MT07.08J-Integral in FEM Analysis of Interfacial Cracks in

Low-K Structure of Microsystems Packaging

Gerben van den Oord

Professor: Kuoning Chiang

Coach: Chien-Chia Chiu (Luke)

Internship at the National Tsing Hua University, TaiwanAdvanced Microsystem Packaging and Nano Mechanics Research Lab

December 16th, 2006

Abstract

In integrated circuits, interfacial delamination poses a big problem for their reliability. Topredict the growth of such delamination, fracture mechanics is employed. In particular, theJ-integral has been gaining popularity because the ease of computation in a Finite Elementenvironment. However, in interfacial problems, the value of the J-integral is dependent onthe path chosen for integration. In this research, the path and mesh dependency of theJ-integral is studied for the case of a crack between two layers of dielectric materials inan integrated circuit. A fictive model of a Ball Grid Array assembly is modeled usingthe Finite Element Method, using five different mesh densities in the arrea of interestand calculating the J-integral for every possible path. It is shown that, judging from thegeneral stress state of the model, a path can be chosen that returns a J value that is moreor less stable for small changes in the path or mesh.

Contents

1 Problem Definition 3

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Research Topic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Geometry and Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Finite Element Simulation 7

2.1 Equation and Simulation Settings . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 General Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.2 Mesh in the Crack Tip Region . . . . . . . . . . . . . . . . . . . . . 11

2.3 J-Integral Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Simulation Results 15

3.1 General Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 J-Integral Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 Crack Tip Vicinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.2 Variations in b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.3 Variations in a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.4 Mesh Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.5 J-Integral Path Selection . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Conclusion 25

A J-Integral Computation Code 26

B Results Overview 28

C Direct Calculation of G 30

Bibliography 31

1

List of Figures

1.1 Geometry of the structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Layers of materials of low dielectric constants (low K layers) . . . . . . . . . 6

2.1 Overview of the meshed geometry . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Meshed low K layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Dense mesh around the crack tip . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Node numbers defined according to relative position to the crack tip . . . . 12

2.5 Paths for determining the J-integral . . . . . . . . . . . . . . . . . . . . . . 14

3.1 Deformed 10 nm model, deformation scaled ×10 . . . . . . . . . . . . . . . 16

3.2 Deformed geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Deformed geometry in the crack region . . . . . . . . . . . . . . . . . . . . . 16

3.4 Crack region of the deformed 10 nm model . . . . . . . . . . . . . . . . . . 17

3.5 Detailed view of von Mises stress around crack tip, 5 nm model . . . . . . . 18

3.6 Strain Energy Density of area around the crack tip, 5 nm model . . . . . . 18

3.7 contour plot showing the value of J . . . . . . . . . . . . . . . . . . . . . . . 19

3.8 J values plotted against width a, for constant b . . . . . . . . . . . . . . . . 20

3.9 J values plotted against width a, for constant b, coarser meshes . . . . . . . 22

3.10 J values plotted against width a, for constant b, duplicate paths . . . . . . 22

3.11 J values plotted against aspect ratio b/a, coarser meshes . . . . . . . . . . . 24

B.1 overview of calculation results of parameters used in the computation of theJ-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2

Chapter 1

Problem Definition

1.1 Introduction

In microsystems, interfacial cracks pose a big problem for the reliability of the systems.Microsystems are build up of thin layers of different materials, that each behave differentlyunder thermal or mechanical loading. However, microsystems need the (electrical) connec-tion between their different components in order to function, so a crack in the connectionbetween two materials can cause the system to fail.Because of this, a lot of research is put in fracture problems in microsystems. To predictthe growth of (existing) cracks, the field of Fracture Mechanics for a part focusses on theenergy that is released when a crack grows: the strain energy release rate G:

G = −Ua+∆a − Ua

B∆a(1.1)

With U the total strain energy, a the length of a crack and B the thickness of the model.However, this property is not easily experimentally measured. It could be determinedusing two Finite Element simulations, one for crack length a and one for a + ∆a, but thisrequires double calculation times, and ∆a is so small that the difference between Ua+∆a

and Ua could be lost in numerical precision (in Appendix C, this analysis has been carriedout for this problem, not rendering useful results).

There exist various methods of estimating the strain energy release rate throughother calculations, of which the Crack Tip Opening Displacement Method (CTOD) is themost famous. An increasingly popular alternative is the J-integral; a path integral aroundthe crack tip incorporating the stress field σ(= σij), the displacement field ~u(= ui) andthe strain energy density W (= 1

2σijεij) around the crack. The reason the J-integral is

gaining popularity, is because it is easily calculated from standard results from a singleFinite Element simulation.In index format, the J-integral reads [1]:

J =

∫

Γ

(

Wn1 − Ti∂ui

∂x1

)

ds (1.2)

For cracks in a single linear elastic material, the value of the J-integral is the same as Gand is independent of the chosen path. For interfacial cracks, this is not the case: the value

3

1.2 Research Topic 4

of the J-integral shows dependence of the chosen path.This has spawned some research that investigated the influence of the path and gavesuggestions for paths that gave the most accurate results. For example Shi [2], who statesthat for interfacial cracks a rectangular path around the crack tip should be chosen (witha length a in the crack direction and a width b perpendicular to that) so, that b → 0 andb/a → 0. Based on this, Bulu [3] researched the application of the J-integral for a crackbetween a microchip and its underfill, and suggested b/a ≈ 0.1 and b to contain at least 5or 6 node layers to approach G determined through CTOD.Unfortunately, it seems that different problems require different paths. In this report, theinfluence of the path en mesh on the J-integral for yet another problem will be researched;namely the interfacial delamination between two layers of dielectric materials in IntegratedCircuits. This topic will be discussed in the next section.

