Atomistic calculations of the mechanical properties cu sn intermetallic compounds

The properties of intermetallic compounds formed at thesolder joint need to be known accurately so that macro-scale computer modellingcan take place to evaluate the reliability of compon

ATOMISTIC CALCULATIONS OF THE MECHANICALPROPERTIES OF Cu-Sn INTERMETALLIC COMPOUNDS

LEE TIONG SENG NORMAN

NATIONAL UNIVERSITY OF SINGAPORE

2008

LEE TIONG SENG NORMAN

(B.Eng(Hons),NUS)

A THESIS SUBMITTEDFOR THE DEGREE OF DOCTOR OF PHILOSOPHYDEPARTMENT OF MECHANICAL ENGINEERINGNATIONAL UNIVERSITY OF SINGAPORE

2008

In my time, I tried to educate our people in an understanding of thedignity of human life and their right as fellow human beings, and youthwas not only interested but excited about what I consider things thatmatter Things of the spirit, the development of a human being tohis true potential in accordance with his own personal genius in thecontext of equal rights of others.

David Saul Marshall (1908-1995)

Over the past six years, my supervisors, Dr Lim Kian Meng, Dr Vincent Tan and

Dr Zhang Xiao Wu have provided valuable guidance, advice and support Veryoften, they would ask probing questions that spurred me to think deeper into thetopic at hand or my interpretation of the research results They have also inspired

me to explore new research areas, often bringing me out of my comfort zone

My mum has also been a pillar of support I would come home after takingthe last bus to find fruits or food on my desk

I acknowledge the financial support from the following sources: NUS researchscholarship (2002 - 2004), NUSNNI(2005 - 2006), Institute of Microelectronics(2002 - 2004) Credit must also be given to Centre for Science and Mathematics,Republic Polytechnic for employment (2008) and their understanding on the occa-sions when I was unable to fulfill my duties Financial support came also from themany opportunities for teaching from the Department of Mechanical Engineering(especially with Prof CJ Tay and Prof Cheng Li), Professional Activities Centreand the Bachelor of Technology department

The facilities provided by NUS have been excellent The staff at the ScienceLibrary have been most friendly and helpful I have also made extensive use of theresources provided by the Supercomputing and Visualization Unit (SVU), and Ithank the staff, Dr Zhang Xinhuai and Mr Yeo Eng Hee for their excellent service

I would also like to thank the University Health and Wellness centre

My thanks also goes out to the following members of the scientific community,

Dr Alexander Goldberg of Accelrys Inc and Prof Lee Ming-Hsien of TamkangUniversity, Taiwan Although the need to credit their contributions in the maintext of this thesis did not arise, I appreciate their willingness to respond to myemail queries

I owe special thanks to my colleagues and lab officers Regardless of the time

of day, they would provide useful words of advice and encouragement when thedemands of this research seemed overwhelming Their knowledge, opinions andideas which they shared with me often gave me the needed push to move on Theyalso had to put up with my idiosyncracies In this, I acknowledge Adrian Koh,Zhang Bao, Alvin Ong, Dr Zhang YingYan, Dr Dai Ling, Dr Deng Mu and DrYew Yong Kin, as well as lab officers Mr Joe Low, Mr Alvin Goh and Mr ChiamTow Jong

My thanks goes out to my friends for their encouragement and their advice inthe decisions that I have made Talking to them always helped in seeing thingsclearer I am sure that their prayers helped a lot Last of all, there were manyoccasions when serendipity and decisions that I made in the past (e.g taking basicGerman lessons) played a role in getting the research work done I recognize therole of the Creator in all that has happened Gloria in altissimis Deo

