Atomistic calculations of the mechanical properties cu sn intermetallic compounds

ATOMISTIC CALCULATIONS OF THE MECHANICAL
PROPERTIES OF Cu-Sn INTERMETALLIC COMPOUNDS
LEE TIONG SENG NORMAN
NATIONAL UNIVERSITY OF SINGAPORE
2008
ATOMISTIC CALCULATIONS OF THE MECHANICAL
PROPERTIES OF Cu-Sn INTERMETALLIC COMPOUNDS
LEE TIONG SENG NORMAN
(B.Eng(Hons),NUS)
A THESIS SUBMITTED
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
DEPARTMENT OF MECHANICAL ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2008
In my time, I tried to educate our people in an understanding of the
dignity of human life and their right as fellow human beings, and youth
was not only interested but excited about what I consider things that
matter. Things of the spirit, the development of a human being to
his true potential in accordance with his own personal genius in the
context of equal rights of others.
David Saul Marshall (1908-1995)
iv
Acknowledgements
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. Very
often, they would ask probing questions that spurred me to think deeper into the
topic 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 taking
the last bus to find fruits or food on my desk.
I acknowledge the financial support from the following sources: NUS research

scholarship (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 the
many opportunities for teaching from the Department of Mechanical Engineering
(especially with Prof CJ Tay and Prof Cheng Li), Professional Activities Centre
and the Bachelor of Technology department.
The facilities provided by NUS have been excellent. The staff at the Science
Library have been most friendly and helpful. I have also made extensive use of the
resources provided by the Supercomputing and Visualization Unit (SVU), and I
thank 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,
v
Dr Alexander Goldberg of Accelrys Inc. and Prof Lee Ming-Hsien of Tamkang
University, Taiwan. Although the need to credit their contributions in the main
text of this thesis did not arise, I appreciate their willingness to respond to my
email 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 the
demands of this research seemed overwhelming. Their knowledge, opinions and
ideas which they shared with me often gave me the needed push to move on. They
also 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 Dr
Yew Yong Kin, as well as lab officers Mr Joe Low, Mr Alvin Goh and Mr Chiam
Tow Jong.
My thanks goes out to my friends for their encouragement and their advice in
the decisions that I have made. Talking to them always helped in seeing things
clearer. I am sure that their prayers helped a lot. Last of all, there were many
occasions when serendipity and decisions that I made in the past (e.g. taking basic

German lessons) played a role in getting the research work done. I recognize the
role of the Creator in all that has happened. Gloria in altissimis Deo.
vi
Contents
List of Tables xv
List of Figures xix
List of Symbols and Acronyms xxiv
1 Introduction 1
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
CONTENTS viii
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 Cu
6
Sn
5
. 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
CONTENTS x
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 K
IC
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 K
IC
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
Bibliography 180
CONTENTS xi
A Elasticity Formulas 208
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 Cu
6
Sn
5
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
CONTENTS xii
F Review of experimental work 227
F.1 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . 227
F.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
xiii
Summary
Intermetallic compounds such as Cu

6
Sn
5
play a vital role in the reliability of
electronic components. The properties of intermetallic compounds formed at the
solder joint need to be known accurately so that macro-scale computer modelling
can take place to evaluate the reliability of components during the design process.
However, experiments to determine the mechanical properties of Cu
6
Sn
5
, such as
the Young’s Modulus and the critical mode I fracture toughness, have been sparse
and have not produced any definitive value yet. Hence, there is a need to clarify
what 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 inter-
metallic compounds Cu
3
Sn and Cu
6
Sn
5
to obtain the elastic properties of these
materials. Using a single unit cell, their lattice constants are calculated and shown
to be in agreement with experimental data. The hitherto unknown single-crystal
elastic constants are then calculated. Using these values, the orientation depen-
dence of the single-crystal Young’s Modulus is evaluated. The direction of the
largest value is found to coincide with closely-packed planes. The bounds on the

polycrystalline elastic moduli are also evaluated using the methods of Hill and
Hashin-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
CONTENTS xiv
compounds.
Following which, an interatomic potential for Cu-Sn interactions in the Modi-
fied Embedded Atom Method formalism tailored to the properties of Cu
6
Sn
5
in the
NiAs crystal structure is developed. Using this interatomic potential, Molecular
Dynamics simulations of the fracture of Cu
6
Sn
5
are conducted with thousands of
atoms. Atomic behaviour corresponding to brittle fracture are seen in the simu-
lations. This shows that it is feasible to develop an interatomic potential for the
NiAs crystal structure, and to conduct realistic simulations that can reproduce
qualitatively the properties and behaviour of the brittle intermetallic compound.
xv
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 Cu
6
Sn
5
. . . . . . . . . . . . . . . . . . . 51
LIST OF TABLES xvi
2.13 Positional Parameters for Ni
3
Sn
4
. . . . . . . . . . . . . . . . . . . 52
2.14 Positional parameters for Cu
3
Sn . . . . . . . . . . . . . . . . . . . 52
2.15 Positional parameters for Ag
3
Sn . . . . . . . . . . . . . . . . . . . 52

