.

Deformation in thin SiN membranes due to residual stresses

B.G.R. van Schaik, ID:0491100

November 9, 2005

MT05.48

Ecole Polytechnique Federal Lausanne

LMIS1 - Microsystems and nanoengineering

Coach: M.A.F. van den Boogaart

AbstractFor the patterning on ’unconventional’ surfaces and/or non-IC applications new micro- and nanopat-terning technologies are developed. One of these techniques is the localized material deposi-tion through ultra-miniature shadow masks (nanostencils). Nanostencil lithography is a patternmethod, which uses a membrane to shield the substrate from material flux during deposition ona substrate. In this project a partial solution will be given for one of the known problems withinthis stencil lithography method. This report focusses on the problem of the existence of residualstresses in the SiN membranes and film material after an evaporation process.

These residual stresses are causing deformations in the membrane through which the gap size be-tween substrate and membrane will increase. These different gap sizes are influencing the sharpnessand sizes of the deposited structures on the substrate. To decrease the deformations in the mem-brane stabilization techniques are developed. In this project the deformations in a membrane,caused by residual stresses, are described within a finite element model (FEM). Second the stabi-lization techniques are implemented in the model to make an optimized design for the nanostencils.

First the experiments, done on a SiN membrane, are analyzed. From these experiments the valuesof the residual stresses present in the materials are calculated. The results of an experiment on amembrane, wherein cantilever structures are present, are used as verification when developing 2Dand 3D models. Cantilever shapes are used, because theoretical verification and relative simplemodeling is possible with these structures.

To obtain a model that is describing the experiments accurate, different methods for the imple-mentation of residual stresses into the models are developed and compared with the experimentalresults. Four methods are evaluated (an ANSYS option, surfacestress and two different temper-ature difference methods). The method wherein the stresses are described by implementing atemperature difference between two materials has got the smallest error comparing the resultswith the experiments (±20%).

The 2D cantilever models are expanded to a 3D model. In this 3D model an element type is usedthat is still giving an accurate solution when using large sizes and aspect ratio’s in the models.The behavior of the element type is verified. An optimalization of the geometry of the model isdone and the optimized model is compared with the experimental values. The differences of the3D models are within 10% of the experimental results.

A 2D silicon stabilization cantilever model is developed to see the reduction of the deflections com-pared to the unstabilized cantilevers. In this stabilized model a simple calculation of the stressworking on the silicon is made and this stress is implemented by calculating the temperature dif-ferences. The approach is verified with the use of different simulations. The verified 2D models areexpanded to 3D models. An optimalization is done for 2D and 3D models. This optimal designdata can be used as guideline for the optimalization of whole membranes.

The developed modeling techniques are used in the models of whole membranes. Tetrahedralelements are used to mesh the complex geometry of the membranes. Because of a to large mem-brane size (compared to the element size) the element solutions aren’t accurate enough. A siliconstabilized model is made to verify the reduction values between stabilized and unstabilized sim-ulations with the experiments. The absolute values obtained in the simulations aren’t the sameas the experiments, but the reduction values when stabilizing are giving a good description. Thedeveloped guideline isn’t verified with the results from the simulations, because a lack of time.However first simulations are giving a good impression for the verification of this guideline.

Contents

1 Introduction 31.1 Nanostencil lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Standard stencil fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Stabilization methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Goal of the project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Experimental determination of membrane deformation 92.1 Residual stresses in thin films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Analysis of experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Measurement of cantilever deflections . . . . . . . . . . . . . . . . . . . . . 112.3 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Initial models in ANSYS 133.1 Corrugated cantilevers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Theoretical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Description of initial models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.1 Geometry of 3D cantilevers . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3.2 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.3 Modeling aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4 Comparison and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Methods for applying residual stresses in 2D 184.1 Setup of 2D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Nonlinearity in the models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 ISTRESS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.4 Surfacestress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.5 Temperature: method 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.6 Temperature: method 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.7 Comparison and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Modeling from 2D to 3D models 275.1 Modeling aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.1.1 Elementtype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.1.2 Ratio dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.2 Behavior and optimalization of cantilever geometry . . . . . . . . . . . . . . . . . . 295.3 Comparison of the results between 2D and 3D . . . . . . . . . . . . . . . . . . . . . 30

6 Stabilization of 2D and 3D cantilevers 326.1 Silicon stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.1.1 Geometry of 2D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.1.2 Theoretical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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6.2 Implementation of silicon residual stress in ANSYS . . . . . . . . . . . . . . . . . . 346.2.1 Verifications of implementation . . . . . . . . . . . . . . . . . . . . . . . . . 356.2.2 Results of 2D stabilized models . . . . . . . . . . . . . . . . . . . . . . . . . 36

6.3 Stabilization in 3D models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.3.1 Results of 3D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6.4 Optimalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

7 Model of a membrane 437.1 Modeling of a membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7.1.1 Free versus mapped meshing . . . . . . . . . . . . . . . . . . . . . . . . . . 447.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7.2.1 Dependency of results on model size . . . . . . . . . . . . . . . . . . . . . . 457.3 Optimalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7.3.1 Geometry of stabilized model . . . . . . . . . . . . . . . . . . . . . . . . . . 477.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487.3.3 Verification of the optimalization guideline . . . . . . . . . . . . . . . . . . 50

8 Conclusions and recommendations 51

A Calculation of deflection and moment of inertia 55A.1 Derivation of deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55A.2 Derivation of moment of inertia for cantilevers . . . . . . . . . . . . . . . . . . . . 56

B ANSYS input files for unstabilized 2D models 57

C ANSYS input files for unstabilized 3D models 61

D ANSYS input files for stabilized 2D and 3D models 65

E ANSYS input files for stabilized and unstabilized membrane 71

2

Chapter 1

Introduction

Novel micro- and nanopatterning technologies are needed for the increasing demand for pattern-ing on ’unconventional’ surfaces and/or non-IC applications within a micro- and nanometer scale.Within an European project (NaPa) different new and alternative patterning technologies aredeveloped. New methods are for example soft-lithography [11], nano-imprinting lithography [12]and localized material deposition through ultra-miniature shadow masks (nanostencils) [1]. Thisreport focuses on the stencil lithography method and provides a partial solution for one of theknown problems within this technology.

First an introduction of the nanostencil lithography technique is given with a summation of theadvantages and disadvantages followed by some examples. In the second paragraph the fabricationprocess is described for standard stencil fabrication. A closer look to membrane deformations, oneof the problems in the stencils, is made. Practical stabilization methods are described and analyzedfrom which the problem definition and goal of the project can be deduced.

1.1 Nanostencil lithography

Nanostencil lithography is a pattern method, which uses a membrane (nanostencil) to shield thesubstrate from material flux during deposition on a substrate. In this technique the membraneis used as a shadow mask for depositing structures on for example a Si wafer (figure 1.1). Thismethod can be used to deposited structures with a size of 10-100 nm.

Si

Metal

Stencil

Figure 1.1: Principle of masking with the membrane, substrate and the evaporated metal

The main value for this method is that structures can be deposited on a substrate without the needof photoresist processes. Compared to other patterning technologies this method has the advantageof a non-contact application. A material independent deposition can be made and there is no needfor flat surfaces to deposit on. This makes the stencil method applicable to surfaces that are either

3

mechanically unstable, such as cantilevers and membranes, or functionalised for e.g. bio-sensorapplications. This method is also a potential for multiple layer deposition without breaking thevacuum. In figure 1.2 an example is given of a possible applications of the nanostencil. This figureshows a nanopattern on a CMOS chip.

Figure 1.2: Application of a nanostencil: a nanoresonator on a CMOS chip

1.2 Disadvantages

The resulting surface structure deposition through a nanostencil on a substrate depends on theaperture size and the gap. This is the distance between the stencil and the substrate (figure1.3a).During the evaporation of a material through a stencil different effects occur, which affect the gapof the membrane and distance to the substrate. Therefore it also affects the result on the sub-strate. One effect is the clogging effect. In this case an opening in the stencil will slowly be closedby the accumulation of evaporated material in the neighborhood of the apertures. This effect de-creases the aperture size and less material is present on the substrate near the edges of an aperture.

Another effect is the existence of residual stresses in the evaporated material during an evaporatingprocess. These stresses are resulting in a deformation of the membrane. This deformation resultsin a different distance between the stencil and substrate, shown in figure 1.3b. Resulting surfacepatterns deposited with different gapsize distances are shown in figure 1.4, where one can clearlysee the difference in sharpness of the images. In this figure the results of different distances betweenmembrane and substrate are shown. A large difference in size and sharpness of the images can beseen, because of the change in distance between the membrane and substrate. The reason for theexistence of the residual stresses in the material will be explained in chapter 2.

1.3 Standard stencil fabrication

The fabrication of the stencil can be divided in different process steps. A schematical overview ofthe fabrication process is given in figure 1.5.

In the first step the membrane material (SiN, thickness: 500 nm ) will be deposited on the bothsides of the wafer. Than the photoresist will be placed on the membrane material after which thepatterns are written in the resist with a DUV exposure process. The pattern is transferred to themembrane using a dry etch method. The photoresist will be removed and the membrane lookslike shown in figure 1.5d.

With a wet KOH etch the Silicon will be removed. This etch process always use the 111 planeof Silicon to etch. This is the reason for the typical slope that can be seen in the silicon (fig-

4

aperture

Figure 1.3: (a)Influence of aperture and gap size on the size of the structure on the substrate. (b)Influence of the residual stress to the aperture size and distance to the substrate

Figure 1.4: Results of deposited surface patterns with different distances between membrane andsubstrate

ure 1.5e). After the stencil is finished, the stencil needs to be turned over before a material canbe evaporated. This turn is an important aspect while modeling the patterns, because everypattern should be designed in a reflected view. With this technique up to 800 membranes of 1mm by 1 mm are present in one stencil (1.6) on a 100 mm wafer is used with a thickness of 380 µm.

Next to the standard fabrication method, methods are developed for fabricating stabilized mem-brane designs, like silicon supported and corrugated stencils [13, 14].

1.4 Stabilization methods

To decrease the deformation in the stencil and therefore get a better result on the substrate, stabi-lization techniques are developed within the EPFL. These techniques will improve the mechanicalstability of the membrane resulting in an increased surface pattern definition. The techniques arebased on the principal that more stiffness can be gained to a bending structure by increasing the

5

Figure 1.5: Fabrication process of a nanostencil and the turned membrane, during evaporation

Figure 1.6: Completed 100mm wafer with the 1 mm by 1 mm membranes

moment of inertia of that structure. This principal is used for the nanostencil in two differentways; corrugated stencil membranes [13] and silicon supported membranes [14]. In figure 1.7a anexample is given for the corrugated stencil and in 1.7b for a silicon supported stencil.

In the case of a silicon supported membrane a block of Si is put onto the SiN membrane. Figure1.8 shows the difference in distance from the membrane to the substrate between a stabilized andan unstabilized stencil with a Si supported method.

Within this figure one can see that the stabilization techniques have got a large influence onthe result. However it is not clear how and where exactly the membranes should be stabilized

6

(a) (b)

Figure 1.7: (a) corrugated stencil (b) silicon supported stencil

for a minimal deformation of the membrane and an optimal result on the substrate. To predictwhere the membranes should be stabilized and what the result is of this stabilization a modelshould be made to predict the deformations. Developing a model of the membrane to predict thedeformations is preferred, because doing experiments for optimalization is a very time-consumingand expensive business. This report focus on the silicon stabilization.

20 µm

gap increase

original membrane position

500-nm thick SiN membrane

Cr (50 nm) on Si

A

B

40 µm

(a)

20 µm

20-µm thick Si support

500-nm thick SiN membrane

Cr (50-nm) on Si

original membrane position

reduced gap increase

A

B10 µm

46 µm

40 µm

(b)

Figure 1.8: Difference between supported(b) and non supported(a) membrane with Si

1.5 Goal of the project

As discussed in the previous paragraphs residual stresses are present in the materials. Experimentshave proven that these deformations can be minimized by using different stabilization techniques.

The goal of this project is to describe the deformations in a membrane, caused by residual stresses,within a finite element model (FEM). Second the stabilization technique will be implemented inthe model to make an optimized design for the nanostencils.

To make a FEM model for the nanostencils first the experimental data should be analyzed. Thisdata has to be used to make a good description of the model. Some initial models will be made to

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learn and validate ANSYS as FEM tool within this project. Simulations of the corrugated stencilmembranes are done successfully by the Tyndall National Institute (TNI) in Cork [4]. Thereforethe corrugated stencils will be used to make an initial model. The implementation of residualstresses in 2D models is the next step in the development of the model. Afterwards this residualstress is implemented in a simple 3D model. Followed by modeling the silicon stabilization intosimple 2D and 3D models. Finally a whole membrane is modeled and optimized.

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Chapter 2

Experimental determination ofmembrane deformation

For the development of a finite element model of the membrane it is important to know more aboutthe behavior aspects. Experiments can be used to observe this behavior of the membrane. Ananalysis of the experimental data can help to decribe the behavior of the membrane. Afterwardsthis information will be implemented in a FEM model. For a good validation of the finite elementmodel a membrane with structures, that easily can be verified, is used. In this case cantileverstructures are produced. This has the advantage that the behavior can be described with use ofknown theories about these structures, resulting in experimental data that is used in the models.

In this chapter the results of the experiments done with a membrane are discussed. First atheoretical review is given to the origin of residual stresses in the membrane. An analysis of theexperiments done with a membrane, containing cantilever structures, will be used to calculate theinput parameters for the model. An explanation of the measurement method used to obtain thisdata is given. In the last paragraph the material properties are discussed.

2.1 Residual stresses in thin films

During the fabrication process and when evaporating a metal on a substrate through a membraneresidual stresses appear in the material [2]. This is a result from mechanical effects in thin films.Two sources of the appearance of residual stresses in thin films can be pointed. One source isa temperature change during a process (for example an evaporation). The difference in thermalexpansion coefficients of the materials results in a thermal stress. Another source is the intrinsicstress in the material, which is build up during film formation. The existence of the intrinsic stressis caused by for example the lattice spacing mismatch.

The stress present in the membrane (and the additional film) can be either compressive or tensile.A tensile stress occurs when the thermal expansion coefficient of the substrate is smaller than ofthe deposited film. The substrate is in tension and vice versa a compressive stress occurs. Infigure 2.1 the difference between tensile and compressive stresses is shown.

The intrinsic stress build up during the film formation is process specific and depends on depositionvariables. It is hard to describe this stress as a function of these variables in a model. On thecontrary, an estimation for the thermal stresses can be provided when the thermomechanicalparameters are known.

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Figure 2.1: (a) tensile residual stress in a film and (b) compressive residual stress in a film

2.2 Analysis of experimental data

For the determination of the values of the residual stresses in the material, test structures aremade. Cantilever structures within the membrane are developed , because these structure arerelatively simple to model, simple to measure and different theories exist to analyze and calculatethe behavior of the cantilevers. In figure 2.2 a membrane after the evaporation of a metal is showncontaining cantilevers with different lengths. In this figure the deflections of the cantilevers can beseen after depositing a metal film (50 nm thick chrome) on the silicon nitride (SiN). From thesecantilevers the deflections can be measured.

100 µm

large deflection

unstabilized cantilevers

membrane

Figure 2.2: Test structures for analyzing the behavior of the membrane after metal evaporation

A theory about residual stresses in materials is described by Stoney [3]. This theory describeshow a residual stress can be calculated when the deflection of a cantilever is known. In figure2.3 a schematically analysis of the film and substrate is given from which the so called Stoneyformula (equation 2.1) can be derived. With use of this figure the approach to get to equation 2.1is globally described. One starts with the calculation of the summation of the moments (betweenthe membrane and film), that is set equal to zero. The maximum tension (σm) can be expressedin terms of the beam radius of curvature (R) and angle (θ) Hooke’s law is implemented into themoment equation. Taking into account that the thickness of the substrate (ds) is much larger thanof the film (df ), so the contribution of the film to the total stiffness of the film is neglected. From

10

Figure 2.3: Stress analysis of film and substrate

here the Stoney equation can be derived. The residual stress in the film (σf ) can be calculated bymeasuring the beam radius of curvature and the thickness of the substrate and film. In the nextparagraph the method for measuring the beam radius curvature will be explained.

σf =1

6R

Esd2s

(1− νs)df(2.1)

2.2.1 Measurement of cantilever deflections

In this paragraph the method to measure the beam radius curvature and deflections from thecantilevers is described. The cantilever deflection is determined by measuring the maximum out-of plane deflection of each cantilever with respect to the membrane surface by both scanningelectron microscope (SEM, LEO) and an optical surface profiler (Vecco NTI100). The meandeflection values of several measurements after evaporating the metal are given in table 2.1.

Table 2.1: Measured deflections for different cantilever lengths after evaporating 50 nm Cr on theSiN

Cantilever length (µm) 50 100 150 200 250 300Deflection (µm) 10,3 28,1 57,0 93,3 135,2 181,0

To obtain the value for the residual stress the wafer curvature method [2] is used for measuringthe beam radius curvature used in equation 2.1. This method measures the curvature of a silicon

11

substrate before and after depositing of the film material. The difference between these two valuesis used for calculating the beam radius curvature ( 1

R = 1Rsubstrate

− 1Rfilm

). The values obtainedfor the stress in layers are depending on the thickness of the evaporated film. In this report afilm thickness for the chrome of 50 nm and for the silicon nitride of 500 nm is used. The valuesobtained for the two layers are:

σSiN = 200MPaσCr = 1230MPa

2.3 Material properties

In this section all the materials used in this report are discussed and the typical values for thethermal and mechanical properties are given. In literature a lot of different properties for thematerials are given. The reason for a large deviation in properties for the materials is that on amicroscale the behavior of the material is more depending on the molecular structures (with latticemismatch spacings etc.) compared to the behavior on a macroscale. The molecular structure ofthe material is sensitive to the different processes one uses to produce the material. In this modelonly single values for the properties are used. This assumption makes theoretical verification ofthe results easier, which enables to understand typical phenomena more easy.