1.1 Current Trends in the Electronics Industry 1

1.2 Computational Methods 3

1.3 Objectives 6

1.4 Outline of the thesis 6

2 Elastic Properties of Intermetallic Compounds 8 2.1 Introduction 8

2.1.1 Motivation 8

2.1.2 Experimental Studies of the Elastic Properties of the Inter-metallic Compounds 9

2.1.3 Overview 10

2.2 Literature Review 13

2.2.1 Introduction 13

2.2.2 Geometry Optimization 14

2.2.3 Making use of the energy 15

CONTENTS vii

2.2.4 Elastic Constants 16

2.2.5 DFT Calculations of Intermetallic compounds 17

2.2.6 Conclusion 18

2.3 Theory 19

2.3.1 Crystal structure 19

2.3.2 Density Functional Theory 22

2.3.3 Elasticity of Single Crystals 27

2.3.4 Bounds on Polycrystalline Elastic Moduli 30

2.4 Methodology 33

2.4.1 Crystal structure of the Intermetallic Compounds 33

2.4.2 Software 38

2.4.3 Computational Resources 38

2.4.4 Geometry Optimization 38

2.4.5 Accuracy Settings 38

2.4.6 Elastic Constants 41

2.4.7 Validation 42

2.4.8 Summary 43

2.5 Properties of Monoatomic Metals 44

2.5.1 Introduction 44

2.5.2 Lattice Constants and Elastic Constants of Ag Ni and Cu 45 2.5.3 Lattice Constants and Elastic Constants of Sn 45

2.5.4 Effect of using GGA-generated pseudopotentials 47

2.5.5 Conclusions 48

2.6 Properties of the Intermetallic Compounds 49

2.6.1 Introduction 49

2.6.2 Lattice Constants 49

2.6.3 Internal Crystal Parameters 51

2.6.4 Elastic Constants 51

2.6.5 Elastic Anisotropy 54

2.6.6 Bounds on Polycrystalline Elastic Moduli 57

2.6.7 Summary 64

2.7 Discussion 64

2.7.1 Limitations of the DFT calculations performed 64

2.7.2 Comparison of polycrystalline bounds with nanoindentation data 65

2.8 Conclusions 69

3 Molecular Dynamics Potential For Cu-Sn 70 3.1 Introduction 70

3.1.1 Motivation 70

3.1.2 Molecular Dynamics Method 71

3.1.3 Types of Interatomic Potentials 73

3.1.4 Challenges with developing an interatomic potential for Cu-Sn 75 3.1.5 Overview 78

3.2 Literature Review 78

3.2.1 Introduction 78

3.2.2 Interatomic Potentials 79

3.2.3 MD simulation of materials with complex structures 81

3.2.4 MD simulation of materials with two or more atomic species 83 3.2.5 Conclusion 85

3.3 Theory 86

3.3.1 Notation and Definitions 86

3.3.2 Modified Embedded Atom Method 89

3.3.3 Optimization Methods 100

3.4 Potential Fitting 103

3.4.1 Predictions of existing parameters 103

CONTENTS ix

3.4.2 Potential Fitting Strategy 105

3.4.3 Density Functional Theory Calculations 108

3.4.4 Fitting database 110

3.4.5 Choice of functions and parameters 114

3.4.6 Optimization Methodology 119

3.5 Results and Validation 123

3.5.1 Parameters Obtained 123

3.5.2 Calculation Method 124

3.5.3 Minimum Energy Structure 125

3.5.4 Elastic Constants 127

3.5.5 Surface Energy 133

3.5.6 Other Structures 135

3.6 Discussion 138

3.6.1 Introduction 138

3.6.2 Choice of Potential Parameters 139

3.6.3 Limitations 140

3.7 Summary 141

4 MD Simulations of Fracture 142 4.1 Introduction 142

4.1.1 Motivation 142

4.1.2 Experimental studies of the fracture toughness of Cu6Sn5 143 4.1.3 Overview 144

4.2 Literature Review 145

4.2.1 Introduction 145

4.2.2 Fracture of Metals and Semiconductors 145

4.2.3 Fracture of Intermetallic Compounds 147

4.2.4 Conclusions 148

4.3 Simulation Procedure 149

4.3.1 Interatomic Potential 149

4.3.2 Visualization 149

4.3.3 MD Software 149

4.3.4 Unit Cell 153

4.4 KIC calculation using the tensile loading on a periodic crack 154

4.4.1 Method 154

4.4.2 Results 156

4.4.3 Calculating the fracture toughness 160

4.4.4 Summary 160

4.5 KIC calculation using a crack-tip displacement field 161

4.5.1 Method 161

4.5.2 ac plane 164

4.5.3 Basal plane 165

4.5.4 Summary 168

4.6 Discussion 170

4.6.1 Comparing the two simulation methods 170

4.6.2 Comparison with experimental results 171

4.6.3 Qualitative features of the simulations 172

4.6.4 How realistic is the interatomic potential? 173

4.7 Summary 174

5 Conclusions 175 5.1 Conclusions 175

5.2 Recommendations for Future Work 177

CONTENTS xi

A.1 Orientation Dependance Of the Young’s Modulus 208

A.1.1 Cubic Crystal 208

A.1.2 Monoclinic Crystal 209

B Internal Relaxations In a Crystal 210 B.1 Honeycomb Lattice 210

C MEAM Formulas 212 C.1 MEAM Energy 212

C.2 Cu-Sn Interatomic Distance 212

C.3 MEAM Lattice Sums for the NiAs Crystal 213

C.3.1 Sn atom 213

C.3.2 Cu atom 214

C.3.3 Conclusion 215

C.4 MEAM Forces 216

C.4.1 Derivative of the Pairwise energy 217

C.4.2 Derivative of the screening function 218

C.4.3 Derivative of the Embedding Energy 218

C.4.4 Derivative of the partial electron densities 219

C.4.5 Derivative of Γ 220

D Formation Energy of Cu6Sn5 with DFT 221 E Negative Elastic Constants for a monoclinic crystal 223 E.1 Introduction 223