1.2 Research Topic

The origin of the type of delamination that is considered here goes back to the manu-facturing process of the individual microchip from a silicon wafer structure ([4]). On thewafer, through the process of lithography, a structure of metallic connections embeddedwithin layers of dielectric materials (low K layers) is created. After that, the whole disc isseparated into individual integrated circuits (IC, or die) through the process of dicing. Inthis process a diamond saw cuts through the wafer disc and its delicate structure. Duringthis sawing action it is inevitable that, on the edges of the IC, microscopic interfacialcracks are created.In this research, the IC will be packaged in a Ball Grid Array (BGA) assembly. In this,the IC is glued to a substrate. When mounting the packaged IC, for example to a printedcircuit board (PCB), it is this substrate that is actually soldered (using balls of solder in arectangular array) to the PCB. To make the electrical connection between the IC and thesubstrate, wire bonds are soldered to the top of the die and are connected to the substrate.To protect the die and wire bonds from the environment, they are sealed by molding aheated polymer around them, creating a protective casing. However, this process takesplace at high temperatures (200 ◦C) and subsequent cooling down to room temperature(25 ◦C) creates a thermal strain. The differences in coefficients of thermal expansion (CTE)of the different materials involved then cause stresses and deformations in the structures.Under this thermal loading, the existing interfacial cracks in the low K structure of theIC might grow to such an extend as to cause actual failure of the IC.

4

1.3 Geometry and Materials 5

Figure 1.1: Geometry of the structure

1.3 Geometry and Materials

The geometry studied here is a fictional (but representable) model of a BGA assemblyshown in figure 1.1, with a more close view of the low K layers in figure 1.2. Omitted inthis model are the wire bonds that connect the in and output connections of the IC tothe substrate. Also left out are the nano scale connections of copper or aluminium thatrun through the low K layers to make the connections between the semiconductors on thesilicon surface and the wire bonds. In the model, a crack of 3 µm is considered along theinterface between the lower surface of the M1 dielectric layer and the corresponding EtchStop Layer (ESL). The properties of the materials used in each layer can be found in table1.1.Using the Finite Element Method, a temperature step of ∆T = −175◦C for this modelwill be simulated, which will be discussed in chapter 2.

Table 1.1: Material propertiesMaterials E (GPa) CTE (ppm/◦C) Poisson Ratio

Silicon chip 169 3 0.28ILD 60 3 0.3ESL 60 3 0.32

M1-M3 8 20 0.31M4-M6 12 20 0.31

USG (M7-M8) 80 1.5 0.3SiN 13 12 0.31SiOx 79.5 12 0.3

Molding compound 24 20 0.3BT substrate 26 15.7 0.36

Die-attach epoxy 6 100 0.35

5

1.3 Geometry and Materials 6

SiN 800nm

SiOX 1200nm

SiN 75nm

SiOX 600nm

SiN 95nm

M8 1570nm

ESL 70nm

M7 1700nm

ESL 70nm

M6 700nm

ESL 70nm

M5 700nm

ESL 70nm

M4 700nm

ESL 70nm

M3 600nm

ESL 70nm

M2 600nm

ESL 70nm

M1 500nm

ESL 70nm

ILD 1000nm

Figure 1.2: Layers of materials of low dielectric constants (low K layers)

6

Chapter 2

Finite Element Simulation

In this chapter, the manner of solving the problem posed in chapter 1 will be discussed.First, the general equation that has to be solved will be discussed, along with the mainanalysis settings in the FEM calculation program Ansys. Next, the meshes that are usedfor the geometry from section 1.3 will be shown and finally, the way of determining thevalue of the J-integral in Ansys will be explained.

2.1 Equation and Simulation Settings

The governing equations for this problem is just the equilibrium equation:

~∇ · σ + ~q = ~0 (2.1)

were σ is the stress tensor, determined by the deformation and the constitutive relations.In this case, linear elastic material properties are assumed, and the stress tensor can bedetermined through σ = [E]ε, where [E] is the stiffness matrix, whose components arefunctions of the Young’s modulus E and the Poison’s ratio ν. ε is the strain tensor, afunction of the displacements.Also in equation 2.1, ~q are body forces acting on the geometry. In this case, there are nobody forces, so ~q = ~0.

In computation by Ansys, the components of both the stress and strain tensor arestored in vector representations, {σ} and {ε} respectively (for 2D):

{σ} = [σx σy σxy]T (2.2)

{ε} = [εx εy εxy]T (2.3)

and are related through a modified version of the stiffness matrix as

{σ} = [D]{εel} (2.4)

7

2.2 Mesh 8

In this, the elastic strain {εel} is used, which is the part of the total strain that actuallycauses stresses:

{εel} = {ε} − {εth} (2.5)

It is here that the temperature enters the equation, through the thermal strain vector:

{εth} = ∆T [αsex αse

y 0]T (2.6)

In which the α’s are the secant coefficient of thermal expansion from table 1.1, and ∆Tis the temperature difference between the temperature and the reference temperature,T − Tref .

Plugging all this into the equilibrium equation, and applying further principles ofthe finite element method, this leads to the following set of equations:

[K] {u} = {f} (2.7)

In this, [K] is the total stiffness matrix, assembled through the element stiffness matrices.{u} is the nodal displacement vector, and {f} is the load vector, which is were the thermalstrains have gone, according to the element thermal load vector:

{f the } =

∫

V[B]T [D]{εth}dV (2.8)

In which [B] is the strain displacement matrix (a function of the element shape functions)and V is the volume of the element.

For these calculations, the large deformation option in Ansys is flagged on, makingthe system of equations non linear ( [K] can become a function of {u}). To find a solution,a full Newton-Raphson iteration scheme is used, with an L2 convergence norm for thedisplacements and a tolerance factor of 1.000 · 10−3. Further more, the total tempera-ture change from 473 K (reference temperature) to 298 K is performed in a single load step.