From results of Tyndall National Institute in Cork [4], who works successfully on the FEM mod-eling of the corrugated nanostencils, material properties for the silicon nitride and chrome arefound. The typical values for SiN, Cr and Si [5] are given in table 2.2:

Table 2.2: Material properties of used materialsSiN Cr Si

Youngs modulus [GPa] 276 277 170Poisson ratio [-] 0.27 0.3 0.26

Thermal expansion coefficient [µK−1] 2.8 7.3 5.2

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Chapter 3

Initial models in ANSYS

In chapter two the behavior of the membrane is experimentally determined and structures aredeveloped which can be calculated with known analytical theories and modeled with simulations.Now a finite element model can be made to simulate the experimental behavior. The finite ele-ment software ANSYS will be used for the simulations. To learn and validate this FEM programinitial models will be made. In chapter two the cantilever structures are used as test structures.As Tyndall National Institute has shown, the implementation of the cantilevers into a model [4]with Conventor as FEM tool has been successfully carried out. For a good validation of ANSYSsimilar models are made and the behavior of the cantilevers is checked roughly. A validationtest on ANSYS is carried out, because there are no experimental data for the stabilization withsilicon. Experimental data is present for the corrugated membranes. By doing this check a betterconsideration of the results for the silicon stabilization can be made.

In this chapter the results of the experiments with the corrugated cantilevers are discussed. Thisis done, because Tyndall National Institute simulated next to the unstabilized, also the corru-gated cantilevers. A theoretical calculation for the deflections of the different cantilevers is made.Followed by a description of the FEM simulation models. Finally the results of the three ap-proaches (experiments, models, theory) are compared and conclusions for the further modeling ofthe membrane are made.

3.1 Corrugated cantilevers

In chapter two a membrane is made wherefrom the deflections for unstabilized cantilevers aremeasured. In this paragraph a focus to membranes with corrugated cantilevers is made. In figure3.1 (left) a cross section of a corrugation in a cantilever is shown. This U-shaped structure addsmore stiffness to a cantilever structure by increasing the moment of inertia, resulting in a lowerdeflection caused by the residual stresses after evaporating a chrome film.

Within an experiment the corrugated shapes are used, to increase the stiffness of the cantilevers,like described above. In this experiment cantilevers where made with a corrugated shape over thefull length of a cantilever. Cantilevers with a different number of corrugated shapes (RIM’s) overthe width are developed to see the influence on the deflection. An example of a membrane contain-ing cantilevers with one RIM is shown in figure 3.1. The deflections of these corrugated cantileversare measured and will be used to compare with the theoretical calculations and simulation results.The following structures will be modeled:

1. Unstabilized cantilever (see chapter 2)

2. Corrugated cantilever with one RIM

3. Corrugated cantilever with three RIM’s

13

2 µm

2 µm high

Figure 3.1: left, a cross section of a corrugation in a cantilever, right, Experimental results forcorrugated cantilevers with one RIM

3.2 Theoretical calculations

To calculate the deflection of the (corrugated) cantilevers the Stoney equations are used. Thedifference between the stabilized and unstabilized cantilevers within an analytical calculation isgiven by calculating a different moment of inertia for the cantilevers. To simplify the calculationsthe assumption ds À df (where ds is the thickness of the silicon nitride and df of the chrome)is made, so the contribution to the deflection from the film layer is neglected. The error madewith this assumption is 10%[2]. Looking to the uncertainties in the experimental data (materialproperties, residual stress) this error is accepted. The deflection for a cantilever with a givenlength (l), width (w), thickness (df , ds) and moment of inertia (Is) is given by equation 3.1. Inappendix A the derivation for the deflection and the moment of inertia of the different cantileverstructures is given.

δ =σf (1− νc)wdfdsl

2

4EsIs(3.1)

In figure 3.2 the results of the calculation of the moment of inertia for the different cantilevers aregiven. In the left the values of the moment of inertia are shown. Looking to equation 3.1 and theincreasing moment of inertia one expects an increasing stiffness and a smaller deflection with anincreasing number of RIMS. This can be seen in the right picture of figure 3.2, where the influenceof the moment of inertia on the deflection for different cantilever lengths is shown.

One should consider that equation 3.1 is a derivation of Stoney equation, where easily the momentof inertia of the different shape can be implemented. In principal Stoney equation describes thesituation of film layer on a rectangular cantilevers. Within the corrugated stabilized cantileversthe film layer, causing the stresses, is divided in a different way. Therefore extra forces andmoments are introduced on the corrugated membranes and the calculation of the deflection ofthose cantilevers is just a rough estimation.

3.3 Description of initial models

An initial model can be made when using the experimental data for a description of the geometry,boundary conditions and applied loads. The models made for the validation of ANSYS are 3Dmodels. The reason to use 3D models is that the experimental results of corrugated and un-stabilized cantilevers show different deflections over the width and length of the cantilevers. Bymodeling in 3D a general idea of the behavior in two directions can be observed.

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0.5

50 0.5

50

3

1

cross-

section

[µm]

Moment

of Inertia

[µm ]

Unstabilized

membraneCorrugated

membrane

1 rim: 2.05

3 rims: 4.470.52

450 100 150 200 250 300

0

20

40

60

80

100

120

140

160

180Deflection values for different cantilever lengths calculated with Stoney theory

different cantilever lengths

he

igh

t o

f d

efl

ect

ion

unstabilized

one RIM

three RIM’s

Unstabilized

one RIM

3 RIM’s

d

e

fl

e

c

ti

o

n

Figure 3.2: left, moments of inertia for the different cantilever structures, right, the result ofincreased stiffness to the deflection for different cantilever lengths

3.3.1 Geometry of 3D cantilevers

The geometry of the 3D model is copied from the cantilever shape seen in the membranes used forthe different experiments. In first case it is to complex to model the whole membrane with the can-tilevers. Besides the deflection of the cantilevers, the deformation in the membrane is neglectable.Therefore is chosen to model the different cantilevers (with the different lengths) separately. Itis possible small deformations occur before the beginning of the cantilevers structure and thisaspect should be taken into account when developing a geometry. To simulate a deflection in thecantilever, two layers (SiN and Cr) should be present in the model.

Implementing all described aspects, the geometry of an unstabilized cantilever will become likeshown in figure 3.3. Assuming that the behavior of the cantilever is equal on both sides symmetricconditions are taken through the middle of the cantilever (no displacement in the direction ofthe width). Making this assumption results in a decrease of number of elements needed for thecalculation and therefore a decrease in computing time. The other boundary conditions are fixedin all directions, because the deformation at the edges are assumed to be neglectable. The modelsare designed in such a way that the lengths can be easily changed.

Cantilever le

ngth

0.5 * cantilever width

elementCr

SiN

Symmetric conditions

BC, no displacement in all directions

Figure 3.3: Element plot of the geometry and boundary conditions of a symmetric 3D unstabilizedmodel

15

For the corrugated models the same assumptions as for the unstabilized model are made andtherefore no further attention is paid to geometry aspects of these models. The geometry of thesemodels can be seen in figure 3.4.

Figure 3.4: From left to right: volume plots of 1 RIM model, 3 RIM’s model and a cross sectionof the 3 RIM model

3.3.2 Scaling

Modeling thin films demands working on nano and micro size level. Finite element programs,like ANSYS, have got a limited amount of decimal values where they can calculate with. Whencalculations are done, mistakes can be made due to a wrong round off of the micro- and nanosizevalues. To avoid problems with forces and displacements, that are very small and maybe lie inthis area of the finite element program, the problem is scaled.

When forces and displacements are taken in µm, the resulting stress will be in MPa, shown inequation 3.2. Scaling the Youngs modulus results in: E[GPa] = Escaled[kPa]

[σ]MPa =[F ]µN

[x](µm)2= 106 =

10−6

10−12= [σscaled]Pa =

[Fscaled]N[xscaled](m)2

(3.2)

The thermal expansion coefficient doesn’t need to be scaled, because the implementation in themodel is only via the strain which is dimensionless (α×4T = ε [-])

3.3.3 Modeling aspects

Next to the description of the geometry a finite element model needs an element type (which isthe basis of a FEM calculation) and an applied load, that causes a deformation in the model. Inthe case of the modeling of the membrane the applied load is the residual stress.

To implement the residual stress in the cantilever a standard ANSYS input is used. This input(ISTRESS) can be used to give every element a internal stress in different directions. For now, nofurther research is done to the behavior of the function. It is just used to give an impression ofthe behavior of the cantilevers in ANSYS.

By using this input option some restrictions occur. Only a couple of element types can be usedand the residual stress can only be applied to one material. Like given in chapter 2 the chromecauses the largest effect. Therefore the value of σ = 1230MPa is the input for these models. Theelementtype used is a 3-D 20-node structural solid element (SOLID186)[6]. The description andbehavior of this element will be discussed in chapter 5. The results of these models are discussedin the next paragraph.

16

3.4 Comparison and conclusions

An overview of the results is given in figure 3.5. In this figure the three different measurements(experiments, theory, simulations) for the three cantilevers (unstabilized, 1 RIM and 3 RIM’s) arecompared. The displayed data is taken at the tip of the cantilever, where the maximum deflectionoccurs. A logarithmic scale is taken on the y-axis to give a clear overview of the three differentstructures. The different lengths of the cantilevers are shown on the x-axis of the plot.

50 100 150 200 250 300 350 400 450 500 550

10−1

100

101

102

Comparison model/experiments/analytical

length of bars (µm)

heig

ht o

f def

lect

ion

(µm

)

ANSYS modelexperimentanalytical

unstabilized

1 RIM

3 RIM

Figure 3.5: Comparison between experiments/theory/simulations for three kinds of cantilevers

It can be seen that large differences are present between the experiments, theory and simulationsfor the cantilevers. These difference can be explained by several reasons; uncertainties in theexperiments, the theoretical calculation is a rough estimation and the ANSYS models are justgiving a first impression about the possibilities. However the results are all in the same order ofdeflection values. This is the reason the conclusion is made that ANSYS can be used as a FEM tool.

The goal is that ANSYS models will describe the experiments as accurate as possible and thereforethese ANSYS models should be improved. Especially the choice for the implementation of theresidual stress is an aspect, that can be improved. In the further research attention should bepaid to the input of the residual stresses in the model.

17

Chapter 4

Methods for applying residualstresses in 2D

As concluded in chapter 3 the implementation of the residual stresses in the model is an issue thathas to be improved for a good description of the model. In this chapter this subject is discussedin more detail. Different ways of implementing the residual stress in the model are discussed andcompared with the experiments. A decision is made with help of this comparison, which techniquewill be used in future work for modeling the membrane. The methods are validated with use ofa 2D model. A 2D geometry is used to simplify the model resulting in for example a decrease ofcomputing time.

The following methods are discussed in this chapter:

• ISTRESS (standard ANSYS option)

• Surfacestress

• Temperature: Method 1 (giving each individual body an own temperature difference)

• Temperature: Method 2 (giving the whole body one temperature difference)

This chapter starts with a short description of the 2D model. In the second paragraph the aspectof non-linearity in a model is discussed, because this has a large influence on the modeling results.The paragraphs where the different methods are described start with the general idea of themethod, followed by some specific modeling aspects. In every paragraph the results are given.Finally a comparison of the four different methods is made and a decision will be made, whichmethod will be used in further work.

4.1 Setup of 2D model

The 2D model is a simplification of the 3D model described in chapter 3. No attention is paid tothe deflection over the width of the cantilever. The extra part of the membrane included in the 3Dcantilevers is neglected. Because of these two simplifications the geometry of the 2D model will belike shown in figure 4.1. It consists of two layers: silicon nitride (with a thickness of 500 nm) andchrome (with a thickness of 50 nm). The situation after evaporation is modeled, because beforeno bending occurs in the cantilever. In the model the different lengths of the structures can beeasily changed, so the different cantilevers can be modeled with the same model. The assumptionfor the boundary condition is there is no displacement in all directions (x, y, z)at the left side ofthe cantilever. The general input file for the 2D model in ANSYS is given in appendix B.

18

50 nm Cr

500 nm SiN

Figure 4.1: Geometry and boundary conditions of 2D unstabilized model

4.2 Nonlinearity in the models

In the methods for applying a residual stress two different ways of calculating the result will beused. Within the ANSYS software results can be obtained by using a linear or a nonlinear calcu-lation method. Large differences in the results of the two methods can be seen, when calculationsare done. For understanding what the finite element program is doing and to do a good valida-tion of the results it is important to know what the differences are between linear and nonlinearcalculations within ANSYS. In this paragraph a theoretical review is done on the methods.

If in a structure large displacements or rotations occur it changes the geometric configuration andthe structure responds nonlinearly. For example, when one increases the force on a long cantileverthe displacement will not increase linearly. This phenomenon can be seen in the experiments withthe cantilevers. When having large deformations (deflection À thickness) one should expect anonlinear behavior and in that case use nonlinear solving techniques to get an accurate result [9].

ANSYS uses the Newton-Rhapson (NR) approach to solve nonlinear problems. In this approachthe load is subdivided into a serie of load increments. In every load increment convergence criteriashould be reached to get the right solution. This approach (in one direction) is sketched in figure4.2. In the left one iteration step is shown and in the right the whole iteration process is sketched.A detailed explanation will be given to understand the nonlinear solution technique.

Figure 4.2: Sketch of working principle of Newton Rhapson method [10]

19

The finite element discretization process gives a set of equations and can be written like equation4.1. Inhere [K] is the stiffness (or system) matrix, [u] is the unknown displacement and [Fa] isthe vector with the applied forces. Within NR the problem is divided in increment steps. Onecan state that when adding a small step ∆ui at increment step ui, the matrix equation for thisstep is written in equation 4.2 and 4.3. Inhere Ki and Fi are the matrices at step i. When[Fa] − [Fi] ≤ convergence criteria the solution should have reached the applied load [Fa] andthe corresponding displacement is the nonlinear solution. The algorithm one uses to solve thisiteration process is devided in five steps and listed below.

[K][u] = [Fa] (4.1)

Ki∆ui = [Fa]− [Fi] (4.2)

ui+1 = ui + ∆ui (4.3)

1. Assume u0, which is the converged solution from the previous time step

2. Calculate the stifness matrix Ki and the restoring load Fi

3. Calculate [∆ui] with use of equation 4.2

4. Calculate equation 4.3 to obtain the next approximation

5. Repeat step 2-4 until the convergence criteria is reached

The difference between a nonlinear and a linear calculation method is that in a linear methodno updates are done on the approximated values (step 3-5 in nonlinear process). The unknowndisplacement vector [u] in equation 4.1 is just calculated one time for the given applied loads andstiffness matrix. In figure 4.2 the stiffness matrix is shown as Ki. The linear solution will be thevalue of [u] where [Fa] and Ki ’cross’ each other in the figure. This value can be very differentfrom the nonlinear solution.

4.3 ISTRESS

The first method which is analyzed is the ISTRESS function (also used in chapter 3). It is afunction within the ANSYS software and puts a given initial stress to every element. This methodis easy to use and the ISTRESS command can be easily adjusted (with help of an input file)to give some specific details/differences to the structures. A disadvantage is that it only can beimplemented in one layer. The stress is put into the chrome layer, because it has got the highestvalue for the residual stress.

The element chosen for this model is the PLANE42 element. This plane element has got fournodes and two degrees of freedom (δx and δy). The element is described with plane stress with agiven thickness (cantilever width), because the specimen can’t be seen as a thin material (planestress:σz = 0) or as a plane strain situation (εz = 0) [15]. Within these models a linear and non-linear calculation is done. The extra input for the ISTRESS method in the ANSYS file is givenin appendix B. The results are given in figure 4.3. On the x-axis the lengths of the six differentcantilevers are given.

The results of the ISTRESS method are lying below the experimental values. This phenomenacould also be seen in the corrugated 3D models in chapter 3. The error made in this model is tolarge to continue working with this method. A more detail comparison and an errorplot are givenin the last paragraph of this chapter.

20

50 100 150 200 250 3000

20

40

60

80

100

120

140

160

180

200Comparison of ISTRESS method with experiments

cantilever length (µm)

heig

ht o

f def

lect

ion

(µm

)

experimentsISTRESS linearISTRESS non linear

Figure 4.3: Comparison between experiments, ISTRESS method (for linear and non-linear)

4.4 Surfacestress

Another method is by replacing the chrome layer by a surface stress, like shown in figure 4.4. Inreality the chrome layer is very thin (50 nm). Therefore the contribution to the stiffness of thecantilever is not very large. In this way the chrome layer can be replaced by an external load on allthe nodes. The advantages of this method are the reduction of elements in the model (especiallyimportant, when looking to whole membranes) and no elements are present with large differentratio’s. The size of a chrome element hasn’t got a 1 by 1 ratio(chrome is a thin layer), what canbe seen in the figures of the geometry of the 2D and 3D model. Large ratio differences can causewrong solutions. To check the influence of the ratio on the result more research should be doneto the ratio independency of the element type.