E.2 Literature Review 223

E.3 Eigenvalues 224

E.4 Implications 224

E.5 Summary 226

F Review of experimental work 227F.1 Experimental Methods 227F.2 Discussion 229

Summary

Intermetallic compounds such as Cu6Sn5 play a vital role in the reliability ofelectronic components The properties of intermetallic compounds formed at thesolder joint need to be known accurately so that macro-scale computer modellingcan take place to evaluate the reliability of components during the design process.However, experiments to determine the mechanical properties of Cu6Sn5, such asthe Young’s Modulus and the critical mode I fracture toughness, have been sparseand have not produced any definitive value yet Hence, there is a need to clarifywhat the values of these properties are Therefore, this study investigates the use

of calculations at the atomistic level in order to obtain the mechanical properties

of Cu-Sn intermetallic compounds

Density Functional Theory (DFT) calculations are performed with the metallic compounds Cu3Sn and Cu6Sn5 to obtain the elastic properties of thesematerials Using a single unit cell, their lattice constants are calculated and shown

inter-to be in agreement with experimental data The hitherinter-to unknown single-crystalelastic constants are then calculated Using these values, the orientation depen-dence of the single-crystal Young’s Modulus is evaluated The direction of thelargest value is found to coincide with closely-packed planes The bounds on thepolycrystalline elastic moduli are also evaluated using the methods of Hill andHashin-Shtrikman These bounds are found to be on the upper range of experi-mental results that are currently available This shows that DFT calculations are

a feasible means of predicting the polycrystalline elastic properties of intermetallic

Following which, an interatomic potential for Cu-Sn interactions in the fied Embedded Atom Method formalism tailored to the properties of Cu6Sn5 in theNiAs crystal structure is developed Using this interatomic potential, MolecularDynamics simulations of the fracture of Cu6Sn5 are conducted with thousands ofatoms Atomic behaviour corresponding to brittle fracture are seen in the simu-lations This shows that it is feasible to develop an interatomic potential for theNiAs crystal structure, and to conduct realistic simulations that can reproducequalitatively the properties and behaviour of the brittle intermetallic compound

List of Tables

1.1 A comparison of the three computational methods 5

2.1 Experimental data on intermetallic compounds investigated in this Chapter 12

2.2 Constraints on the elastic constants for different crystal systems 28

2.3 Crystal structure data of the four intermetallic compounds 37

2.4 Main computational resources used 38

2.5 Filenames of CASTEP pseudopotentials used 41

2.6 Strain Patterns used in calculating elastic constants 42

2.7 Calculated lattice constants and elastic constants (GPa) for Ag, Cu and Ni using LDA-generated pseudopotentials 46

2.8 Lattice Constants and Elastic Constants (GPa) for α-Sn and β-Sn using LDA-generated pseudopotentials (Sn 00.usp) 47

2.9 GGA calculations of the lattice constants and elastic constants (GPa) of Cu and α-Sn with GGA-generated pseudopotentials (Cu 00PBE.usp and Sn 00PBE.usp) 48

2.10 Calculated lattice constants 50

2.11 Computational time in CPU-days for the lattice constants For each xc-functional, the values are arranged in order of increasing accuracy setting 50

2.12 Positional Parameters for Cu6Sn5 51

2.13 Positional Parameters for Ni3Sn4 52

2.14 Positional parameters for Cu3Sn 52

2.15 Positional parameters for Ag3Sn 52

2.16 Elastic Constants of Cu3Sn calculated at different accuracy setting All values are given in GPa 54

2.17 Elastic Constants of Ni3Sn4 calculated at different accuracy setting All values are given in GPa 55

2.18 Elastic Constants calculated for Ag3Sn calculated at different accu-racy setting All values are given in GPa 55

2.19 Computational time in CPU-days for the elastic constants For each xc-functional, the values are arranged in order of increasing accuracy setting 55

2.20 Maximum single-crystal Young’s Modulus and their directions 57

2.21 Bounds on the Polycrystalline Moduli (GGA) 62

2.22 Bounds on the Polycrystalline Moduli (LDA) 63

3.1 Physical constants and values used in calculating the Thermal de Broglie wavelength of Cu 72

3.2 The different summation symbols in use 88

3.3 Parameters for Cu-Sn by Aguilar et al 103

3.4 Screening parameters used by Aguilar et al 103

3.5 Elastic Constants calculated Calculations are performed with the GGA 108

3.6 The fitting database 113

3.7 Screening parameters 115

3.8 Available experimental values of the enthalpy of formation 119

3.9 Constraints placed on the fitting parameters 121

3.10 Weights for the items in the fitting database 122

LIST OF TABLES xvii

3.11 Parameters for CuSn obtained from the potential fitting process Set P1 1233.12 Parameters for Cu-Sn obtained from the potential fitting process -Set P2 1243.13 Coordinates of the atoms in Cell A They are expressed as a fraction

-of the cell vectors The cell lengths are a ×√3a × c 1263.14 Coordinates of the atoms in Cell B They are expressed as a fraction

of the cell vectors The cell lengths are a ×√3a × c 1263.15 Lattice constants of the B8 unit cell predicted by the two sets of in-teratomic potential parameters using the threshold acceptance cal-culations For each, three trials are performed 1273.16 Strain patterns for the various elastic constants 1283.17 Elastic Constants calculated All values are given in GPa †c13 iscalculated from the combined elastic constants 1303.18 Elastic Constants requiring relaxation All values are given in GPa