2.16 Elastic Constants of Cu
3
Sn calculated at different accuracy setting.
All values are given in GPa. . . . . . . . . . . . . . . . . . . . . . . 54
2.17 Elastic Constants of Ni
3
Sn
4
calculated at different accuracy setting.
All values are given in GPa. . . . . . . . . . . . . . . . . . . . . . . 55
2.18 Elastic Constants calculated for Ag
3
Sn 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 Cu-Sn obtained from the potential fitting process -
Set P1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.12 Parameters for Cu-Sn obtained from the potential fitting process -
Set P2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.13 Coordinates of the atoms in Cell A. They are expressed as a fraction
of the cell vectors. The cell lengths are a ×
√
3a × c. . . . . . . . . 126
3.14 Coordinates of the atoms in Cell B. They are expressed as a fraction
of the cell vectors. The cell lengths are a ×
√
3a × c. . . . . . . . . 126
3.15 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. . . . . . . . . . . . 127
3.16 Strain patterns for the various elastic constants . . . . . . . . . . . 128
3.17 Elastic Constants calculated. All values are given in GPa. †c
13
is
calculated from the combined elastic constants. . . . . . . . . . . . 130
3.18 Elastic Constants requiring relaxation. All values are given in GPa.
†c
11
and c
12
are calculated from the combined elastic constants. . . 131
3.19 Predicted surface energies. All values are in eV/

˚
A
2
. . . . . . . . . 135
3.20 Predicted Energies of other structures . . . . . . . . . . . . . . . . 138
4.1 Experimental data for the fracture toughness K
IC
of Cu
6
Sn
5
. . . . 144
4.2 Atomic-Scale units used in the software . . . . . . . . . . . . . . . 151
4.3 Description of the size of the simulation boxes used in this section.
The pre-crack size and box dimensions are given in terms of unit
cell lengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
4.4 Values of stress and predicted K
IC
for the two cases, assuming that
σ = c
11
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.5 Different box sizes used in the study in this section . . . . . . . . . 164
4.6 Different box sizes used in the study in this section . . . . . . . . . 166
LIST OF TABLES xviii
C.1 Position vector components of the atoms in the coordination poly-
hedron of the Sn atom . . . . . . . . . . . . . . . . . . . . . . . . . 214
C.2 Position vector components of the Cu atoms in the coordination
polyhedron of the Cu atom . . . . . . . . . . . . . . . . . . . . . . . 215
C.3 Position vector components of the Sn atoms in the coordination

polyhedron of the Cu atom . . . . . . . . . . . . . . . . . . . . . . . 215
D.1 DFT energies per atom for Cu (FCC) , Sn (DC) and CuSn (B8).
The energy change per atom (∆E
0
) and cohesive energy (E
coh
) are
then derived from them. . . . . . . . . . . . . . . . . . . . . . . . . 222
F.1 Experimental work on the Young’s Modulus of Cu
6
Sn
5
and Cu
3
Sn.
All values are in GPa. . . . . . . . . . . . . . . . . . . . . . . . . . 231
xix
List of Figures
1.1 Structure of a solder joint . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Illustrating the idea of a lattice with basis. The result is a honey-
comb lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 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. . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Cu
6
Sn
5
in the NiAs-type structure according to Gangulee et al
The larger spheres represent the Sn atoms and the smaller spheres