FnodeFnode Fnode

SiN

Cr

SiN SiN SiN

Cr Cr Cr

FnodeFnode

SiN SiN SiN SiN

Figure 4.4: Replacement of chrome layer by a surface stress

The same element type is used as in the ISTRESS method. The stress applied on the membraneis σCr = 1230MPa. To calculate the replacing force the following steps are made. In reality thestress in the chrome causes a deformation of the chrome layer (ε < 1). This deformation hasgot effect on the silicon nitride layer, causing a bending of the structure. The nodal force willreplace the chrome layer (with the given thickness of 50 nm). To calculate the nodal force thestress should be multiplied by the area of the chrome layer where the stress is working on. This

21

50 100 150 200 250 3000

20

40

60

80

100

120

140

160

180

200Comparison of surfacestress method with experiments

cantilever length (µm)

heig

ht o

f def

lect

ion

(µm

)

experimentssurfacestress linearnon linear: no convergence

Figure 4.5: Comparison between experiments and the surfacestress method

multiplication is between the thickness and the cantilever width of the chrome layer. For this 2Dmodel the nodal force should be divided by the number of nodes the force works on. The area inequation 4.4and the applied force in equation 4.5 become:

A = thickness chrome× cantilever width = 0.05µm× 50µm = 2.5µm2 (4.4)

Fnode =σCrA

number of nodes=

1230MPa× 2.5µm2

1 + cantilever lengthelementsize

(4.5)

The value of the force changes for the different lengths of the cantilever and the element sizes.The results of this method are shown in figure 4.5. No results for the non-linear calculations arepresent, because the models didn’t converge. The calculations done with the non-linear method arevery force sensitive. By adding a very small value to the nodal force, the model will not convergeanymore. Probably the force applied in this case is to high and causes in the non-linear modevery large deflections. This is causing the problem of the non converging model. An explanationfor this behavior can be found when looking to linear and nonlinearity in ANSYS.

The non converging behavior of the nonlinear solution of the surfacestress can be explained. Whenusing a nonlinear solution control the force is applied in substeps. Every substep the cantileverwill bend a little bit, however the forces (implemented in x-direction) are not bending with theshape of the cantilever. The force will be divided in a vector along the surface and one 90 degreeson the surface. This causes strange deflections and cantilever shapes or a non converging solution.To solve this problem the force should be implemented in a different way. In general the problemof non converging can be solved to switch to smarter iteration schemes, like the arc-length methodor a decrease in elementsize.In appendix B the extra input for this method in the ANSYS file is given. The error of the linearmethod and the experiments is already decreased compared to the ISTRESS method. Because ofthe non converging behavior of the non-linear model and the difficulties implementing this conceptin a whole membrane (different nodal forces) this method is not used.

22

4.5 Temperature: method 1

In reality temperature differences during the process are a contribution to the residual stress. Amethod to describe the stress is to put temperature differences in the model . The temperaturedifferences are causing a strain in each layer and the cantilever will bend. This principle is basedon the idea of bi-metals [7]. The stress in each layer is described with the Youngs modulus (E),thermal expansion coefficient (α) and a ∆T :

σlayer = Elayerε = Elayerαlayer∆T (4.6)

By rewriting this equation ∆T can be calculated for each layer. The temperature difference iscalculated between a reference temperature (T = Tenvironment = 0) and the temperature of thematerial. A body temperature is given to each layer. The following values for the temperatureare calculated:

∆TCr =σCr

ECrαCr=

1230MPa

277GPa× 7.3µK= 608K (4.7)

∆TSiN =σSiN

ESiNαSiN=

200MPa

276GPa× 2.8µK= 259K (4.8)

The results are shown in figure 4.6. Both the linear and non-linear results are close to the experi-ments. In appendix B the ANSYS input is given. Unclear is the behavior of the body temperaturesbetween the layers, because a different ∆T occurs between the layers. No reference temperatureis between the SiN and Cr, this results in a lower stress between the materials.

50 100 150 200 250 3000

20

40

60

80

100

120

140

160

180

200Comparison of method 1: temperature with experiments

cantilever length (µm)

heig

ht o

f def

lect

ion

(µm

)

experimentstemperature 1: lineartemperature 1: non linear

Figure 4.6: Comparison between experiments and the temperature method 1: individual temper-ature differences to the bodies

4.6 Temperature: method 2

By using method 1, described above, an important boundary conditions isn’t implemented inthe calculation. This boundary condition is the demand that the film-substrate system stays

23

intact [8]. This means that both tips of the cantilever have the same length after a process step(compatibility requirement). Because αCr > αSiN , it means that if the film is cooled down froma stress-free state, the film will contract more than the substrate, resulting in a tensile stress inthe substrate and a compressive stress in the film. This concept is shown in figure 4.7, where δT

is the deformation caused by the temperature and δM caused by the compatibility requirement.

Figure 4.7: Concept for compatibility between the two lengths of the layers

Writing this concept in terms of an equation, results in equation 4.9. The deformation causedby the temperature is equal to equation 4.10 and the extra deformation, because of the stress, isgiven in equation 4.11

δT (SiN) + δM(SiN) = δT (Cr) − δM(Cr) (4.9)

δT = αl∆T (4.10)

δM = lε = lσ

E(4.11)

Implementing equation 4.10 and 4.11 in 4.9 results in equation 4.12. Rewriting this equation inthe form that ∆T is the only unknown results in equation 4.13.

αSiN l∆T +lσSiN

ESiN= αCrl∆T − lσCr

ECr(4.12)

∆T =σSiN

ESiN+ σCr

ECr

αCr − αSiN= 1148K (4.13)

The result is one temperature difference, that should be implemented in the model. The advantageof one temperature difference to describe the stress in both layers is, no attention is needed towhat is happening between the layers. This behavior was unclear in the first method. The resultsare shown in figure 4.8. The results are closer to the experimental values than method 1. Inthis method one implements an extra boundary condition in the model, which causes an extracontribution to the deflection.

Comparing the two temperature methods an overall error of the deflection of 6% is present. This isthe result of the extra boundary condition. In chapter 2 the theoretical analysis of the calculationof the deflection is given. Inhere a linear relation between temperature and deflection is present.For this reason it is possible to compensate method 1 with a factor of 6% to obtain the same, moreaccurate, results as method 2. This concept will be used when modeling stabilized cantilevers.

24

50 100 150 200 250 3000

50

100

150

200

250Comparison of method 2: temperature with experiments

cantilever length (µm)

heig

ht o

f def

lect

ion

(µm

)

experimentstemperature 2: lineartemperature 2: non linear

Figure 4.8: Comparison between experiments and the temperature method 2: one temperaturedifference to whole body

4.7 Comparison and conclusions

In this paragraph the four methods are compared by giving a plot of the errors. Explanations forthe differences between the models and the experiments are given. Finally a choice will be madefor the method which is going to be used in further work considering the different modeling aspects.

The different methods are compared with the experiments. A choice has been made to plot theerror of the nonlinear solutions (except for the surfacestress method, only a linear solution isobtained). In figure 4.9 the errors between the experiments and every method for the differentcantilever lengths are shown. The line which is lying the closest to the zero percentage error isthe most accurate method. In this case the temperature method 2 is the best method to use. Thesurfacestress is lying closer to the experimental results, but the method can’t be described easily.

All results are below the experimental values, this can be partly explained by the two restrictionmade with the 2D model. The first restriction is there is no extension of the cantilever at thebeginning, resulting in less deflection and no deflection over the width of the cantilever is takeninto account in the 2D model. Further explanations for the differences will be given in chapter 5.

The differences in the models and the experiments can be explained, because of several reasons.The errors made in the measurement and materialdata are causing one part of it. A larger con-tribution to the error is the description of the model. In this model the assumption is made thata homogenous deviation of the chrome is on the silicon nitride. In reality the metal is evaporatedand during the evaporation the membrane starts bending so the material is not evaporated ho-mogenous.

One can conclude that the behavior of the cantilevers isn’t very nonlinear, because the differencesbetween the linear and nonlinear result is not so big. However a nonlinear result is more accurate

25

50 100 150 200 250 300−100

−80

−60

−40

−20

0

20

40Error in 4 models compared to experiments (x−axis)

cantilever length

err

or

in %

of

sma

ller

de

fle

ctio

ns

in m

od

els

different temp to materialsone temperatureistresssurfacestress

Figure 4.9: Errors of four methods in comparison with the experimental results

and for that reason this option should be used. At the moment whole membranes are going to bemodeled other nonlinear effects can occur, because of strange shapes in the structure.

The modeling aspects that are considered for the choice of the method developed in this chapter,are the error between the method and the experiments and the computing time needed for thecalculations. The computing time is mostly depending on the number of elements and the use ofnonlinear solving method. The fastest method is the method using surfacestress, because half thenumber of elements is used. A disadvantage of this method is the implementation of the nodalforces (depending on the structure geometry). The other three methods have the same amount ofelements, resulting in no faster computing time between the methods. Looking to the error thetemperature method 2 (nonlinear) is giving the best results. In further work this method will beused, remembering that temperature method 1

26

Chapter 5

Modeling from 2D to 3D models

In chapter 4 different methods are developed for applying residual stresses in a 2D model. Themethod, which describes the experiments the most accurate is used in further research. Next stepin the development of modeling a whole membrane is to implement this method in a 3D model.The results of this 3D models should be comparable with the 2D models and the experiments.When comparable results are obtained within a 3D model one can continue with implementingstabilization methods in the model and work on the model for a whole membrane.

In this chapter the development of a 3D model is discussed. The first paragraph contains themodeling aspects. Especially in this section attention is paid to the behavior of the element typethat is used in the 3D model, because this plays an important role. The second section is usedfor the optimalization of the geometry of the model. Inhere also the deflection along the width ofthe cantilever is checked with the experimental results. Finally the results are shown and a closerlook is given to the differences between the 2D and 3D models.

5.1 Modeling aspects

The geometry of the 3D model used for the modeling of the cantilevers is the same as the geometryof the initial models, described in chapter 2. The method where different temperatures (∆T ) tothe materials are given to describe the residual stresses is implemented in the model. This methodwith compensation factor gives the same accurate results as the one body temperature differencemethod. Like in the 2D model, the temperature difference is applied by giving the materials abody load (see appendix C for ANSYS input files).

5.1.1 Elementtype

Before starting the simulations an elementtype is chosen for the 3D model. The element needs thequalities that it has degrees of freedom in all deformation directions (x,y,z) and a temperaturedifference can be implemented as an applied load. Another quality is the independency on thesize and shape of the element. This should be considered before working on whole membranes,because in those structures the elements will have large size and ratio’s (large difference betweenlength, width and height of the element).

An element is build up out of nodes. During a calculation within an element an elementmatrix isformed, by calculating the nodal results by interpolating between the nodes. When implementingall separate elementmatrices in one large matrix the stiffness matrix is build. The interpolationbetween the nodes can be done with linear or higher order interpolation schemes. A linear solutionis less accurate than a higher order element. Advantage of a linear element is that it has relativefew nodes, which decreases the size of the stiffness matrix corresponding with a shorter computing

27

time. Choosing an elementtype the lowest number of nodes in an element is preferred, becausethe computing time will decrease. However the independency on element size and ratio is moreimportant, because when an element has to be very small for an accurate solution, the advantageof a small number of nodes disappears.

In this paragraph three different elementtypes are compared. All three types have the qualities ofhaving the wanted degrees of freedom and applied load options. A short description of the threeelementtypes is given below:

1. SOLID45: 3D Structural Solid

2. SOLID185: 3D 8 Node Structural Solid

3. SOLID186: 3D 20 Node Structural Solid

Simulations are done to check the behavior of the different elementtypes with the 3D modeldescribed above. The first element (SOLID45) has the advantage that it has 8 nodes and thereforea linear interpolation between the nodes is done. Unfortunately the elementtype is extremelydepending on the size and no accurate solution can be obtained when the elements are to large.The SOLID185 shows the same behavior as the SOLID45 element and therefore is not useful inmodeling the cantilevers or membranes. The results of this element behavior are given in table5.1. Inhere the deflections are compared to a reference model with a very fine mesh and an errorin the deflection is given. In the next paragraph this approach is discussed in more detail. TheSOLID186, in contrary to the other elementtypes, is not highly depending on the elementsize andtherefore it can be used in the models. Reason for the independency is the large number of nodes(20) in the element, which give accurate element solutions. In the next paragraph a focus on theelement SOLID186 is made, doing a quantitative analysis of the ratio dependency of the element.

Table 5.1: Errors and differences in deflection for SOLID45 and SOLID185 elementselement type elementsize (µm) deflection reference model (µm) deflection (µm) error (%)

SOLID45 2.5 10.064 0.087 À 100SOLID185 2.5 10.064 0.0074 À 100

5.1.2 Ratio dependency

The layout of the SOLID186 element is shown in figure 5.1. The behavior of the element canbe checked by making models with different ratio’s and sizes. Assuming that a model gives anaccurate result when decreasing the elementsize, a model with very small elementsizes is used asreference model to compare the results of increased elementsize and large ratio’s.

The behavior is checked by changing the height/length ratio (ratio = h/l) and the elementsize(width × length × height) of the element. A cantilever with a length of 50µm is used as refer-ence model. The elementsize of the reference model is 0.5µm × 0.5µm × 0.5µm for the elementscontaining SiN material properties and 0.05µm × 0.5µm × 0.5µm for the elements containing Crmaterial properties (ratio = 1/10). Smaller elements aren’t possible, because of the restrictionsin ANSYS University Edition to the total number of elements. The results of the element behaviorare listed in tabel 5.2.

The errors aren’t higher than 3.2 % (with elements which have enormous sizes). It can be concludedthat the element SOLID186 is not highly ratio and size dependent. Therefore different sizes andratio’s can be used in the cantilevers and membranes models, knowing the elementtype is notcausing large errors.

28

height

length wid

th

Figure 5.1: Layout of elementtype SOLID186

Table 5.2: Errors and differences in deflection changing elementsize and ratio of SOLID186 elementdeflection reference model (µm) deflection (µm) error (%)

h/l = 1/20 10.064 10.061 −0.03h/l = 1/40 10.064 10.062 -0.02h/l = 1/80 10.064 10.055 -0.09

w × l × h = 2× 2× 2 10.064 10.062 -0.02w × l × h = 5× 5× 5 10.064 10.025 -0.4

w × l × h = 10× 10× 10 10.064 9.743 -3.2

5.2 Behavior and optimalization of cantilever geometry

Already in chapter 2 a note was made that the measured cantilever deflection is a summation ofthe cantilever deformation and the deformation of a small part where the cantilever is attachedto the membrane. In this paragraph the contribution of the membrane to the total deflection ismodeled using an ANSYS simulation. The membrane deformation can be modeled by implement-ing a large part of the membrane in the geometry of the cantilever (figure 5.2). A simulation ofthis model is done and the deflections through the different cross sections are measured.

Cross sectio

n 1

Cross section 2

Cross section 3

Figure 5.2: Extended model for measuring the contribution of the membrane to the cantileverdeflection

29

The results of the deflection through cross sections one and two are shown in figure 5.3. Thesefigures show that only a small part of the membrane contributes to the deflection of the cantilever.When developing 3D models only this part has to be implemented in the geometry. This resultsin a decrease of elements and therefore a decrease in computing time.

0 10 20 30 40 50 60 70 80 90 100−2

0

2

4

6

8

10

12Shape of cantilever in z−direction to check influence of membrane to the total deformation

length along cantilever

he

igh

t o

f d

efl

ect

ion

Cross section 1

Cantilever length

0 5 10 15 20 25 30 35 40 45−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4Shape of cantilever in x−direction at beginning of the cantilever

length along cantilever(SL)

height of deflection

Cross section 2

Cantilever width

Figure 5.3: Deflection of the extended model through the cross sections, (left) cross section 1,(right) cross section 2

When modeling with 3D models another aspect, which has to be checked, is the behavior of thedeflection over the cantilever width (cross section 3 in figure 5.2). In the experiments a differencein deflection at the tip between the middle and the most right point was observed. An averagevalue in the middle of 10.3µm and at the rigth point of 10.9µm was measured for a cantileverof 50µm. This result from the experiments has to be compared with the deflection simulated inANSYS. In figure 5.4 the values for the deflection through cross section 3 (from figure 5.2) areshown. The absolute value of the deflection in the model lies 0.4µm above the deflection of theexperiments. However the difference between the two points is the same in the experiments and inthe model (0.6µm). It can be concluded that the same behavior over the width of the cantileveris described with help of the simulations.

5.3 Comparison of the results between 2D and 3D

With use of the optimized and verified model from the previous paragraph simulations are doneto model the cantilevers with the different lengths in 3D. The maximum deflections of thosemodels are compared with the values for the deflections found in the 2D models. In figure 5.5athe experimental, 2D and the 3D modeling results are plotted. In figure 5.5b the differencesbetween the 2D and 3D model compared to the experimental values is plotted in terms of anerrorpercentage. For the 3D models the error is within ±10% of the experimental results.

A difference in deflection values is present between the 2D and the 3D model. This difference canpartly be explained by the different geometry and boundary conditions discussed in the previousparagraph. In a 2D model no deflection over the width of the cantilever is taken into account. Ina cantilever (with a length of 50µm) in 3D this results in around 5% extra deflection comparedto a 2D model. An extra contribution to the deflection in a 3D model is caused by the extensionof the cantilever. In a cantilever (with a length of 50µm) this results in around 5% extra deflec-tion. The other differences in 2D and 3D models occur, because different calculations are donewithin ANSYS. More research into the calculations in ANSYS should be done to understand thisdifference.