2.2 Mesh

2.2.1 General Mesh

With the basic equation and simulation settings discussed in section 2.1, all that is left forthe simulation of the temperature step response is to divide the geometry into elements.This will be discussed in this section.

The mesh for this problem is one with very large scale differences: whereas the to-tal model size (half of the chip size, due to symmetry) is in the order of 10 mm, the layerthickness in de low K layer area can be as little 70 nm. Furthermore, the focus on themodel is in the part with the smallest scales, requiring these small features to be meshed

8

2.2 Mesh 9

with elements of even smaller scale.

For the Finite Element simulation, because of the general shape of the model (consistingof thin rectangular shapes) and to allow for easy meshing, it is chosen to use quadrilateralelements, oriented along the global x and y axes, with 4 nodes throughout the model (fullintegration, Plane42 in Ansys). This implies that the singularity in strain in the crack tipcan not be picked up by the model. For the calculation of the J-integral, that uses a patharound the crack tip, this does not pose a problem.To prevent the crack faces from penetrating each other, contact elements are used(elements targe169 and conta172 in Ansys) on the crack faces. Also, Ansys is programmedto automatically reduce the initial penetration (in the order of 10−16 mm) that exists dueto numerical errors, by flagging on the reduce initial penetration option.

If the geometry is meshed as a single component, areas that are to be meshed us-ing very small elements (i.e. the crack region) also control one dimension of the size of allelements directly above and below, or left and right of that area. Because of this, and thelarge scale difference in the model, it is difficult to mesh this structure without causinga number of elements to have high aspect ratios. One way of solving this problem wouldbe to use a very dense mesh (defined by the smallest features) throughout the model.This is undesirable because of the large number of elements this would create, makingthe problem to big to handle on a normal computer.To cope with this problem, at least for a part, the single component of the model is dividedinto smaller sections. The sections that mark the transition between a large scale regionand a small scale region are meshed using the spacing ratio option in Ansys, meaningthat the size of the elements along a direction gradually changes. This decreases thenumber of elements with high aspect ratios, but does not solve the problem completely.However, all the high aspect ratio elements lie outside the area of particular interest andtheir deformation (not considering translation) will be small enough to not cause anyfurther problems. An overview of the mesh can be seen in figure 2.1, and a closer view onthe thin low K layers at the die edge in 2.2.

9

2.2 Mesh 10

Figure 2.1: Overview of the meshed geometry

Figure 2.2: Meshed low K layers

10

2.2 Mesh 11

2.2.2 Mesh in the Crack Tip Region

After simulation, various rectangular paths will be selected for the calculation of theJ-integral. For these paths, a region of square elements around the crack tip is created.The size of this region defines the largest paths that can be created; the mesh densitydefines the results resolution and total number of paths. For the computation of theJ-integral, paths of symmetrical shape, as proposed by Shi [2] (subsection 2.3, that crossonly one material interface will be selected. Within these limits, the area of this densemesh is created (figure 2.3) that consists the thickness of the M1 layer and relevant EtchStop Layer, stretching out 1000 nm, both to the left and right of the crack tip. The widthof 2× 1000 nm could be extended to 2× 3000 nm (the total crack length), but this wouldcreate too much elements and would only add path possibilities with very high aspectratios.

In order to select the paths for the J-integral afterwards automatically by use of a Fortranscript, the mesh in this region is created by another script that places the nodes accordingto a logical pattern. This way, paths can later be selected by node numbers easily. Thestructure of numbering used is shown in figure 2.4.In horizontal direction, the nodes are counted from the crack tip in numbers smallerthan 999 (a sufficiently high number for the mesh densities that are considered) withodd numbers going in the right direction, and even numbers going to te left, the cracktip being 1. In vertical direction, nodes are counted in a similar way, but this number isrepresented by numbers of thousand, odd numbers going up, even numbers going down,the crack tip having 1000. On the crack face coincident nodes exist, either belonging tothe M1 layer or the ESL. To be able to refer to them, the upper crack face is countedlike stated before with numbers in the 1000 range. The lower face is then counted inthe 0000 range, which does not exist anywhere else. The combination of the two “coor-dinates” gives every node a unique node number that also represents its place in the mesh.

Further more, to check the influence of the mesh density, models with different meshdensities around the crack tip are built. The smallest element size used in these variationsis 5 nm, giving 14 elements across the thickness of the relevant Etch Stop Layer and200 elements on either horizontal side of the crack tip. This creates a total number of13 × 199 = 2587 possible paths for this model. For the best comparison between modelswith different mesh densities, it should be possible to select the exact same path. Therefore,also models with element sizes of 10 nm and 20 nm are built. However, the thickness theof ESL is 70 nm, which can not be divided into 20 nm elements. This means that acrossthe ESL four elements of 17.5 nm are used.To complete the set of models, the thickness of the ESL is also subdivided into three andtwo elements. For the former, a basic element size of 25 nm is used (scaled to 23.33 nmalong the thickness of the ESL) and for the latter, a basic element size of 35 nm is used(scaled to 34.48 nm in horizontal direction).All five models in the remainder of this report will be referred to by their mean elementsizes (5, 10, 20, 25, 35 nm).

11

2.2 Mesh 12

Figure 2.3: Dense mesh around the crack tip

Figure 2.4: Node numbers defined according to relative position to the crack tip

12

2.3 J-Integral Computation 13

2.3 J-Integral Computation

Using the programming functions in Ansys, the computation of the J-integral can be au-tomated through great extend (Ansys documentation, [5]). First, the J-integral is writtenfor a crack in the (x, y)-plane:

J =

∫

Γ

(

Wnx − Ti∂ui

∂x

)

ds (2.9)

J =

∫

Γ

Wdy −

∫

Γ

(

tx∂ux

∂x+ ty

∂uy

∂x

)

ds (2.10)

J =

∫

Γ

Wdy −

∫

Γ

(

(σxnx + σxyny)∂ux

∂x+ (σyny + σxynx)

∂uy

∂x

)

ds (2.11)

In this:Γ = any path surrounding the crack tipW = strain energy densitytx = traction vector along x-axis = σxnx + σxyny

ty = traction vector along y-axis = σyny + σxynx

σ = component stressn = unit outer normal vector to path Γu = displacement vectors = distance along the path Γ

The programming code (from the Verification Manual, VM143 or Ansys, [5]) used to getall these variables from the standard results in Ansys and to compute the J-integral, canbe found in the table in Appendix A.