†c11 and c12 are calculated from the combined elastic constants 1313.19 Predicted surface energies All values are in eV/˚A2 1353.20 Predicted Energies of other structures 1384.1 Experimental data for the fracture toughness KIC of Cu6Sn5 1444.2 Atomic-Scale units used in the software 1514.3 Description of the size of the simulation boxes used in this section.The pre-crack size and box dimensions are given in terms of unitcell lengths 1564.4 Values of stress and predicted KIC for the two cases, assuming that

σ = c11 1604.5 Different box sizes used in the study in this section 1644.6 Different box sizes used in the study in this section 166

C.1 Position vector components of the atoms in the coordination hedron of the Sn atom 214C.2 Position vector components of the Cu atoms in the coordinationpolyhedron of the Cu atom 215C.3 Position vector components of the Sn atoms in the coordinationpolyhedron of the Cu atom 215D.1 DFT energies per atom for Cu (FCC) , Sn (DC) and CuSn (B8).The energy change per atom (∆E0) and cohesive energy (Ecoh) arethen derived from them 222F.1 Experimental work on the Young’s Modulus of Cu6Sn5 and Cu3Sn.All values are in GPa 231

List of Figures

1.1 Structure of a solder joint 22.1 Illustrating the idea of a lattice with basis The result is a honey-comb lattice 212.2 Illustrating the various definitions of the unit cells using the (100)plane of the FCC crystal Different possibilities for the lattice vec-tors are also shown 212.3 Cu6Sn5 in the NiAs-type structure according to Gangulee et al The larger spheres represent the Sn atoms and the smaller spheresrepresent the Cu atoms The excess Cu atoms fill the interstitialsites to make up the 6:5 stoichiometry 342.4 Unit Cell of Cu6Sn5 according to Larsson et al The larger spheresrepresent the Sn atoms and the smaller spheres represent the Cuatoms 352.5 Unit Cell of Cu3Sn according to Burkhardt et al The larger spheresrepresent the Sn atoms and the smaller spheres represent the Cuatoms 362.6 Unit Cell of Ni3Sn4 according to Jeitschko et al The larger spheresrepresent the Sn atoms and the smaller spheres represent the Niatoms 36

2.7 Unit Cell of Ag3Sn according to Fairhurst et al The larger spheresrepresent the Sn atoms and the smaller spheres represent the Agatoms 362.8 Documentation Provided in LDA-generated pseudopotential files for

Cu (‘Cu 00.usp’) and Sn (‘Sn 00.usp’) 402.9 Flowchart showing the methodology of the calculations in this chap-ter 442.10 Stiffness matrix for Cu6Sn5 calculated with the GGA at 320 eVcutoff and 4 × 4 × 3 k-point mesh All values are given in GPa 542.11 Relation of the coordinate axes to the crystal axes 562.12 Young’s Modulus as a function of crystallographic direction for

Cu3Sn The colours represent the magnitude of the Young’s ulus while the x, y, z coordinates represent the crystallographic di-rections 582.13 Young’s Modulus as a function of crystallographic direction for

Mod-Cu6Sn5 The colours represent the magnitude of the Young’s ulus while the x, y, z coordinates represent the crystallographic di-rections 582.14 Young’s Modulus as a function of crystallographic direction of vari-ous planes for Cu3Sn Angles are measured anti-clockwise from thefirst-mentioned axis in each plane 592.15 Young’s Modulus as a function of crystallographic direction of var-ious planes for Cu6Sn5 Angles are measured anti-clockwise fromthe first-mentioned axis in each plane 592.16 The plane corresponding to 55.8◦ anti-clockwise from the positivex-axis in Cu3Sn 602.17 The plane corresponding to 147.6◦ anti-clockwise from the positivex-axis in Cu6Sn5 61

Mod-LIST OF FIGURES xxi

2.18 A two-dimensional view of the plane marked in Figure 2.17 in Cu6Sn5.61

2.19 Cu6Sn5: Cumulative Distribution Function for the single-crystalYoung’s Modulus of Cu6Sn5 682.20 Cu3Sn: Cumulative Distribution Function for the single-crystal Young’sModulus 683.1 FCC lattice in 2D projection “First neighbour” and “second neigh-bour” shells of shaded atom are indicated 883.2 Illustrating the screening in the MEAM 923.3 Two cases to explain the MEAM Screening Procedure For Case 1,