30

0 5 10 15 20 2510.7

10.8

10.9

11

11.1

11.2

11.3Shape of cantilever in x−direction

length along cantilever

he

igh

t o

f d

efl

ect

ion

Cross section 3

Figure 5.4: Shape of the cantilever through cross section 3

50 100 150 200 250 3000

50

100

150

200

250Deflection at tip of the cantilevers for different lengths in 2D and 3D model

length of the cantilevers

he

igh

t o

f d

efl

ect

ion

2D model3D modelexperiments

(a) (b)

50 100 150 200 250 300−50

−40

−30

−20

−10

0

10

20Error in 2D and 3D model compared to experiments

length of the cantilever

err

or

% o

f sm

alle

r d

efl

ect

ion

in m

od

els

error in 2D modelerror in 3D model

2D

3D

Figure 5.5: (a)Comparison of deflection of different cantilevers between exp.,2D and 3D models,(b) Error (in %) of 2D and 3D models compared to the experimental values

31

Chapter 6

Stabilization of 2D and 3Dcantilevers

Unstabilized cantilever models are simulated in ANSYS and compared with results from the ex-periments. The results of the simulations are describing the experiments within an acceptablerange (10 %). Next step in the development of FEM models of whole (stabilized) membranes isthe implementation of the stabilization method into the model. In this report the focus on thesilicon stabilization is made. The development of a silicon stabilized simple model (cantilever) in2D and 3D is described in this chapter.

First the experiments done with silicon stabilization are studied to develop a geometry of a stabi-lized model. No experimental data is present for cantilevers with silicon stabilization, therefore inthe first paragraph a theoretical calculation is done to give an impression of the behavior of siliconstabilized cantilevers. In the second paragraph the implementation of the silicon is discussed inmore detail, some verifications on the model are done and the results of the 2D models are given.In the following paragraph the 2D models are extended to 3D models. Followed by the resultsand an impression of an optimalization of membrane deflections with use of the cantilever modelsgiven. Finally conclusions are made out of the done simulations and optimalizations.

6.1 Silicon stabilization

In chapter two and three experiments for unstabilized membranes and corrugated stabilized mem-branes are discussed. No membranes with cantilever structures are produced with silicon stabiliza-tion on it. Only a whole membrane is produced with silicon stabilization on it. Therefore directverification of the experiments with FEM simulations of the cantilevers is not possible. Makinga stabilized whole membrane simulation model to compare the results with the experiments isto complex at the moment. Verifications can be done with help of theoretical calculations andverifications in the FEM model, described later in this chapter.

The silicon stabilization is based on the same principle as the corrugated stabilization. A siliconblock is deposited on the SiN membrane (figure 6.1). This silicon structure adds more stiffness toa cantilever structure by increasing the moment of inertia, resulting in a lower deflection causedby the residual stresses after evaporating a chrome film. When developing the stabilization on themembranes the following condition need to be considered. The stabilization structures should notinterfere with the material transport from the source through the apertures, i.e. the line-of-sightof the apertures to the material source should not be obstructed by the stabilization structures.The results of the maximum ratio and line-of-sight is shown in figure 6.1.

32

Si-support

SiN membrane

Ratio max. 7:1

α > 16 °

Figure 6.1: Schematic representation of a membrane containing a silicon stabilization with themaximum line of sight and ratio

6.1.1 Geometry of 2D model

For the development of a silicon stabilized simulation a simple 2D model (cantilever) should bemade. Within this 2D model techniques can be tested and verified, which can be used in 3Dmodels. The structure of a silicon stabilized cantilever can be seen in figure 6.1. The geometry ofthe 2D model is made out of this 3D configuration by taking the cross section over the length ofthe cantilever. In this way deflections over the width of the cantilever are not taken into account.The length of the cantilever is taken without an extension, like done in the 3D models of theprevious chapter.

Another aspect for the stabilization of the model is the adaptability of the silicon geometry on themembrane for an optimalization of different membrane apertures (obtaining minimum deflections).This should be implemented in the model with an easily changeable silicon geometry. This is doneby giving the model a variable free edge (FREE) from the silicon to the end of the cantilever(beginning of the aperture in the membranes) and a variable stabilization height (Stab.H.), likeshown in figure 6.2. Other modeling aspects are the same as the model described in chapter 4,except for the implementation of the residual stress in the stabilized model.

Chrome

Silicon Nitride

Silicon

Chrome

FREEStab. H.

Cantilever lenght

U(x)=0

U(y)=0

Figure 6.2: Geometry and boundary conditions of 2D silicon stabilized model with variable freeedge and stabilization heigth

6.1.2 Theoretical calculations

To calculate the deflection of a silicon stabilized cantilever the theory described in chapter 3 (para-graph 3.2: Theoretical calculations) is used. The difference comparing equation 3.1 for corrugatedand silicon stabilized cantilevers is the moment of inertia (Is) which is different. The calculationof the moment of inertia for the silicon is different, because two different materials are used inthe cantilever structure. To calculate a moment of inertia for the structure a mathematical trickis done. The silicon material block is replaced by a SiN material block with the same stiffness.A compensation factor between the two material blocks is used (ESi/ESiN ) to obtain the samestiffness by decreasing the width of the silicon block. The moment of inertia can now be calculatedfor the SiN material properties.

33

For the geometry of the silicon stabilization on the cantilevers different shapes and sizes can besimulated. Different values for the free edge and the stabilization height (figure 6.2) can be taken.Therefore an optimalization of the geometry to obtain minimum deflections in a silicon stabilizedmodel can be made by changing this geometry. Optimalization of the silicon block is possibleby looking to different ratio’s and/or varying the distance of the free edge. In the case of thetheoretical calculation for the verification of the behavior a free edge of 5µm and a stabilizationheight of 5µm are taken (ratio is Stab.H./FREE = 1/1). In this report the choice is made todo optimalizations with changing the free edge and stabilization height with a ratio 1/1 to give agood impression of the dependency of the silicon geometry on the result.

In figure 6.3a the values for the moment of inertia for a silicon stabilized cantilever (with a freeedge of 5µm is given. In figure 6.3b the deflection calculated with Stoney is shown for differentcantilever lengths. The deflection values are very low, because of the large moment of inertia (thedeflection of the free edge is not taken into account). When verifying the simulation results thedeflections should be measured at the end of the silicon stabilization and not on the tip of thecantilever.

0.5

50

0.5

50

40

5

cross-

section

[µm]

Moment

of Inertia

[µm ]

Unstabilized

membrane

Si supported

membrane

0.524

361.40

(a) (b)

50 100 150 200 250 3000

0.05

0.1

0.15

0.2

0.25

0.3

0.35Deflection values for different cantilever lengths calculated with Stoney theory

different cantilever lengths ( µm)

he

igh

t o

f d

efl

ect

ion

(

µm

)

silicon stabilization

Figure 6.3: (a)moments of inertia for the different cantilever structures (b) the result of increasedstiffness to the deflection for different cantilever lengths

6.2 Implementation of silicon residual stress in ANSYS

The geometry and boundary conditions for a 2D silicon stabilized model are developed in theprevious paragraph. The choice is made to look to a ratio of 1/1 for the relation between freeedge and stabilization height. To complete the 2D stabilized model only the residual stresses inthe silicon material and the implementation in the model should be considered.

In the experiments crystalline silicon [14] is used as stabilization. Normally this material is resid-ual stress free and no extra implementation into the model is needed. However stresses from thechrome and silicon nitride are working on the silicon stabilization block. These stresses are causinga deformation in the silicon. Because of the compatibility requirement that the silicon should de-form as much as the Cr and the SiN a given deformation in the silicon is obtained (see paragraph4.5). A simple calculation can be made to get an impression how to describe the deformation in

34

the silicon to satisfy this compatibility requirement.

In figure 6.4a the situation is sketched, where the stress of the Cr and SiN is working on the siliconblock. A compressive chrome stress is working on the silicon, because the thermal expansioncoefficient of Cr is larger than of Si (αCr > αSi). A tensile SiN stress is working on the Si, becauseαSiN < αSi. In figure 6.4b a linear stress distribution is sketched on the silicon block in case ofthe applied stresses from figure 6.4a.

Silicon

Stress in Chrome

Stress in SiN

Silicon

(a)(b) (c)

1200 MPa

200 MPa

Silicon

500 MPa

Figure 6.4: (a)Stress of SiN and Cr working on Si (b) Linear distribution of stress in Si, causedby stresses of SiN and Cr (c) An average of the stress in silicon

In the bilayer cantilevers a temperature difference in each layer is used to describe the residualstresses. The method of giving two layers one temperature difference isn’t used for three layers,because calculations for three layers (like done in paragraph 4.6) are much more complicated. Inthe case of the extra stabilization layer the technique of giving a temperature to each body is used.To give the silicon a value for the temperature, to describe the stress, in figure 6.4c an averagevalue for the stress in silicon is sketched. With use of equation 6.1 a temperature can be calculatedwhen the stress is known. In this case an average compressive stress of 500MPa is present in thesilicon. The temperature difference to implement in the silicon material in the model is given inequation 6.1. No changes in the input values for the SiN and Cr are done.

∆TSi =σSi

ESiαSi= − 500MPa

170GPa× 5.2µK= −566K (6.1)

6.2.1 Verifications of implementation

Before a body load temperature is given to the silicon stabilization in the models different checksshould be done to verify if the simple approach of describing the stress in the silicon block is valid.This verification is done in three different ways:

1. Comparing the theory with ANSYS results

2. Simulating a bilayer cantilever with a three layer cantilever

3. Changing the boundary conditions to see behavior of the (unstabilized) tip of the cantilever

First the theoretical results will be compared with simulations of the stabilized 2D model. A modelis made for the different lengths with a silicon stabilization of 5µm on top of the SiN cantilever.The deflection is measured at the end of the stabilization block, so it can be compared with thetheory. However the deflection values of the simulations are an order of 101 higher than of thetheoretical values. This large difference is also seen when lower values of the silicon stress are used

35

for the FEM modeling of the stabilization.

Therefore the input of the residual stress isn’t the cause for this difference. One can doubt aboutthe theoretical results. The theoretical calculation only takes into account the stress of the chromeon top of the structure. The other stresses are not taken into account. In FEM simulations it canbe seen that the influence of the chrome on top of the silicon stabilization is not the dominantcontribution to the deflection. In this way it is hard to compare these results. It can be concludedthat the comparison between the used theory and the simulations isn’t a good method to verifythe input of the stress in the silicon.

Another method which is developed for the verification of the silicon stress input value is a modelwhere a very small layer of silicon is taken as stabilization. With a very small stabilization height(dSi ¿ dSiN ) the model should approach the situation of a bilayer model, because the contributionof the stiffness of the Si should be very small. A comparison between simulations with 2D bilayermodels and a three layer with a very small silicon layer is shown in figure 6.5. The values of thedeflections are very close to eachother for the two different models. Based on this check a goodapproximation of the silicon input stress is found.

50 100 150 200 250 3000

20

40

60

80

100

120

140

160

180Check for value silicon delta T

cantilever length ( µm)

heig

ht o

f def

lect

ion

of c

antil

ever

s ( µ

m)

BilayerTemp = 566 K

Figure 6.5: Comparison between a bilayer model and a three layer model with a very thin siliconlayer to check the input value of the silicon stress

Finally a check is done on the behavior of the tip of the cantilever. At the tip of a stabilized modelno stabilization is present. This situation can be seen as a short bilayer cantilever. Thereforethe deflection of a short bilayer cantilever should give the same results as the deflection of onlya cantilever tip with the same length (with the boundary conditions (ux = uy = 0) at the endof the stabilization). With this method a check can be done on the influence of the large siliconstabilization block on the calculation made in ANSYS. In figure 6.6 the geometry, boundaryconditions and result of this check is shown. It can be seen that the deflection values are the same.It can be concluded that no calculation errors in ANSYS are made, because of the existence ofthe large silicon stabilization.

6.2.2 Results of 2D stabilized models

After the 2D model is verified simulations can be done to calculate the deflections of this siliconstabilized model. In paragraph 1.1.2 the choice is made to make a model with ratio 1/1 with a

36

0.228567

Figure 6.6: top, a deformed bilayer cantilever (length = 10 µm), under, a deformed stabilizedmodel with different BC to check the behavior of the model on the tip (length = 10 µm)

free edge and stabilization height of 5µm. In the following results the same ratio and size for thestabilization is used. The deflection at the tip of the stabilized cantilever is measured for differentlengths. To give an impression of the influence of the stabilization on the cantilever the results arecompared with the deflections of the unstabilized cantilevers in figure 6.7. In this figure a largedecrease of the deflection of the stabilized cantilever compared to the unstabilized cantilever canbe seen.

50 100 150 200 250 3000

20

40

60

80

100

120

140

160

180Comparison of stabilized 2D models with unstabilized 2D model

different cantilever lengths (µm)

heig

ht o

f def

lect

ion

(µm

)

stabilizationunstabilized

Figure 6.7: Comparison between a stabilized 2D model (free edge and stab. height is 5 µm) andan unstabilized 2D model

37

6.3 Stabilization in 3D models

A 2D stabilized model is made. The next step is the development of a 3D stabilized model. Inthe experiments a silicon stabilized membrane is produced (figure 1.7b). In this figure one can seethat a silicon block is present at a fixed position from the edge of the apertures. A fixed distanceof 10µm is taken in the example of figure 1.7b. The reason for this fixed distance is the maximumline-of-sight that has to be obtained, like discussed in the first paragraph of this chapter. In thegeometry of a 3D model a fixed distance for the stabilization block to the edge of the membraneshould be taken.

To compare the results of the 2D models with the 3D models a cantilever structure is used for thegeometry of a 3D model. A 3D unstabilized model (figure 3.3)is used as basis for the geometryand a silicon block is build on top of it. The edge of the silicon stabilization block has got a fixeddistance to the edge of the cantilever. In figure 6.8 a stabilized cantilever model with a ratio of 1/1with a free edge and stabilization height of 5µm is shown. No silicon block is present on the wholeextension of the cantilever. Looking to the discussed experiments with a silicon stabilization on it,it should be present. In chapter 5 the deflection over the width of the cantilever is discussed. Atthe position, where in the geometry of figure 6.8 no silicon is present, in the unstabilized cantilevera negligible deformation can be seen. Therefore the influence of the silicon block at that positionisn’t to large. To decrease the number of elements in the model the choice is made to only do asilicon stabilization block over the length of the cantilever.

In the 3D stabilized geometry a chrome layer is present on top of the silicon and on the siliconnitride. A temperature difference of T = −566K is implemented in the model to describe thestresses in the silicon. The stresses in silicon nitride and chrome are also described with a tem-perature difference. Like in the unstabilized models, the SOLID186 element is used in this model.The boundary conditions of this model are the same as taken in the unstabilized model.

Cr

SiN

Si

Cr

Free

Stab. Height

Cantilever le

ngth

0.5 Cantilever width

Figure 6.8: Geometry of 3D model (volumes) of a silicon stabilized cantilever

38

6.3.1 Results of 3D model

The results of a silicon stabilized 3D model with a ratio 1/1 and a size of 5 µm are shown in figure6.9. In this figure the deflections of models with different cantilever lengths, which are simulated,are shown. These results are compared with the deflection values of the 2D models. The deflectionvalues of the 3D models are higher than of the 2D models. A absolute difference between 0.4µmand 4µm is present. Take into account that the extra deflection of the extended part in the 3Dmodel is smaller compared to the unstabilized version, because of the silicon block. Therefore thesame order difference can be seen as between the unstabilized 2D and 3D models. In paragraph5.3 a partial explanation for these differences was given.

50 100 150 200 250 3000

2

4

6

8

10

12

14

16Comparison between a stabilized 3D model and a 2D model

different cantilever lengths (µm)

heig

ht o

f def

lect

ion

(µm

)

2D model3D model

Figure 6.9: Comparison between a stabilized 3D model (free edge and stab. height is 5 µm) anda 2D model

6.4 Optimalization

In the previous paragraphs results are given for a stabilization height and free edge of 5µm for dif-ferent cantilever lengths. In this paragraph different sizes of the free edge and stabilization heightare simulated. The total deflection of a silicon stabilized model is a summation of the deflectionof the tip and the deflection of the stabilized part. When a ratio of 1/1 between the size of thefree edge and the stabilization height is given, the summation of the two parts can be optimized.A small free edge results in a small deflection at the tip, but will result in a larger deflection of thesilicon stabilized part (because it has a small stabilization height). Therefore an optimalizationbetween the free edge and stabilization height sizes is made.

An optimalization for the different cantilever lengths (from length 50 µm to 300 µm) and fordifferent free edge sizes and stabilization heights (from 1 µm to 25 µm) is done. In figure 6.10this optimalization is shown for the 2D stabilized models. The deflections are measured at thetip of the cantilever. It can be seen that the point of minimum deflection is changing for thedifferent cantilever lengths. This is explained by looking to the two contributions of the totaldeflection. This is the summation of the deflection of the unstabilized tip and the stabilized part.The deflection of the tip has, for an equal taken free edge, for every length the same contribution.However the deflection of the stabilized part is increasing with a increasing cantilever length.Therefore the optimalization point (min. defl.) for different cantilever lengths is changing.

39

0 5 10 15 20 25

100

101

Optimalization plot for 2D cantilever with Si stabalization

height of stabilazation = free edge length (µm)

heig

ht o

f def

lect

ion

(µm

)

L50L100L150L200L250L300

Figure 6.10: Optimalization plot for 2D cantilever with Si stabalization for 6 different lengths

In figure 6.11 an optimalization is shown for 2D cantilevers with a ratio of 2:1. Inhere the sta-bilization height is two times larger as the free edge size. In this figure it can be seen, thatlarger stabilization heights result in a large decrease of the deflection at the cantilever tips for thedifferent cantilever lengths. Using a stabilization ratio of 2:1 will result in less deflection of themembrane, like expected, because the moment of inertia is increased in the stabilized part.