For the paths needed for these calculations, only rectangular paths are considered,as suggested by Shi [2] and shown in figure 2.5. In this, the path has dimension a inhorizontal (x) direction, and dimension b in vertical (y) direction. The position of thepath is such that the crack tip is at a/2 and b/2: the symmetric center of the path.To select these paths, on the basis of node numbers, a Fortran script is written thatmakes use of the structure of numbers shown in subsection 2.2.2. Automating the processof selecting the nodes for the J-integral allows for the calculation of much more differentpaths than would be the case for hand picked paths. This way, more data can be retrieved,which will be discussed in chapter 3

13

2.3 J-Integral Computation 14

a

Figure 2.5: Paths for determining the J-integral

14

Chapter 3

Simulation Results

After the simulation, described in chapter 2, the outcome will be described in this chapter.First, the general deformation is discussed to check if the model behaved in a way thatwas te be expected. After that, for different paths en meshes, the value of the J-integralwill be determined and discussed, in order to give a suggestion for the path selection insimilar problems.

3.1 General Deformation

Simulation of the temperature step in Ansys reveals that in the deformation (figure 3.1),the thin layers of low K materials, and the epoxy that binds the die to the substrate,don’t have much influence on the general deformation. The general deformation is almostcompletely governed by the three main parts; being the substrate and molding compound,which have a relatively large coefficient of thermal expansion (CTE), and the die, whichhas a relatively small value (table 1.1).With the Molding compound having a slightly larger CTE and a little more material thanthe substrate, this makes the model bend slightly towards the side of the molding com-pound, as shown in figure 3.2. Also, at the edge of the structure, there is an extra “lump” ofthe molding compound that shrinks, while remaining attached to the die edge en substrate.

For the greatest part, these large scale deformations also determine the behavior in thecrack region (figure 3.3). The shrinking of the overlapping lump of the molding compoundcauses the crack to be closed near the die edge, whereas the top part of the moldingcompound (that bends the structure upwards and towards the symmetry axis) slightlyopens the crack and shears the top face towards the symmetry axis.Between the two layers of concern here, the M1 layer has a far greater CTE than theESL, where the ESL is much more stiff. This will lead to larger deformations in the M1layer.Actual simulation results, for the 10 nm model, are shown in figure 3.4. In this, on theleft hand, the deformation is scaled up ten times. This does not only scale up the openpart of the crack (near the crack tip) for better visibility, but also exaggerates the small

15

3.1 General Deformation 16

Figure 3.1: Deformed 10 nm model, deformation scaled ×10. Plotted is the equivalent vonMises stress, in MPa

Molding compound: 20 ppm/°C

Die: 3 ppm/°C

Substrate: 15.7 ppm/°C

∆T = -175 °C

Figure 3.2: Deformed geometry

Molding compound

Die

Low K Layers

Crack

Figure 3.3: Deformed geometry in the crackregion

16

3.1 General Deformation 17

Figure 3.4: Crack region of the deformed 10 nm model, plotted is the equivalent von Misesstress, in MPa. On the left, the deformation is scaled ×10, on the right no scaling is used.

numerical deviations on the right part. These deviations are, however, small enough tostate that the deformation of figure 3.3 is representable for the actual deformation infigure 3.4.

Studying the numerical results, it can be noticed that the equivalent von Mises stressat the die edge, where the crack faces bond with the molding compound, is even higherthan at the actual crack tip. This can be explained by the fact that on that point threematerials, each with very different CTE’s and different main shrinkage directions arebonded together, whereas at the considered crack tip only two materials are joined. Thedifference would be even greater if smaller elements would have been used at that part ofthe crack.For the study of the J-integral, only the crack tip on the left is of concern. A moredetailed view of the equivalent von Mises stress is shown in figure 3.5. Apart from thestress patterns, the strain energy density is of importance. This is calculated by dividingthe standerd Ansys outputs of strain energy and element volume (for unit thickness). Ascan be seen from figure 3.6, the strain energy density is only an important factor nearthe crack tip, and only in the M1 layer, because of the larger deformations there. Anoverview of other calculated results used for the J-integral computation, for the 5 nmmodel, is shown in Appendix B.

17

3.2 J-Integral Results 18

Figure 3.5: Detailed view of von Misesstress around crack tip, 5 nm model. vonMises stress shown in MPa

Figure 3.6: Strain Energy Density W ofarea around the crack tip, 5nm model, Win MPa

3.2 J-Integral Results

Using the methods described in section 2.3, values of the J-integral for all possible paths,in all five models are determined. This data is then entered in Matlab for data processing.Because the way the paths are defined (rectangular and symmetric in the local x and yaxis through the crack tip) every node in the relevant part of the mesh acts as the cornerof one path only. This gives the opportunity to plot the calculated J value of every pathon its corner nodes to give a general overview of the J value on different paths. In figure3.7, this is what is done. Because of symmetry, only the first quadrant (being the partabove the crack face, to the right of the crack tip) is plotted. The 5 nm model is used dueto the higher number of data points available.This plot is a good way to get a general idea about the path-dependency for the J-integralin this case. In the next subsections, this dependency will be discussed.

