9 Canada Abstract This paper examines the modelling of a square four layer MEMS membrane for a combined thermal and structural analysis. The membrane consists of four dissimilar layers, two polymer and two metal layers. Only the bottom polymer layer is fixed to the substrate and only around the perimeter. The second polymer layer is sandwiched between two metal layers. Dielectric loss in this polymer induced by a high frequency AC voltage between the metal layers heats the membrane and deforms it. Due to the extreme aspect ratios of the membrane, up to 3,500:1 for the metal layers, solid modelling proves inefficient and alternatives are examined in this paper. It is shown that for this specific membrane set-up axisymmetric modelling yields results that are sufficient for parameter optimization. The shortcomings of this method are presented as well. Introduction Traditional (MEMS) micro-electro-mechanical systems devices are fabricated using silicon, either etched in the bulk material as with wet and dry bulk etching, etched in a device layer for silicon on insulator (SOI) devices, or deposited in polycrystalline form on a wafer surface. While silicon has proved to be a viable material for the fabrication of MEMS structures, devices tend to be limited to several microns of motion and limited to in-plane 2-D motion. It has been proposed that the fabrication of devices using polymer materials can offer larger ranges of motion and allow out-of-plane motion. There is a substantial amount of work in the literature involving bimorphed and multi-morphed structures for the realization of motion with MEMS devices. Much of it involves the use of either pseudo bi-morph structures (by using geometry to concentrate the heat in one portion of a structure; Hickey et al., 2001) or by using dissimilar materials (Rashidian and Allen, 1993). This paper will present an analysis technique for optimizing a finite element model, using ANSYS, to maximize the displacement of a multi-layer out of plane actuator subject to power and geometric constraints. The structure is formed by the deposition of two layers of polymer materials (with the properties determined from the simulation) and two metal layers. The actuation mechanism is similar to a bi-morph; however, the heat is provided by dielectric heating in a lossy polymer. The advantages of structures such as those proposed here include: 1. The structures allow out-of-plane motion, 2. Can be more readily fabricated than silicon MEMS, and 3. Can operate at a lower power-displacement metric (mw/µm) than typical silicon MEMS.

2 Multi-layer Actuator Membrane A model of an actuator formed from four layers of dissimilar material is desired (see Figure 1). When one layer of the actuator is heated (in this case the lossy polymer), the actuator bows out as shown in Figure 2. This effect is similar to the thermostat effect (Boley, 1960). It is desirable to optimize the maximum displacement d by finding the optimal thickness t j, the relative thermal coefficient of expansion (TCE j ), and material Young s Modulus E j for each layer, as well as the opening size b, for a given power consumption P and overall dimension a. Metal Lossy Polymer Metal Base Polymer Silicon Substrate } V~ Figure 1 - Schematic of layers used for analysis of multi-morph membrane actuator structure. Note that the alternating voltage is applied between the metal layers encasing the lossy dielectric. t4 t3 d t2 t1 b Figure 2 - Schematic of multi-layer structure showing the pertinent dimensions and deflected shape of the structure due to a heat load applied to the lossy dielectric (layer 3). Dielectric Heating Typical polymer materials experience dielectric losses in the presence of an alternating electric field due to the alternating orientation change of dipoles and the movement of charge carriers (Rashidian and Allen, 1993). These losses are realized as heat generated in the material. In the models presented here, the heat is generated in the devices by dielectric loss in the third (second polymer) layer by applying an alternating voltage across the two metal plates as shown in Figure 1. The specific heat generated in this layer is given as (Rashidian and Allen, 1993): Where P = Power generated per unit volume [W/m 3 ] 1 2 P = ω ε ε0 E (1) 2