0 5 10 15 20 25

10−1

100

101

Optimalization plot for 2D cantilever with Si stabalization and ratio − 2:1

height of stabilazation = 2 * free edge length ( µm)

he

igh

t o

f d

efl

ect

ion

(µ

m)

L50L100L150L200L250L300

(stabilization height = 2*free edge)

Figure 6.11: Optimalization plot for 2D cantilever with Si stabalization for 6 different lengths witha ratio of 2:1 (Stab.Height:free edge)

The optimalization is done for 2D models, but it is important to check if a 3D model will givethe same optimalization results for the different cantilevers. Therefore the same optimaliza-

40

tion should be done for 3D models. To simulate all the different cantilevers in 3D results ina very large computing time. Therefore only an optimalization is done on three cantilever lengths(50µm, 100µm and 150µm). The results of this optimalization are found in figure 6.12. Thedeflection values of the 3D models are higher than of the 2D models, like seen in all the othersimulations.

0 2 4 6 8 10 12 14 16 18

100

101

Optimalization plot for 3D cantilever with Si stabalization

height of stabilazation = free edge length (µm)

heig

ht o

f def

lect

ion

(µm

)

L50 3DL100 3DL150 3D

Figure 6.12: Optimalization plot for 3D cantilever with Si stabilization for 3 different lengths withratio 1:1

6.5 Conclusions

In this chapter a silicon stabilized cantilever model is developed. A simple approach is used forthe implementation of stress in the silicon stabilization material. This approach is verified withthe use of different simulations, because of the lack of experimental data. An optimalization isdone on the total deflection of the cantilever for different sizes of free edge and stabilization height.The minimum deflection for different cantilever lengths is found by different sizes of stabilizationheight and free edge sizes.

In figure 6.13 the optimal points for the different cantilever lengths and models are shown. In thisfigure it can be seen that the optimalization points (length of the free edge) of the 2D and 3Dmodels for the ratio 1:1 are almost the same. Only the height of the deflection is larger (thesedifferences are seen in all 2D and 3D models). Therefore the 2D model optimalization results canbe used to find an optimalization point for 3D models.

This aspect is important when developing an optimal result in a whole membrane. In a wholemembrane only one stabilization height is possible, because of the production process. Howeverthe membrane consists of different parts, which can be seen as different length cantilever shapes.A simple optimalization guideline can be made.

First study the membrane and get the length of the largest cantilever shape (or the length in themost interesting part of the membrane). Check the optimal design point of such a cantilever lengthwith help of figure 6.10. This gives the optimal design sizes of the membrane, using a ratio of 1:1.In the optimalization of the membrane some parts will not obtain a minimum deflection, whenoptimizing. It is important to decide which parts are the most important and at those positions

41

5 10 15 20 250

0.5

1

1.5

2

2.5Optimal line for 2D and 3D ratio 1:1 models

free edge length ( µm)

he

igh

t o

f d

efl

ect

ion

(µ

m)

optimalization points of 2Doptimalization points of 3D

rati

o 1

:1 -

3D

mo

de

l

ratio 1

:1 - 2

D m

odel

200

250

300

50

100

100

50

150

150

Figure 6.13: Optimal points for 2D and 3D (ratio 1:1) cantilever lengths with Si stabilization

an optimalization should be done. This should be done by doing a trade-off with the rest of themembrane deflection.

In the optimalization plot of figure 6.10 not a smooth curve is obtained. It appears that the resultsof some stabilization heights are not exact. This is caused by the relative large elementsize, whichcauses a small error in the final result. To prevent long computing times in the 2D models noelementrefinement is done.

42

Chapter 7

Model of a membrane

In the previous chapters different simulations were done to describe the behavior of the nanos-tencils. Silicon stabilized cantilever models were developed to find the optimal design to obtainminimal deflections. In this chapter the techniques developed in the previous chapters will beused to simulate a whole membrane. These models will give a prediction of the deflections in themembrane. This information can be used to make an optimal design of a whole membrane.

This chapter starts with a description of the different modeling aspects needed to make a modelof an unstabilized membrane. In the following paragraph the results of an unstabilized membranewill be discussed and an explanation will be given to understand the results given by ANSYS. Inthe last paragraph stabilized models are developed. These models will be used to make an optimaldesign of the membrane. Optimalization rules will be discussed and verified with those models.

7.1 Modeling of a membrane

In chapter 1 of this report an example is given of an experiment of a membrane with, and without,a silicon stabilization (figure 1.8). The membrane used in this experiment is the only producedmembrane with silicon stabilization on it. Therefore a model is made which will describe theseexperiments to compare the simulations with the experimental data. In this way it can be verifiedif the developed techniques in this report can be used in a whole membrane simulation.

The geometry of the described membrane is shown in figure 7.1. In this figure only 1/4 of the fullmembrane is shown. The full membrane (1 mm by 1 mm) consists of four parts, all with the sameshape like shown in figure 7.1. For the development of the geometry of the model a symmetryline can be drawn through the middle of the membrane (figure 7.1). Decreasing the size of themembrane results in a decrease of elements and therefore a decrease in calculating time. At thetop, the membrane is fixed to the silicon surrounding of the membrane. Therefore the boundaryconditions at the top are fixed in all directions. The two unconstrained sides are fixed to therest of the membrane and cannot move in the plane directions and are constrained in those twodirections. Deformation in the out-of plane direction is possible.

The deformations of the membrane are partly caused by a Cr-layer with a thickness of 35 nmand a residual stress of 1930MPa in it. For the implementation in the model this residual stressshould be changed into a temperature difference. Equation 4.6 is used to develop a temperaturewith a given stress. For this membrane it results in a temperature difference of T = 954K. Thisvalue is without the compensation factor of 6% described in chapter 4. Therefore the implementedtemperature in the model should be T = 1011K. The other part of the deformation is causedby residual stresses in SiN and these stresses are the same as used in the cantilever models. Thegeometry of the membrane used in the model is directly imported from the design software (Expert,

43

45

0 m

icro

me

ter

275 micrometer

Symmetry lineno disp. in all directions

no

dis

p. i

n p

lan

e d

ir.

Figure 7.1: Geometry of a 1/4 membrane, that is used in the experiments

in CIF-format), used to design the nanostencils. In appendix E the input file for this model canbe found.

7.1.1 Free versus mapped meshing

After discussing the geometry, boundary conditions and the applied loads of the model the meshingis discussed. Because of curved shapes and a changing area width, the geometry of the membraneis complex compared to the cantilever structures . This complex mesh is the reason, that it is verydifficult to develop a mesh by using hexahedron elements. Hexahedron elements are cubic-shapedelements. Those elements are normally used, because they can easily deform to a given load,because of their shape. A rectangular shape easily deforms with an applied load (small stiffness).This causes elements which adapt well to a given applied load. Therefore the results of modelsusing those elements are relative accurate. The disadvantage of this shape is that it results indifficult fitting into a complex geometry. The implementation of the right sizes and shapes into ageometry by hand is called mapped meshing and should be done when using hexahedron elements.

An option to avoid time consuming meshing of the membrane geometry is using tetrahedral ele-ments. A tetrahedral element has got four triangular shaped area’s. This shape can more easily befitted into a complex mesh. This can be done by a computer algoritme and the meshing process istherefore much faster. This type of meshing is called free meshing. A disadvantage of this elementshape is the larger stiffness it got. A triangular shape will deform less when a load is applied toit. In theory this results in a less accurate result using these elements.

However in the described membrane the implementation of hexahedron elements is to complexand therefore the tetrahedral elements have to be used. To verify the accuracy of these elementsa mapped meshed cantilever model is compared to a free meshed cantilever model, with the sameelement sizes. Both models were giving the same deflection value. It can be concluded that a free

44

meshed model can be used for the simulation of the membranes.

Another point of attention developing the mesh for the membrane are the element sizes. A smallelement size is preferred to obtain an accurate solution (like discussed in chapter 5). Because ofthe size of the membrane it is not possible to mesh the entire membrane with small elements.This will result in a model, which takes very long computing time. It is also possible that thesemodels will become to large for the present computer memory. Therefore a trade-off should bemade between element sizes, computing time and the accuracy of the results. In general smallelement sizes should be used in area’s with large deflections. In this membrane no large differencesare expected at the edges of the membrane and therefore large elements should be used at thosepositions. An example of a meshed model is given in figure 7.2.

Figure 7.2: Meshed membrane model with tetrahedral elements

7.2 Results

The model described above is used for the simulation of the unstabilized membrane. First a linearsolution is done and studied. This linear result doesn’t take a lot of computing time, because thestiffness matrix is calculated one time. However this linear result gives a good impression wherethe large deflections will appear in the model. This model can be used to do a mesh refinementin some specific places.

After the linear solution is obtained and used for a mesh refinement in the interesting places anon-linear solution is done. The out-of-plane deflections of this model are shown in figure 7.3. Inthis figure the deflection at the same point (A in figure 7.3) is measured as done in figure 1.8. Thedeflection in the experiment at this point is 47µm. In this model the deflection is 13, 4µm. Anexplanation for the difference between the experiments and the simulations are discussed in thefollowing paragraph.

7.2.1 Dependency of results on model size

To understand why the deflections of the membrane simulations are much smaller than of theexperiments research is done to the behavior of the elements. The results of the cantilever sim-

45

A

Figure 7.3: Deformations in the out-of plane direction in a model of the membrane with a non-linear solution

ulations are close to the experimental values. In the simulations of the membrane elements aretaken with the same element size or smaller, as used in the cantilever models. For this reason theinaccuracy can’t be caused by a to large element size.

Because the cantilever models are giving a good result and the much larger membrane isn’t,research is done to the influence of the size of the membrane to the cantilever deflection. Thereforedifferent models are made, wherein the influence of the model size can be measured. In figure 7.4these models are shown. In model A an expansion is made, compared with the reference modelfrom chapter 5. Model B is an expansion of model A. All those three models have got thesame elementsizes. In model C a element refinement is done on the cantilever, because the largedeformations are appearing in this area. In table 7.1 the deflections are given for the differentmodels. Model C2 is the same as model C, but it is meshed with tetrahedral elements.

A B C

Figure 7.4: Three models with an increasing model size to study deflection behavior dependencyon model size

From the results of table 7.1 it can be seen that a simulation on an expanded model results in

46

Table 7.1: Deflection values for cantilever models with different sizesmodel name deflection (µm) noteref. model 10.91model A 9.96model B 4.915model C 10.91 element refinement on the cantilevermodel C2 9.358 model C, meshed with tetrahedral elements

lower deflection values, when using same element sizes. With the expansion of model B, thisresults already in a 55% reduction of the deflection. Implementing smaller elements in area’s withlarge deformations (mesh refinement) will result in the original deflection values. In model C2 asimulation is done with tetrahedral elements. Looking to the deflection of this model, it can beconcluded that results of simulations using tetrahedral elements are less accurate, but don’t causethe large differences obtained in the membrane models. This is caused by the models size of themembrane. To obtain a good result for the membrane model a much more refined mesh should bebuild. With the restriction of the computer memory this isn’t possible in the scope of this project.

7.3 Optimalization

The results of the unstabilized membrane model aren’t describing the experiments accurate enough,as discussed in the previous paragraph. However in this paragraph an attempt is done to do anoptimalization on this model of the membrane. Silicon stabilized models will be developed toobtain a general idea of the most optimal design to get minimal deflections in the membrane.These stabilized membrane models can be used to verify the optimalization rules developed inchapter 6 and in further research these models can be used to obtain a more exact solution. Inthis paragraph the geometry of the stabilized model is discussed. Afterwards the results of themodels are compared to the experimental results. In the last part the optimalization rules ofchapter 6 will be verified and, when necessary, corrected.

7.3.1 Geometry of stabilized model

To build a silicon stabilization block on the membrane developed in paragraph 7.1 the sameconditions should be considered as discussed in chapter 6.1. The stabilization structures shouldnot interfere with the material transport from the source through the apertures, i.e. the line-of-sight of the apertures to the material source should not be obstructed by the stabilizationstructures. This results in a fixed distance of the beginning of the silicon stabilization to allthe edges of the membrane, depending on the height of the stabilization. The distance betweenthe edge and the silicon is shown in figure 7.5. To obtain this same distance over the wholemembrane a relative complex stabilization geometry should be developed. At corner points anarc-shape should be made and points at the tips of the membrane should be replaced, which isa difficult task. Therefore some simplifications are made designing the silicon geometry. Thesesimplifications can be seen in figure 7.5.

In figure 7.6 a cross-section (through line 1 (figure 7.5) is shown for the stabilized membrane. Fourdifferent kind of layers can be seen. The layer on the bottom is the SiN layer. On top of thislayer are the Si-layer and the Cr-layer (on the unstabilized free edges). On top of the Si-block isa Cr-layer present.

A relative complex geometry is present to build a good mesh for the stabilized model. This iscaused by the sharp edges of the silicon block (area’s at the tip have got a very sharp angle betweeneachother of ¿ 90o), which causing difficult tetrahedral implementation. The different shapes ofthe four layers also result in difficult meshing, because the nodes of the elements at the edges of

47

FREE

FREE

Line 1

sin 43 FREE

cos 43 x

x

FREE

FREE

FREE

Stabilization

Free edges

Figure 7.5: Top view of silicon stabilized membrane model, where a fixed distance between mem-brane edge and beginning of the silicon block is shown

FREE

Sta

b. H

eig

ht

Cr

Si SiN

FREE FREEStab. H

Cr

Si SiN

Figure 7.6: Cross-section of membrane through line 1 (figure 7.5)

the layers are used by elements of the two layers and therefore should be positioned at the samelocation. These meshing aspects are causing a mesh, where relative small elements are needed.Again, this will result in models that are becoming to large for the computer memory. However infigure 7.7 a meshed stabilization membrane model is shown. In this model the stabilization heightand free edge are changeable, so an optimalization is possible for the membrane.

7.3.2 Results

Simulations are done with the stabilized membrane model. A result of the out of plane deflectionsof the membrane is shown in figure 7.8. In the experiments the deflections of two points of themembrane are measured. These deflections can be compared with the same points in the membrane

48

Figure 7.7: A meshed silicon stabilization membrane model with a stab. height: free edge of20µm:10µm

models (point A and B in figure 7.8). When looking to the reduction values (in percents) of thedeflections between the stabilized and unstabilized membranes, an idea of the usability of thesemodels can be given. The values of the deflections in point A and B for the experiments andmodels are given in table 7.2.

A

B

A

B

Figure 7.8: Result of the out-of-plane deformation of the stabilized membrane (stab.height =20µm and free edge = 10µm))

49

Table 7.2: Deflection values for (stabilized and unstabilized) membrane models and experimentsat points A and B

unstabilized deflection (µm) stabilized deflection (µm) reductionexperiment point A 47 15 68 %

model point A 13,36 1,37 89 %experiment point B no absolute value present (n.a.v.p.) n.a.v.p. 89 %

model point B 7,03 0,61 91 %

Large differences between the absolute values of the experiments and the simulations in table 7.2are present. The reason for these differences is explained in the previous paragraph. However thevalues for the reduction of the deflection are in the same order. Therefore it can be concludedthat the membrane models aren’t giving an absolute accurate result, but it is giving qualitativelya good description of the influence of the stabilization on the membranes.

7.3.3 Verification of the optimalization guideline

In chapter 6 an optimalization guideline is developed. This guideline can be verified with the helpof the results of different silicon stabilized models. Using this guideline, in general the longestcantilever like shape should be taken out of the membrane and the length should be measured. Inthe optimalization plot of the cantilever models this length should be compared with a cantileverwith the same length. The optimal design point for the cantilever should also be the optimaldesign point for the membrane.

This guideline can be verified by looking to the modeled membrane. In this membrane cantilevershapes can be found with a length of around 150µm. In the results of the unstabilized modelthe largest deflection was at the tip, where two cantilever shapes come together (point A). Thedeflection of this tip is probably a bit larger than of a 150µm cantilever, because of the contribu-tion of two cantilever shapes to the deflection. In figure 6.10 (2D cantilevers) the optimal designpoint can be found between the optimum of 150µm and the 200µm-cantilevers. In figure 6.12 (3Dcantilever models) no models are present for the 200µm cantilevers and therefore the 150µm-lineshould be taken to observe the deflections.

Three different stabilized membrane models are build with a free edge and stabilization height of5µm, 12µm and 15µm. Taking into account that the deflection at point A is a bit larger thanof a 150µm cantilever, figure 6.12 shows that the model with the free edge of 15µm should bethe optimal design (looking to ratio 1:1 stabilizations). In table 7.3 it can be seen that, with thetested models, the stabilized membrane models are describing a decrease of the deflection until the15µm stabilization model. Because a lack of time no further stabilized membrane models couldbe simulated. Therefore it can be concluded that the developed guideline in chapter 6 and 7 amethod can’t be verified at this moment, but some first results are giving a good impression forthe verification of this guideline.

Table 7.3: Deflection values for stabilized membrane models and experiments at point Aratio sizes (µm) deflection point A (µm) deflection cantilever (3D) tip 150µm (µm)

5:5 10,2 4,312:12 3,15 2,115:15 2,5 2,2

50

Chapter 8

Conclusions and recommendations

Conclusion In this project finite element models are developed to describe residual stresses inthin SiN membranes. These models are developed in the FEM software ANSYS. These models aredesigned to describe the deformation of the membranes caused by the residual stresses appearingwhen evaporating a material on the SiN membranes (nanostencils). In this project stabilizationmethods to decrease the deformations in the membrane are simulated and the results can be usedfor an optimalization of the membrane design.

