2002-Int-ANSYS-Conf-84.PDF

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Thermo-Mechanical Analysis of a Multi-Layer MEMSMembrane

Heiko Fettig, PhD

James Wylde, PhDNortel 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 frequencyAC 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 shortcom-ings 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 mo-

tion and allow out-of-plane motion. There is a substantial amount of work in the literature involving bi-

morphed and multi-morphed structures for the realization of motion with MEMS devices. Much of it in-

volves the use of either pseudo bi-morph structures (by using geometry to concentrate the heat in one por-

tion 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, tomaximize the displacement of a multi-layer out of plane actuator subject to power and geometric con-

straints. The structure is formed by the deposition of two layers of polymer materials (with the propertiesdetermined 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/m) than typical silicon MEMS.


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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 dis-

placement dby finding the optimal thickness tj, the relative thermal coefficient of expansion (TCEj), and

material Youngs ModulusEj for each layer, as well as the opening size b, for a given power consumptionPand overall dimension a.

Metal

Lossy Polymer

MetalBase Polymer

Silicon Substrate

V~}

Figure 1 - Schematic of layers used for analysis of multi-morph membrane actuator struc-ture. Note that the alternating voltage is applied between the metal layers encasing the

lossy dielectric.

t4

t3

t2

t1

b

d

Figure 2 - Schematic of multi-layer structure showing the pertinent dimensions and de-flected 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 isgenerated 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):

2

02

1EP = (1)

Where

P=

Power generated per unit volume [W/m3]


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=

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 'and the loss tangent (or dissi-pation factor) given as:

=tan (2)

0=

Permittivity of free space equal to 8.8510-12F/m

E=

The electric field between the two metal layers in the lossy dielectric

Bartinkas (2000) gives a more thorough description of dielectric losses.

Analysis

Analysis Theory

Due to the number of variables (thickness, TCE, Youngs 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 thesolution 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 dis-placement dof 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 vol-umes. 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


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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 maximumtemperature and the maximum deflection of the top surface of the stack. It was desired to enhance thissolve 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 largermembranes (a> 1000m), 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 ele-

ments proved to result in a higher number of elements and DOFs than in the quarter model. This also re-

sulted 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 anaxisymmetric model of half the membranes cross-section using axisymmetric PLANE78/PLANE83 ele-ments 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 mem-

brane.

Analysis Results & Discussion

Figure 3 shows the displacement of a membrane with a side length a= 800m, an opening b= 480m withan input power of 200mW as calculated using a quarter solid model. Figure 4 shows the same displacementas calculated using an axisymmetric model (quarter model symmetric expansion was used).


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Figure 3 - Plot of deflected shape of multi-layer membrane simulated using a quarter solidmodel.

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 byadjusting the geometry shown in Figure 2. The sample plot is for a membrane with a side length a=

1100m, an opening b= 1000m with an input power of 200mW.


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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 t1= 8.46m.

Although the actual calculated displacements show a difference of 23% between the results from the axi-symmetric 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.

5 6 7 8 9 10 11 12 13

-7.4

-7.2

-7.0

-6.8

-6.6

-6.4

-6.2

-6.0

-5.8

-5.6

-5.4

-5.2

-5.0

-4.8

-4.6

8.4

6m

MembraneDisplacement[m]

Base Polymer Thickness [m]

Axisymmetric Quarter Solid

Fit (AS) Fit (QS)

Figure 6 - Plot of calculated membrane deflection for various base polymer thicknesses.


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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 relativelylow 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 axisymmetricand a quarter solid model. If a maximum allowable membrane temperature of T= 75C over ambient isassumed it can be seen from Figure 7 that the maximum allowable input power isPQS= 187mW for the

solid model andPAS= 215mW for the axisymmetric model. Using these power values in Figure 8 the

maximum possible membrane deflections are found to be dQS= -6.7m and dAS= -5.9m. This means that

although the maximum allowable power is over-predicted by the axisymmetric model the maximum

achievable displacement is still under-predicted.

0 25 50 75 100 125 150 175 200 225 250

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

-5.9m

215mWM

embraneDisplacement

[m]

Power Consumption [mW]


Fit (AS) Fit (QS)

-6.7m

187mW

Figure 7 - Plot of calculated membrane temperature versus input power.


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0 25 50 75 100 125 150 175 200 225 250

0

10

20

30

40

50

60

70

80

90

100

215mW

75C


Fit (AS) Fit (QS)

MembraneTemperatureAboveAmbie

nt[C]

Power Consumption [mW]

187mW

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 numberof 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 parame-

ters, like thickness and size, the axisymmetric model yields the same optimal parameter values as a quartersolid model, in considerably less time.

However, it was also shown, that the axisymmetric model constantly under-predicts the results of the quar-

ter solid model for membrane temperature and displacement. Although the under-estimate of the maximumachievable displacement is welcome in a conservative design approach, the under-estimate of the mem-

brane 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 influ-

ences and the use of a quarter solid model to calculate the temperature/displacement versus power curvesfor a given optimized membrane.

References

1. Bartinkas, R. 2000. Dielectrics and Insulators. In The Electrical Engineering Handbook. Boca Raton:

CRC Press.

2. Boley, B.A. 1960. Theory of Thermal Stresses. New York: John Wiley and Sons, Inc.

3. Hickey, R. M., M. R. Kujath, and T. J. Hubbard. 2002. Heat Transfer Analysis and Optimization of

MEMS Thermal Actuators.Journal of Vacuum Science and Technology A20 (2).


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4. Rashidian, B., and M. G. Allen, 1993, Electrothermal Microactuators Based on Dielectric Loss Heat-

ing. InProceedings of IEEE MEMS 93, Fort Lauderdale, FL, February 1993. San Diego: IEEE. 24-

29.

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2002-Int-ANSYS-Conf-84.PDF