3.2.1 Crack Tip Vicinity

In figure 3.7, it can be seen that the highest values of J can be found for paths in thedirect vicinity of the crack tip. This of course has to do with the singular point on thecrack tip itself, that causes very high stresses. (towards infinitely if singular elements wereused). The high stresses have a great influence on the second integral in equation 2.11.For the first term in that equation, the strain energy density W is important. Due to highstresses and the high strains in the part of the M1 layer near the crack tip (the lowerright hand plot in Appendix B), W is very high there. This also causes the high values ofJ for paths near the crack tip.Also, the narrowest paths, in both vertical and horizontal direction, render higher valuesof J than paths that have aspect ratios closer to one. This still is largely because thosenarrow paths go through two of the eight nodes surrounding the crack tip. This is way,as will be shown later, paths that are only two elements wide (either in horizontal orvertical orientation) all return high values of J , in all mesh densities.

18

3.2 J-Integral Results 19

J value for combinations of a and b, mean elementlength 5nm

number of elements in a/2

nu

mb

er

of

ele

me

nts

in

b/2

20 40 60 80 100 120 140 160 180

2

4

6

8

10

12

number of elements in a/2

nu

mb

er

of

ele

me

nts

in

b/2

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

2.2

2.4

2.6

2.8

Figure 3.7: contour plot showing the value of J on the corner node of its correspondingpath. J in J/m2

For other variations of J with different paths, it is less easy to find an explana-tion. This is because of the number of variables involved and the difficulty to judge theredirect influence on J , because of differences in magnitude and the difficulties involvedwith signs and directions of the path. However, in the following two subsections, theinfluence of both a and b will be discussed

3.2.2 Variations in b

In figure 3.8 the same data (of all possible paths in the 5 nm model) is plotted in a graph.Again, it is clear that for the smallest possible b, values of J are highest, as well as forthe smallest possible a. However, it is also clear that even further increasing b, graduallylowers the value of J even more; all be it not as much as changing b from 2 to 4 elements.This effect seems to be slightly stronger for lower values of a, which leads to believe thatthe effect is caused on the vertical parts of the path (for large a, the vertical parts ofthe path have relatively less influence, due to larger horizontal parts). If this is true, thedifference for different values of b should be made in the parts of the computation wereny is zero and nx is 1.From equation 2.11, the parts associated with nx are σx and σxy. Because the effect isobserved over the entire range of a’s used, effects that only occur near the crack tip canbe ruled out as the cause. From the plots in Appendix B, it seems that (for a paths withlarge enough a’s to clear the disturbances of the crack tip) especially σxy is the mostimportant variable that shows any change for increasing b. Therefore, σxy seems to be themain cause of the dependency of b.

19

3.2 J-Integral Results 20

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10−6

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

a [m]

J [

J/m

2]

J value plotted against lenght a, mean elementlength 5nm

b = 10nm

b = 20nm

b = 30nm

b = 40nm

b = 50nm

b = 60nm

b = 70nm

b = 80nm

b = 90nm

b = 100nm

b = 110nm

b = 120nm

b = 130nm

Figure 3.8: J values plotted against width a, for constant b, 5 nm model

Note that for the largest b in this case, 130 nm, the path runs only 5 nm from an-other interface: the one between the Inter Layer Dielectric and the Etch Stop Layer ofour concern. However, results show no evidence of any disturbances from this.

3.2.3 Variations in a

Looking at the values of the J-integral for constant b’s and varying a, different parts canbe identified. First, the peak for very small a’s that was discussed in subsection 3.2.1.For slightly larger a, a small dip is found (which is deeper for larger b) and that growsto a peak value at approximately a = 0.15µm. After that, J steadily decreases with in-creasing a until 1.2 µm and rises again up until the largest values of a used in this research.

As can be seen in figure 3.7, the dip in the J-value occurs very distinct on a/2 = 2elements (or a = 4 elements) and only for b larger than 2 elements. This means thatthe dip occurs on the smallest value for a possible without using the four elements thatsurround the crack tip (i.e.: the second smallest a possible).Because the region of concern here is so close to the crack tip, it is hard to single out theone variable in the computation that causes the dip in J : all plots in Appendix B showlarge gradients in that area. However, when taking a look at the equivalent von Misesresults in figure 3.5, it can be observed that the paths of concern run through the narrowcenter of the ∞-shape in the von Mises stress respons. This means that those paths passthrough areas which have a generally lower stress state than paths for slightly larger a.This might be the cause of the lower J-value.

20

3.2 J-Integral Results 21

In subsection 3.2.4 it can be seen that for coarser meshes, the dip in the J value is stillat the second smallest possible value of a

For the small peak in the J value observed in figure 3.8, it is equally difficult tofind a single cause. Because the peak shifts to values of higher a for increasing b, there isreason to believe that this respons also is due to the general stress state. The paths thatgive the small peak in J seem to correspond to the paths that follow the total ∞-shape inthe von Mises results the best (considering the size of the path). This means that thosepaths include as much of the region with higher stress levels as possible, resulting in ahigh value of J .For greater values of a, in the range of 0.2 µm to 1.2 µm, the influence of the higher stressstates, symbolized by the ∞-shape, decreases. This is partly because the distance to thecrack tip increases, but also because of the fact that the paths can no longer match the∞-shape, which extends outside the area available for the J-integral paths.

Finally, for values of a from 1.2 µm until 2 µm, the value of J increases again.Most notably in the results for these larger a’s is that

∂uy

∂xchanges in sign in the upper

right quadrant of the results, as can be seen in Appendix B. Also, |∂ux

∂x| decreases in that

same quadrant. This decreases the influence of σxnx + σxyny in equation 2.11, which

might be the reason the sign change in∂uy

∂xshows up in the J values as an increasing trend.