By doing an analysis of experiments, done on a membrane, the values of the residual stresses arecalculated. A membrane is produced, wherein cantilever structures are present. Because of theshape of a cantilever these structures can be easily modeled and these cantilevers are used for thedevelopment of the model. A theoretical calculation to verify the results from the simulations isdone, because theories are present to calculate deflections of a cantilever. The material propertiesused in the models are obtained from simulations done on the SiN membranes by Tyndall NationalInstitute.

Different methods for the implementation of residual stresses into the models are developed andcompared with the experimental results by measuring the deflections of 2D cantilever models. Fourmethods are evaluated (an ANSYS option, stress replacement by nodal forces and two differenttemperature difference methods). The method wherein the stresses are described by implementinga temperature difference between two materials has got the smallest error comparing the resultswith the experiments (±20%). The characteristic of this temperature difference method is thata compatibility requirement is implemented to obtain the same length of the two layers whendeforming. The deflections of the cantilevers are large and therefore the models should be solvedwith non-linear solution options in it.

The 2D cantilever models are expanded to a 3D model. In this 3D model an element type is usedthat is still giving an accurate solution when using large sizes and aspect ratio’s in the models.The behavior of the element type is verified by changing the sizes and aspect ratio’s in differentsimulations. This kind of element should be used, because in whole membranes large elements areneeded. This is needed to prevent to many elements in the model and therefore to large computingtimes. An optimalization of the geometry of the model is done and the results of the optimizedmodel are compared with the experimental values. The differences of the 3D models are within10% of the experimental results and can be more accurate when f.e. better material data is present.

A 2D silicon stabilization cantilever model is developed to see the reduction of the deflectionscomparing stabilized and unstabilized cantilevers. In this stabilized model a simple calculationof the stress working on the silicon is made and this stress is implemented by calculating thetemperature differences. No experiments are done on silicon stabilized cantilevers and thereforethe models can’t be verified with the experimental results. The approach is verified with the use

51

of different simulations. The verified 2D models are expanded to 3D models. An optimalization isdone for 2D and 3D models by changing the stabilization heights and free edges. The 2D and 3Dmodels are giving the same optimalization design points. This optimal design data can be usedas guideline for the optimalization of whole membranes. In a membrane the longest cantileverlike shape should be measured. The optimal design point for this length should be found in theoptimal cantilever design data and the found stabilization sizes should be implemented in thesilicon stabilized membrane to obtain minimal out of plane deflections.

The developed modeling techniques in this project are used in the modeling of whole membranes.Tetrahedral elements are used to mesh the complex geometry of the membranes. The differencesin accuracy of the results when using the tetrahedral element type instead of hexahedron elementsisn’t large. Because of a to large membrane size the element solutions aren’t accurate enough.Therefore these simulations results can’t be compared with experiments done on these membranes.A silicon stabilized model is made to verify the reduction values between stabilized and unstabilizedsimulations with the experiments. The absolute values obtained in the simulations aren’t the sameas the experiments, but the reduction values when stabilizing are giving a good description. Thedeveloped guideline couldn’t be verified, because a lack of time. However some first results aregiving a good impression for the verification of this guideline.

Recommendations Improvements in the simulations and further research are important to de-velop an advanced method to obtain an optimal design for a membrane, without time consumingmodeling work. At the moment no accurate results are obtained for the whole model membranes.Smaller element sizes should be used to obtain better results, resulting in large models. The sim-ulations can be done by using more computer memory to run those large models.

In the field of the accuracy of the results better material data should be obtained, by doing exper-iments (like nano indentation) to get better values for the Youngs Modulus etc.. Using anisotropicmaterial data instead of isotropic data can also improve the accuracy of the simulations, whencomparing the results with the experiments. Experiments on a membrane, containing silicon sta-bilized cantilever structures, can be done to verify the obtained silicon optimalization points.

To decrease the long computing times, research can be done to the implementation of other el-ement types (f.e. SOLID185) in no large deflection area’s in the membrane (f.e. in the siliconstabilization). Using shell elements for the Cr-layer can also result in better results. Anotherpossibility is to model only the interesting parts of the membrane. In this way less elements areneeded and this effort can be used to use smaller elements to obtain an accurate solution.

At the moment the development of a stabilized membrane model is a very time-consuming work.Research should be done to some automatization of this modeling process. In this way optimizedmembrane designs can be easily obtained. The described optimizing guideline can be used as firstimpression for the optimal point. To use the developed guideline for designing a membrane itshould first be verified.

Another interesting topic for further research is the stress behavior in the membranes. To preventcracks and plasticity in the membrane, because of to large stresses, research should be done to thestress distribution during the evaporation processes. Doing the described research can lead to anadvanced, accurate, FEM model of a random optimal designed membrane.

52

Acknowledgement

In this acknowledgement I want to thank the entire LMIS1-group for the nice time four months Ihad in Switzerland. Especially I want to thank Marc van den Boogaart for all the support duringmy project and helping me to continue my project, when I run into some problems. Also I wantto thank Lianne Doeswijk for arranging my stay, showing me around in the EPFL and gettingme started the first couple of weeks in Lausanne. Thanks to David Mendels for answering andhelping me with the different questions I had with the ANSYS software.

Other people who I want to thank for the good time I had in the LMIS1 group:

• The post-docs, for the many coffee-breaks

• Vahid, for his many jokes

• Jeroen, for his discussions about proteins and cycling

• Christophe and Kevin, for the time I spend with them in the office

• Juergen, for being the professor of a very nice group and the tips he gave for the bike rides

• Schahrazede, for the attempt to improve my French skills

53

Bibliography

[1] G.M.Kim,M.A.F. van den Boogaart, and J.Brugger,’Fabrication and application of a fullwafer size micro/nanostencil for multiple length-scale surface patterning’,Microelectronicengineering,67-68,p.609-614

[2] M. Ohring, The Material Science of Thin Films;Academic Press, 1992

[3] G. G. Stoney, ”The Tension of Metallic Films Deposited by Electrolysis,” Proc. Roy. Soc.London, A82(553), 1909, pp. 172

[4] M.Lishchynska: titel , jaartal

[5] Brandes, E.A., Brook, G.B., Smithells Metals Reference Book, seventh edition, ISBN 0-7506-1020-4

[6] http://www1.ansys.com/customer/

[7] S.Timoshenko, Analysis of bi-metal thermostats,Proc.Roy.Soc. London, A107,1925,pp. 291-309

[8] Solid mechanics chapter 9,Lecture nodes Thin Film Mechanics TU/e, Jaap den Toonder, pp.135-136

[9] ANSYS Structural Guide, ch 8, Non Linear Structural Analysis

[10] ANSYS Theoretical Referency Guide, ch 15, Newton Rhapson Procedure

[11] B.Michel, et al.,’Printing meets lithography:Soft approach to high resolution patterning’,IBMJ.Res.Dev.,45(5),2001,pp697-179

[12] S.Y.Chou,P.R.Krauss and P.J.Renstrom,’Nanoimprint lithography’,J.Vac.Sci.Tech.B,14(6),pp4129-4133

[13] M.A.F.van den Boogaart, M.Lishchynska,L.M.Doeswijk,J.C.Greer and J.Brugger,’Corrugatedmembranes for improved large-area and high-density nanostencil lithogra-phy’,Microelectromechanical Systems,submitted,2005

[14] M.A.F.van den Boogaart, L.M.Doeswijk and J.Brugger,’Silicon supported membranes forimproved large-area and high density nanostencil lithography’,Microelectromechanical Sys-tems,submitted,2005

[15] M.G.D. Geers: Fundamentals of Deformation and Linear Elasticity, 2001

54

Appendix A

Calculation of deflection andmoment of inertia

In this appendix the calculation is made for the deflection and the moment of inertia’s of theunstabilized and corrugated cantilevers. In the first section the derivation of the deflection is given.In the second section the calculation for the moment of inertia for the corrugated cantilevers isgiven.

A.1 Derivation of deflection

The derivation of the deflection starts with Stoney equation posed in Chapter 2 [3].

σf =1

6R

Esd2s

(1− νs)df(A.1)

First the curvature (R) is rewritten into a deflection. The curvature is related to the secondderivative of the beam displacement [2], like:

1R

=δ2y

δx2(A.2)

after the following integration (because of boundary conditions c1 = c2 = 0):

δy

δx=

∫ l

0

1R

dx =l

R+ c1 (A.3)

y =∫ l

0

(l

R)dx =

12l2

1R

=l2

R= δ(deflection) (A.4)

R =l2

2δ(A.5)

Implementing R in the Stoney equation results (A.6) and rewriting the equation A.7:

σf =δ

3l2Esd

2s

(1− νs)df(A.6)

δ =σf (1− νs)df3l2

Esds(A.7)

55

The equation A.7 is derived for the situation of a rectangular cantilever with a moment of inertiaof Is = 1

12wd3s. Rewriting the moment of inertia such that Is can be put into the equation:

4Is =wdfd3

s

3df(A.8)

14Is

=3df

wdfd3s

(A.9)

wdfds

4Is=

3df

d2s

(A.10)

Put equation A.10 into equation A.7 results in equation A.11. This equation is used to calculatethe different deflections for the different moments of inertia

δ =σf (1− νs)wdfdsl

2

4EsIs(A.11)

A.2 Derivation of moment of inertia for cantilevers

REMARK: IDEAL SITUATION

STB _ W

STB _ H

Cantilever _ W

Membrane

Thickness

(1)

(2)

(3)y: neutral line

∑∑ ⋅

=A

Ayy

∑ += )( 2AdII

Figure A.1: corrugated structure for calculating moment of inertia

The formula’s used for the calculation of the moment of inertia are shown in figure A.1. The Itotal

is calculated by the summation of the three different parts:

ΣIi = Itotal = I1 + I2 − I3 (A.12)

First the value of ytotal is calculated (A.13), afterwards the values di , which are the distancesbetween ytotal − yi. Now every separate value for Ii + Ad is calculated. The three different valuesare implemented in equation A.12 and the moment of inertia for the structure in figure A.1 isknown.

ytotal =Σy1A1 + y2A2 − y3A3

ΣA1 + A2 + A3(A.13)

The moment of inertia for the one RIM cantilever is known. The calculation for the three RIM’scantilever is calculated in the same way. Three RIM’s are present so the value of the cantileverwidth decreases (figure A.1) and the I1 decreases. The total moment of inertia is therefore notthree times larger than the 1 RIM.

56

Appendix B

ANSYS input files for unstabilized2D models

A general ANSYS input file is given for the 2D model used in Chapter 4. The different methodsare implemented in this file. The commands for the methods can be found below this general file.First the file with the input data is given.

Filename: macro3Dv1.mac

:MATERIALESIN = 276e3 ! MATERIAL 1 YOUNG’S MODULUSECR = 277e3 ! MATERIAL 2 YOUNG’S MODULUSESI = 170e3 ! MATERIAL 3 YOUNG’S MODULUSPOISSONSIN = 0.27 ! MATERIAL 1 POISSON’S RATIOPOISSONCR = 0.3 ! MATERIAL 2 POISSON’S RATIOPOISSONSI = 0.26 ! MATERIAL 3 POISSON’S RATIOALPXSIN = 2.8E-6 ! MATERIAL 1 ALPHAALPXCR = 7.3E-6 ! MATERIAL 2 ALPHAALPXSI = 5.2E-6 ! MATERIAL 3 ALPHA

MP,EX,1,ESINMP,PRXY,1,POISSONSINMP,EX,2,ECRMP,PRXY,2,POISSONCRMP,EX,3,ESIMP,PRXY,3,POISSONSIMP,ALPX,1,ALPXSINMP,ALPX,2,ALPXCRMP,ALPX,3,ALPXSI

MP,EY,1,ESINMP,PRYZ,1,POISSONSINMP,EY,2,ECRMP,PRYZ,2,POISSONCRMP,EY,3,ESIMP,PRYZ,3,POISSONSIMP,ALPY,1,ALPXSINMP,ALPY,2,ALPXCRMP,ALPY,3,ALPXSI

MP,EZ,1,ESINMP,PRXZ,1,POISSONSINMP,EZ,2,ECRMP,PRXZ,2,POISSONCRMP,EZ,3,ESIMP,PRXZ,3,POISSONSIMP,ALPZ,1,ALPXSINMP,ALPZ,2,ALPXCRMP,ALPZ,3,ALPXSI

/EOF

:PARAMETERSC_L = 50 ! VALUE FOR CANTILEVER LENGHTC_W = 50 ! VALUE FOR CANTILEVER WIDTH (NOT THE HALF)

57

S_L = 10 ! EXPANDED LENGHT VALUES_W = 5 ! EXPANDED WIDTH VALUEI_L = 5 ! LENGHT OF INHAMI_W = 5 ! WIDTH OF INHAM

M_T = 0.5 ! VALUE FOR SIN MEMBRAME THICKNESSMAT_T = 0.05 ! VALUE FOR CR MEMBRAME THICKNESS

VERH = 1 ! RATIO BETWEEN FREE EDGE AND STAB HEIGHTFREE = 5 ! FREE EDGE FOR SILICIUMSTAB_HEIGHT = FREE*VERH ! STABILIZATION HEIGHT FOR SILICIUM

RATIO = 40 ! SIZE OF ELEMENTS RATIO FOR 2D MODELSELEMSIZE = 50 ! SIZE OF ELEMENTS RATIO FOR 3D MODELSTH = 1 ! THICKNESS RATIO OF CR

COMPENSATION = 1.06 ! COMPENSATION DESCRIBED IN CHAPTER 4DELTATEMPCR = 608*COMPENSATION ! TEMPERATURE TO CHROMEDELTATEMPSIN = 259*COMPENSATION ! TEMPERATURE TO SILICON NITRIDEDELTATEMPSI = 566 ! TEMPERATURE IN SILICON

/EOF

Filename: twolayerclv2.mac

J_ARRAY=6*DIM ,C_L ,,J_ARRAY! CANTILEVER LENGHTC_L(1) = 50, 100, 150, 200, 250, 300! START DO LOOP FOR CANTILEVER LENGHTS*DO ,J ,1 ,J_ARRAY ,1

/TITLE, RESIDUAL STRESS IN SIN AND CR 2D MODEL

/PREP7

ALLSEL,ALLACLEAR,ALLADELE,ALL,,,1KDELE,ALL

/COM, MATERIALS DATA/COM,/INPUT, macro, mac, ’C:\DOCUME~1\vanschai\MYDOCU~1\ANSYSM~1\TEMPERATURE’ ,:MATERIAL, 1

/COM, GEOMETRY OF FINITE ELEMENT MODEL/COM,

/INPUT, macro, mac, ’C:\DOCUME~1\vanschai\MYDOCU~1\ANSYSM~1\TEMPERATURE’ ,:PARAMETERS, 1

K,1,0,0 ! BUILDING GEOMETRY WITH KEYPOINTSK,2,C_L(J),0K,3,C_L(J),M_TK,4,0,M_TK,5,C_L(J),M_T+MAT_T ! KEYPOINT 5 AND 6 NOT NEEDED IN METHOD OF SURFACESTRESSK,6,0,M_T+MAT_T

A,1,2,3,4,1 ! AREA 1 SINA,4,3,5,6,4 ! AREA 2 CHROME

NUMMRG,ALL

/COM, ELEMENTTYPES

ET,1,PLANE42,,,1 ! PLANE42 ELEMENTTYPE SELECTEDKEYOPT,1,3,3 ! PLANE STRAIN FUNCTIONR,1,C_W ! THICKNESS OF MEMBRANE

LESIZE,ALL,ELEMSIZE ! SIZE OF ELEMENTS EQUALS 0.5 MICROMETERTYPE,1

ASEL, S, AREA, ,1, , ,1CM,SIN,AREAAATT,1 ! MATERIAL DATA 1

ASEL, S, AREA, ,2, , ,1CM,CR,AREAAATT,2 ! MATERIAL DATA 2

ALLSEL,ALL

58

AMESH,ALL

/COM, BOUNDARY AND SYMMETRIC CONDITIONS

NSEL,S,LOC,X,0D,ALL,UX,0 !SYMMETRIC CONDITIONSD,ALL,UY,0 !SYMMETRIC CONDITIONS

ALLSEL,ALL!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

! IMPLEMENTATION OF THE DIFFERENT BODY FORCES TO DESCRIBE THE STRESS

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!ALLSEL,ALL

FINISH

/SOLUTION

ANTYPE,STATIC ! STATIC ANALYSISNLGEOM, ON ! NON LINEAR EFFECTSNSUB,15 ! NUMBER OF ITERATION STEPSSOLVE

FINISH

/POST1

*GET,DEFL,NODE,NODE(C_L(J),M_T+MAT_T,0),UY*CFOPEN,reporttemp2nlOFF,mac,’C:\Documents and Settings\vanschai\’,APPEND*VWRITE,%C_L(J)%,DEFL(E12.5,’ ’,E12.5)*CFCLOS

PLNSOL, U,Y, 0,1.0!/IMAGE, SAVE, twolayer2D, bmp,’C:\Documents and Settings\vanschai’

FINISH*ENDDO

The input for the implementation of the residual stresses for the four different methods:

ISTRESS:twolayerclistress.mac (Implementation in solution environment)

ISTRESS,1230,1230,1230, , , ,2

Output data NLGEOM ON and OFF:

0.50000E+02 0.35705E+010.10000E+03 0.12473E+020.15000E+03 0.23216E+020.20000E+03 0.52970E+020.25000E+03 0.67422E+020.30000E+03 0.11262E+03

0.50000E+02 0.20646E+010.10000E+03 0.82582E+010.15000E+03 0.18581E+020.20000E+03 0.33033E+020.25000E+03 0.51613E+020.30000E+03 0.74324E+02

Surfacestress: twolayerclsurface.mac

NSEL,S,LOC,Y,M_TFAPPL = (STRESSCR*C_W*MAT_T)/(1+(C_L(J)/(ELEMSIZE))) ! MAGNITUDE OF APPLIED FORCE

F,ALL,FX,-FAPPL ! MINUS SIGN COMPRESSIVE STRESS

Output data NLGEOM OFF:

0.50000E+02 0.44454E+010.10000E+03 0.17804E+020.15000E+03 0.40075E+020.20000E+03 0.71261E+020.25000E+03 0.11135E+030.30000E+03 0.16036E+03

59

Temperature method 1:twolayerclv2.mac

TREF,0

BF,SIN,TEMP,-DELTATEMPSIN ! BODY TEMP TO SINBF,CR,TEMP,-DELTATEMPCR ! BODY TEMP TO CR

Output data NLGEOM ON and OFF:

0.50000E+02 0.53199E+010.10000E+03 0.21001E+020.15000E+03 0.46256E+020.20000E+03 0.80168E+020.25000E+03 0.12098E+030.30000E+03 0.16697E+03

0.50000E+02~~0.53720E+010.10000E+03~~0.21474E+020.15000E+03~~0.48306E+020.20000E+03~~0.85872E+020.25000E+03~~0.13415E+030.30000E+03~~0.19300E+03

Temperature method 2

TREF,0

BF,SIN,TEMP,-DELTATEMP ! BODY TEMP TO SINBF,CR,TEMP,-DELTATEMP ! BODY TEMP TO CR

Output data NLGEOM ON and OFF:

0.50000E+02 0.57386E+010.10000E+03 0.22622E+020.15000E+03 0.49731E+020.20000E+03 0.85688E+020.25000E+03 0.12833E+030.30000E+03 0.17572E+03

0.50000E+02 0.58487E+010.10000E+03 0.23371E+020.15000E+03 0.52569E+020.20000E+03 0.93451E+020.25000E+03 0.14595E+030.30000E+03 0.20975E+03

Figure B.1: top, the elementdivision of the model, under, a result of a simulation with a cantileverof length 50 µm

60

Appendix C

ANSYS input files for unstabilized3D models

A general ANSYS input file is given for the 3D model used in Chapter 5. The method with the temperature difference isimplemented in this file.