represent the Cu atoms. The excess Cu atoms fill the interstitial
sites to make up the 6:5 stoichiometry. . . . . . . . . . . . . . . . . 34
2.4 Unit Cell of Cu
6
Sn
5
according to Larsson et al The larger spheres
represent the Sn atoms and the smaller spheres represent the Cu
atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5 Unit Cell of Cu
3
Sn according to Burkhardt et al The larger spheres
represent the Sn atoms and the smaller spheres represent the Cu
atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6 Unit Cell of Ni
3
Sn
4
according to Jeitschko et al The larger spheres
represent the Sn atoms and the smaller spheres represent the Ni
atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
LIST OF FIGURES xx
2.7 Unit Cell of Ag
3
Sn according to Fairhurst et al The larger spheres
represent the Sn atoms and the smaller spheres represent the Ag
atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.8 Documentation Provided in LDA-generated pseudopotential files for
Cu (‘Cu 00.usp’) and Sn (‘Sn 00.usp’) . . . . . . . . . . . . . . . . 40
2.9 Flowchart showing the methodology of the calculations in this chap-

ter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.10 Stiffness matrix for Cu
6
Sn
5
calculated with the GGA at 320 eV
cutoff and 4 × 4 × 3 k-point mesh. All values are given in GPa . . 54
2.11 Relation of the coordinate axes to the crystal axes . . . . . . . . . . 56
2.12 Young’s Modulus as a function of crystallographic direction for
Cu
3
Sn. The colours represent the magnitude of the Young’s Mod-
ulus while the x, y, z coordinates represent the crystallographic di-
rections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.13 Young’s Modulus as a function of crystallographic direction for
Cu
6
Sn
5
. The colours represent the magnitude of the Young’s Mod-
ulus while the x, y, z coordinates represent the crystallographic di-
rections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.14 Young’s Modulus as a function of crystallographic direction of vari-
ous planes for Cu
3
Sn. Angles are measured anti-clockwise from the
first-mentioned axis in each plane. . . . . . . . . . . . . . . . . . . 59
2.15 Young’s Modulus as a function of crystallographic direction of var-
ious planes for Cu
6

Sn
5
. Angles are measured anti-clockwise from
the first-mentioned axis in each plane. . . . . . . . . . . . . . . . . 59
2.16 The plane corresponding to 55.8
◦
anti-clockwise from the positive
x-axis in Cu
3
Sn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.17 The plane corresponding to 147.6
◦
anti-clockwise from the positive
x-axis in Cu
6
Sn
5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
LIST OF FIGURES xxi
2.18 A two-dimensional view of the plane marked in Figure 2.17 in Cu
6
Sn
5
.
61
2.19 Cu
6
Sn
5
: Cumulative Distribution Function for the single-crystal

Young’s Modulus of Cu
6
Sn
5
. . . . . . . . . . . . . . . . . . . . . . 68
2.20 Cu
3
Sn: Cumulative Distribution Function for the single-crystal Young’s
Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.1 FCC lattice in 2D projection. “First neighbour” and “second neigh-
bour” shells of shaded atom are indicated. . . . . . . . . . . . . . . 88
3.2 Illustrating the screening in the MEAM. . . . . . . . . . . . . . . . 92
3.3 Two cases to explain the MEAM Screening Procedure. For Case 1,
C = 3 and for case 2, C =
1
3
. . . . . . . . . . . . . . . . . . . . . . 93
3.4 Relationship between the pairwise, EAM and MEAM potential func-
tionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.5 Evolution of the energy of Cu-Sn B8 unit cell predicted using the
Aguilar MEAM potential . . . . . . . . . . . . . . . . . . . . . . . 104
3.6 Evolution of the lattice constants a and c of Cu-Sn B8 unit cell
using the Aguilar MEAM potential . . . . . . . . . . . . . . . . . . 105
3.7 Energy-Volume calculations for B8 Cu-Sn with DFT. The lines are
a guide to the eye. . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.8 Energy-Volume calculations for B8 Cu-Sn with DFT, enlarged to
show points near equilibrium volume. . . . . . . . . . . . . . . . . 111
3.9 Variation of equilibrium c/a ratio with volume. The lines are a
guide to the eye. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.10 Coordination Polyhedra for Cu and Sn in the B8 unit cell. The basis

atoms are indicated with yellow letters and the first neighbours are
joined with yellow lines to show the coordination polyhedron. . . . 115
LIST OF FIGURES xxii
3.11 Alternative orthogonal unit cells that have the same symmetry as
the B8 unit cell. The c-direction is perpendicular to the plane of the
diagram. Both Cell A and Cell B have sides of length a ×
√
3a × c. 125
3.12 Evolution of the lattice constants during the threshold acceptance
process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.13 Strain patterns for the various elastic constants illustrated by show-
ing how the unit cell deforms. The strains for c
11
−c
12
and c
44
destroy
the hexagonal symmetry. . . . . . . . . . . . . . . . . . . . . . . . 129
3.14 Unit cell to calculate c
44
. . . . . . . . . . . . . . . . . . . . . . . . 132
3.15 Illustrating the Surface Energy Calculation. Two simulation cells of
different boundary conditions are considered, one with fully periodic
boundary conditions, the other with a surface. The surface energy
= (E
1
− E
0
)/A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