C = 3 and for case 2, C = 13 933.4 Relationship between the pairwise, EAM and MEAM potential func-tionals 983.5 Evolution of the energy of Cu-Sn B8 unit cell predicted using theAguilar MEAM potential 1043.6 Evolution of the lattice constants a and c of Cu-Sn B8 unit cellusing the Aguilar MEAM potential 1053.7 Energy-Volume calculations for B8 Cu-Sn with DFT The lines are

a guide to the eye 1103.8 Energy-Volume calculations for B8 Cu-Sn with DFT, enlarged toshow points near equilibrium volume 1113.9 Variation of equilibrium c/a ratio with volume The lines are aguide to the eye 1113.10 Coordination Polyhedra for Cu and Sn in the B8 unit cell The basisatoms are indicated with yellow letters and the first neighbours arejoined with yellow lines to show the coordination polyhedron 115

3.11 Alternative orthogonal unit cells that have the same symmetry asthe B8 unit cell The c-direction is perpendicular to the plane of thediagram Both Cell A and Cell B have sides of length a ×√3a × c 1253.12 Evolution of the lattice constants during the threshold acceptanceprocess 1283.13 Strain patterns for the various elastic constants illustrated by show-ing how the unit cell deforms The strains for c11−c12and c44destroythe hexagonal symmetry 1293.14 Unit cell to calculate c44 1323.15 Illustrating the Surface Energy Calculation Two simulation cells ofdifferent boundary conditions are considered, one with fully periodicboundary conditions, the other with a surface The surface energy

= (E1− E0)/A 1333.16 Unit cells used in the DFT calculation of the surface energy (a)surface normal x (b) surface normal z The height of the insertedvacuum is about 8 ˚A 1343.17 Unit cells for different structures with 1:1 stoichiometry For eachstructure, the Strukturbericht designation is given followed by thechemical formula of a typical compound 1374.1 Schematic diagram of cracks forming from the corners of the Vickersindentation The crack length C is measured from the centre of theindentation 1444.2 Flowchart showing the organization of RidgeMD 1524.3 Description of the basal plane and the “ac plane” 1534.4 Periodic Crack 1544.5 Schematic diagram of the periodic crack simulation box The side

n of the pre-crack is parallel to the c-axis 155

LIST OF FIGURES xxiii

4.6 Configuration PC1 Snapshot at 20000 time steps with an initialstrain of  = 0.20 applied throughout The sample cleaves betweenthe immobile layer and the mobile atoms and no crack propagatesfrom the pre-crack 1574.7 Configuration PC2 with a initial strain of  = 0.20 applied through-out Snapshots at (a) 6500 time steps (b) 20000 time steps 1574.8 Configuration PC3 with an initial strain of  = 0.12 applied through-out Snapshots at (a) 1200 time steps (b) 20000 time steps 1594.9 Configuration PC3 Snapshot at 1200 time steps representing theatoms as circles All the atoms are superimposed on each other.The plane at which the crack propagates can be clearly seen 1594.10 (a) Array of atoms (b) The simulation box after Sih’s equation isapplied to the atoms 1614.11 ac plane: Snapshots of atomic configuration for size Small at 20000time steps at 300 K for (a) KI = 0.40 MPa√

m(b) KI = 0.64 MPa√

m1664.12 ac plane: Snapshots of atomic configuration at 20000 time steps at

20000 time steps at 300 K for (a) KI = 0.48 MPa√

m(b) KI = 0.64MPa√

m 1694.15 Basal plane: snapshots of atomic configuration for Size Large at

20000 time steps at 300 K for various values of KI = 0.64 MPa√

m.(a) 3100 time steps (b) 20000 time steps 1694.16 Basal plane: Size Large, KI = 0.64 MPa√

m using circles to sent the atoms (a) without tails (b) with tails 170

repre-B.1 Part of the honeycomb lattice The (x, y) coordinates are givenbeside each particle (a) Before shear, all particles are unit distanceaway from each other (b) after shear The arrows beside particles 3and 4 shows how they have been displaced 211E.1 (a) (010) plane of Cu6Sn5(b) Schematic diagram of the (010) plane.The dashed lines represent the planes of atoms seen in (a) (c) Uponapplication of the shear strain 13, the unit cell deforms according tothe red dotted line The angle β is reduced (d) Bottom left corner

of the unit cell BC is shortened when β is reduced 225F.1 (a) nanoindentation load-displacement curve (b) Berkovich indenter 228