3.2.4 Mesh Dependency

In the last subsection, only results for the 5 nm mesh were discussed. As stated in section2.2, also coarser meshes around the crack tip are built, the results of which are shown infigure 3.9. In this figure, the line style shows the mean element size round the crack tip,the line color shows the number of elements in b.At first glance, the coarser meshes show good correspondence with the finer ones,especially for higher values of a. In that region, the value of J for a given value of a isstill largely determined by the total value of b. However, comparing the results for pathsthat exist in more than one mesh (for example b = 6× 20 and b = 12× 10) it is clear thatmesh density also plays a role in this.For better comparison, three paths that exist in three different meshes are plotted infigure 3.10. In this, lines of the same color show data for the same path, and lines of thesame style show data for the same mesh. Figure 3.10 shows that a finer mesh resultsin a generally lower J value. This is because in a fine mesh there are more elements inbetween the singularity on the crack tip and the path itself, so that the high gradientscan be better mapped.Secondly, coarsening the mesh has an effect at low values of a: coarser meshes will havetheir dip in the J value at higher values of a than finer meshes. Indeed, this is because(as stated in subsection 3.2.3) this dip occurs at the second smallest value of a possiblefor a particular mesh. For coarser meshes, with larger elements, the second smallest valueof a is of course larger than for finer meshes.

21

3.2 J-Integral Results 22

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10−6

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

a [m]

J [

J/m

2]

J value plotted against length a, various mean elementlenghts

b = 2*35nm

b = 2*25nm

b = 4*25nm

b = 2*20nm

b = 4*20nm

b = 6*20nm

b = 2*10nm

b = 4*10nm

b = 6*10nm

b = 8*10nm

b = 10*10nm

b = 12*10nm

Figure 3.9: J values plotted against width a, for constant b, 10,20,25,35 nm models

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10−6

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

a [m]

J [

J/m

2]

J value plotted against length a, coincident paths in different meshes

b = 2*20nm

b = 4*10nm

b = 8*5nm

b = 4*20nm

b = 8*10nm

b = 16*5nm

b = 6*20nm

b = 12*10nm

b = 24*5nm

Figure 3.10: J values plotted against width a, for constant b. Three paths that exist in the5, 10 and 20 nm model are plotted, to show mesh dependency

22

3.2 J-Integral Results 23

3.2.5 J-Integral Path Selection

Having studied the effect of both path and mesh changes on the value of J-integral, thiscan be used to set up a suggestion for choosing an appropriate path in similar problems(note that there is no other calculation of G available for comparison). Ideally, this pathwould have to be chosen based on the results for the stress respons of such a problem.For such a suggestion, it is desirable that the value of the J-integral is stable for smallchanges in both mesh and path. Because of this, paths that run too close to the crack tipare rejected, because in those areas high gradients exists, that are also dependant on themean element size.Judging from figure 3.8, the smallest change in J for a change in b is found at the higherb’s. Further more, figure 3.10 shows that larger a b also decreases mesh dependency.The smallest change in J for a change in a is found at the level parts of figure 3.8: thesmall peak at approximately a = 0.15 µm and the minimum at a = 1.2 µm.Because the paths that cause the small peak are more easily recognizable (their shapecorresponds to the shape in the equivalent von Mises stress response), this would be theeasier one to pick.This way, a suggestion for choosing J-integral paths in interfacial cracks in low K layersin microsystems would be: the largest possible path that most closely resembles the shape

of the contour lines in the equivalent von Mises response.

In the research of Bulu [3], a different suggestion is given, based on the aspect ra-tio b/a of the path: it is stated there that paths with aspect ratio b/a ≈ 0.1 with bconsisting of at least 5/6 node layer. However Bulu [3] deals with a whole differentproblem, still some comparison can be made. To that extend, in figure 3.11 the value ofthe J-integral is plotted against b/a.In [3], the suggestion of b/a ≈ 0.1 is made because in his results, for different paths,the J-integral seems to result in similar values, for aspect ratios of b/a approaching 0.However, the problem researched here does not show such a convergence; in fact foraspect ratios b/a approaching zero, the value of the J-integral shows an increase, becauseof the effect at larger a’s. At b/a ≈ 0.1, the minimum corresponding with a ≈ 1.2µm canbe seen. This minimum is dependent on b/a however (because actually, this minimumis found at a ≈ 1.2µm for all b) and can therefore, in this research, not be chosen foran appropriate suggestion of path selection. However, it is interesting to see that, whenplotting the data in terms of b/a (figure 3.11), choosing a higher b gives a flatter responcein J . In [3], this effect was the reason to suggest higher values of b, and that seems tohold for this problem too.

23

3.2 J-Integral Results 24

0 1 2 3 4 5 6 72

2.2

2.4

2.6

2.8

b/a

J [J/m

2]

J value plotted against b/a, various mean elementlenghts

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.52.05

2.1

2.15

2.2

2.25

2.3

2.35

2.4

2.45

b/a

J [J/m

2]

b = 2*35nm

b = 2*25nm

b = 4*25nm

b = 2*20nm

b = 4*20nm

b = 6*20nm

b = 2*10nm

b = 4*10nm

b = 6*10nm

b = 8*10nm

b = 10*10nm

b = 12*10nm

Figure 3.11: J values plotted against aspect ratio b/a, 10,20,25,25 nm models. Notice thatplotting against b/a, instead of a, means that the graph of figure 3.9 flips and scales, butessentially still shows the same trend

24

Chapter 4

Conclusion

In this report, the cooling down of a microchip with a delamination in the low K layerswas simulated. For various mesh sizes and path shapes, the J-integral that plays animportant role in Fracture Mechanics as a measure for the strain energy release rate wasevaluated. The paths that were evaluated were rectangular paths with the crack tip attheir symmetric center.