3.16 Unit cells used in the DFT calculation of the surface energy (a)
surface normal x (b) surface normal z. The height of the inserted
vacuum is about 8
˚
A. . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.17 Unit cells for different structures with 1:1 stoichiometry. For each
structure, the Strukturbericht designation is given followed by the
chemical formula of a typical compound. . . . . . . . . . . . . . . . 137
4.1 Schematic diagram of cracks forming from the corners of the Vickers
indentation. The crack length C is measured from the centre of the
indentation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.2 Flowchart showing the organization of RidgeMD. . . . . . . . . . . 152
4.3 Description of the basal plane and the “ac plane”. . . . . . . . . . . 153
4.4 Periodic Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.5 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 initial
strain of  = 0.20 applied throughout. The sample cleaves between
the immobile layer and the mobile atoms and no crack propagates
from the pre-crack. . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.7 Configuration PC2 with a initial strain of  = 0.20 applied through-
out. Snapshots at (a) 6500 time steps (b) 20000 time steps . . . . . 157
4.8 Configuration PC3 with an initial strain of  = 0.12 applied through-
out. Snapshots at (a) 1200 time steps (b) 20000 time steps . . . . . 159
4.9 Configuration PC3. Snapshot at 1200 time steps representing the
atoms as circles. All the atoms are superimposed on each other.
The plane at which the crack propagates can be clearly seen. . . . 159
4.10 (a) Array of atoms (b) The simulation box after Sih’s equation is
applied to the atoms . . . . . . . . . . . . . . . . . . . . . . . . . . 161

4.11 ac plane: Snapshots of atomic configuration for size Small at 20000
time steps at 300 K for (a) K
I
= 0.40 MPa
√
m(b) K
I
= 0.64 MPa
√
m166
4.12 ac plane: Snapshots of atomic configuration at 20000 time steps at
300 K for K
I
= 0.80 MPa
√
m. (a) Size Small (b) Size Large. . . . 167
4.13 ac plane: Size Large, K
I
= 0.80 MPa
√
m using circles to represent
the atoms. (a) without tails (b) with tails. . . . . . . . . . . . . . 167
4.14 Basal plane: snapshots of atomic configuration for size Small at
20000 time steps at 300 K for (a) K
I
= 0.48 MPa
√
m(b) K
I
= 0.64

MPa
√
m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.15 Basal plane: snapshots of atomic configuration for Size Large at
20000 time steps at 300 K for various values of K
I
= 0.64 MPa
√
m.
(a) 3100 time steps (b) 20000 time steps . . . . . . . . . . . . . . . 169
4.16 Basal plane: Size Large, K
I
= 0.64 MPa
√
m using circles to repre-
sent the atoms. (a) without tails (b) with tails. . . . . . . . . . . . 170
LIST OF FIGURES xxiv
B.1 Part of the honeycomb lattice. The (x, y) coordinates are given
beside each particle. (a) Before shear, all particles are unit distance
away from each other (b) after shear. The arrows beside particles 3
and 4 shows how they have been displaced . . . . . . . . . . . . . . 211
E.1 (a) (010) plane of Cu
6
Sn
5
(b) Schematic diagram of the (010) plane.
The dashed lines represent the planes of atoms seen in (a) (c) Upon
application of the shear strain 
13
, the unit cell deforms according to

the red dotted line. The angle β is reduced (d) Bottom left corner
of the unit cell. BC is shortened when β is reduced. . . . . . . . . 225
F.1 (a) nanoindentation load-displacement curve (b) Berkovich indenter 228
xxv
List of Symbols and Acronyms
Latin Symbols
A (MEAM) Parameter in Embedding Function
A Surface area
A Amplitude of plane wave
A Indentation Projected Contact Area
a crack half-length
a lattice constant
a
ij
direction cosine between original axis j and rotated axis i
a Primitive lattice vector
B Bulk Modulus
B
0
Hashin-Shktrikman bulk modulus for a hypothetical material
B
R
Reuss’ average of the Bulk Modulus
B
V
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
C
min
(MEAM) Lower limit on C
C
max
(MEAM) Upper limit on C
C Elastic stiffness matrix