A Amplitude of plane wave

A Indentation Projected Contact Area

a crack half-length

aij direction cosine between original axis j and rotated axis i

~a Primitive lattice vector

B0 Hashin-Shktrikman bulk modulus for a hypothetical material

BR Reuss’ average of the Bulk Modulus

BV Voight’s average of the Bulk Modulus

b integer value that defines the points in the reciprocal lattice

~b Reciprocal lattice vector

C Vickers Crack length

C (MEAM) Parameter in screening function

Cmin (MEAM) Lower limit on C

Cmax (MEAM) Upper limit on C

C Elastic stiffness matrix

ck,G Coefficient for planewaves in the one-electron wavefunction

cij elastic constant

< cij > average over all crystal orientations of cij

D Parameter in Morse potential

d(h)j (MEAM) h-th atomic electron density due to neighbour j

E Energy from Schr¨odinger’s Equation

Ecoh Cohesive Energy

Ecoh,Cu Cohesive Energy of Copper

Ed Total energy calculated by Density Functional Theory

Ei Young’s Modulus of the indenter

Er Indentation Modulus

Esurf Surface Energy

Eu Rose’s Universal Equation of State

Exc Exchange-correlation energy functional

F Vickers Indentation Load

Fi Embedding Function of atom i

fi Total force on atom i

fi Occupancy of the i-th one-electron wavefunction

fa,x Total force on atom a in the x-direction

fij force between atoms i and j

G (MEAM) Function used in calculating electron density

LIST OF SYMBOLS AND ACRONYMS xxvii

G0 Hashin-Shktrikman shear modulus for a hypothetical material

GR Reuss’ average of the Shear Modulus

GV Voight’s average of the Shear Modulus

i summation index for atoms

j summation index for an atom’s neighbours

KI Mode I fracture toughness

K1 Function in Hashin-Shtrikman Bounds

K2 Function in Hashin-Shtrikman Bounds

KIC Critical Mode I fracture toughness

k summation index when three-body forces are involved

N (MEAM) Number of interactions for each pair of atomic species

n integer value that defines the points in the conventional lattice

~nij Unit vector of the position vector betweeom atoms i and j

P Number of neighbours for each atom pair

p Anisotropic crack-tip displacement field parameter

q Anisotropic crack-tip displacement field parameter

r Interatomic distance

r Distance in polar coordinates

ri Position of atom i

rij Distance between atoms i and j

ro Equilibrium distance between first neighbours in a crystal

~r Position vector

S Elastic compliance matrix

sij Elastic compliance

Sij Elastic compliance adjusted for plane strain

Sij (MEAM) Combined screening parameter between atoms i and j

sijk (MEAM) Screening parameter between atoms i,j and k

s1 Anisotropic crack-tip displacement field parameter

U Electron-electron interactions energy functional

u Displacement due to the anisotropic crack-tip displacement field

V Electron-nucleus interactions energy functional

Vo Equilibrium Volume per atom

vs Effective potential of a system of non-interacting electrons

v speed of stress wave

LIST OF SYMBOLS AND ACRONYMS xxix

W Width of crack sample

Xik (MEAM) Parameter required in the calculation of the screening

parameter C

xa x-coordinate of atom a

xij x component of the position vector between atom i and atom j

xα Direction cosine in the axis direction α

ya y-coordinate of atom a

yij y component of the position vector between atom i and atom j

Z (MEAM) Number of first neighbours

Z Objective function

za z-coordinate of atom a

zij z component of the position vector between atom i and atom j

Greek Symbols

α Parameter in Morse potential function

α Parameter in Rose’s Universal Equation of State

α Function in Hashin-Shtrikman Bounds

β lattice constant (angle between a and c)

β Function in Hashin-Shtrikman Bounds

β(h) (MEAM) Parameter in atomic electron density

Γ (MEAM) Function that combines the partial electron densities

∆E0 Energy change due to the formation of a compound

∆H Enthalpy of formation of a compound

θ Angle in polar coordinates

κ (MEAM) Parameter introduced into Embedding function

Λ De Broglie wavelength

νi Poisson’s ratio of the indenter

ρ density of the material

¯i (MEAM) electron density of atom i

ρ(h) (MEAM) h-th partial electron density

BCC Body-centred cubic unit cell

DC Diamond cubic unit cell

DFT Density Functional Theory

FCC Face-centred cubic unit cell

FEM Finite Element Method

f.u Formula unit

GGA Generalized Gradient Approximation

LIST OF SYMBOLS AND ACRONYMS xxxi

MEAM Modified Embedded Atom Method

pbc Periodic boundary condition

RNG Random Number Generator

SW Stillinger-Weber

tu time unit equivalent to ≈ 1.02 × 10−14s

UES Universal Equation of State

VI Vickers Indentation

VRH Voight-Reuss-Hill

xc exchange-correlation

Chapter 1

Introduction

Portable electronic products have become ubiquitous in today’s society The sumer demands that such products be small and packed with more features, yetremain reliable when subject to daily use Also, as these products continue to beimproved and packed with more features, older products get thrown away Suchdemands of the consumer drives two major trends in the electronics industry thatpresents new challenges in the field of materials science - the use of lead-free solderand reliability to drop impact

con-For many years, Pb-Sn solder alloy was the main solder in use in products.However, as the amount of electronic waste in landfills increases, concerns grewthat the lead, which is toxic to humans and wildlife, would contaminate the envi-ronment As such, legislation has been passed in the European Union to ban leadfrom electronics products [1, 2]

Over the years, many lead-free solders have been developed [3, 4], often withnovel compositions involving rare-earth metals [5] Unlike eutectic PbSn solderwhich has been well-studied due to its extensive use in the past, these new sol-der alloys require extensive testing to understand their mechanical and reliability