As was expected, the value of the J-integral showed a considerable dependency onthe path of integration and mesh of the model. Reason’s for higher or lower values of Jwere sought in the stress and displacement results of the simulation, comparing them withthe definition of the J-integral. It was proposed that the minimum and small maximumrelatively close to the crack tip could be predicted by the general stress response of themodel. It was also speculated that the minimum and change in slope further from thecrack tip could be linked to changes in the stress results and deformation field, althoughthis was not verified.It was found that paths should run far enough from the actual crack tip to stay clear of thehigh peak in the response caused by the singularity. Also, a finer mesh will give results thatapproximate the singular peak better and thus cause the effects of the singular point to bemore localized. Because of this, for a denser mesh, the value of the J-integral for the samepath, returns a lower value. This effects decreases with increasing distance to the crack tip.

Based on these results, a suggestion was made for the selection of the integralpaths in similar problems in layered structures. In order to get a value for the J-integralthat has a reasonable small path and mesh dependence, the path should be as thickas possible (perpendicular to the crack) and the length should be such that the totalpath shape corresponds the best to the contour shapes in the general stress state of themodel. Although this returns a stable value for J , this value was not compared to analternatively found value for the strain energy release rate.

25

Appendix A

J-Integral Computation Code

RSYS,SOLU read solution resultsAVPRIN,0,0 ”AVRES,2, ”/EFACET,1 ”LAYER,0 ”FORCE,TOTAL ”

AVPRIN,0,0, define element table elementsETABLE, ,S,EQV ”AVPRIN,0,0, ”ETABLE,StrainE,SENE, store strain energy per elementAVPRIN,0,0, ”ETABLE,Volume,VOLU, store volume per element

SEXP,W,STRAINE,VOLUME,1,-1, divide strain energy by volume to give Wpdef,wpath,etab,w,avg interpolate W on the pathpcalc,intg,j1,wpath,yg,1 integrate W on the path with respect to y*get,ja,path,,last,j1 store the value for the left part of 2.11

pdef,clear clear path itemspvect,norm,nx,ny,nz create normal vectors on pathpdef,intr,sx,sx map the component stresses onto the pathpdef,intr,sy,sy ”pdef,intr,sxy,sxy ”pcalc,mult,sxnx,sx,nx,1 multiply sx and nx

pcalc,mult,sxyny,sxy,ny,1 multiply sxy and ny

pcalc,mult,syny,sy,ny,1 multiply sy and ny

pcalc,mult,sxynx,sxy,nx,1 multiply sxy and nx

pcalc,add,tx,sxnx,sxyny add sxnx to sxyny to gain tx

pcalc,add,ty,syny,sxynx add syny to sxynx to gain ty

26

27

*set,dx,2e-6 set a small ∂x to use for computation of differentialspcalc,add,xg,xg,,1,1,-dx/2 shift path ∂x to the leftpdef,intr,ux1,ux get value of ux at new position, ux1

pdef,intr,uy1,uy get value of uy at new position, uy1

pcalc,add,xg,xg,,1,1,dx shift path ∂x to the rightpdef,intr,ux2,ux get value of ux at new position, ux2

pdef,intr,uy2,uy get value of uy at new position, uy2

pcalc,add,xg,xg,,1,1,-∂x/2 shift path back to original position*set,c,1/dx store 1/∂x as a new variable cpcalc,add,c1,ux2,ux1,c,-c,, calculate ∂ux/∂x through ux2/∂x - ux1/∂xpcalc,add,c2,uy2,uy1,c,-c,, calculate ∂uy/∂x through uy2/∂x - uy1/∂xpcalc,mult,txc1,tx,c1,1 multiply tx and ∂ux/∂xpcalc,mult,tyc2,ty,c2,1 multiply ty and ∂uy/∂xpcalc,add,work,txc1,tyc2,1,1,, add previous resultspcalc,intg,j2,work,s,1, integrate result*get,jb,path,,last,j2 store the value for the right part of 2.11*set,Jintegral,(Ja-Jb) compute J-integral from left and right parts

27

Appendix B

Results Overview

On the next page, an overview of the simulation results that are used for the computationof the J-integral of the 5 nm model is plotted. Shown are the different stress componentssx, sy, sxy, the displacements ux, uy, strain energy density W and the equivalent von Misesstress. Stresses are displayed in MPa, displacements in mm.The area of which the results are plotted consists of all elements available for the com-putation of the J-integral: 1000nm left and right of the crack tip, and 70nm above andbelow the crack tip.

28

29

Figure B.1: overview of calculation results of parameters used in the computation of theJ-integral, for the 5 nm model

29

Appendix C

Direct Calculation of G

For verification, it was tried to calculate the Strain Energy Release by a differentcomputation than using the J-integral. Although, as stated in section 1.1, the directcalculation of the Strain Energy Release Rate through equation 1.1 would probablyrender incorrect results, this method was attempted nonetheless, because of the ease ofcalculation.

According to the Ansys Manual [5], a crack extension of 0.5% to 2% should beused. In this case, extensions ranging from 20 nm (or 0.66%) to 140 nm (or 4.66%)were used. Also, the Strain Energy Release Rates from extended crack lengths to otherextended crack lengths were computed; making the total number of different combinations(to compute G) 10, when for each model 5 different total crack lengths are simulated.The cracks were extended by creating a node 0001 along with the correct number of evennumbered nodes in the 0000 range (section 2.2.2). The crack tip is thus no longer at nodenumber 1001, but for example at 1002 or 1004. Other than the extended crack length,the simulation in Ansys was exactly the same as in chapter 2.

In Ansys, summing up all the Element Strain Energy (SENE) for all elementsgives the Total Strain Energy, Ua (original crack length) or Ua+∆a (extended cracklength). The total amount of Strain Energy for each of the simulations is shown in tablesC.1 and C.2, from were it becomes clear that the differences are indeed very small.Already in these tables, it can be seen that for some crack extensions, the Total StrainEnergy goes up, rather than down by crack growth. This is not the expected behav-ior for this problem; in which crack growth should be favorable from an energy stand point.