3 ω = ε" = ε 0 = E = Circular frequency of applied voltage [rad/s] Imaginary part of the dielectric permittivity, sometimes called dielectric loss factor. Note that many materials are defined in terms of the dielectric constant ε' and the loss tangent (or dissipation factor) given as: Permittivity of free space equal to F/m ε δ = ε The electric field between the two metal layers in the lossy dielectric Bartinkas (2000) gives a more thorough description of dielectric losses. tan (2) Analysis Analysis Theory Due to the number of variables (thickness, TCE, Young s Modulus, dimensions) and because the analysis is a coupled thermal-structural analysis, it is desirable to optimize the device by developing a numerical model using ANSYS and examining the effect of each variable. Furthermore it is desirable to reduce the solution time such that a large range and number of cases for each variable can be examined. The types of models are outlined below. The models were analyzed using six steps: 1. Applying a heat load (through HGEN) to the lossy dielectric layer of the stack, 2. Performing a steady-state thermal analysis to determine the temperature distribution throughout the membrane, 3. Re-meshing the model with structural elements (through ETCHG), 4. Loading the temperature results of the previous analysis as body loads for the structural analysis, 5. Solving for the deflected shape of the membrane, and 6. Extracting the deflection curve along the centreline of the structure. As boundary condition, an isothermal surface at the interface between the bottom layer and the substrate was modeled. The heat load per unit volume was calculated using Equation (1), with the power dissipation calculated from the voltage, frequency, and dielectric properties (input as parameters, using *SET) and the known geometry. The output from the model was the maximum temperature in the device and the displacement d of the centreline as shown in Figure 2. Solid Model A solid model of the smallest of the membranes in question was constructed using four square block volumes. The model was meshed with more than 40,000 SOLID90/SOLID95 elements yielding approximately 150,000 DOF for the thermal model and 450,000 DOF for the mechanical model. These large numbers are the result of the high aspect ratios that are present in this model. Some of the structures to be simulated

4 feature overall aspect ratios of up to 200:1 (lateral dimension to thickness) with aspect ratios of some of the layers being up to 3,500:1. To avoid running into shape violations flagged by ANSYS the maximum aspect ratio of the brick elements has to be 20:1 or less. Therefore the maximum element size in lateral direction is governed by the thickness of the thinnest layer. The thinner the layer, the smaller the lateral element size, the more elements are needed. Solve time on a 650MHz desktop PC with 384MB RAM running Windows 2000 was approximately 1000 minutes for the combined thermal and structural analysis. The outputs from the model were the maximum temperature and the maximum deflection of the top surface of the stack. It was desired to enhance this solve time to speed up the optimization process for the model. Several methods for enhancing the solve time are presented. Performance Enhancement Symmetric Model Observation of the solid model reveals that it is symmetric about 2 axes. A quarter model was developed with similar boundary conditions applied along two edges and symmetric boundary conditions applied along the other two edges. This reduced the number of elements and DOF by a factor of four. In return the solve time per iteration was reduced to 60 minutes. The advantage of the symmetric model is that the model is developed similarly to the full 3-D model by entering the geometry into PREP7, meshing with brick elements, and solving. For examination of larger membranes (a > 1000µm), however, even the quarter model with 8-node brick elements (SOLID45/70) resulted in models with over 400,000 DOF. These models were not economic for large number of variation optimization runs. A one-eighth model of the membrane was also considered but the meshing of the triangle with brick elements proved to result in a higher number of elements and DOFs than in the quarter model. This also resulted in a longer solve time and was therefore not examined any further. Axisymmetric Model Although the quadratic membrane examined in this paper is not axisymmetric, initial tests showed that an axisymmetric model of half the membrane s cross-section using axisymmetric PLANE78/PLANE83 elements produced results under-estimating the solid model by 10-30% (lower for larger membranes). In spite of the fact that the axisymmetric model describes a circular membrane rather than a square membrane, the general trends for the behaviour of the membrane were found to be the same. The axisymmetric model, however, solves in about one minute, which makes the calculation of a wide variety of scenarios possible. Using the axisymmetric model it was possible to quickly examine variations in layer thickness, membrane size, overlap size and layer materials, in order to derive guidelines for the behaviour of the square membrane. Analysis Results & Discussion Figure 3 shows the displacement of a membrane with a side length a = 800µm, an opening b = 480µm with an input power of 200mW as calculated using a quarter solid model. Figure 4 shows the same displacement as calculated using an axisymmetric model (quarter model symmetric expansion was used).