1.1 CURRENT TRENDS IN THE ELECTRONICS INDUSTRY 2

properties In particular, using Pb-free solder poses a different set of challenges.Firstly, previous designs optimized for use with Pb-Sn solder cannot be used forthe new solders [6] For example, newer Sn-rich solders have higher melting pointsthan Pb-Sn solders, requiring a higher reflow temperature, thus exacerbating thethermal mismatch stresses [7] The choice of materials in the rest of the packagemust take into account this fact With Pb-Sn solder, one can draw upon 40 years

of experimental data With the novel lead-free solders that have been developed,there is no such benefit and much work has to be carried out to understand theproperties of these material systems and their components

Secondly, with different choices of materials, the interfacial reactions are ferent for each type of solder A vastly-simplified schematic diagram of a typicalsolder joint is shown in Figure 1.1 Along each material interface is a layer ofintermetallic compounds that are formed when the two layers bond together

dif-Figure 1.1: Structure of a solder joint

Frear identified the reactions at these interfaces as a key issue in consumerelectronics applications [6], as they have an effect on the reliability and strength

of the solder joint It was shown by Frear et al that different lead-free solders

produced different morphology and microstructure at the solder joint, and resulted

in different fatigue behaviour [8] Ho et al mentioned that when the solder joint

is subjected to thermal aging, growth of the intermetallic compound Cu3Sn wasaccompanied by voids at the intermetallic layer [9] Shear and pull tests haveshown that these voids weaken the solder joint [10]

This weakening of the solder joint is inimical to the reliability of electronicpackages to drop impact, which is an important concern due to the consumerdemand that portable electronic devices remain working when dropped The in-termetallic compounds that form at the solder joint play an important role in dropimpact reliability They are known to be brittle, and studies over the years haveshown that when electronic packages are subjected to drop impact test, cracks arefound at the interface between the intermetallic compounds and the underlyingmetallization [11, 12] It was also demonstrated that fracture occurred throughthe voids (caused by growth of Cu3Sn) after thermal aging [13]

With so many factors that influence the reliability of package designs, there is aclear need for tools to enable designs to be evaluated Experimental testing isexpensive and can only provide a limited amount of data [14] As such, computa-tional methods play an integral role in addressing these challenges

The Finite Element Method (FEM) has established itself to be integral tothe design and development of electronic packaging Early dynamic FEM studiessimulating the drop-impact behaviour of the solder joint did not include the in-termetallic layer [15] However, as the role of the intermetallic layer became clear,studies taking into account the intermetallic layer have appeared A FEM modelwas developed to study the fracture behaviour of a solder joint and it was shownthat the stress was highest in the intermetallic layer[16] In the study involving

1.2 COMPUTATIONAL METHODS 4

design optimization, Syed et al used an FEM model that included an lic layer and showed that designs that reduced the stress in the intermetallic layercorrespondingly proved to be more reliable in drop-testing[17] Chen et al stud-ied the thermal cycling behaviour of a solder joint, showing that the locations ofcracks obtained experimentally were correlated with high-stress locations in thesolder[18]

intermetal-Essential to successful modelling with FEM is knowledge of accurate materialproperties These studies previously described made use of properties reported byFields et al [19] Unfortunately, later studies of the Young’s Modulus did notagree with their values, and the reported values show a wide range This is alsotrue for the critical mode I fracture toughness Details of these studies will begiven in the following chapters

At the same time, as the length scale in electronic devices continue to be duced, atomic-scale effects will begin to dominate For example, it was found thatthe yield stress of micron-scale Ni wires depended on the sample dimensions [20]

re-As such, computational tools need to be developed so that predictions of such fects can be made Consequently, the challenges that the International TechnologyRoadmap for Semiconductors identifies are the need for “Computational materi-als science tools to describe material properties”, and especially “Linkage withfirst principle computation and reduced model (classical MD or thermodynamiccomputation)” [21]

ef-It is now possible to predict material properties from atomic-level tions.One of these tools is calculations based on Density Functional Theory (DFT),which is essentially a means of solving the Schr¨odinger’s Equation [22] There is

calcula-a growing body of evidence thcalcula-at DFT ccalcula-an obtcalcula-ain the elcalcula-astic constcalcula-ants of existingmaterials that correlate very well with those obtained experimentally Examples

of such works include studies on TiB2 and MgB2 [23] and BeO [24] Some studieshave also made predictions for newly-synthesised materials, like Ti3SiC2 [25], and

Table 1.1: A comparison of the three computational methods

the possible crystal structures of theoretical materials [26]

However, DFT is computationally intensive and calculations typically takeweeks and months on desktop computers to complete, especially for complex struc-tures with tens of atoms Most atomic-level processes involve much more atomsthan this, for example, the fracture of a material

Molecular Dynamics simulation (MD) can bridge the gap between DFT andFEM It is essentially a classical mechanics treatment of the forces between atomsand this reduces the computational effort considerably A linear spring joiningtwo particles has the potential energy φ(x) = 12kx2 and the force between the twoatoms as F = −dφdx = −kx