From different combinations of these energies, the Energy Release Rate G has beencalculated according to equation 1.1; G = −

Ua+∆a−Ua

B∆a. In this, B = 1 mm, because the

models were built in the millimeter scale. The results are shown in tables C.3 and C.4.The results show that, for the finer meshes of the 5 and 10 nm models, great variation isfound in the results, as well as some improbable negative values. Probably, this is causedby the larger number of degrees of freedom in those models: the more elements, and thusdegrees of freedom, the more possible calculation results (approximations to the realresult) fall within the tolerance band used. The fluctuations within the tolerance bandmight be the cause of the Total Strain Energies also fluctuating, hence causing the large

30

31

variations in Energy Release Rates. This shows exactly the problem of using this directapproach for calculating the Strain Energy Release Rate: because the interesting digitsare in the sixth and further decimals of the Total Strain Energy, the tolerance should beset strict enough.For the coarser meshed models, the 20, 25 and 35 nm models, the calculated values of Gshow a better reproducibility, with only small variations. Also, the values of G are closeto the values of J calculated in the main body of this report. However, the found valuesof G show a large dependence on the mesh density; larger than using the J-integral. Forthe larger meshes, because of the smaller number of degrees of freedom, there are littleapproximations possible within the tolerance band; thus causing the good reproducibility.Judging from the large differences in G between the models, the simulation results differquite a lot between the models. To make the difference smaller, the tolerance for thesemodels should also be set more strict. However, because the smaller number of degreesof freedom, possibly the iteration scheme would then be unable to converge; as noapproximation might be possible within the tolerance band.

In conclusion it can be said that, when using the direct approach for calculatingthe Strain Energy Release Rate, the tolerance level in the Finite Element Simulationshould be more strict to get useful results. The tolerance should be strict enough to makethe possible fluctuations within the tolerance band small enough to not cause any trouble.This means that, apart from having to do two simulations, the simulations would alsotake longer, as there will be more iteration steps needed to reach the desired numericalprecision. That is, if the model converges at all within the necessary tolerance band.

Table C.1: Strain Energies U for original crack length 3000 nm and various crack extensions∆a; 5,10 and 20 nm models; U in 10−3 J

5 nm model 10 nm model 20 nm model

Ua [10−3 J] 3.993220792156306320 3.993220475649119638 3.993220360328408880

Ua+20nm [10−3 J] 3.993220768617614436 3.993220389262326364 3.993220318956489212

Ua+40nm [10−3 J] 3.993220789024205075 3.993220403764965010 3.993220277464312851

Ua+60nm [10−3 J] 3.993220903025156421 3.993220368135421694 3.993220236511693777

Ua+80nm [10−3 J] 3.993221346229904434 3.993220331158194192 3.993220195787817506

Table C.2: Strain Energies U for original crack length 3000 nm and various crack extensions∆a; 25 and 35 nm model; U in 10−3 J

25 nm model 35 nm model

Ua [10−3 J] 3.993220324792523090 Ua [10−3 J] 3.993220293648793273

Ua+25nm [10−3 J] 3.993220270388735837 Ua+35nm [10−3 J] 3.993220214927526435

Ua+50nm [10−3 J] 3.993220216555145363 Ua+70nm [10−3 J] 3.993220136755865735

Ua+75nm [10−3 J] 3.993220162719949506 Ua+105nm [10−3 J] 3.993220058924273008

Ua+100nm [10−3 J] 3.993220110337673656 Ua+140nm [10−3 J] 3.993219981497929449

31

32

Table C.3: Strain Energy Release Rates G calculated between different crack extensions∆a; 5, 10 and 20 nm models; G in J/m2

5 nm model 10 nm model 20 nm model

G0→20nm 1.1769 4.3193 2.0686G20nm→40nm -1.0203 -0.7251 2.0746G40nm→60nm -5.7000 1.7815 2.0476G60nm→80nm -22.1602 1.8489 2.0362

G0→40nm 0.0783 1.7971 2.0716G20nm→60nm -3.3602 0.5282 2.0611G40nm→80nm -13.9301 1.8152 2.0419

G0→60nm -1.8478 1.7919 2.0636G20nm→80nm -9.6269 0.9684 2.0528

G0→80nm -6.9259 1.8061 2.0568

Table C.4: Strain Energy Release Rate G calculated between different crack extensions∆a; 25 and 35 nm models; G in J/m2

25 nm model 35 nm model

G0→25nm 2.1762 G0→35nm 2.2492G25nm→50nm 2.1533 G35nm→70nm 2.2335G50nm→75nm 2.1534 G70nm→105nm 2.2238G75nm→100nm 2.0953 G105nm→140nm 2.2122

G0→50nm 2.1647 G0→70nm 2.2413G25nm→75nm 2.1534 G35nm→105nm 2.2286G50nm→100nm 2.1243 G70nm→140nm 2.2180

G0→75nm 2.1610 G0→105nm 2.2355G25nm→100nm 2.1340 G35nm→140nm 2.2231

G0→100nm 2.1445 G0→140nm 2.2296

32

Bibliography

[1] J.R Rice. A path independent integral and the approximate analysis of strain concen-trations by notches and cracks. Journal of Applied Mechanics, 35:379–386, 1968.

[2] Shi Weichen and Zhen-Bang Kuang. J-integral of dissimilar anisoptropic media. In-

ternational journal of fracture, 96(4):37–42, 1999.

[3] Bulu Xu, Xai Cai, Weidong Huang, and Zhaonian Cheng. Research of underfill de-lamination in frlip chip by the j-integral method. Journal of Electronic Packaging,126:94–99, March 2004.

[4] Rao R. Tummala. Fundamentals of Microsystems Packaging. McGraw-Hill, interna-tional edition edition, 2001.

[5] ANSYS, Inc. Release 10.0 Documentation for Ansys, r10.0 edition, 2005.

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