Filename: cantilever3Dv3.mac

/TITLE, 3D MODEL OF CANTILEVER WITH RESIDUAL STRESSES

/UNITS, SI/VIEW,1,1,1,1

/PREP7

! CLEAR ALL DONE WORKALLSEL,ALLACLEAR,ALLADELE,ALL,,,1VCLEAR ,ALLVDELE ,ALL , , ,1

/COM, MATERIALS DATA/COM,

! READING MATERIALDATA/COM,/INPUT, macro3Dv1, mac, ’C:\DOCUME~1\vanschai\MYDOCU~1\ANSYSM~1\MODELS3D’ ,:MATERIAL, 1

/COM, INPUT PARAMETERS/COM,

/COM,/INPUT, macro3Dv1, mac, ’C:\DOCUME~1\vanschai\MYDOCU~1\ANSYSM~1\MODELS3D’ ,:PARAMETERS, 1

/COM, GEOMETRY OF FINITE ELEMENT MODEL/COM,

! DEFINITION OF UNIT VOLUMEK,1,0,0,0K,2,1,0,0K,3,1,1,0K,4,0,1,0

A,1,2,3,4,1VEXT,1,,,0,0,1,0,0,0CM,UNITVOLUME,VOLU

! START MAKING DIFFERENT COMPONENTS FOR THE MODEL

!VLSCALE,UNITVOLUME,,,RX,RY,RZ,0,0,0VLSCALE,UNITVOLUME,,,0.5*C_W,M_T,S_L,0,0,0 ! VOLUME 2

VLSCALE,UNITVOLUME,,,0.5*C_W,M_T,C_L,0,0,0 ! VOLUME 3VGEN,2,3,,,0,0,S_L,,0,1

VLSCALE,UNITVOLUME,,,I_W,M_T,S_L,0,0,0 ! VOLUME 4VGEN,2,4,,,0.5*C_W,0,0,,0,1

61

VLSCALE,UNITVOLUME,,,S_W,M_T,S_L,0,0,0 ! VOLUME 5VGEN,2,5,,,0.5*C_W+I_W,0,0,,0,1

VLSCALE,UNITVOLUME,,,S_W,M_T,I_L,0,0,0 ! VOLUME 6VGEN,2,6,,,0.5*C_W+I_W,0,S_L,,0,1

! DELETE UNIT VOLUMEVCLEAR,1VDELE,1,,,1

! MAKE CR LAYER

ALLSEL,ALL

CM,SIN,VOLU

VLSCALE,SIN,,,1,MAT_T/M_T,1,0,0,0 ! VOLUME 7 TO 10VGEN,2,7,10,,0,M_T,0,,0,1VGEN,2,1,,,0,M_T,0,,0,1

VSEL,S,VOLU,,7,10VSEL,A,VOLU,,1

CM,CR,VOLU

ALLSEL,ALL

NUMMRG, ALL

/COM, ELEMENTTYPE DEFINITION/COM,

ET,1,SOLID186

/COM, MESHING OF VOLUMES/COM,

NDX = C_W/ELEMSIZENDY = 10*MAT_TNDZ = C_L/ELEMSIZE

LSEL,S,LOC,Y,0.5*M_T ! LINES IN Y DIRECTIONLSEL,A,LOC,Y,0.5*MAT_T+M_TLESIZE,ALL,NDY

LSEL,S,LOC,Z,0 ! LINES IN X DIRECTIONLSEL,A,LOC,Z,S_LLSEL,A,LOC,Z,S_L+I_LLSEL,A,LOC,Z,C_L+S_LLSEL,U,LOC,Y,0.5*M_TLESIZE,ALL,NDX

LSEL,S,LOC,X,0 ! LINES IN Z DIRECTIONLSEL,A,LOC,X,0.5*C_WLSEL,A,LOC,X,0.5*C_W+I_WLSEL,A,LOC,X,0.5*C_W+I_W+S_WLSEL,U,LOC,Y,0.5*M_TLESIZE,ALL,NDZ

VSEL,S,VOLU,,2,6VATT,1,,1,0

VSEL,S,VOLU,,7,10VSEL,A,VOLU,,1VATT,2,,1,0

ALLSEL,ALL

TYPE,1MSHKEY, 1 ! KEY 1 MAPPED MESHINGVMESH, ALL

/COM, BOUNDARY AND SYMMETRIC CONDITIONS/COM,

! SYMMETRIC AND BOUNDARY CONDITIONSNSEL,S,LOC,X,0 ! SYMMETRICDSYM,SYMM,X

62

ASEL,S,AREA,,32CM,BC,AREA

NSEL,S,LOC,Z,0 ! BOUNDARYNSEL,A,LOC,X,0.5*C_W+I_W+S_WNSEL,A,LOC,,BCD,ALL,UX,0D,ALL,UY,0D,ALL,UZ,0

ALLSEL,ALL

! APPLY FORCES AND TEMPERATURES

! TEMPERATURES IN SIN AND CRTREF,0

CM,SIN,VOLUBF,SIN,TEMP,-DELTATEMPSINBF,CR,TEMP,-DELTATEMPCR

ALLSEL,ALL

FINISH

/COM, SOLVE STEP/COM,

/SOLU

ANTYPE,STATIC ! STATIC ANALYSISNLGEOM, ON ! NON LINEAR EFFECTSNSUB,15 ! NUMBER OF ITERATION STEPS, LAST VALUE MAX. ITERATIONS

SOLVE

FINISH

/COM, POSTPROCESSING STEP/COM,

/POST1

Output data NLGEOM ON:

0.50000E+02 0.92564E+010.10000E+03 0.30769E+020.15000E+03 0.63206E+020.20000E+03 0.10430E+030.25000E+03 0.15116E+030.30000E+03 0.20061E+03

63

A

Figure C.1: left, the elementdivision of the model, right, a result of a simulation with a cantileverof length 50 µm

64

Appendix D

ANSYS input files for stabilized2D and 3D models

A general ANSYS input file is given for the 2D and 3D silicon stabilized model used in Chapter 6. Starting with the 2Dmodel.

Filename 2D model: threelayerv4.mac

! OPTIMALIZATION LOOP FOR FREEI_ARRAY=1J_ARRAY=6

*DIM,FREE,,I_ARRAY*DIM,C_L,,J_ARRAY

FREE(1) = 0.01C_L(1) = 50,100,150,200,250,300

*DO ,B ,1 ,J_ARRAY,1*DO ,A ,1 ,I_ARRAY,1

/TITLE, RESIDUAL STRESS IN SIN AND CR

/PREP7

ALLSEL,ALLACLEAR,ALLADELE,ALL,,,1

/COM, MATERIALS DATA/COM,

/COM,/INPUT, macro, mac, ’C:\DOCUME~1\vanschai\MYDOCU~1\ANSYSM~1\TEMPERATURE’ ,:MATERIAL, 1 ! READING MATERIALDATA

/INPUT, macro, mac, ’C:\DOCUME~1\vanschai\MYDOCU~1\ANSYSM~1\TEMPERATURE’ ,:PARAMETERS, 1

STAB_HEIGHT = FREE(A)*VERH ! STABILIZATION HEIGHT FOR SILICIUM

/COM, GEOMETRY OF FINITE ELEMENT MODEL/COM,

K,1,0,0K,2,C_L(B)-FREE(A),0K,3,C_L(B)-FREE(A),M_TK,4,0,M_TK,5,C_L(B),0K,6,C_L(B),M_TK,7,C_L(B)-FREE(A),M_T+STAB_HEIGHTK,8,0,M_T+STAB_HEIGHTK,9,C_L(B),M_T+MAT_TK,10,C_L(B)-FREE(A),M_T+MAT_TK,11,C_L(B)-FREE(A),M_T+MAT_T+STAB_HEIGHTK,12,0,M_T+MAT_T+STAB_HEIGHT

A,1,2,3,4,1 ! AREA 1 SINA,2,5,6,3,2 ! AREA 2 SIN

65

A,4,3,7,8,4 ! AREA 3 SIA,3,6,9,10,3 ! AREA 4 CRA,8,7,11,12,8 ! AREA 5 CR

NUMMRG,ALL

/COM, ELEMENTTYPES/COM,

ET,1,PLANE42,,,1 ! PLANE42 DOES NOT NEED REAL CONSTANTSKEYOPT,1,3,3 ! PLANE STRAIN FUNCTIONR,1,C_W ! THICKNESS OF MEMBRANE

LESIZE,ALL,RATIO*MAT_T ! SIZE OF ELEMENTS EQUAL TO CHROME THICKNESSTYPE,1 ! PLANE77 ELEMENT

ASEL, S, AREA, ,1,2 , ,1CM,SIN,AREAAATT,1 ! MATERIAL DATA 1

ALLSEL,ALL

ASEL,S, AREA, ,3, , ,1CM,SI,AREAAATT,3 ! MATERIAL DATA 2

ASEL,S, AREA, ,4,5 , ,1CM,CR,AREAAATT,2 ! MATERIAL DATA 2

ALLSEL,ALL

AMESH,ALL

/COM, BOUNDARY AND SYMMETRIC CONDITIONS/COM,

NSEL,S,LOC,X,0D,ALL,UX,0 !SYMMETRIC CONDITIONSD,ALL,UY,0 !SYMMETRIC CONDITIONS

ALLSEL,ALL

! APPLYING RESIDUAL STRESS TO SI AND SIN

TREF,0

BF,SIN,TEMP,-DELTATEMPSIN ! BODY TEMP TO SINBF,SI,TEMP,-DELTATEMPSI ! BODY TEMP TO SIBF,CR,TEMP,-DELTATEMPCR ! BODY TEMP TO CR

ALLSEL,ALL

FINISH

/SOLUTION

ANTYPE,STATIC ! STATIC ANALYSISNLGEOM, ON ! NON LINEAR EFFECTSNSUB,15 ! NUMBER OF ITERATION STEPSSOLVE

FINISH

/POST1

FINISH

*ENDDO*ENDDO

Filename 3D model: cantilever3Dstabv2.mac

I_ARRAY=2J_ARRAY=1

*DIM,FREE,,I_ARRAY*DIM,C_L,,J_ARRAY

FREE(1) = 9,10 1,2,3,4,5,6,7,8,

66

C_L(1) = 100 ,150,200,250,300

*DO ,B ,1 ,J_ARRAY,1*DO ,A ,1 ,I_ARRAY,1

/TITLE, 3D MODEL OF CANTILEVER WITH RESIDUAL STRESSES

/UNITS, SI/VIEW,1,1,1,1

/PREP7

! CLEAR ALL DONE WORKALLSEL,ALLVCLEAR ,ALLVDELE ,ALL , , ,1ACLEAR,ALLADELE,ALL,,,1

/COM, MATERIALS DATA/COM,

! READING MATERIALDATA/COM,/INPUT, macro3Dv1, mac, ’C:\DOCUME~1\vanschai\MYDOCU~1\ANSYSM~1\MODELS3D’ ,:MATERIAL, 1

/COM, INPUT PARAMETERS/COM,

/COM,/INPUT, macro3D, mac, ’C:\DOCUME~1\vanschai\MYDOCU~1\ANSYSM~1\MODELS3D’ ,:PARAMETERS, 1

/COM, GEOMETRY OF FINITE ELEMENT MODEL/COM,

! DEFINITION OF UNIT VOLUMEK,1,0,0,0K,2,1,0,0K,3,1,1,0K,4,0,1,0

A,1,2,3,4,1VEXT,1,,,0,0,1,0,0,0CM,UNITVOLUME,VOLU

! START MAKING DIFFERENT COMPONENTS FOR THE MODEL

!VLSCALE,UNITVOLUME,,,RX,RY,RZ,0,0,0VLSCALE,UNITVOLUME,,,(0.5*C_W)-FREE(A),M_T,S_L,0,0,0 ! VOLUME 2

VLSCALE,UNITVOLUME,,,FREE(A),M_T,S_L,0,0,0 ! VOLUME 3VGEN,2,3,,,(0.5*C_W)-FREE(A),0,0,,0,1

VLSCALE,UNITVOLUME,,,I_W,M_T,S_L,0,0,0 ! VOLUME 4VGEN,2,4,,,0.5*C_W,0,0,,0,1

VLSCALE,UNITVOLUME,,,S_W,M_T,S_L,0,0,0 ! VOLUME 5VGEN,2,5,,,0.5*C_W+I_W,0,0,,0,1

VLSCALE,UNITVOLUME,,,(0.5*C_W)-FREE(A),M_T,C_L(B)-FREE(A),0,0,0 ! VOLUME 6VGEN,2,6,,,0,0,S_L,,0,1

VLSCALE,UNITVOLUME,,,FREE(A),M_T,C_L(B)-FREE(A),0,0,0 ! VOLUME 7VGEN,2,7,,,(0.5*C_W)-FREE(A),0,S_L,,0,1

VLSCALE,UNITVOLUME,,,S_W,M_T,I_L,0,0,0 ! VOLUME 8VGEN,2,8,,,0.5*C_W+I_W,0,S_L,,0,1

VLSCALE,UNITVOLUME,,,(0.5*C_W)-FREE(A),M_T,FREE(A),0,0,0 ! VOLUME 9VGEN,2,9,,,0,0,S_L+C_L(B)-FREE(A),,0,1

VLSCALE,UNITVOLUME,,,FREE(A),M_T,FREE(A),0,0,0 ! VOLUME 10VGEN,2,10,,,(0.5*C_W)-FREE(A),0,S_L+C_L(B)-FREE(A),,0,1

! DELETE UNIT VOLUMEVCLEAR,1VDELE,1,,,1

! MAKE SIN LAYER

67

ALLSEL,ALLCM,SIN,VOLU

! MAKE CR LAYERALLSEL,ALL

VSEL,S,VOLU,,3,10VSEL,U,VOLU,,6

CM,SINCR,VOLU

ALLSEL,ALL

VLSCALE,SINCR,,,1,MAT_T/M_T,1,0,0,0 ! VOLUME 1 + 11 TO 16VGEN,2,11,16,,0,M_T,0,,0,1VGEN,2,1,,,0,M_T,0,,0,1

VSEL,S,VOLU,,11,16VSEL,A,VOLU,,1

CM,CR,VOLU

ALLSEL,ALL

! MAKE SI LAYER 1

VSEL,S,VOLU,,2VSEL,A,VOLU,,6

CM,SINSI1,VOLU

ALLSEL,ALL

VLSCALE,SINSI1,,,1,MAT_T/M_T,1,0,0,0 ! VOLUME 17 + 18VGEN,2,17,18,,0,M_T,0,,0,1

VSEL,S,VOLU,,17,18

CM,SI1,VOLU

ALLSEL,ALL

! MAKE SI LAYER 2

VLSCALE,SINSI1,,,1,(FREE(A)-MAT_T)/M_T,1,0,0,0 ! VOLUME 19 + 20VGEN,2,19,20,,0,M_T+MAT_T,0,,0,1

VSEL,S,VOLU,,19,20

CM,SI2,VOLU

ALLSEL,ALL

! MAKE CR LAYER 2

VLSCALE,SINSI1,,,1,MAT_T/M_T,1,0,0,0 ! VOLUME 19 + 20VGEN,2,21,22,,0,M_T+FREE(A),0,,0,1

VSEL,S,VOLU,,21,22

CM,CR2,VOLU

ALLSEL,ALL

NUMMRG, ALL

/COM, ELEMENTTYPE DEFINITION/COM,

ET,1,SOLID186

/COM, MESHING OF VOLUMES/COM,

NDX = 2*C_W/ELEMSIZENDY = 30*MAT_TNDZ = C_L(B)/ELEMSIZE

LSEL,S,LOC,Y,0.5*M_T ! LINES IN Y DIRECTION

68

LSEL,A,LOC,Y,0.5*MAT_T+M_TLSEL,A,LOC,Y,0.5*FREE(A)+M_TLSEL,A,LOC,Y,0.5*FREE(A)+MAT_T+M_TLSEL,A,LOC,Y,0.5*MAT_T+FREE(A)+M_TLSEL,A,LINE,,218LSEL,A,LINE,,220LSEL,A,LINE,,222LSEL,A,LINE,,224LSEL,A,LINE,,234LSEL,A,LINE,,236LSEL,U,LOC,X,0.5*(0.5*C_W-FREE(A))LSEL,U,LOC,Z,0.5*S_LLSEL,U,LOC,Z,S_L+0.5*(C_L(B)-FREE(A))LESIZE,ALL,NDY