Much of MD simulation work involves finding suitable expressions for φ(x)

so that certain essential properties of materials are reproduced (eg elastic stants) and then subsequently conducting simulations with the potential obtained.The process of finding suitable expressions of φ(x) involves arbitrary choices as tothe choice of function and which essential properties to reproduce, which wouldindicate a reduced predictive capability and accuracy of MD simulations Never-theless, for a simulation on the scale of thousands of atoms, there are no betteralternatives, and much simulation work has shown promise

con-Putting it all in perspective, the characteristics of the three methods discussedare summarized in Table 1.1 As these methods are done at different length scales,they complement each other DFT can calculate the energy of a system accu-rately but the computational cost restricts the system size to the order of tens of

1.3 OBJECTIVES 6

atoms MD simulations involve thousands of atoms and can simulate atomic-levelphenomenon, but the process behind obtaining the interatomic potential is oftensubject to many arbitrary choices

The situation regarding the intermetallic compounds in the electronics industryhas been discussed and the computational methods that work at the atomic levelhave also been described The objectives of this thesis are as follows

1 The elastic properties of intermetallic compounds found in electronic ing show little agreement, with the values obtained depending on the exper-imental method used Methods in computational materials science, such asDFT, can now calculate the properties of materials DFT calculations of theelastic properties of Cu6Sn5, Cu3Sn, Ag3Sn and Ni3Sn4 will be performed

packag-2 The vast majority of MD simulations are concentrated in the cubic metalsand semiconductors, due to the simplicity of the crystal structure and tech-nological importance Comparatively less effort has been put in to conductsimulations of intermetallic compounds An interatomic potential for Cu6Sn5

in the NiAs structure will be obtained

3 The interatomic potential obtained will be used to perform MD simulations

of the fracture of Cu6Sn5

Having described the intermetallic compounds and their importance to the mance of the solder joint, this thesis will describe calculations performed to obtainthe mechanical properties of Cu6Sn5 and Cu3Sn with computational methods

perfor-Calculations to predict the Young’s Modulus of Cu6Sn5 and Cu3Sn using sity Functional Theory are described in Chapter 2 For the single crystal, predic-tions are made for the structural properties and elastic properties for Cu6Sn5 and

Den-Cu3Sn It will be shown how these results are validated by comparison with perimental results from polycrystalline specimens Results for Ag3Sn and Ni3Sn4

ex-are also included for comparison

In order to facilitate MD simulations of Cu6Sn5, the process of developing aninteratomic potential for Cu6Sn5 in the NiAs-structure is described in Chapter 3.Parameters in the Modified Embedded Atom Method are obtained by means ofoptimizing their values to reproduce a set of energies in a database

Using the potential developed in Chapter 3, MD simulations of the fracture of

Cu6Sn5 are described in Chapter 4

These results are summed up in Chapter 5, where possibilities for further search are also laid out

Ag3Sn and Ni3Sn4.

These intermetallic compounds feature prominently in modern life Apart frombeing found in electronic packaging, both Cu6Sn5 and Ag3Sn are found in dentalamalgam [27, 28] Along with Cu6Sn5 [29, 30], Ni3Sn4 [31, 32, 33] is also beingconsidered as anode materials for lithium-ion batteries The desire to eliminatelead because of its toxicity has led to the use of Cu-Sn frangible bullets, withinwhich Cu6Sn5 and Cu3Sn are present[34] It has also been recently demonstratedthat it is feasible to conduct electroless plating (ie a method of coating that does

not require passing a current through the set-up) of Sn on Cu substrates, with thethin Cu-Sn alloy layer formed consisting of Cu6Sn5 and Cu3Sn [35, 36, 37].

Within the solder joint, Ni3Sn4 forms at the interface between Sn-based solderand Ni bonding pads, while Ag3Sn forms within the Sn-Ag solder during solderreflow During the solder reflow the solder reacts with a Cu substrate to form Cu-

Sn intermetallic compounds Studies of the effect of temperature and duration onthe formation of intermetallic compounds have shown that only Cu6Sn5 is formedinitially, while Cu3Sn is formed after a longer reaction time [38, 39]

Intermetallic Compounds

The formation of these intermetallic compounds are necessary for good bonding.However, being brittle, they compromise the reliability of the joint if present inexcessive amounts This also becomes more drastic with the miniaturization ofcomponents, as the fraction of the intermetallic layer to the total size of the solderjoint increases

These material systems are still not well understood with regards to theirproperties and their effects on the overall joint reliability Researchers employingFinite Element Method studies are beginning to include the intermetallic layer inthe solder joint Accurate knowledge of their properties is critical if modelling is

to be employed to predict the behaviour of these systems

There have also been a fair number of studies on the polycrystalline elastic uli of these intermetallic compounds Results from these studies are summarized

mod-in Table 2.1 Early studies, such as those by Fields et al [19] and Subrahmanyan[40] made use of bulk specimens More recent studies were conducted on cross-sectioned diffusion couples and the solder joint itself These studies are discussed

in further detail in Appendix F