5 Figure 3 - Plot of deflected shape of multi-layer membrane simulated using a quarter solid model. Figure 4 - Plot of deflected shape of multi-layer membrane using an axisymmetric model. Figure 5 shows a plot of the displacement along the centreline of a deflected membrane (extracted from an ANSYS result with PATH commands). For design purposes, it is desirable to maximize this deflection by adjusting the geometry shown in Figure 2. The sample plot is for a membrane with a side length a = 1100µm, an opening b = 1000µm with an input power of 200mW.

6 Figure 5 - Plot of deflected shape along the centreline of the multi-layer membrane In order to optimize the base polymer thickness of this membrane for maximum deflection, a number of variations were calculated using both an axisymmetric and a quarter solid model. As can be seen from the graph shown in Figure 6 both models show a maximal deflection for a base polymer thickness t 1 = 8.46µm. Although the actual calculated displacements show a difference of 23% between the results from the axisymmetric and the quarter solid model, both models give the same optimal base polymer thickness for the membrane. This effect was observed for other parameters as well. Membrane Displacement [µm] µm Base Polymer Thickness [µm] Axisymmetric Fit (AS) Quarter Solid Fit (QS) Figure 6 - Plot of calculated membrane deflection for various base polymer thicknesses.

7 Comparisons between axisymmetric and quarter solid models have shown that the axisymmetric model constantly under-predicts the results of the quarter solid model. For the displacement result this is not a problem, in fact it is rather welcome since it makes the estimate more conservative. The under-prediction in the temperature result, however, can lead to problems since the temperature of the device for a given power will actually be higher than the value predicted by the axisymmetric model. Since polymers have relatively low melting points an overheating of the membrane could mean its destruction. Figures 7 and 8 show graphs of membrane displacement and temperature versus power for an axisymmetric and a quarter solid model. If a maximum allowable membrane temperature of T = 75 C over ambient is assumed it can be seen from Figure 7 that the maximum allowable input power is P QS = 187mW for the solid model and P AS = 215mW for the axisymmetric model. Using these power values in Figure 8 the maximum possible membrane deflections are found to be d QS = -6.7µm and d AS = -5.9µm. This means that although the maximum allowable power is over-predicted by the axisymmetric model the maximum achievable displacement is still under-predicted Membrane Displacement [µm] µm µm mW 215mW Power Consumption [mw] Axisymmetric Fit (AS) Quarter Solid Fit (QS) Figure 7 - Plot of calculated membrane temperature versus input power.

8 100 Membrane Temperature Above Ambient [ C] C 187mW 215mW Power Consumption [mw] Axisymmetric Fit (AS) Quarter Solid Fit (QS) Figure 8 - Plot of calculated membrane deflection versus input power. Conclusion The modelling of a multi-layered square membrane structure for the purpose of optimizing its geometry to yield maximum deflection was examined in this paper. It was found that the cycle time for a given number of optimization iterations could be reduced by a factor of 16 by simulating the solid model as a 1/4 model and by a factor of 1,000 by using an axisymmetric model. It was shown that for the optimization of parameters, like thickness and size, the axisymmetric model yields the same optimal parameter values as a quarter solid model, in considerably less time. However, it was also shown, that the axisymmetric model constantly under-predicts the results of the quarter solid model for membrane temperature and displacement. Although the under-estimate of the maximum achievable displacement is welcome in a conservative design approach, the under-estimate of the membrane temperature for given power can lead to over-heating and therefore destruction of the membrane. Given that the purpose of the simulation was to understand the influence of geometric parameters on the membrane deflection, the authors recommend the use of an axisymmetric model to examine these influences and the use of a quarter solid model to calculate the temperature/displacement versus power curves for a given optimized membrane. References 1. Bartinkas, R Dielectrics and Insulators. In The Electrical Engineering Handbook. Boca Raton: CRC Press. 2. Boley, B.A Theory of Thermal Stresses. New York: John Wiley and Sons, Inc. 3. Hickey, R. M., M. R. Kujath, and T. J. Hubbard Heat Transfer Analysis and Optimization of MEMS Thermal Actuators. Journal of Vacuum Science and Technology A 20 (2).

9 4. Rashidian, B., and M. G. Allen, 1993, Electrothermal Microactuators Based on Dielectric Loss Heating. In Proceedings of IEEE MEMS 93, Fort Lauderdale, FL, February San Diego: IEEE