LSEL,S,LOC,X,0.5*(0.5*C_W-FREE(A)) ! LINES IN X DIRECTIONLSEL,A,LOC,X,0.5*FREE(A)+(0.5*C_W-FREE(A))LSEL,A,LOC,X,0.5*I_W+FREE(A)+(0.5*C_W-FREE(A))LSEL,A,LOC,X,0.5*S_W+I_W+FREE(A)+(0.5*C_W-FREE(A))LESIZE,ALL,NDX

LSEL,S,LOC,Z,0.5*S_L ! LINES IN Z DIRECTIONLSEL,A,LOC,Z,0.5*(C_L(B)-FREE(A))+S_LLSEL,A,LOC,Z,0.5*I_L+S_LLSEL,A,LOC,Z,0.5*FREE(A)+(C_L(B)-FREE(A))+S_LLESIZE,ALL,NDZ

! GIVE ELEMENTTYPE AND MAT TO VOLUVSEL,S,VOLU,,2,10VATT,1,,1,0

VSEL,S,VOLU,,11,16VSEL,A,VOLU,,1VSEL,A,VOLU,,21,22VATT,2,,1,0

VSEL,S,VOLU,,17,20VATT,3,,1,0

ALLSEL,ALL

TYPE,1MSHKEY, 1 ! KEY 1 MAPPED MESHINGVMESH, ALL

/COM, BOUNDARY AND SYMMETRIC CONDITIONS/COM,

! SYMMETRIC AND BOUNDARY CONDITIONSNSEL,S,LOC,X,0 ! SYMMETRICDSYM,SYMM,X

ASEL,S,AREA,,44ASEL,S,AREA,,80CM,BC,AREA

NSEL,S,LOC,Z,0 ! BOUNDARYNSEL,A,LOC,X,0.5*C_W+I_W+S_WNSEL,A,LOC,,BCD,ALL,UX,0D,ALL,UY,0D,ALL,UZ,0

ALLSEL,ALL

! APPLY FORCES AND TEMPERATURES

! TEMPERATURES IN SIN AND CRTREF,0

BF,SIN,TEMP,-DELTATEMPSINBF,SI1,TEMP,-DELTATEMPSIBF,SI2,TEMP,-DELTATEMPSIBF,CR,TEMP,-DELTATEMPCRBF,CR2,TEMP,-DELTATEMPCR

ALLSEL,ALL

FINISH

69

/COM, SOLVE STEP/COM,

/SOLU

ANTYPE,STATIC ! STATIC ANALYSISNLGEOM, ON ! NON LINEAR EFFECTSNSUB,15 ! NUMBER OF ITERATION STEPS, LAST VALUE MAX. ITERATIONS

SOLVE

FINISH

/COM, POSTPROCESSING STEP/COM,

/POST1

*ENDDO*ENDDO

To many results to show in this appendix are obtained modeling the stabilization cantilevers.Therefore only a small part of the data is listed below. The results of the 3D models canbe found in: cantileverstabv6.mac. The results of the 2D models with a ratio of 1:1 can befound in: threelayerv4alllengths566Kv1.mac. The 2D results of the ratio 2:1 can be found in:verh2tot1fr1stab2.mac

Plots of 2D stabilized model

Figure D.1: left, the elementdivision of the model, right, a result of a simulation with a cantileverof length 50 µm

Plots of 3D stabilized model

Figure D.2: left, the elementdivision of the model, right, a result of a simulation with a cantileverof length 50 µm

70

Appendix E

ANSYS input files for stabilizedand unstabilized membrane

A general ANSYS input file is given for the whole (stabilized) membrane model used in Chapter7. Starting with the unstabilized membrane model.

Filename Unstabilized membrane model: inputCr35nmv1.mac

/TITLE, WHOLE MEMBRANE IMPORTING

/UNITS, SI

IGESIN,’inputv2’,’iges’,’..\..\..\..\ANSYS\’

/PREP7

/COM, MATERIALS DATA/COM,

! READING MATERIALDATA/COM,/INPUT, macro3Dv1, mac, ’C:\ANSYS\’ ,:MATERIAL, 1

/COM, INPUT PARAMETERS/COM,

! READING INPUT PARAMETERS/COM,/INPUT, macro3Dv1, mac, ’C:\ANSYS\’ ,:PARAMETERS, 1

! RELOCATING THE AREAAGEN, ,2, , ,5000,-5000,0, , ,1

! REDIMENSION THE MEMBRANEARSCALE,2, , ,0.01,0.01,1, ,0,1

! EXTRUDE THE AREA INTO A VOLUMEVEXT,2, , ,0,0,M_T,1,1,1,

! MAKE A COMPONENT OF VOLUMEALLSEL,ALLCM,SIN,VOLU

! MAKE A CR LAYER

VLSCALE,SIN,,,1,1,TH*MAT_T/M_T,0,0,0VGEN, ,2, , ,0,0,M_T, , ,1

VSEL,S,VOLU,,2CM,CR,VOLU

ALLSEL,ALL

! DEFENITION OF ELEMENTTYPEET,1,SOLID186

71

! MESHING OF VOLUMES/COM, MESHING OF VOLUMES/COM,

! ADJUSTING MATERIAL PROPERTIESVSEL,S,VOLU,,1,12VATT,1,,1,0

ALLSEL,ALL

VSEL,S,VOLU,,13,24VATT,2,,1,0

ALLSEL,ALL

! BEFORE FREE MESHING GIVE THE MEMBRANE A GLOBAL IDEA OF THE ELEMENTSIZES

ASEL,S,AREA,,1,2ASEL,A,AREA,,32,33

LSLA,S

LSEL,U,LOC,X,0LSEL,U,LOC,Y,0LSEL,U,LOC,Y,450LSEL,U,LOC,X,275LSEL,U,LOC,X,35.26LSEL,U,LOC,X,203.26LSEL,U,LOC,Y,32.28LSEL,U,LOC,Y,418.32LSEL,U,LOC,X,95.26LSEL,U,LOC,Y,226.32LSEL,U,LOC,X,155.26

LESIZE,ALL,1

LSEL,INVELSEL,U,LOC,X,0LSEL,U,LOC,Y,0LSEL,U,LOC,Y,450

LESIZE,ALL,6

LSEL,S,LOC,X,0LSEL,A,LOC,Y,0LSEL,A,LOC,Y,450

LESIZE,ALL,15

ALLSEL,ALL

NUMMRG,ALL

MSHKEY,0 ! LINE USED FOR FREE MESHINGMSHAPE,1,3D ! 0 FOR QUAD AND 1 FOR TRIANGLE

TYPE,1VMESH,ALL

ALLSEL,ALL

/COM, BOUNDARY AND SYMMETRIC CONDITIONS/COM,

! SYMMETRIC AND BOUNDARY CONDITIONSASEL,S,AREA,,4ASEL,A,AREA,,19ASEL,A,AREA,,35ASEL,A,AREA,,50NSLA,S ! SYMMETRICDSYM,SYMM,X

ALLSEL,ALL

! BOUNDARY CONDITIONS

NSEL,S,LOC,X,0NSEL,A,LOC,Y,450NSEL,A,LOC,Y,0D,ALL,UX,0

72

D,ALL,UY,0D,ALL,UZ,0

ALLSEL,ALL

! APPLY FORCES AND TEMPERATURES

! TEMPERATURES IN SINTREF,0

BF,SIN,TEMP,-DELTATEMPSINBF,CR,TEMP,-DELTATEMPCR

ALLSEL,ALL

FINISH

/COM, SOLVE STEP/COM,

/SOLU

ANTYPE,STATIC ! STATIC ANALYSISNLGEOM, ON ! NON LINEAR EFFECTSNSUB,25 ! NUMBER OF ITERATION STEPS, LAST VALUE MAX. ITERATIONS

SOLVE

FINISH

/COM, POSTPROCESSING STEP/COM,

SAVE,’resultCr35’,db, ’C:\ANSYS\’

Images and results of the meshed model and the solution can be found in chapter 7 (figure 7.2and 7.3).

Filename Stabilized membrane model: inputstabv7.mac

RESUME,’areaSIN’,db, ’C:\ANSYS\STAB\’

/PREP7

FR = 0STAB = 20ASP = 0.01

! TO DEVELOP STAB AREA

K,1,275,22.28-FR,0K,2,193.26-FR,22.28-FR,0K,3,193.26-FR,180.625,0K,4,239.06-FR*0.731,223.37-FR*0.682+ASP,0K,44,239.06-FR*0.731,223.37-FR*0.682-ASP,0K,5,165.26+FR,211.92-FR,0K,6,165.26+FR,96.32-FR,0K,7,25.26-FR,96.32-FR,0K,8,25.26-FR,236.32+FR,0K,9,149.31+FR,236.32+FR,0K,10,229.755-FR*0.979,252.88+ASP,0K,100,229.755-FR*0.979,252.88-ASP,0K,11,204.61-FR,264.32-FR,0K,12,85.26-FR,264.32-FR,0K,13,85.26-FR,428.32+FR,0K,14,249.26+FR,428.32+FR,0K,15,249.26+FR,293.44+FR,0K,16,275,260.72+FR,0K,17,275,450,0K,18,0,450,0K,19,0,0,0K,20,275,0,0K,21,0,22.28-FR,0 ! IMPLEMENTATION OF EXTRA BECAUSE MAX. NODE NR’SK,22,0,22.28-FR,0 ! TO AVOID PROBLEMS WITH EXTRUDING AREA’S = K,21K,23,193.26-FR,22.28-FR,0 ! TO AVOID PROBLEMS WITH EXTRUDING AREA’S = K,2

! TO DEVELOP CR LAYER IN FRONT

73

! AREA 4-5-6-7, START AREA 4

K,120,193.26-FR,22.28-FR,0K,121,275,22.28-FR,0K,122,275,32.28,0K,123,203.26,32.28,0K,124,203.26,176.28,0K,125,273.82,242.135,0K,126,239.06-FR*0.731,223.37-FR*0.682+ASP,0K,127,193.26-FR,180.625,0K,128,239.06-FR*0.731,223.37-FR*0.682-ASP,0

! KEYPOINTS AREA 5

K,130,239.06-FR*0.731,223.37-FR*0.682+ASP,0K,131,273.82,242.135,0K,132,273.82,243.26,0K,133,272.85,244.23,0K,134,272.82,244.28,0K,135,268.54,243.495,0K,136,268.54,242.535,0K,137,243.88,234.24,0K,138,155.26,220.485,0K,139,155.26,106.32,0K,140,35.26,106.32,0K,141,35.26,226.32,0K,142,25.26-FR,236.32+FR,0K,143,25.26-FR,96.32-FR,0K,144,165.26+FR,96.32-FR,0K,145,165.26+FR,211.92-FR,0

! KEYPOINTS AREA 6

K,150,25.26-FR,236.32+FR,0K,151,35.26,226.32,0K,152,150.33,226.32,0K,153,243.88,245.58,0K,154,268.54,243.495,0K,155,272.82,244.28,0K,156,206.78,274.32,0K,157,95.26,274.32,0K,158,95.26,418.32,0K,159,85.26-FR,428.32+FR,0K,160,85.26-FR,264.32-FR,0K,161,204.61-FR,264.32-FR,0K,162,229.755-FR*0.979,252.88+ASP,0K,163,149.31+FR,236.32+FR,0K,164,229.755-FR*0.979,252.88-ASP,0

! KEYPOINTS AREA 7

K,170,85.26-FR,428.32+FR,0K,171,95.26,418.32,0K,172,239.26,418.32,0K,173,239.26,289.62,0K,174,239.5,289.67,0K,175,272.68,247.5,0K,176,275,247.5,0K,177,275,260.72+FR,0K,178,249.26+FR,293.44+FR,0K,179,249.26+FR,428.32+FR,0

! KEYPOINTS AREA 8

K,180,249.26+FR,428.32+FR,0K,181,249.26+FR,293.44+FR,0K,182,275,260.72+FR,0K,183,275,450,0

A,2,3,44,4,5,6,7,8,9,100,10,11,12,13,14,17,18,21,2 ! AREA 1A,19,20,1,23,22,19 ! AREA 3A,120,121,122,123,124,125,126,128,127,120 ! AREA 4A,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,130 ! AREA 5A,150,151,152,153,154,155,156,157,158,159,160,161,162,164,163,150 ! AREA 6A,170,171,172,173,174,175,176,177,178,179,170 ! AREA 7A,180,181,182,183,180 ! AREA 8

! REPLACE AREA’S BEFORE EXTRUDING

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!AGEN, ,1, , ,0,0,M_T, , ,1!AGEN, ,3, , ,0,0,M_T, , ,1

! EXTRUDE AREA’S TO SIN AND STAB VOL MEMBRANE

VEXT,1, , ,0,0,STAB,1,1,1,VEXT,2, , ,0,0,M_T,1,1,1,VEXT,3, , ,0,0,STAB,1,1,1,VEXT,4,7, ,0,0,MAT_T,1,1,1,VEXT,8, , ,0,0,STAB,1,1,1,

! SAVE volzonCr.db

! MAKE COMPONENTS FOR APPLYING TEMPERATURE

VSEL,S,VOLU,,1VSEL,A,VOLU,,3VSEL,A,VOLU,,8CM,SI,VOLU

VSEL,S,VOLU,,2CM,SIN,VOLU

ALLSEL,ALL

! MAKE CR LAYER AT TOPVLSCALE,SI,,,1,1,MAT_T/STAB,0,0,0 ! VOLUME 9 AND 11

! REPLACING VOLUMES

VGEN, ,1, , ,0,0,M_T, , ,1VGEN, ,3, , ,0,0,M_T, , ,1VGEN, ,8, , ,0,0,M_T, , ,1VGEN, ,9,11,,0,0,STAB+M_T,,0,1VGEN, ,4,7,,0,0,M_T,,0,1

! SAVE volmetCR.db

VSEL,S,VOLU,,4,7VSEL,A,VOLU,,9,11CM,CR,VOLU

ALLSEL,ALL

! DEFENITION OF ELEMENTTYPE/COM, ELEMENTTYPE DEFINITION/COM,

ET,1,SOLID186

! IMPLEMENTING MATERIAL PROPERTIESVSEL,S,VOLU,,1VSEL,A,VOLU,,3VSEL,A,VOLU,,8VATT,3,,1,0 ! SI

ALLSEL,ALL

VSEL,S,VOLU,,2VATT,1,,1,0 ! SIN

VSEL,S,VOLU,,4,7VSEL,A,VOLU,,9,11VATT,2,,1,0 ! CR

ALLSEL,ALL

! SAVE formeshv3.db better: formeshv2.db

ASLV,SLSLA,S

LESIZE,ALL,5

NUMMRG,ALL

MSHKEY,0 ! LINE USED FOR MAPPED MESHINGMSHAPE,1,3D ! 0 FOR QUAD AND 1 FOR TRIANGLE

75

TYPE,1VMESH,ALLALLSEL,ALL

! BOUNDARY CONDITIONS

ASEL,S,AREA,,146ASEL,A,AREA,,60ASEL,A,AREA,,30ASEL,A,AREA,,66ASEL,A,AREA,,154ASEL,A,AREA,,121ASEL,A,AREA,,45ASEL,A,AREA,,114NSLA,S ! SYMMETRICDSYM,SYMM,X

ALLSEL,ALL

! BOUNDARY CONDITIONS

NSEL,S,LOC,X,0NSEL,A,LOC,Y,450NSEL,A,LOC,Y,0D,ALL,UX,0D,ALL,UY,0

NSEL,S,LOC,Y,450D,ALL,UZ,0

ALLSEL,ALL

! APPLY FORCES AND TEMPERATURES

! TEMPERATURES IN SINTREF,0

BF,SIN,TEMP,-DELTATEMPSINBF,CR,TEMP,-DELTATEMPCRBF,SI,TEMP,-DELTATEMPSI

ALLSEL,ALL

ESEL,,STRAIGHTENEDEPLOT,ALL

ALLSEL,ALL

FINISH

SAVE,’prepstabSTAB%STAB%FR%FR%’,db, ’C:\ANSYS\STAB\RES\’

/SOLU

ANTYPE,STATIC ! STATIC ANALYSISNLGEOM, ON ! NON LINEAR EFFECTSNSUB,15 ! NUMBER OF ITERATION STEPS, LAST VALUE MAX. ITERATIONS

SOLVE

FINISH

/COM, POSTPROCESSING STEP/COM,

SAVE,’resstabSTAB%STAB%FR%FR%’,db, ’C:\ANSYS\STAB\RES\’

Plots of stabilized membrane model

Images and results of the meshed model and the solution can be found in chapter 7 (figure 7.7and 7.8).

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